1911 Encyclopædia Britannica/Geometry
GEOMETRY, the general term for the branch of mathematics which has for its province the study of the properties of space. From experience, or possibly intuitively, we characterize existent space by certain fundamental qualities, termed axioms, which are insusceptible of proof; and these axioms, in conjunction with the mathematical entities of the point, straight line, curve, surface and solid, appropriately defined, are the premises from which the geometer draws conclusions. The geometrical axioms are merely conventions; on the one hand, the system may be based upon inductions from experience, in which case the deduced geometry may be regarded as a branch of physical science; or, on the other hand, the system may be formed by purely logical methods, in which case the geometry is a phase of pure mathematics. Obviously the geometry with which we are most familiar is that of existent space—the threedimensional space of experience; this geometry may be termed Euclidean, after its most famous expositor. But other geometries exist, for it is possible to frame systems of axioms which definitely characterize some other kind of space, and from these axioms to deduce a series of noncontradictory propositions; such geometries are called nonEuclidean.
It is convenient to discuss the subjectmatter of geometry under the following headings:
I. Euclidean Geometry: a discussion of the axioms of existent space and of the geometrical entities, followed by a synoptical account of Euclid’s Elements. 
II. Projective Geometry: primarily Euclidean, but differing from I. in employing the notion of geometrical continuity (q.v.)—points and lines at infinity. 
III. Descriptive Geometry: the methods for representing upon planes figures placed in space of three dimensions. 
IV. Analytical Geometry: the representation of geometrical figures and their relations by algebraic equations. 
V. Line Geometry: an analytical treatment of the line regarded as the space element. 
VI. NonEuclidean Geometry: a discussion of geometries other than that of the space of experience. 
VII. Axioms of Geometry: a critical analysis of the foundations of geometry. 
Special subjects are treated under their own headings: e.g. Projection, Perspective; Curve, Surface; Circle, Conic Section; Triangle, Polygon, Polyhedron; there are also articles on special curves and figures, e.g. Ellipse, Parabola, Hyperbola; Tetrahedron, Cube, Octahedron, Dodecahedron, Icosahedron; Cardioid, Catenary, Cissoid, Conchoid, Cycloid, Epicycloid, Limaçon, Oval, Quadratrix, Spiral, &c.
History.—The origin of geometry (Gr. γῆ, earth, μέτρον, a measure) is, according to Herodotus, to be found in the etymology of the word. Its birthplace was Egypt, and it arose from the need of surveying the lands inundated by the Nile floods. In its infancy it therefore consisted of a few rules, very rough and approximate, for computing the areas of triangles and quadrilaterals; and, with the Egyptians, it proceeded no further, the geometrical entities—the point, line, surface and solid—being only discussed in so far as they were involved in practical affairs. The point was realized as a mark or position, a straight line as a stretched string or the tracing of a pole, a surface as an area; but these units were not abstracted; and for the Egyptians geometry was only an art—an auxiliary to surveying.^{[1]} The first step towards its elevation to the rank of a science was made by Thales (q.v.) of Miletus, who transplanted the elementary Egyptian mensuration to Greece. Thales clearly abstracted the notions of points and lines, founding the geometry of the latter unit, and discovering per saltum many propositions concerning areas, the circle, &c. The empirical rules of the Egyptians were corrected and developed by the Ionic School which he founded, especially by Anaximander and Anaxagoras, and in the 6th century B.C. passed into the care of the Pythagoreans. From this time geometry exercised a powerful influence on Greek thought. Pythagoras (q.v.), seeking the key of the universe in arithmetic and geometry, investigated logically the principles underlying the known propositions; and this resulted in the formulation of definitions, axioms and postulates which, in addition to founding a science of geometry, permitted a crystallization, fractional, it is true, of the amorphous collection of material at hand. Pythagorean geometry was essentially a geometry of areas and solids; its goal was the regular solids—the tetrahedron, cube, octahedron, dodecahedron and icosahedron—which symbolized the five elements of Greek cosmology. The geometry of the circle, previously studied in Egypt and much more seriously by Thales, was somewhat neglected, although this curve was regarded as the most perfect of all plane figures and the sphere the most perfect of all solids. The circle, however, was taken up by the Sophists, who made most of their discoveries in attempts to solve the classical problems of squaring the circle, doubling the cube and trisecting an angle. These problems, besides stimulating pure geometry, i.e. the geometry of constructions made by the ruler and compasses, exercised considerable influence in other directions. The first problem led to the discovery of the method of exhaustion for determining areas. Antiphon inscribed a square in a circle, and on each side an isosceles triangle having its vertex on the circle; on the sides of the octagon so obtained, isosceles triangles were again constructed, the process leading to inscribed polygons of 8, 16 and 32 sides; and the areas of these polygons, which are easily determined, are successive approximations to the area of the circle. Bryson of Heraclea took an important step when he circumscribed, in addition to inscribing, polygons to a circle, but he committed an error in treating the circle as the mean of the two polygons. The method of Antiphon, in assuming that by continued division a polygon can be constructed coincident with the circle, demanded that magnitudes are not infinitely divisible. Much controversy ranged about this point; Aristotle supported the doctrine of infinite divisibility; Zeno attempted to show its absurdity. The mechanical tracing of loci, a principle initiated by Archytas of Tarentum to solve the last two problems, was a frequent subject for study, and several mechanical curves were thus discovered at subsequent dates (cissoid, conchoid, quadratrix). Mention may be made of Hippocrates, who, besides developing the known methods, made a study of similar figures, and, as a consequence, of proportion. This step is important as bringing into line discontinuous number and continuous magnitude.
A fresh stimulus was given by the succeeding Platonists, who, accepting in part the Pythagorean cosmology, made the study of geometry preliminary to that of philosophy. The many discoveries made by this school were facilitated in no small measure by the clarification of the axioms and definitions, the logical sequence of propositions which was adopted, and, more especially, by the formulation of the analytic method, i.e. of assuming the truth of a proposition and then reasoning to a known truth. The main strength of the Platonist geometers lies in stereometry or the geometry of solids. The Pythagoreans had dealt with the sphere and regular solids, but the pyramid, prism, cone and cylinder were but little known until the Platonists took them in hand. Eudoxus established their mensuration, proving the pyramid and cone to have onethird the content of a prism and cylinder on the same base and of the same height, and was probably the discoverer of a proof that the volumes of spheres are as the cubes of their radii. The discussion of sections of the cone and cylinder led to the discovery of the three curves named the parabola, ellipse and hyperbola (see Conic Section); it is difficult to overestimate the importance of this discovery; its investigation marks the crowning achievement of Greek geometry, and led in later years to the fundamental theorems and methods of modern geometry.
The presentation of the subjectmatter of geometry as a connected and logical series of propositions, prefaced by Ὅροι or foundations, had been attempted by many; but it is to Euclid that we owe a complete exposition. Little indeed in the Elements is probably original except the arrangement; but in this Euclid surpassed such predecessors as Hippocrates, Leon, pupil of Neocleides, and Theudius of Magnesia, devising an apt logical model, although when scrutinized in the light of modern mathematical conceptions the proofs are riddled with fallacies. According to the commentator Proclus, the Elements were written with a twofold object, first, to introduce the novice to geometry, and secondly, to lead him to the regular solids; conic sections found no place therein. What Euclid did for the line and circle, Apollonius did for the conic sections, but there we have a discoverer as well as editor. These two works, which contain the greatest contributions to ancient geometry, are treated in detail in Section I. Euclidean Geometry and the articles Euclid; Conic Section; Apollonius. Between Euclid and Apollonius there flourished the illustrious Archimedes, whose geometrical discoveries are mainly concerned with the mensuration of the circle and conic sections, and of the sphere, cone and cylinder, and whose greatest contribution to geometrical method is the elevation of the method of exhaustion to the dignity of an instrument of research. Apollonius was followed by Nicomedes, the inventor of the conchoid; Diocles, the inventor of the cissoid; Zenodorus, the founder of the study of isoperimetrical figures; Hipparchus, the founder of trigonometry; and Heron the elder, who wrote after the manner of the Egyptians, and primarily directed attention to problems of practical surveying.
Of the many isolated discoveries made by the later Alexandrian mathematicians, those of Menelaus are of importance. He showed how to treat spherical triangles, establishing their properties and determining their congruence; his theorem on the products of the segments in which the sides of a triangle are cut by a line was the foundation on which Carnot erected his theory of transversals. These propositions, and also those of Hipparchus, were utilized and developed by Ptolemy (q.v.), the expositor of trigonometry and discoverer of many isolated propositions. Mention may be made of the commentator Pappus, whose Mathematical Collections is valuable for its wealth of historical matter; of Theon, an editor of Euclid’s Elements and commentator of Ptolemy’s Almagest; of Proclus, a commentator of Euclid; and of Eutocius, a commentator of Apollonius and Archimedes.
The Romans, essentially practical and having no inclination to study science qua science, only had a geometry which sufficed for surveying; and even here there were abundant inaccuracies, the empirical rules employed being akin to those of the Egyptians and Heron. The Hindus, likewise, gave more attention to computation, and their geometry was either of Greek origin or in the form presented in trigonometry, more particularly connected with arithmetic. It had no logical foundations; each proposition stood alone; and the results were empirical. The Arabs more closely followed the Greeks, a plan adopted as a sequel to the translation of the works of Euclid, Apollonius, Archimedes and many others into Arabic. Their chief contribution to geometry is exhibited in their solution of algebraic equations by intersecting conics, a step already taken by the Greeks in isolated cases, but only elevated into a method by Omar al Hayyami, who flourished in the 11th century. During the middle ages little was added to Greek and Arabic geometry. Leonardo of Pisa wrote a Practica geometriae (1220), wherein Euclidean methods are employed; but it was not until the 14th century that geometry, generally Euclid’s Elements, became an essential item in university curricula. There was, however, no sign of original development, other branches of mathematics, mainly algebra and trigonometry, exercising a greater fascination until the 16th century, when the subject again came into favour.
The extraordinary mathematical talent which came into being in the 16th and 17th centuries reacted on geometry and gave rise to all those characters which distinguish modern from ancient geometry. The first innovation of moment was the formulation of the principle of geometrical continuity by Kepler. The notion of infinity which it involved permitted generalizations and systematizations hitherto unthought of (see Geometrical Continuity); and the method of indefinite division applied to rectification, and quadrature and cubature problems avoided the cumbrous method of exhaustion and provided more accurate results. Further progress was made by Bonaventura Cavalieri, who, in his Geometria indivisibilibus continuorum (1620), devised a method intermediate between that of exhaustion and the infinitesimal calculus of Leibnitz and Newton. The logical basis of his system was corrected by Roberval and Pascal; and their discoveries, taken in conjunction with those of Leibnitz, Newton, and many others in the fluxional calculus, culminated in the branch of our subject known as differential geometry (see Infinitesimal Calculus; Curve; Surface).
A second important advance followed the recognition that conics could be regarded as projections of a circle, a conception which led at the hands of Desargues and Pascal to modern projective geometry and perspective. A third, and perhaps the most important, advance attended the application of algebra to geometry by Descartes, who thereby founded analytical geometry. The new fields thus opened up were diligently explored, but the calculus exercised the greatest attraction and relatively little progress was made in geometry until the beginning of the 19th century, when a new era opened.
Gaspard Monge was the first important contributor, stimulating analytical and differential geometry and founding descriptive geometry in a series of papers and especially in his lectures at the École polytechnique. Projective geometry, founded by Desargues, Pascal, Monge and L. N. M. Carnot, was crystallized by J. V. Poncelet, the creator of the modern methods. In his Traité des propriétés des figures (1822) the line and circular points at infinity, imaginaries, polar reciprocation, homology, crossratio and projection are systematically employed. In Germany, A. F. Möbius, J. Plücker and J. Steiner were making farreaching contributions. Möbius, in his Barycentrische Calcul (1827), introduced homogeneous coordinates, and also the powerful notion of geometrical transformation, including the special cases of collineation and duality; Plücker, in his Analytischgeometrische Entwickelungen (1828–1831), and his System der analytischen Geometrie (1835), introduced the abridged notation, line and plane coordinates, and the conception of generalized space elements; while Steiner, besides enriching geometry in numerous directions, was the first to systematically generate figures by projective pencils. We may also notice M. Chasles, whose Aperçu historique (1837) is a classic. Synthetic geometry, characterized by its fruitfulness and beauty, attracted most attention, and it so happened that its originally weak logical foundations became replaced by a more substantial set of axioms. These were found in the anharmonic ratio, a device leading to the liberation of synthetic geometry from metrical relations, and in involution, which yielded rigorous definitions of imaginaries. These innovations were made by K. J. C. von Staudt. Analytical geometry was stimulated by the algebra of invariants, a subject much developed by A. Cayley, G. Salmon, S. H. Aronhold, L. O. Hesse, and more particularly by R. F. A. Clebsch.
The introduction of the line as a space element, initiated by H. Grassmann (1844) and Cayley (1859), yielded at the hands of Plücker a new geometry, termed line geometry, a subject developed more notably by F. Klein, Clebsch, C. T. Reye and F. O. R. Sturm (see Section V., Line Geometry).
Noneuclidean geometries, having primarily their origin in the discussion of Euclidean parallels, and treated by Wallis, Saccheri and Lambert, have been especially developed during the 19th century. Four lines of investigation may be distinguished:—the naïvesynthetic, associated with Lobatschewski, Bolyai, Gauss; the metric differential, studied by Riemann, Helmholtz, Beltrami; the projective, developed by Cayley, Klein, Clifford; and the criticalsynthetic, promoted chiefly by the Italian mathematicians Peano, Veronese, BuraliForte, Levi Civittà, and the Germans Pasch and Hilbert. (C. E.*)
I. Euclidean Geometry
This branch of the science of geometry is so named since its methods and arrangement are those laid down in Euclid’s Elements.
§ 1. Axioms.—The object of geometry is to investigate the properties of space. The first step must consist in establishing those fundamental properties from which all others follow by processes of deductive reasoning. They are laid down in the Axioms, and these ought to form such a system that nothing need be added to them in order fully to characterize space, and that nothing may be omitted without making the system incomplete. They must, in fact, completely “define” space.
§ 2. Definitions.—The axioms of Euclidean Geometry are obtained from inspection of existent space and of solids in existent space,—hence from experience. The same source gives us the notions of the geometrical entities to which the axioms relate, viz. solids, surfaces, lines or curves, and points. A solid is directly given by experience; we have only to abstract all material from it in order to gain the notion of a geometrical solid. This has shape, size, position, and may be moved. Its boundary or boundaries are called surfaces. They separate one part of space from another, and are said to have no thickness. Their boundaries are curves or lines, and these have length only. Their boundaries, again, are points, which have no magnitude but only position. We thus come in three steps from solids to points which have no magnitude; in each step we lose one extension. Hence we say a solid has three dimensions, a surface two, a line one, and a point none. Space itself, of which a solid forms only a part, is also said to be of three dimensions. The same thing is intended to be expressed by saying that a solid has length, breadth and thickness, a surface length and breadth, a line length only, and a point no extension whatsoever.
Euclid gives the essence of these statements as definitions:—
Def. 1, I. A point is that which has no parts, or which has no magnitude. 
Def. 2, I. A line is length without breadth. 
Def. 5, I. A superficies is that which has only length and breadth. 
Def. 1, XI. A solid is that which has length, breadth and thickness. 
It is to be noted that the synthetic method is adopted by Euclid; the analytical derivation of the successive ideas of “surface,” “line,” and “point” from the experimental realization of a “solid” does not find a place in his system, although possessing more advantages.
If we allow motion in geometry, we may generate these entities by moving a point, a line, or a surface, thus:—
The path of a moving point is a line. 
And we may then assume that the lines, surfaces and solids, as defined before, can all be generated in this manner. From this generation of the entities it follows again that the boundaries—the first and last position of the moving element—of a line are points, and so on; and thus we come back to the considerations with which we started.
Euclid points this out in his definitions,—Def. 3, I., Def. 6, I., and Def. 2, XI. He does not, however, show the connexion which these definitions have with those mentioned before. When points and lines have been defined, a statement like Def. 3, I., “The extremities of a line are points,” is a proposition which either has to be proved, and then it is a theorem, or which has to be taken for granted, in which case it is an axiom. And so with Def. 6, I., and Def. 2, XI.
§ 3. Euclid’s definitions mentioned above are attempts to describe, in a few words, notions which we have obtained by inspection of and abstraction from solids. A few more notions have to be added to these, principally those of the simplest line—the straight line, and of the simplest surface—the flat surface or plane. These notions we possess, but to define them accurately is difficult. Euclid’s Definition 4, I., “A straight line is that which lies evenly between its extreme points,” must be meaningless to any one who has not the notion of straightness in his mind. Neither does it state a property of the straight line which can be used in any further investigation. Such a property is given in Axiom 10, I. It is really this axiom, together with Postulates 2 and 3, which characterizes the straight line.
Whilst for the straight line the verbal definition and axiom are kept apart, Euclid mixes them up in the case of the plane. Here the Definition 7, I., includes an axiom. It defines a plane as a surface which has the property that every straight line which joins any two points in it lies altogether in the surface. But if we take a straight line and a point in such a surface, and draw all straight lines which join the latter to all points in the first line, the surface will be fully determined. This construction is therefore sufficient as a definition. That every other straight line which joins any two points in this surface lies altogether in it is a further property, and to assume it gives another axiom.
Thus a number of Euclid’s axioms are hidden among his first definitions. A still greater confusion exists in the present editions of Euclid between the postulates and axioms so called, but this is due to later editors and not to Euclid himself. The latter had the last three axioms put together with the postulates (αἰτήματα), so that these were meant to include all assumptions relating to space. The remaining assumptions, which relate to magnitudes in general, viz. the first eight “axioms” in modern editions, were called “common notions” (κοιναὶ ἔννοιαι). Of the latter a few may be said to be definitions. Thus the eighth might be taken as a definition of “equal,” and the seventh of “halves.” If we wish to collect the axioms used in Euclid’s Elements, we have therefore to take the three postulates, the last three axioms as generally given, a few axioms hidden in the definitions, and an axiom used by Euclid in the proof of Prop. 4, I, and on a few other occasions, viz. that figures may be moved in space without change of shape or size.
§ 4. Postulates.—The assumptions actually made by Euclid may be stated as follows:—
(1) Straight lines exist which have the property that any one of them may be produced both ways without limit, that through any two points in space such a line may be drawn, and that any two of them coincide throughout their indefinite extensions as soon as two points in the one coincide with two points in the other. (This gives the contents of Def. 4, part of Def. 35, the first two Postulates, and Axiom 10.)
(2) Plane surfaces or planes exist having the property laid down in Def. 7, that every straight line joining any two points in such a surface lies altogether in it.
(3) Right angles, as defined in Def. 10, are possible, and all right angles are equal; that is to say, wherever in space we take a plane, and wherever in that plane we construct a right angle, all angles thus constructed will be equal, so that any one of them may be made to coincide with any other. (Axiom 11.)
(4) The 12th Axiom of Euclid. This we shall not state now, but only introduce it when we cannot proceed any further without it.
(5) Figures maybe freely moved in space without change of shape or size. This is assumed by Euclid, but not stated as an axiom.
(6) In any plane a circle may be described, having any point in that plane as centre, and its distance from any other point in that plane as radius. (Postulate 3.)
The definitions which have not been mentioned are all “nominal definitions,” that is to say, they fix a name for a thing described. Many of them overdetermine a figure.
§ 5. Euclid’s Elements (see Euclid) are contained in thirteen books. Of these the first four and the sixth are devoted to “plane geometry,” as the investigation of figures in a plane is generally called. The 5th book contains the theory of proportion which is used in Book VI. The 7th, 8th and 9th books are purely arithmetical, whilst the 10th contains a most ingenious treatment of geometrical irrational quantities. These four books will be excluded from our survey. The remaining three books relate to figures in space, or, as it is generally called, to “solid geometry.” The 7th, 8th, 9th, 10th, 13th and part of the 11th and 12th books are now generally omitted from the school editions of the Elements. In the first four and in the 6th book it is to be understood that all figures are drawn in a plane.
Book I. of Euclid’s “Elements.”
§ 6. According to the third postulate it is possible to draw in any plane a circle which has its centre at any given point, and its radius equal to the distance of this point from any other point given in the plane. This makes it possible (Prop. 1) to construct on a given line AB an equilateral triangle, by drawing first a circle with A as centre and AB as radius, and then a circle with B as centre and BA as radius. The point where these circles intersect—that they intersect Euclid quietly assumes—is the vertex of the required triangle. Euclid does not suppose, however, that a circle may be drawn which has its radius equal to the distance between any two points unless one of the points be the centre. This implies also that we are not supposed to be able to make any straight line equal to any other straight line, or to carry a distance about in space. Euclid therefore next solves the problem: It is required along a given straight line from a point in it to set off a distance equal to the length of another straight line given anywhere in the plane. This is done in two steps. It is shown in Prop. 2 how a straight line may be drawn from a given point equal in length to another given straight line not drawn from that point. And then the problem itself is solved in Prop. 3, by drawing first through the given point some straight line of the required length, and then about the same point as centre a circle having this length as radius. This circle will cut off from the given straight line a length equal to the required one. Nowadays, instead of going through this long process, we take a pair of compasses and set off the given length by its aid. This assumes that we may move a length about without changing it. But Euclid has not assumed it, and this proceeding would be fully justified by his desire not to take for granted more than was necessary, if he were not obliged at his very next step actually to make this assumption, though without stating it.
§ 7. We now come (in Prop. 4) to the first theorem. It is the fundamental theorem of Euclid’s whole system, there being only a very few propositions (like Props. 13, 14, 15, I.), except those in the 5th book and the first half of the 11th, which do not depend upon it. It is stated very accurately, though somewhat clumsily, as follows:—
If two triangles have two sides of the one equal to two sides of the other, each to each, and have also the angles contained by those sides equal to one another, they shall also have their bases or third sides equal; and the two triangles shall be equal; and their other angles shall be equal, each to each, namely, those to which the equal sides are opposite.
That is to say, the triangles are “identically” equal, and one may be considered as a copy of the other. The proof is very simple. The first triangle is taken up and placed on the second, so that the parts of the triangles which are known to be equal fall upon each other. It is then easily seen that also the remaining parts of one coincide with those of the other, and that they are therefore equal. This process of applying one figure to another Euclid scarcely uses again, though many proofs would be simplified by doing so. The process introduces motion into geometry, and includes, as already stated, the axiom that figures may be moved without change of shape or size.
If the last proposition be applied to an isosceles triangle, which has two sides equal, we obtain the theorem (Prop. 5), if two sides of a triangle are equal, then the angles opposite these sides are equal.
Euclid’s proof is somewhat complicated, and a stumblingblock to many schoolboys. The proof becomes much simpler if we consider the isosceles triangle ABC (AB = AC) twice over, once as a triangle BAC, and once as a triangle CAB; and now remember that AB, AC in the first are equal respectively to AC, AB in the second, and the angles included by these sides are equal. Hence the triangles are equal, and the angles in the one are equal to those in the other, viz. those which are opposite equal sides, i.e. angle ABC in the first equals angle ACB in the second, as they are opposite the equal sides AC and AB in the two triangles.
There follows the converse theorem (Prop. 6). If two angles in a triangle are equal, then the sides opposite them are equal,—i.e. the triangle is isosceles. The proof given consists in what is called a reductio ad absurdum, a kind of proof often used by Euclid, and principally in proving the converse of a previous theorem. It assumes that the theorem to be proved is wrong, and then shows that this assumption leads to an absurdity, i.e. to a conclusion which is in contradiction to a proposition proved before—that therefore the assumption made cannot be true, and hence that the theorem is true. It is often stated that Euclid invented this kind of proof, but the method is most likely much older.
§ 8. It is next proved that two triangles which have the three sides of the one equal respectively to those of the other are identically equal, hence that the angles of the one are equal respectively to those of the other, those being equal which are opposite equal sides. This is Prop. 8, Prop. 7 containing only a first step towards its proof.
These theorems allow now of the solution of a number of problems, viz.:—
To bisect a given angle (Prop. 9).
To bisect a given finite straight line (Prop. 10).
To draw a straight line perpendicularly to a given straight line through a given point in it (Prop. 11), and also through a given point not in it (Prop. 12).
The solutions all depend upon properties of isosceles triangles.
§ 9. The next three theorems relate to angles only, and might have been proved before Prop. 4, or even at the very beginning. The first (Prop. 13) says, The angles which one straight line makes with another straight line on one side of it either are two right angles or are together equal to two right angles. This theorem would have been unnecessary if Euclid had admitted the notion of an angle such that its two limits are in the same straight line, and had besides defined the sum of two angles.
Its converse (Prop. 14) is of great use, inasmuch as it enables us in many cases to prove that two straight lines drawn from the same point are one the continuation of the other. So also is
Prop. 15. If two straight lines cut one another, the vertical or opposite angles shall be equal.
§ 10. Euclid returns now to properties of triangles. Of great importance for the next steps (though afterwards superseded by a more complete theorem) is
Prop. 16. If one side of a triangle be produced, the exterior angle shall be greater than either of the interior opposite angles.
Prop. 17. Any two angles of a triangle are together less than two right angles, is an immediate consequence of it. By the aid of these two, the following fundamental properties of triangles are easily proved:—
Prop. 18. The greater side of every triangle has the greater angle opposite to it;
Its converse, Prop. 19. The greater angle of every triangle is subtended by the greater side, or has the greater side opposite to it;
Prop. 20. Any two sides of a triangle are together greater than the third side;
And also Prop. 21. If from the ends of the side of a triangle there be drawn two straight lines to a point within the triangle, these shall be less than the other two sides of the triangle, but shall contain a greater angle.
§ 11. Having solved two problems (Props. 22, 23), he returns to two triangles which have two sides of the one equal respectively to two sides of the other. It is known (Prop. 4) that if the included angles are equal then the third sides are equal; and conversely (Prop. 8), if the third sides are equal, then the angles included by the first sides are equal. From this it follows that if the included angles are not equal, the third sides are not equal; and conversely, that if the third sides are not equal, the included angles are not equal. Euclid now completes this knowledge by proving, that “if the included angles are not equal, then the third side in that triangle is the greater which contains the greater angle”; and conversely, that “if the third sides are unequal, that triangle contains the greater angle which contains the greater side.” These are Prop. 24 and Prop. 25.
§ 12. The next theorem (Prop. 26) says that if two triangles have one side and two angles of the one equal respectively to one side and two angles of the other, viz. in both triangles either the angles adjacent to the equal side, or one angle adjacent and one angle opposite it, then the two triangles are identically equal.
This theorem belongs to a group with Prop. 4 and Prop. 8. Its first case might have been given immediately after Prop. 4, but the second case requires Prop. 16 for its proof.
§ 13. We come now to the investigation of parallel straight lines, i.e. of straight lines which lie in the same plane, and cannot be made to meet however far they be produced either way. The investigation which starts from Prop. 16, will become clearer if a few names be explained which are not all used by Euclid. If two straight lines be cut by a third, the latter is now generally called a “transversal” of the figure. It forms at the two points where it cuts the given lines four angles with each. Those of the angles which lie between the given lines are called interior angles, and of these, again, any two which lie on opposite sides of the transversal but one at each of the two points are called “alternate angles.”
We may now state Prop. 16 thus:—If two straight lines which meet are cut by a transversal, their alternate angles are unequal. For the lines will form a triangle, and one of the alternate angles will be an exterior angle to the triangle, the other interior and opposite to it.
From this follows at once the theorem contained in Prop. 27. If two straight lines which are cut by a transversal make alternate angles equal, the lines cannot meet, however far they be produced, hence they are parallel. This proves the existence of parallel lines.
Prop. 28 states the same fact in different forms. If a straight line, falling on two other straight lines, make the exterior angle equal to the interior and opposite angle on the same side of the line, or make the interior angles on the same side together equal to two right angles, the two straight lines shall be parallel to one another.
Hence we know that, “if two straight lines which are cut by a transversal meet, their alternate angles are not equal”; and hence that, “if alternate angles are equal, then the lines are parallel.”
The question now arises, Are the propositions converse to these true or not? That is to say, “If alternate angles are unequal, do the lines meet?” And “if the lines are parallel, are alternate angles necessarily equal?”
The answer to either of these two questions implies the answer to the other. But it has been found impossible to prove that the negation or the affirmation of either is true.
The difficulty which thus arises is overcome by Euclid assuming that the first question has to be answered in the affirmative. This gives his last axiom (12), which we quote in his own words.
Axiom 12.—If a straight line meet two straight lines, so as to make the two interior angles on the same side of it taken together less than two right angles, these straight lines, being continually produced, shall at length meet on that side on which are the angles which are less than two right angles.
The answer to the second of the above questions follows from this, and gives the theorem Prop. 29:—If a straight line fall on two parallel straight lines, it makes the alternate angles equal to one another, and the exterior angle equal to the interior and opposite angle on the same side, and also the two interior angles on the same side together equal to two right angles.
§ 14. With this a new part of elementary geometry begins. The earlier propositions are independent of this axiom, and would be true even if a wrong assumption had been made in it. They all relate to figures in a plane. But a plane is only one among an infinite number of conceivable surfaces. We may draw figures on any one of them and study their properties. We may, for instance, take a sphere instead of the plane, and obtain “spherical” in the place of “plane” geometry. If on one of these surfaces lines and figures could be drawn, answering to all the definitions of our plane figures, and if the axioms with the exception of the last all hold, then all propositions up to the 28th will be true for these figures. This is the case in spherical geometry if we substitute “shortest line” or “great circle” for “straight line,” “small circle” for “circle,” and if, besides, we limit all figures to a part of the sphere which is less than a hemisphere, so that two points on it cannot be opposite ends of a diameter, and therefore determine always one and only one great circle.
For spherical triangles, therefore, all the important propositions 4, 8, 26; 5 and 6; and 18, 19 and 20 will hold good.
This remark will be sufficient to show the impossibility of proving Euclid’s last axiom, which would mean proving that this axiom is a consequence of the others, and hence that the theory of parallels would hold on a spherical surface, where the other axioms do hold, whilst parallels do not even exist.
It follows that the axiom in question states an inherent difference between the plane and other surfaces, and that the plane is only fully characterized when this axiom is added to the other assumptions.
§ 15. The introduction of the new axiom and of parallel lines leads to a new class of propositions.
After proving (Prop. 30) that “two lines which are each parallel to a third are parallel to each other,” we obtain the new properties of triangles contained in Prop. 32. Of these the second part is the most important, viz. the theorem, The three interior angles of every triangle are together equal to two right angles.
As easy deductions not given by Euclid but added by Simson follow the propositions about the angles in polygons, they are given in English editions as corollaries to Prop. 32.
These theorems do not hold for spherical figures. The sum of the interior angles of a spherical triangle is always greater than two right angles, and increases with the area.
§ 16. The theory of parallels as such may be said to be finished with Props. 33 and 34, which state properties of the parallelogram, i.e. of a quadrilateral formed by two pairs of parallels. They are—
Prop. 33. The straight lines which join the extremities of two equal and parallel straight lines towards the same parts are themselves equal and parallel; and
Prop. 34. The opposite sides and angles of a parallelogram are equal to one another, and the diameter (diagonal) bisects the parallelogram, that is, divides it into two equal parts.
§ 17. The rest of the first book relates to areas of figures.
The theory is made to depend upon the theorems—
Prop. 35. Parallelograms on the same base and between the same parallels are equal to one another; and
Prop. 36. Parallelograms on equal bases and between the same parallels are equal to one another.
As each parallelogram is bisected by a diagonal, the last theorems hold also if the word parallelogram be replaced by “triangle,” as is done in Props. 37 and 38.
It is to be remarked that Euclid proves these propositions only in the case when the parallelograms or triangles have their bases in the same straight line.
The theorems converse to the last form the contents of the next three propositions, viz.: Props, 40 and 41.—Equal triangles, on the same or on equal bases, in the same straight line, and on the same side of it, are between the same parallels.
That the two cases here stated are given by Euclid in two separate propositions proved separately is characteristic of his method.
§ 18. To compare areas of other figures, Euclid shows first, in Prop. 42, how to draw a parallelogram which is equal in area to a given triangle, and has one of its angles equal to a given angle. If the given angle is right, then the problem is solved to draw a “rectangle&rrdquo; equal in area to a given triangle.
Next this parallelogram is transformed into another parallelogram, which has one of its sides equal to a given straight line, whilst its angles remain unaltered. This may be done by aid of the theorem in
Prop. 43. The complements of the parallelograms which are about the diameter of any parallelogram are equal to one another.
Thus the problem (Prop. 44) is solved to construct a parallelogram on a given line, which is equal in area to a given triangle, and which has one angle equal to a given angle (generally a right angle).
As every polygon can be divided into a number of triangles, we can now construct a parallelogram having a given angle, say a right angle, and being equal in area to a given polygon. For each of the triangles into which the polygon has been divided, a parallelogram may be constructed, having one side equal to a given straight line and one angle equal to a given angle. If these parallelograms be placed side by side, they may be added together to form a single parallelogram, having still one side of the given length. This is done in Prop. 45.
Herewith a means is found to compare areas of different polygons. We need only construct two rectangles equal in area to the given polygons, and having each one side of given length. By comparing the unequal sides we are enabled to judge whether the areas are equal, or which is the greater. Euclid does not state this consequence, but the problem is taken up again at the end of the second book, where it is shown how to construct a square equal in area to a given polygon.
Prop. 46 is: To describe a square on a given straight line.
§ 19. The first book concludes with one of the most important theorems in the whole of geometry, and one which has been celebrated since the earliest times. It is stated, but on doubtful authority, that Pythagoras discovered it, and it has been called by his name. If we call that side in a rightangled triangle which is opposite the right angle the hypotenuse, we may state it as follows:—
Theorem of Pythagoras (Prop. 47).—In every rightangled triangle the square on the hypotenuse is equal to the sum of the squares of the other sides.
And conversely—
Prop. 48. If the square described on one of the sides of a triangle be equal to the squares described on the other sides, then the angle contained by these two sides is a right angle.
On this theorem (Prop. 47) almost all geometrical measurement depends, which cannot be directly obtained.
Book II.
§ 20. The propositions in the second book are very different in character from those in the first; they all relate to areas of rectangles and squares. Their true significance is best seen by stating them in an algebraic form. This is often done by expressing the lengths of lines by aid of numbers, which tell how many times a chosen unit is contained in the lines. If there is a unit to be found which is contained an exact number of times in each side of a rectangle, it is easily seen, and generally shown in the teaching of arithmetic, that the rectangle contains a number of unit squares equal to the product of the numbers which measure the sides, a unit square being the square on the unit line. If, however, no such unit can be found, this process requires that connexion between lines and numbers which is only established by aid of ratios of lines, and which is therefore at this stage altogether inadmissible. But there exists another way of connecting these propositions with algebra, based on modern notions which seem destined greatly to change and to simplify mathematics. We shall introduce here as much of it as is required for our present purpose.
At the beginning of the second book we find a definition according to which “a rectangle is said to be ‘contained’ by the two sides which contain one of its right angles”; in the text this phraseology is extended by speaking of rectangles contained by any two straight lines, meaning the rectangle which has two adjacent sides equal to the two straight lines.
We shall denote a finite straight line by a single small letter, a, b, c, . . . x, and the area of the rectangle contained by two lines a and b by ab, and this we shall call the product of the two lines a and b. It will be understood that this definition has nothing to do with the definition of a product of numbers.
We define as follows:—
The sum of two straight lines a and b means a straight line c which may be divided in two parts equal respectively to a and b. This sum is denoted by a + b.
The difference of two lines a and b (in symbols, a−b) means a line c which when added to b gives a; that is,
The product of two lines a and b (in symbols, ab) means the area of the rectangle contained by the lines a and b. For aa, which means the square on the line a, we write a^{2}.
§ 21. The first ten of the fourteen propositions of the second book may then be written in the form of formulae as follows:—
Prop.  1.  a (b + c + d + ... ) = ab + ac + ad + ... 
”  2.  ab + ac = a^{2} if b + c = a. 
”  3.  a (a + b) = a^{2} + ab. 
”  4.  (a + b)^{2} = a^{2} + 2ab + b^{2}. 
”  5.  (a + b)(a − b) + b^{2} = a^{2}. 
”  6.  (a + b)(a − b) + b^{2} = a^{2}. 
”  7.  a^{2} + (a − b)^{2} = 2a (a − b) + b^{2}. 
”  8.  4(a + b)a + b^{2} = (2a + b)^{2}. 
”  9.  (a + b)^{2} + (a − b)^{2} = 2a^{2} + 2b^{2}. 
”  10.  (a + b)^{2} + (a − b)^{2} = 2a^{2} + 2b^{2}. 
It will be seen that 5 and 6, and also 9 and 10, are identical. In Euclid’s statement they do not look the same, the figures being arranged differently.
If the letters a, b, c, ... denoted numbers, it follows from algebra that each of these formulae is true. But this does not prove them in our case, where the letters denote lines, and their products areas without any reference to numbers. To prove them we have to discover the laws which rule the operations introduced, viz. addition and multiplication of segments. This we shall do now; and we shall find that these laws are the same with those which hold in algebraical addition and multiplication.
§ 22. In a sum of numbers we may change the order in which the numbers are added, and we may also add the numbers together in groups and then add these groups. But this also holds for the sum of segments and for the sum of rectangles, as a little consideration shows. That the sum of rectangles has always a meaning follows from the Props. 4345 in the first book. These laws about addition are reducible to the two—
a + b = b + a  (1), 
a + (b + c) = a + b + c  (2); 
or, when expressed for rectangles,
ab + ed = ed + ab  (3), 
ab + (cd + ef) = ab + cd + ef  (4). 
The brackets mean that the terms in the bracket have been added together before they are added to another term. The more general cases for more terms may be deduced from the above.
For the product of two numbers we have the law that it remains unaltered if the factors be interchanged. This also holds for our geometrical product. For if ab denotes the area of the rectangle which has a as base and b as altitude, then ba will denote the area of the rectangle which has b as base and a as altitude. But in a rectangle we may take either of the two lines which contain it as base, and then the other will be the altitude. This gives
ab = ba  (5). 
In order further to multiply a sum by a number, we have in algebra the rule:—Multiply each term of the sum, and add the products thus obtained. That this holds for our geometrical products is shown by Euclid in his first proposition of the second book, where he proves that the area of a rectangle whose base is the sum of a number of segments is equal to the sum of rectangles which have these segments separately as bases. In symbols this gives, in the simplest case,
a(b + c) = ab + ac  
and  (b + c)a = ba + ca  (6). 
To these laws, which have been investigated by Sir William Hamilton and by Hermann Grassmann, the former has given special names. He calls the laws expressed in
(1) and (3) the commutative law for addition; (5) the commutative law for multiplication; (2) and (4) the associative laws for addition; (6) the distributive law. 
§ 23. Having proved that these six laws hold, we can at once prove every one of the above propositions in their algebraical form.
The first is proved geometrically, it being one of the fundamental laws. The next two propositions are only special cases of the first. Of the others we shall prove one, viz. the fourth:—
But
and
Therefore
(a + b)^{2}  = aa + ab + (ab + bb) 
= aa + (ab + ab) + bb  
= aa + 2ab + bb 
This gives the theorem in question.
In the same manner every one of the first ten propositions is proved.
It will be seen that the operations performed are exactly the same as if the letters denoted numbers.
Props. 5 and 6 may also be written thus—
Prop. 7, which is an easy consequence of Prop. 4, may be transformed. If we denote by c the line a + b, so that
we get
c^{2} + (c − b)^{2}  = 2c(c − b) + b^{2} 
= 2c^{2} − 2bc + b^{2}. 
Subtracting c^{2} from both sides, and writing a for c, we get
In Euclid’s Elements this form of the theorem does not appear, all propositions being so stated that the notion of subtraction does not enter into them.
§ 24. The remaining two theorems (Props. 12 and 13) connect the square on one side of a triangle with the sum of the squares on the other sides, in case that the angle between the latter is acute or obtuse. They are important theorems in trigonometry, where it is possible to include them in a single theorem.
§ 25. There are in the second book two problems, Props. 11 and 14.
If written in the above symbolic language, the former requires to find a line x such that a(a − x) = x^{2}. Prop. 11 contains, therefore, the solution of a quadratic equation, which we may write x^{2} + ax = a^{2}. The solution is required later on in the construction of a regular decagon.
More important is the problem in the last proposition (Prop. 14). It requires the construction of a square equal in area to a given rectangle, hence a solution of the equation
In Book I., 4245, it has been shown how a rectangle may be constructed equal in area to a given figure bounded by straight lines. By aid of the new proposition we may therefore now determine a line such that the square on that line is equal in area to any given rectilinear figure, or we can square any such figure.
As of two squares that is the greater which has the greater side, it follows that now the comparison of two areas has been reduced to the comparison of two lines.
The problem of reducing other areas to squares is frequently met with among Greek mathematicians. We need only mention the problem of squaring the circle (see Circle).
In the present day the comparison of areas is performed in a simpler way by reducing all areas to rectangles having a common base. Their altitudes give then a measure of their areas.
The construction of a rectangle having the base u, and being equal in area to a given rectangle, depends upon Prop. 43, I. This therefore gives a solution of the equation
where x denotes the unknown altitude.
Book III.
§ 26. The third book of the Elements relates exclusively to properties of the circle. A circle and its circumference have been defined in Book I., Def. 15. We restate it here in slightly different words:—
Definition.—The circumference of a circle is a plane curve such that all points in it have the same distance from a fixed point in the plane. This point is called the “centre” of the circle.
Of the new definitions, of which eleven are given at the beginning of the third book, a few only require special mention. The first, which says that circles with equal radii are equal, is in part a theorem, but easily proved by applying the one circle to the other. Or it may be considered proved by aid of Prop. 24, equal circles not being used till after this theorem.
In the second definition is explained what is meant by a line which “touches” a circle. Such a line is now generally called a tangent to the circle. The introduction of this name allows us to state many of Euclid’s propositions in a much shorter form.
For the same reason we shall call a straight line joining two points on the circumference of a circle a “chord.”
Definitions 4 and 5 may be replaced with a slight generalization by the following:—
Definition.—By the distance of a point from a line is meant the length of the perpendicular drawn from the point to the line.
§ 27. From the definition of a circle it follows that every circle has a centre. Prop. 1 requires to find it when the circle is given, i.e. when its circumference is drawn.
To solve this problem a chord is drawn (that is, any two points in the circumference are joined), and through the point where this is bisected a perpendicular to it is erected. Euclid then proves, first, that no point off this perpendicular can be the centre, hence that the centre must lie in this line; and, secondly, that of the points on the perpendicular one only can be the centre, viz. the one which bisects the parts of the perpendicular bounded by the circle. In the second part Euclid silently assumes that the perpendicular there used does cut the circumference in two, and only in two points. The proof therefore is incomplete. The proof of the first part, however, is exact. By drawing two nonparallel chords, and the perpendiculars which bisect them, the centre will be found as the point where these perpendiculars intersect.
§ 28. In Prop. 2 it is proved that a chord of a circle lies altogether within the circle.
What we have called the first part of Euclid’s solution of Prop. 1 may be stated as a theorem:—
Every straight line which bisects a chord, and is at right angles to it, passes through the centre of the circle.
The converse to this gives Prop. 3, which may be stated thus:—
If a straight line through the centre of a circle bisect a chord, then it is perpendicular to the chord, and if it be perpendicular to the chord it bisects it.
An easy consequence of this is the following theorem, which is essentially the same as Prop. 4:—
Two chords of a circle, of which neither passes through the centre, cannot bisect each other.
These last three theorems are fundamental for the theory of the circle. It is to be remarked that Euclid never proves that a straight line cannot have more than two points in common with a circumference.
§ 29. The next two propositions (5 and 6) might be replaced by a single and a simpler theorem, viz:—
Two circles which have a common centre, and whose circumferences have one point in common, coincide.
Or, more in agreement with Euclid’s form:—
Two different circles, whose circumferences have a point in common, cannot have the same centre.
That Euclid treats of two cases is characteristic of Greek mathematics.
The next two propositions (7 and 8) again belong together. They may be combined thus:—
If from a point in a plane of a circle, which is not the centre, straight lines be drawn to the different points of the circumference, then of all these lines one is the shortest, and one the longest, and these lie both in that straight line which joins the given point to the centre. Of all the remaining lines each is equal to one and only one other, and these equal lines lie on opposite sides of the shortest or longest, and make equal angles with them.
Euclid distinguishes the two cases where the given point lies within or without the circle, omitting the case where it lies in the circumference.
From the last proposition it follows that if from a point more than two equal straight lines can be drawn to the circumference, this point must be the centre. This is Prop. 9.
As a consequence of this we get
If the circumferences of the two circles have three points in common they coincide.
For in this case the two circles have a common centre, because from the centre of the one three equal lines can be drawn to points on the circumference of the other. But two circles which have a common centre, and whose circumferences have a point in common, coincide. (Compare above statement of Props. 5 and 6.)
This theorem may also be stated thus:—
Through three points only one circumference may be drawn; or, Three points determine a circle.
Euclid does not give the theorem in this form. He proves, however, that the two circles cannot cut another in more than two points (Prop. 10), and that two circles cannot touch one another in more points than one (Prop. 13).
§ 30. Propositions 11 and 12 assert that if two circles touch, then the point of contact lies on the line joining their centres. This gives two propositions, because the circles may touch either internally or externally.
§ 31. Propositions 14 and 15 relate to the length of chords. The first says that equal chords are equidistant from the centre, and that chords which are equidistant from the centre are equal;
Whilst Prop. 15 compares unequal chords, viz. Of all chords the diameter is the greatest, and of other chords that is the greater which is nearer to the centre; and conversely, the greater chord is nearer to the centre.
§ 32. In Prop. 16 the tangent to a circle is for the first time introduced. The proposition is meant to show that the straight line at the end point of the diameter and at right angles to it is a tangent. The proposition itself does not state this. It runs thus:—
Prop. 16. The straight line drawn at right angles to the diameter of a circle, from the extremity of it, falls without the circle; and no straight line can be drawn from the extremity, between that straight line and the circumference, so as not to cut the circle.
Corollary.—The straight line at right angles to a diameter drawn through the end point of it touches the circle.
The statement of the proposition and its whole treatment show the difficulties which the tangents presented to Euclid.
Prop. 17 solves the problem through a given point, either in the circumference or without it, to draw a tangent to a given circle.
Closely connected with Prop. 16 are Props. 18 and 19, which state (Prop. 18), that the line joining the centre of a circle to the point of contact of a tangent is perpendicular to the tangent; and conversely (Prop. 19), that the straight line through the point of contact of, and perpendicular to, a tangent to a circle passes through the centre of the circle.
§ 33. The rest of the book relates to angles connected with a circle, viz. angles which have the vertex either at the centre or on the circumference, and which are called respectively angles at the centre and angles at the circumference. Between these two kinds of angles exists the important relation expressed as follows:—
Prop. 20. The angle at the centre of a circle is double of the angle at the circumference on the same base, that is, on the same arc.
This is of great importance for its consequences, of which the two following are the principal:—
Prop. 21. The angles in the same segment of a circle are equal to one another;
Prop. 22. The opposite angles of any quadrilateral figure inscribed in a circle are together equal to two right angles.
Further consequences are:—
Prop. 23. On the same straight line, and on the same side of it, there cannot be two similar segments of circles, not coinciding with one another;
Prop. 24. Similar segments of circles on equal straight lines are equal to one another.
The problem Prop. 25. A segment of a circle being given to describe the circle of which it is a segment, may be solved much more easily by aid of the construction described in relation to Prop. 1, III., in § 27.
§ 34. There follow four theorems connecting the angles at the centre, the arcs into which they divide the circumference, and the chords subtending these arcs. They are expressed for angles, arcs and chords in equal circles, but they hold also for angles, arcs and chords in the same circle.
The theorems are:—
Prop. 26. In equal circles equal angles stand on equal arcs, whether they be at the centres or circumferences;
Prop. 27. (converse to Prop. 26). In equal circles the angles which stand on equal arcs are equal to one another, whether they be at the centres or the circumferences;
Prop. 28. In equal circles equal straight lines (equal chords) cut off equal arcs, the greater equal to the greater, and the less equal to the less;
Prop. 29 (converse to Prop. 28). In equal circles equal arcs are subtended by equal straight lines.
§ 35. Other important consequences of Props. 2022 are:—
Prop. 31. In a circle the angle in a semicircle is a right angle; but the angle in a segment greater than a semicircle is less than a right angle; and the angle in a segment less than a semicircle is greater than a right angle;
Prop. 32. If a straight line touch a circle, and from the point of contact a straight line be drawn cutting the circle, the angles which this line makes with the line touching the circle shall be equal to the angles which are in the alternate segments of the circle.
§ 36. Propositions 30, 33, 34, contain problems which are solved by aid of the propositions preceding them:—
Prop. 30. To bisect a given arc, that is, to divide it into two equal parts;
Prop. 33. On a given straight line to describe a segment of a circle containing an angle equal to a given rectilineal angle;
Prop. 34. From a given circle to cut off a segment containing an angle equal to a given rectilineal angle.
§ 37. If we draw chords through a point A within a circle, they will each be divided by A into two segments. Between these segments the law holds that the rectangle contained by them has the same area on whatever chord through A the segments are taken. The value of this rectangle changes, of course, with the position of A.
A similar theorem holds if the point A be taken without the circle. On every straight line through A, which cuts the circle in two points B and C, we have two segments AB and AC, and the rectangles contained by them are again equal to one another, and equal to the square on a tangent drawn from A to the circle.
The first of these theorems gives Prop. 35, and the second Prop. 36, with its corollary, whilst Prop. 37, the last of Book III., gives the converse to Prop. 36. The first two theorems may be combined in one:—
If through a point A in the plane of a circle a straight line be drawn cutting the circle in B and C, then the rectangle AB.AC has a constant value so long as the point A be fixed; and if from A a tangent AD can be drawn to the circle, touching at D, then the above rectangle equals the square on AD.
Prop. 37 may be stated thus:—
If from a point A without a circle a line be drawn cutting the circle in B and C, and another line to a point D on the circle, and AB.AC = AD^{2}, then the line AD touches the circle at D.
It is not difficult to prove also the converse to the general proposition as above stated. This proposition and its converse may be expressed as follows:—
If four points ABCD be taken on the circumference of a circle, and if the lines AB, CD, produced if necessary, meet at E, then
and conversely, if this relation holds then the four points lie on a circle, that is, the circle drawn through three of them passes through the fourth.
That a circle may always be drawn through three points, provided that they do not lie in a straight line, is proved only later on in Book IV.
BOOK IV.
§ 38. The fourth book contains only problems, all relating to the construction of triangles and polygons inscribed in and circumscribed about circles, and of circles inscribed in or circumscribed about triangles and polygons. They are nearly all given for their own sake, and not for future use in the construction of figures, as are most of those in the former books. In seven definitions at the beginning of the book it is explained what is understood by figures inscribed in or described about other figures, with special reference to the case where one figure is a circle. Instead, however, of saying that one figure is described about another, it is now generally said that the one figure is circumscribed about the other. We may then state the definitions 3 or 4 thus:—
Definition.—A polygon is said to be inscribed in a circle, and the circle is said to be circumscribed about the polygon, if the vertices of the polygon lie in the circumference of the circle.
And definitions 5 and 6 thus:—
Definition.—A polygon is said to be circumscribed about a circle, and a circle is said to be inscribed in a polygon, if the sides of the polygon are tangents to the circle.
§ 39. The first problem is merely constructive. It requires to draw in a given circle a chord equal to a given straight line, which is not greater than the diameter of the circle. The problem is not a determinate one, inasmuch as the chord may be drawn from any point in the circumference. This may be said of almost all problems in this book, especially of the next two. They are:—
Prop. 2. In a given circle to inscribe a triangle equiangular to a given triangle;
Prop. 3. About a given circle to circumscribe a triangle equiangular to a given triangle.
§ 40. Of somewhat greater interest are the next problems, where the triangles are given and the circles to be found.
Prop. 4. To inscribe a circle in a given triangle.
The result is that the problem has always a solution, viz. the centre of the circle is the point where the bisectors of two of the interior angles of the triangle meet. The solution shows, though Euclid does not state this, that the problem has but one solution; and also,
The three bisectors of the interior angles of any triangle meet in a point, and this is the centre of the circle inscribed in the triangle.
The solutions of most of the other problems contain also theorems. Of these we shall state those which are of special interest; Euclid does not state any one of them.
§ 41. Prop. 5. To circumscribe a circle about a given triangle.
The one solution which always exists contains the following:—
The three straight lines which bisect the sides of a triangle at right angles meet in a point, and this point is the centre of the circle circumscribed about the triangle.
Euclid adds in a corollary the following property:—
The centre of the circle circumscribed about a triangle lies within, on a side of, or without the triangle, according as the triangle is acuteangled, rightangled or obtuseangled.
§ 42. Whilst it is always possible to draw a circle which is inscribed in or circumscribed about a given triangle, this is not the case with quadrilaterals or polygons of more sides. Of those for which this is possible the regular polygons, i.e. polygons which have all their sides and angles equal, are the most interesting. In each of them a circle may be inscribed, and another may be circumscribed about it.
Euclid does not use the word regular, but he describes the polygons in question as equiangular and equilateral. We shall use the name regular polygon. The regular triangle is equilateral, the regular quadrilateral is the square.
Euclid considers the regular polygons of 4, 5, 6 and 15 sides. For each of the first three he solves the problems—(1) to inscribe such a polygon in a given circle; (2) to circumscribe it about a given circle; (3) to inscribe a circle in, and (4) to circumscribe a circle about, such a polygon.
For the regular triangle the problems are not repeated, because more general problems have been solved.
Props. 6, 7, 8 and 9 solve these problems for the square.
The general problem of inscribing in a given circle a regular polygon of n sides depends upon the problem of dividing the circumference of a circle into n equal parts, or what comes to the same thing, of drawing from the centre of the circle n radii such that the angles between consecutive radii are equal, that is, to divide the space about the centre into n equal angles. Thus, if it is required to inscribe a square in a circle, we have to draw four lines from the centre, making the four angles equal. This is done by drawing two diameters at right angles to one another. The ends of these diameters are the vertices of the required square. If, on the other hand, tangents be drawn at these ends, we obtain a square circumscribed about the circle.
§ 43. To construct a regular pentagon, we find it convenient first to construct a regular decagon. This requires to divide the space about the centre into ten equal angles. Each will be 110th of a right angle, or 15th of two right angles. If we suppose the decagon constructed, and if we join the centre to the end of one side, we get an isosceles triangle, where the angle at the centre equals 15th of two right angles; hence each of the angles at the base will be 25ths of two right angles, as all three angles together equal two right angles. Thus we have to construct an isosceles triangle, having the angle at the vertex equal to half an angle at the base. This is solved in Prop. 10, by aid of the problem in Prop. 11 of the second book. If we make the sides of this triangle equal to the radius of the given circle, then the base will be the side of the regular decagon inscribed in the circle. This side being known the decagon can be constructed, and if the vertices are joined alternately, leaving out half their number, we obtain the regular pentagon. (Prop. 11.)
Euclid does not proceed thus. He wants the pentagon before the decagon. This, however, does not change the real nature of his solution, nor does his solution become simpler by not mentioning the decagon.
Once the regular pentagon is inscribed, it is easy to circumscribe another by drawing tangents at the vertices of the inscribed pentagon. This is shown in Prop. 12.
Props. 13 and 14 teach how a circle may be inscribed in or circumscribed about any given regular pentagon.
§ 44. The regular hexagon is more easily constructed, as shown in Prop. 15. The result is that the side of the regular hexagon inscribed in a circle is equal to the radius of the circle.
For this polygon the other three problems mentioned are not solved.
§ 45. The book closes with Prop. 16. To inscribe a regular quindecagon in a given circle. If we inscribe a regular pentagon and a regular hexagon in the circle, having one vertex in common, then the arc from the common vertex to the next vertex of the pentagon is 15th of the circumference, and to the next vertex of the hexagon is 16th of the circumference. The difference between these arcs is, therefore, 15 − 16 = 130th of the circumference. The latter may, therefore, be divided into thirty, and hence also in fifteen equal parts, and the regular quindecagon be described.
§ 46. We conclude with a few theorems about regular polygons which are not given by Euclid.
The straight lines perpendicular to and bisecting the sides of any regular polygon meet in a point. The straight lines bisecting the angles in the regular polygon meet in the same point. This point is the centre of the circles circumscribed about and inscribed in the regular polygon.
We can bisect any given arc (Prop. 30, III.). Hence we can divide a circumference into 2n equal parts as soon as it has been divided into n equal parts, or as soon as a regular polygon of n sides has been constructed. Hence—
If a regular polygon of n sides has been constructed, then a regular polygon of 2n sides, of 4n, of 8n sides, &c., may also be constructed. Euclid shows how to construct regular polygons of 3, 4, 5 and 15 sides. It follows that we can construct regular polygons of
3,  6,  12,  24  sides 
4,  8,  16,  32  ” 
5,  10,  20,  40  ” 
15,  30,  60,  120  ” 
The construction of any new regular polygon not included in one of these series will give rise to a new series. Till the beginning of the 19th century nothing was added to the knowledge of regular polygons as given by Euclid. Then Gauss, in his celebrated Arithmetic, proved that every regular polygon of 2^{n} + 1 sides may be constructed if this number 2^{n} + 1 be prime, and that no others except those with 2^{m} (2^{n} + 1) sides can be constructed by elementary methods. This shows that regular polygons of 7, 9, 13 sides cannot thus be constructed, but that a regular polygon of 17 sides is possible; for 17 = 2^{4} + 1. The next polygon is one of 257 sides. The construction becomes already rather complicated for 17 sides.
Book V.
§ 47. The fifth book of the Elements is not exclusively geometrical. It contains the theory of ratios and proportion of quantities in general. The treatment, as here given, is admirable, and in every respect superior to the algebraical method by which Euclid’s theory is now generally replaced. We shall treat the subject in order to show why the usual algebraical treatment of proportion is not really sound. We begin by quoting those definitions at the beginning of Book V. which are most important. These definitions have given rise to much discussion.
The only definitions which are essential for the fifth book are Defs. 1, 2, 4, 5, 6 and 7. Of the remainder 3, 8 and 9 are more than useless, and probably not Euclid’s, but additions of later editors, of whom Theon of Alexandria was the most prominent. Defs. 10 and 11 belong rather to the sixth book, whilst all the others are merely nominal. The really important ones are 4, 5, 6 and 7.
§ 48. To define a magnitude is not attempted by Euclid. The first two definitions state what is meant by a “part,” that is, a submultiple or measure, and by a “multiple” of a given magnitude. The meaning of Def. 4 is that two given quantities can have a ratio to one another only in case that they are comparable as to their magnitude, that is, if they are of the same kind.
Def. 3, which is probably due to Theon, professes to define a ratio, but is as meaningless as it is uncalled for, for all that is wanted is given in Defs. 5 and 7.
In Def. 5 it is explained what is meant by saying that two magnitudes have the same ratio to one another as two other magnitudes, and in Def. 7 what we have to understand by a greater or a less ratio. The 6th definition is only nominal, explaining the meaning of the word proportional.
Euclid represents magnitudes by lines, and often denotes them either by single letters or, like lines, by two letters. We shall use only single letters for the purpose. If a and b denote two magnitudes of the same kind, their ratio will be denoted by a : b; if c and d are two other magnitudes of the same kind, but possibly of a different kind from a and b, then if c and d have the same ratio to one another as a and b, this will be expressed by writing—
Further, if m is a (whole) number, ma shall denote the multiple of a which is obtained by taking it m times.
§ 49. The whole theory of ratios is based on Def. 5.
Def. 5. The first of four magnitudes is said to have the same ratio to the second that the third has to the fourth when, any equimultiples whatever of the first and the third being taken, and any equimultiples whatever of the second and the fourth, if the multiple of the first be less than that of the second, the multiple of the third is also less than that of the fourth; and if the multiple of the first is equal to that of the second, the multiple of the third is also equal to that of the fourth; and if the multiple of the first is greater than that of the second, the multiple of the third is also greater than that of the fourth.
It will be well to show at once in an example how this definition can be used, by proving the first part of the first proposition in the sixth book. Triangles of the same altitude are to one another as their bases, or if a and b are the bases, and α and β the areas, of two triangles which have the same altitude, then a : b :: α : β.
To prove this, we have, according to Definition 5, to show—

That this is true is in our case easily seen. We may suppose that the triangles have a common vertex, and their bases in the same line. We set off the base a along the line containing the bases m times; we then join the different parts of division to the vertex, and get m triangles all equal to α. The triangle on ma as base equals, therefore, mα. If we proceed in the same manner with the base b, setting it off n times, we find that the area of the triangle on the base nb equals nβ, the vertex of all triangles being the same. But if two triangles have the same altitude, then their areas are equal if the bases are equal; hence mα = nβ if ma = nb, and if their bases are unequal, then that has the greater area which is on the greater base; in other words, mα is greater than, equal to, or less than nβ, according as ma is greater than, equal to, or less than nb, which was to be proved.
§ 50. It will be seen that even in this example it does not become evident what a ratio really is. It is still an open question whether ratios are magnitudes which we can compare. We do not know whether the ratio of two lines is a magnitude of the same kind as the ratio of two areas. Though we might say that Def. 5 defines equal ratios, still we do not know whether they are equal in the sense of the axiom, that two things which are equal to a third are equal to one another. That this is the case requires a proof, and until this proof is given we shall use the :: instead of the sign =, which, however, we shall afterwards introduce.
As soon as it has been established that all ratios are like magnitudes, it becomes easy to show that, in some cases at least, they are numbers. This step was never made by Greek mathematicians. They distinguished always most carefully between continuous magnitudes and the discrete series of numbers. In modern times it has become the custom to ignore this difference.
If, in determining the ratio of two lines, a common measure can be found, which is contained m times in the first, and n times in the second, then the ratio of the two lines equals the ratio of the two numbers m : n. This is shown by Euclid in Prop. 5, X. But the ratio of two numbers is, as a rule, a fraction, and the Greeks did not, as we do, consider fractions as numbers. Far less had they any notion of introducing irrational numbers, which are neither whole nor fractional, as we are obliged to do if we wish to say that all ratios are numbers. The incommensurable numbers which are thus introduced as ratios of incommensurable quantities are nowadays as familiar to us as fractions; but a proof is generally omitted that we may apply to them the rules which have been established for rational numbers only. Euclid’s treatment of ratios avoids this difficulty. His definitions hold for commensurable as well as for incommensurable quantities. Even the notion of incommensurable quantities is avoided in Book V. But he proves that the more elementary rules of algebra hold for ratios. We shall state all his propositions in that algebraical form to which we are now accustomed. This may, of course, be done without changing the character of Euclid’s method.
§. 51. Using the notation explained above we express the first propositions as follows:—
Prop. 1. If
then
Prop. 2. If

then a + e is the same multiple of b as c + f is of d, viz.:—
Prop. 3. If a = mb, c = md, then is na the same multiple of b that nc is of d, viz. na = nmb, nc = nmd.
Prop. 4. If
then
Prop. 5. If
then
Prop. 6. If
then are a − nb and c − nd either equal to, or equimultiples of, b and d, viz. a − nb = (m − n)b and c − nd = (m − n)d, where m − n may be unity.
All these propositions relate to equimultiples. Now follow propositions about ratios which are compared as to their magnitude.
§ 52. Prop. 7. If a = b, then a : c :: b : c and c : a :: c : b.
The proof is simply this. As a = b we know that ma = mb; therefore if
if
if
therefore the first proportion holds by Definition 5.
Prop. 8. If
and
The proof depends on Definition 7.
Prop. 9 (converse to Prop. 7). If
or if
Prop. 10 (converse to Prop. 8). If
and if
Prop. 11. If
and
then
In words, if too ratios are equal to a third, they are equal to one another. After these propositions have been proved, we have a right to consider a ratio as a magnitude, for only now can we consider a ratio as something for which the axiom about magnitudes holds: things which are equal to a third are equal to one another.
We shall indicate this by writing in future the sign = instead of ::. The remaining propositions, which explain themselves, may then be stated as follows:
§ 53. Prop. 12. If
then
Prop. 13. If
then
Prop. 14. If
Prop. 15. Magnitudes have the same ratio to one another that their equimultiples have—
Prop. 16. If a, b, c, d are magnitudes of the same kind, and if
then
Prop. 17. If
then
Prop. 18 (converse to 17). If
then
Prop. 19. If a, b, c, d are quantities of the same kind, and if
then
§ 54. Prop. 20. If there be three magnitudes, and another three, which have the same ratio, taken two and two, then if the first be greater than the third, the fourth shall be greater than the sixth: and if equal, equal; and if less, less.
If we understand by
that the ratio of any two consecutive magnitudes on the first side equals that of the corresponding magnitudes on the second side, we may write this theorem in symbols, thus:—
If a, b, c be quantities of one, and d, e, f magnitudes of the same or any other kind, such that
and if
but if
and if
Prop. 21. If
or if
and if  a > c, then d > f, 
but if  a = c, then d = f, 
and if  a < c, then d < f. 
By aid of these two propositions the following two are proved.
§ 55. Prop. 22. If there be any number of magnitudes, and as many others, which have the same ratio, taken two and two in order, the first shall have to the last of the first magnitudes the same ratio which the first of the others has to the last.
We may state it more generally, thus:
then not only have two consecutive, but any two magnitudes on the first side, the same ratio as the corresponding magnitudes on the other. For instance—
Prop. 23 we state only in symbols, viz.:—
If  a : b : c : d : e : ... = 1/a′ : 1/b′ : 1/c′ : 1/d′ : 1/e′ . . . ,  
then  a : c = c′ : a′,  
b : e = e′ : b′, 
and so on.
Prop. 24 comes to this: If a : b = c : d and e : b = f : d, then
Some of the proportions which are considered in the above propositions have special names. These we have omitted, as being of no use, since algebra has enabled us to bring the different operations contained in the propositions under a common point of view.
§ 56. The last proposition in the fifth book is of a different character.
Prop. 25. If four magnitudes of the same kind be proportional, the greatest and least of them together shall be greater than the other two together. In symbols—
If a, b, c, d be magnitudes of the same kind, and if a : b = c : d, and if a is the greatest, hence d the least, then a + d > b + c.
§ 57. We return once again to the question. What is a ratio? We have seen that we may treat ratios as magnitudes, and that all ratios are magnitudes of the same kind, for we may compare any two as to their magnitude. It will presently be shown that ratios of lines may be considered as quotients of lines, so that a ratio appears as answer to the question, How often is one line contained in another? But the answer to this question is given by a number, at least in some cases, and in all cases if we admit incommensurable numbers. Considered from this point of view, we may say the fifth book of the Elements shows that some of the simpler algebraical operations hold for incommensurable numbers. In the ordinary algebraical treatment of numbers this proof is altogether omitted, or given by a process of limits which does not seem to be natural to the subject.
Book VI.
§ 58. The sixth book contains the theory of similar figures. After a few definitions explaining terms, the first proposition gives the first application of the theory of proportion.
Prop. 1. Triangles and parallelograms of the same altitude are to one another as their bases.
The proof has already been considered in § 49.
From this follows easily the important theorem
Prop. 2. If a straight line be drawn parallel to one of the sides of a triangle it shall cut the other sides, or those sides produced, proportionally; and if the sides or the sides produced be cut proportionally, the straight line which joins the points of section shall be parallel to the remaining side of the triangle.
§ 59. The next proposition, together with one added by Simson as Prop. A, may be expressed more conveniently if we introduce a modern phraseology, viz. if in a line AB we assume a point C between A and B, we shall say that C divides AB internally in the ratio AC : CB; but if C be taken in the line AB produced, we shall say that AB is divided externally in the ratio AC : CB.
The two propositions then come to this:
Prop. 3. The bisector of an angle in a triangle divides the opposite side internally in a ratio equal to the ratio of the two sides including that angle; and conversely, if a line through the vertex of a triangle divide the base internally in the ratio of the two other sides, then that line bisects the angle at the vertex.
Simson’s Prop. A. The line which bisects an exterior angle of a triangle divides the opposite side externally in the ratio of the other sides; and conversely, if a line through the vertex of a triangle divide the base externally in the ratio of the sides, then it bisects an exterior angle at the vertex of the triangle.
If we combine both we have—
The two lines which bisect the interior and exterior angles at one vertex of a triangle divide the opposite side internally and externally in the same ratio, viz. in the ratio of the other two sides.
§ 60. The next four propositions contain the theory of similar triangles, of which four cases are considered. They may be stated together.
Two triangles are similar,—
1. (Prop. 4). If the triangles are equiangular:
2. (Prop. 5). If the sides of the one are proportional to those of the other;
3. (Prop. 6). If two sides in one are proportional to two sides in the other, and if the angles contained by these sides are equal;
4. (Prop. 7). If two sides in one are proportional to two sides in the other, if the angles opposite homologous sides are equal, and if the angles opposite the other homologous sides are both acute, both right or both obtuse; homologous sides being in each case those which are opposite equal angles.
An important application of these theorems is at once made to a rightangled triangle, viz.:—
Prop. 8. In a rightangled triangle, if a perpendicular be drawn from the right angle to the base, the triangles on each side of it are similar to the whole triangle, and to one another.
Corollary.—From this it is manifest that the perpendicular drawn from the right angle of a rightangled triangle to the base is a mean proportional between the segments of the base, and also that each of the sides is a mean proportional between the base and the segment of the base adjacent to that side.
§ 61. There follow four propositions containing problems, in language slightly different from Euclid’s, viz.:—
Prop. 9. To divide a straight line into a given number of equal parts.
Prop. 10. To divide a straight line in a given ratio.
Prop. 11. To find a third proportional to two given straight lines.
Prop. 12. To find a fourth proportional to three given straight lines.
Prop. 13. To find a mean proportional between two given straight lines.
The last three may be written as equations with one unknown quantity—viz. if we call the given straight lines a, b, c, and the required line x, we have to find a line x so that
Prop. 11.
Prop. 12.
Prop. 13.
We shall see presently how these may be written without the signs of ratios.
§ 62. Euclid considers next proportions connected with parallelograms and triangles which are equal in area.
Prop. 14. Equal parallelograms which have one angle of the one equal to one angle of the other have their sides about the equal angles reciprocally proportional; and parallelograms which have one angle of the one equal to one angle of the other, and their sides about the equal angles reciprocally proportional, are equal to one another.
Prop. 15. Equal triangles which have one angle of the one equal to one angle of the other, have their sides about the equal angles reciprocally proportional; and triangles which have one angle of the one equal to one angle of the other, and their sides about the equal angles reciprocally proportional, are equal to one another.
The latter proposition is really the same as the former, for if, as in the accompanying diagram, in the figure belonging to the former the two equal parallelograms AB and BC be bisected by the lines DF and EG, and if EF be drawn, we get the figure belonging to the latter.
It is worth noticing that the lines FE and DG are parallel. We may state therefore the theorem—
If two triangles are equal in area, and have one angle in the one vertically opposite to one angle in the other, then the two straight lines which join the remaining two vertices of the one to those of the other triangle are parallel.
§ 63. A most important theorem is
Prop. 16. If four straight lines be proportionals, the rectangle contained by the extremes is equal to the rectangle contained by the means; and if the rectangle contained by the extremes be equal to the rectangle contained by the means, the four straight lines are proportionals.
In symbols, if a, b, c, d are the four lines, and
if  a : b = c : d, 
then  ad = bc; 
and conversely, if  ad = bc, 
then  a : b = c : d, 
where ad and bc denote (as in § 20), the areas of the rectangles contained by a and d and by b and c respectively.
This allows us to transform every proportion between four lines into an equation between two products.
It shows further that the operation of forming a product of two lines, and the operation of forming their ratio are each the inverse of the other.
If we now define a quotient ab of two lines as the number which multiplied into b gives a, so that
we see that from the equality of two quotients
ab = cd
follows, if we multiply both sides by bd,
abb·d = cdd·b,
But from this it follows, according to the last theorem, that
Hence we conclude that the quotient ab and the ratio a : b are different forms of the same magnitude, only with this important difference that the quotient ab would have a meaning only if a and b have a common measure, until we introduce incommensurable numbers, while the ratio a : b has always a meaning, and thus gives rise to the introduction of incommensurable numbers.
Thus it is really the theory of ratios in the fifth book which enables us to extend the geometrical calculus given before in connexion with Book II. It will also be seen that if we write the ratios in Book V. as quotients, or rather as fractions, then most of the theorems state properties of quotients or of fractions.
§ 64. Prop. 17. If three straight lines are proportional the rectangle contained by the extremes is equal to the square on the mean; and conversely, is only a special case of 16. After the problem, Prop. 18, On a given straight line to describe a rectilineal figure similar and similarly situated to a given rectilineal figure, there follows another fundamental theorem:
Prop. 19. Similar triangles are to one another in the duplicate ratio of their homologous sides. In other words, the areas of similar triangles are to one another as the squares on homologous sides. This is generalized in:
Prop. 20. Similar polygons may be divided into the same number of similar triangles, having the same ratio to one another that the polygons have; and the polygons are to one another in the duplicate ratio of their homologous sides.
§ 65. Prop. 21. Rectilineal figures which are similar to the same rectilineal figure are also similar to each other, is an immediate consequence of the definition of similar figures. As similar figures may be said to be equal in “shape” but not in “size,” we may state it also thus:
“Figures which are equal in shape to a third are equal in shape to each other.”
Prop. 22. If four straight lines be proportionals, the similar rectilineal figures similarly described on them shall also be proportionals; and if the similar rectilineal figures similarly described on four straight lines be proportionals, those straight lines shall be proportionals.
This is essentially the same as the following:—
If  a  : b  = c  : d, 
then  a^{2}  : b^{2}  = c^{2}  : d^{2}. 
§ 66. Now follows a proposition which has been much discussed with regard to Euclid’s exact meaning in saying that a ratio is compounded of two other ratios, viz.:
Prop. 23. Parallelograms which are equiangular to one another, have to one another the ratio which is compounded of the ratios of their sides.
The proof of the proposition makes its meaning clear. In symbols the ratio a : c is compounded of the two ratios a : b and b : c, and if a : b = a′ : b′, b : c = b″ : c″, then a : c is compounded of a′ : b′ and b″ : c″.
If we consider the ratios as numbers, we may say that the one ratio is the product of those of which it is compounded, or in symbols,
a  =  a  ·  b  =  a′  ·  b″  , if  a  =  a′  and  b  =  b″  . 
c  b  c  b′  c″  b  b′  c  c″ 
The theorem in Prop. 23 is the foundation of all mensuration of areas. From it we see at once that two rectangles have the ratio of their areas compounded of the ratios of their sides.
If A is the area of a rectangle contained by a and b, and B that of a rectangle contained by c and d, so that A = ab, B = cd, then A : B = ab : cd, and this is, the theorem says, compounded of the ratios a : c and b : d. In forms of quotients,
a  ·  b  =  ab  . 
c  d  cd 
This shows how to multiply quotients in our geometrical calculus.
Further, Two triangles have the ratios of their areas compounded of the ratios of their bases and their altitude. For a triangle is equal in area to half a parallelogram which has the same base and the same altitude.
§ 67. To bring these theorems to the form in which they are usually given, we assume a straight line u as our unit of length (generally an inch, a foot, a mile, &c.), and determine the number α which expresses how often u is contained in a line a, so that α denotes the ratio a : u whether commensurable or not, and that a = αu. We call this number α the numerical value of a. If in the same manner β be the numerical value of a line b we have
in words: The ratio of two lines (and of two like quantities in general) is equal to that of their numerical values.
This is easily proved by observing that a = αu, b = βu, therefore a : b = αu : βu, and this may without difficulty be shown to equal α : β.
If now a, b be base and altitude of one, a′, b′ those of another parallelogram, α, β and α′, β′ their numerical values respectively, and A, A′ their areas, then
A  =  a  ·  b  =  α  ·  β  =  αβ  . 
A′  a′  b′  α′  β′  α′β′ 
In words: The areas of two parallelograms are to each other as the products of the numerical values of their bases and altitudes.
If especially the second parallelogram is the unit square, i.e. a square on the unit of length, then α′ = β′ = 1, A′ = u^{2}, and we have
A  = αβ or A = αβ·u^{2}. 
A′ 
This gives the theorem: The number of unit squares contained in a parallelogram equals the product of the numerical values of base and altitude, and similarly the number of unit squares contained in a triangle equals half the product of the numerical values of base and altitude.
This is often stated by saying that the area of a parallelogram is equal to the product of the base and the altitude, meaning by this product the product of the numerical values, and not the product as defined above in § 20.
§ 68. Propositions 24 and 26 relate to parallelograms about diagonals, such as are considered in Book I., 43. They are—
Prop. 24. Parallelograms about the diameter of any parallelogram are similar to the whole parallelogram and to one another; and its converse (Prop. 26), If two similar parallelograms have a common angle, and be similarly situated, they are about the same diameter.
Between these is inserted a problem.
Prop. 25. To describe a rectilineal figure which shall be similar to one given rectilinear figure, and equal to another given rectilineal figure.
§ 69. Prop. 27 contains a theorem relating to the theory of maxima and minima. We may state it thus:
Prop. 27. If a parallelogram be divided into two by a straight line cutting the base, and if on half the base another parallelogram be constructed similar to one of those parts, then this third parallelogram is greater than the other part.
Of far greater interest than this general theorem is a special case of it, where the parallelograms are changed into rectangles, and where one of the parts into which the parallelogram is divided is made a square; for then the theorem changes into one which is easily recognized to be identical with the following:—
Of all rectangles which have the same perimeter the square has the greatest area.
This may also be stated thus:—
Of all rectangles which have the same area the square has the least perimeter.
§ 70. The next three propositions contain problems which may be said to be solutions of quadratic equations. The first two are, like the last, involved in somewhat obscure language. We transcribe them as follows:
Problem.—To describe on a given base a parallelogram, and to divide it either internally (Prop. 28) or externally (Prop. 29) from a point on the base into two parallelograms, of which the one has a given size (is equal in area to a given figure), whilst the other has a given shape (is similar to a given parallelogram).
If we express this again in symbols, calling the given base a, the one part x, and the altitude y, we have to determine x and y in the first case from the equations
x  =  p  , 
y  q 
k^{2} being the given size of the first, and p and q the base and altitude of the parallelogram which determine the shape of the second of the required parallelograms.
If we substitute the value of y, we get
(a − x)x =  pk^{2}  , 
q 
or,
where a and b are known quantities, taking b^{2} = pk^{2}/q.
The second case (Prop. 29) gives rise, in the same manner, to the quadratic
The next problem—
Prop. 30. To cut a given straight line in extreme and mean ratio, leads to the equation
This is, therefore, only a special case of the last, and is, besides, an old acquaintance, being essentially the same problem as that proposed in II. 11.
Prop. 30 may therefore be solved in two ways, either by aid of Prop. 29 or by aid of II. 11. Euclid gives both solutions.
§ 71. Prop. 31 (Theorem). In any rightangled triangle, any rectilineal figure described on the side subtending the right angle is equal to the similar and similarlydescribed figures on the sides containing the right angle,—is a pretty generalization of the theorem of Pythagoras (I. 47).
Leaving out the next proposition, which is of little interest, we come to the last in this book.
Prop. 33. In equal circles angles, whether at the centres or the circumferences, have the same ratio which the arcs on which they stand have to one another; so also have the sectors.
Of this, the part relating to angles at the centre is of special importance; it enables us to measure angles by arcs.
With this closes that part of the Elements which is devoted to the study of figures in a plane.
Book XI.
§ 72. In this book figures are considered which are not confined to a plane, viz. first relations between lines and planes in space, and afterwards properties of solids.
Of new definitions we mention those which relate to the perpendicularity and the inclination of lines and planes.
Def. 3. A straight line is perpendicular, or at right angles, to a plane when it makes right angles with every straight line meeting it in that plane.
The definition of perpendicular planes (Def. 4) offers no difficulty. Euclid defines the inclination of lines to planes and of planes to planes (Defs. 5 and 6) by aid of plane angles, included by straight lines, with which we have been made familiar in the first books.
The other important definitions are those of parallel planes, which never meet (Def. 8), and of solid angles formed by three or more planes meeting in a point (Def. 9).
To these we add the definition of a line parallel to a plane as a line which does not meet the plane.
§ 73. Before we investigate the contents of Book XI., it will be well to recapitulate shortly what we know of planes and lines from the definitions and axioms of the first book. There a plane has been defined as a surface which has the property that every straight line which joins two points in it lies altogether in it. This is equivalent to saying that a straight line which has two points in a plane has all points in the plane. Hence, a straight line which does not lie in the plane cannot have more than one point in common with the plane. This is virtually the same as Euclid’s Prop. 1, viz.:—
Prop. 1. One part of a straight line cannot be in a plane and another part without it.
It also follows, as was pointed out in § 3, in discussing the definitions of Book I., that a plane is determined already by one straight line and a point without it, viz. if all lines be drawn through the point, and cutting the line, they will form a plane.
This may be stated thus:—
A plane is determined—
1st, By a straight line and a point which does not lie on it;
2nd, By three points which do not lie in a straight line; for if two of these points be joined by a straight line we have case 1;
3rd, By two intersecting straight lines; for the point of intersection and two other points, one in each line, give case 2;
4th, By two parallel lines (Def. 35, I.).
The third case of this theorem is Euclid’s
Prop. 2. Two straight lines which cut one another are in one plane, and three straight lines which meet one another are in one plane.
And the fourth is Euclid’s
Prop. 7. If two straight lines be parallel, the straight line drawn from any point in one to any point in the other is in the same plane with the parallels. From the definition of a plane further follows
Prop. 3. If two planes cut one another, their common section is a straight line.
§ 74. Whilst these propositions are virtually contained in the definition of a plane, the next gives us a new and fundamental property of space, showing at the same time that it is possible to have a straight line perpendicular to a plane, according to Def. 3. It states—
Prop. 4. If a straight line is perpendicular to two straight lines in a plane which it meets, then it is perpendicular to all lines in the plane which it meets, and hence it is perpendicular to the plane.
Def. 3 may be stated thus: If a straight line is perpendicular to a plane, then it is perpendicular to every line in the plane which it meets. The converse to this would be
All straight lines which meet a given straight line in the same point, and are perpendicular to it, lie in a plane which is perpendicular to that line.
This Euclid states thus:
Prop. 5. If three straight lines meet all at one point, and a straight line stands at right angles to each of them at that point, the three straight lines shall be in one and the same plane.
§ 75. There follow theorems relating to the theory of parallel lines in space, viz.:—
Prop. 6. Any two lines which are perpendicular to the same plane are parallel to each other; and conversely
Prop. 8. If of two parallel straight lines one is perpendicular to a plane, the other is so also.
Prop. 7. If two straight lines are parallel, the straight line which joins any point in one to any point in the other is in the same plane as the parallels. (See above, § 73.)
Prop. 9. Two straight lines which are each of them parallel to the same straight line, and not in the same plane with it, are parallel to one another; where the words, “and not in the same plane with it,” may be omitted, for they exclude the case of three parallels in a plane, which has been proved before; and
Prop. 10. If two angles in different planes have the two limits of the one parallel to those of the other, then the angles are equal. That their planes are parallel is shown later on in Prop. 15.
This theorem is not necessarily true, for the angles in question may be supplementary; but then the one angle will be equal to that which is adjacent and supplementary to the other, and this latter angle will also have its limits parallel to those of the first.
From this theorem it follows that if we take any two straight lines in space which do not meet, and if we draw through any point P in space two lines parallel to them, then the angle included by these lines will always be the same, whatever the position of the point P may be. This angle has in modern times been called the angle between the given lines:—
By the angles between two not intersecting lines we understand the angles which two intersecting lines include that are parallel respectively to the two given lines.
§ 76. It is now possible to solve the following two problems:—
To draw a straight line perpendicular to a given plane from a given point which lies
1. Not in the plane (Prop. 11).
2. In the plane (Prop. 12).
The second case is easily reduced to the first—viz. if by aid of the first we have drawn any perpendicular to the plane from some point without it, we need only draw through the given point in the plane a line parallel to it, in order to have the required perpendicular given. The solution of the first part is of interest in itself. It depends upon a construction which may be expressed as a theorem.
If from a point A without a plane a perpendicular AB be drawn to the plane, and if from the foot B of this perpendicular another perpendicular BC be drawn to any straight line in the plane, then the straight line joining A to the foot C of this second perpendicular will also be perpendicular to the line in the plane.
The theory of perpendiculars to a plane is concluded by the theorem—
Prop. 13. Through any point in space, whether in or without a plane, only one straight line can be drawn perpendicular to the plane.
§ 77. The next four propositions treat of parallel planes. It is shown that planes which have a common perpendicular are parallel (Prop. 14); that two planes are parallel if two intersecting straight lines in the one are parallel respectively to two straight lines in the other plane (Prop. 15); that parallel planes are cut by any plane in parallel straight lines (Prop. 16); and lastly, that any two straight lines are cut proportionally by a series of parallel planes (Prop. 17).
This theory is made more complete by adding the following theorems, which are easy deductions from the last: Two parallel planes have common perpendiculars (converse to 14); and Two planes which are parallel to a third plane are parallel to each other.
It will be noted that Prop. 15 at once allows of the solution of the problem: “Through a given point to draw a plane parallel to a given plane.” And it is also easily proved that this problem allows always of one, and only of one, solution.
§ 78. We come now to planes which are perpendicular to one another. Two theorems relate to them.
Prop. 18. If a straight line be at right angles to a plane, every plane which passes through it shall be at right angles to that plane.
Prop. 19. If two planes which cut one another be each of them perpendicular to a third plane, their common section shall be perpendicular to the same plane.
§ 79. If three planes pass through a common point, and if they bound each other, a solid angle of three faces, or a trihedral angle, is formed, and similarly by more planes a solid angle of more faces, or a polyhedral angle. These have many properties which are quite analogous to those of triangles and polygons in a plane. Euclid states some, viz.:—
Prop. 20. If a solid angle be contained by three plane angles, any two of them are together greater than the third.
But the next—
Prop. 21. Every solid angle is contained by plane angles, which are together less than four right angles—has no analogous theorem in the plane.
We may mention, however, that the theorems about triangles contained in the propositions of Book I., which do not depend upon the theory of parallels (that is all up to Prop. 27), have their corresponding theorems about trihedral angles. The latter are formed, if for “side of a triangle” we write “plane angle” or “face” of trihedral angle, and for “angle of triangle” we substitute “angle between two faces” where the planes containing the solid angle are called its faces. We get, for instance, from I. 4, the theorem, If two trihedral angles have the angles of two faces in the one equal to the angles of two faces in the other, and have likewise the angles included by these faces equal, then the angles in the remaining faces are equal, and the angles between the other faces are equal each to each, viz. those which are opposite equal faces. The solid angles themselves are not necessarily equal, for they may be only symmetrical like the right hand and the left.
The connexion indicated between triangles and trihedral angles will also be recognized in
Prop. 22. If every two of three plane angles be greater than the third, and if the straight lines which contain them be all equal, a triangle may be made of the straight lines that join the extremities of those equal straight lines.
And Prop. 23 solves the problem, To construct a trihedral angle having the angles of its faces equal to three given plane angles, any two of them being greater than the third. It is, of course, analogous to the problem of constructing a triangle having its sides of given length.
Two other theorems of this kind are added by Simson in his edition of Euclid’s Elements.
§ 80. These are the principal properties of lines and planes in space, but before we go on to their applications it will be well to define the word distance. In geometry distance means always “shortest distance”; viz. the distance of a point from a straight line, or from a plane, is the length of the perpendicular from the point to the line or plane. The distance between two nonintersecting lines is the length of their common perpendicular, there being but one. The distance between two parallel lines or between two parallel planes is the length of the common perpendicular between the lines or the planes.
§ 81. Parallelepipeds.—The rest of the book is devoted to the study of the parallelepiped. In Prop. 24 the possibility of such a solid is proved, viz.:—
Prop. 24. If a solid be contained by six planes two and two of which are parallel, the opposite planes are similar and equal parallelograms.
Euclid calls this solid henceforth a parallelepiped, though he never defines the word. Either face of it may be taken as base, and its distance from the opposite face as altitude.
Prop. 25. If a solid parallelepiped be cut by a plane parallel to two of its opposite planes, it divides the whole into two solids, the base of one of which shall be to the base of the other as the one solid is to the other.
This theorem corresponds to the theorem (VI. 1) that parallelograms between the same parallels are to one another as their bases. A similar analogy is to be observed among a number of the remaining propositions.
§ 82. After solving a few problems we come to
Prop. 28. If a solid parallelepiped be cut by a plane passing through the diagonals of two of the opposite planes, it shall be cut in two equal parts.
In the proof of this, as of several other propositions, Euclid neglects the difference between solids which are symmetrical like the right hand and the left.
Prop. 31. Solid parallelepipeds, which are upon equal bases, and of the same altitude, are equal to one another.
Props. 29 and 30 contain special cases of this theorem leading up to the proof of the general theorem.
As consequences of this fundamental theorem we get
Prop. 32. Solid parallelepipeds, which have the same altitude, are to one another as their bases; and
Prop. 33. Similar solid parallelepipeds are to one another in the triplicate ratio of their homologous sides.
If we consider, as in § 67, the ratios of lines as numbers, we may also say—
The ratio of the volumes of similar parallelepipeds is equal to the ratio of the third powers of homologous sides.
Parallelepipeds which are not similar but equal are compared by aid of the theorem
Prop. 34. The bases and altitudes of equal solid parallelepipeds are reciprocally proportional; and if the bases and altitudes be reciprocally proportional, the solid parallelepipeds are equal.
§ 83. Of the following propositions the 37th and 40th are of special interest.
Prop. 37. If four straight lines be proportionals, the similar solid parallelepipeds, similarly described from them, shall also be proportionals; and if the similar parallelepipeds similarly described from four straight lines be proportionals, the straight lines shall be proportionals.
In symbols it says—
Prop. 40 teaches how to compare the volumes of triangular prisms with those of parallelepipeds, by proving that a triangular prism is equal in volume to a parallelepiped, which has its altitude and its base equal to the altitude and the base of the triangular prism.
§ 84. From these propositions follow all results relating to the mensuration of volumes. We shall state these as we did in the case of areas. The startingpoint is the “rectangular” parallelepiped, which has every edge perpendicular to the planes it meets, and which takes the place of the rectangle in the plane. If this has all its edges equal we obtain the “cube.”
If we take a certain line u as unit length, then the square on u is the unit of area, and the cube on u the unit of volume, that is to say, if we wish to measure a volume we have to determine how many unit cubes it contains.
A rectangular parallelepiped has, as a rule, the three edges unequal, which meet at a point. Every other edge is equal to one of them. If a, b, c be the three edges meeting at a point, then we may take the rectangle contained by two of them, say by b and c, as base and the third as altitude. Let V be its volume, V′ that of another rectangular parallelepiped which has the edges a′, b, c, hence the same base as the first. It follows then easily, from Prop. 25 or 32, that V : V′ = a : a′; or in words,
Rectangular parallelepipeds on equal bases are proportional to their altitudes.
If we have two rectangular parallelepipeds, of which the first has the volume V and the edges a, b, c, and the second, the volume V′ and the edges a′, b′, c′, we may compare them by aid of two new ones which have respectively the edges a′, b, c and a′, b′, c, and the volumes V_{1} and V_{2}. We then have
Compounding these, we have
or
V  =  a  ·  b  ·  c  . 
V′  a′  b′  c′ 
Hence, as a special case, making V′ equal to the unit cube U on u we get
V  =  a  ·  b  ·  c  = α·β·γ, 
U  u  u  u 
where α, β, γ are the numerical values of a, b, c; that is, The number of unit cubes in a rectangular parallelepiped is equal to the product of the numerical values of its three edges. This is generally expressed by saying the volume of a rectangular parallelepiped is measured by the product of its sides, or by the product of its base into its altitude, which in this case is the same.
Prop. 31 allows us to extend this to any parallelepipeds, and Props. 28 or 40, to triangular prisms.
The volume of any parallelepiped, or of any triangular prism, is measured by the product of base and altitude.
The consideration that any polygonal prism may be divided into a number of triangular prisms, which have the same altitude and the sum of their bases equal to the base of the polygonal prism, shows further that the same holds for any prism whatever.
Book XII.
§ 85. In the last part of Book XI. we have learnt how to compare the volumes of parallelepipeds and of prisms. In order to determine the volume of any solid bounded by plane faces we must determine the volume of pyramids, for every such solid may be decomposed into a number of pyramids.
As every pyramid may again be decomposed into triangular pyramids, it becomes only necessary to determine their volume. This is done by the
Theorem.—Every triangular pyramid is equal in volume to one third of a triangular prism having the same base and the same altitude as the pyramid.
This is an immediate consequence of Euclid’s
Prop. 7. Every prism having a triangular base may be divided into three pyramids that have triangular bases, and are equal to one another.
The proof of this theorem is difficult, because the three triangular pyramids into which the prism is divided are by no means equal in shape, and cannot be made to coincide. It has first to be proved that two triangular pyramids have equal volumes, if they have equal bases and equal altitudes. This Euclid does in the following manner. He first shows (Prop. 3) that a triangular pyramid may be divided into four parts, of which two are equal triangular pyramids similar to the whole pyramid, whilst the other two are equal triangular prisms, and further, that these two prisms together are greater than the two pyramids, hence more than half the given pyramid. He next shows (Prop. 4) that if two triangular pyramids are given, having equal bases and equal altitudes, and if each be divided as above, then the two triangular prisms in the one are equal to those in the other, and each of the remaining pyramids in the one has its base and altitude equal to the base and altitude of the remaining pyramids in the other. Hence to these pyramids the same process is again applicable. We are thus enabled to cut out of the two given pyramids equal parts, each greater than half the original pyramid. Of the remainder we can again cut out equal parts greater than half these remainders, and so on as far as we like. This process may be continued till the last remainder is smaller than any assignable quantity, however small. It follows, so we should conclude at present, that the two volumes must be equal, for they cannot differ by any assignable quantity.
To Greek mathematicians this conclusion offers far greater difficulties. They prove elaborately, by a reductio ad absurdum, that the volumes cannot be unequal. This proof must be read in the Elements. We must, however, state that we have in the above not proved Euclid’s Prop. 5, but only a special case of it. Euclid does not suppose that the bases of the two pyramids to be compared are equal, and hence he proves that the volumes are as the bases. The reasoning of the proof becomes clearer in the special case, from which the general one may be easily deduced.
§ 86. Prop. 6 extends the result to pyramids with polygonal bases. From these results follow again the rules at present given for the mensuration of solids, viz. a pyramid is the third part of a triangular prism having the same base and the same altitude. But a triangular prism is equal in volume to a parallelepiped which has the same base and altitude. Hence if B is the base and h the altitude, we have
Volume of prism  = Bh, 
Volume of pyramid  = 13Bh, 
statements which have to be taken in the sense that B means the number of square units in the base, h the number of units of length in the altitude, or that B and h denote the numerical values of base and altitude.
§ 87. A method similar to that used in proving Prop. 5 leads to the following results relating to solids bounded by simple curved surfaces:—
Prop. 10. Every cone is the third part of a cylinder which has the same base, and is of an equal altitude with it.
Prop. 11. Cones or cylinders of the same altitude are to one another as their bases.
Prop. 12. Similar cones or cylinders have to one another the triplicate ratio of that which the diameters of their bases have.
Prop. 13. If a cylinder be cut by a plane parallel to its opposite planes or bases, it divides the cylinder into two cylinders, one of which is to the other as the axis of the first to the axis of the other; which may also be stated thus:—
Cylinders on the same base are proportional to their altitudes.
Prop. 14. Cones or cylinders upon equal bases are to one another as their altitudes.
Prop. 15. The bases and altitudes of equal cones or cylinders are reciprocally proportional, and if the bases and altitudes be reciprocally proportional, the cones or cylinders are equal to one another.
These theorems again lead to formulae in mensuration, if we compare a cylinder with a prism having its base and altitude equal to the base and altitude of the cylinder. This may be done by the method of exhaustion. We get, then, the result that their bases are equal, and have, if B denotes the numerical value of the base, and h that of the altitude,
Volume of cylinder  = Bh, 
Volume of cone  = 13Bh. 
§ 88. The remaining propositions relate to circles and spheres. Of the sphere only one property is proved, viz.:—
Prop. 18. Spheres have to one another the triplicate ratio of that which their diameters have. The mensuration of the sphere, like that of the circle, the cylinder and the cone, had not been settled in the time of Euclid. It was done by Archimedes.
Book XIII.
§ 89. The 13th and last book of Euclid’s Elements is devoted to the regular solids (see Polyhedron). It is shown that there are five of them, viz.:—
1. The regular tetrahedron, with 4 triangular faces and 4 vertices;
2. The cube, with 8 vertices and 6 square faces;
3. The octahedron, with 6 vertices and 8 triangular faces;
4. The dodecahedron, with 12 pentagonal faces, 3 at each of the 20 vertices;
5. The icosahedron, with 20 triangular faces, 5 at each of the 12 vertices.
It is shown how to inscribe these solids in a given sphere, and how to determine the lengths of their edges.
§ 90. The 13th book, and therefore the Elements, conclude with the scholium, “that no other regular solid exists besides the five ones enumerated.”
The proof is very simple. Each face is a regular polygon, hence the angles of the faces at any vertex must be angles in equal regular polygons, must be together less than four right angles (XI. 21), and must be three or more in number. Each angle in a regular triangle equals twothirds of one right angle. Hence it is possible to form a solid angle with three, four or five regular triangles or faces. These give the solid angles of the tetrahedron, the octahedron and the icosahedron. The angle in a square (the regular quadrilateral) equals one right angle. Hence three will form a solid angle, that of the cube, and four will not. The angle in the regular pentagon equals 65 of a right angle. Hence three of them equal 185 (i.e. less than 4) right angles, and form the solid angle of the dodecahedron. Three regular polygons of six or more sides cannot form a solid angle. Therefore no other regular solids are possible. (O. H.)
II. Projective Geometry
It is difficult, at the outset, to characterize projective geometry as compared with Euclidean. But a few examples will at least indicate the practical differences between the two.
In Euclid’s Elements almost all propositions refer to the magnitude of lines, angles, areas or volumes, and therefore to measurement. The statement that an angle is right, or that two straight lines are parallel, refers to measurement. On the other hand, the fact that a straight line does or does not cut a circle is independent of measurement, it being dependent only upon the mutual “position” of the line and the circle. This difference becomes clearer if we project any figure from one plane to another (see Projection). By this the length of lines, the magnitude of angles and areas, is altered, so that the projection, or shadow, of a square on a plane will not be a square; it will, however, be some quadrilateral. Again, the projection of a circle will not be a circle, but some other curve more or less resembling a circle. But one property may be stated at once—no straight line can cut the projection of a circle in more than two points, because no straight line can cut a circle in more than two points. There are, then, some properties of figures which do not alter by projection, whilst others do. To the latter belong nearly all properties relating to measurement, at least in the form in which they are generally given. The others are said to be projective properties, and their investigation forms the subject of projective geometry.
Different as are the kinds of properties investigated in the old and the new sciences, the methods followed differ in a still greater degree. In Euclid each proposition stands by itself; its connexion with others is never indicated; the leading ideas contained in its proof are not stated; general principles do not exist. In the modern methods, on the other hand, the greatest importance is attached to the leading thoughts which pervade the whole; and general principles, which bring whole groups of theorems under one aspect, are given rather than separate propositions. The whole tendency is towards generalization. A straight line is considered as given in its entirety, extending both ways to infinity, while Euclid never admits anything but finite quantities. The treatment of the infinite is in fact another fundamental difference between the two methods: Euclid avoids it; in modern geometry it is systematically introduced.
Of the different modern methods of geometry, we shall treat principally of the methods of projection and correspondence which have proved to be the most powerful. These have become independent of Euclidean Geometry, especially through the Geometrie der Lage of V. Staudt and the Ausdehnungslehre of Grassmann.
For the sake of brevity we shall presuppose a knowledge of Euclid’s Elements, although we shall use only a few of his propositions.
§ 1. Geometrical Elements. We consider space as filled with points, lines and planes, and these we call the elements out of which our figures are to be formed, calling any combination of these elements a “figure.”
By a line we mean a straight line in its entirety, extending both ways to infinity; and by a plane, a plane surface, extending in all directions to infinity.
We accept the threedimensional space of experience—the space assumed by Euclid—which has for its properties (among others):—
Through any two points in space one and only one line may be drawn;
Through any three points which are not in a line, one and only one plane may be placed;
The intersection of two planes is a line;
A line which has two points in common with a plane lies in the plane, hence the intersection of a line and a plane is a single point; and
Three planes which do not meet in a line have one single point in common.
These results may be stated differently in the following form:—
I. A plane is determined—  A point is determined— 
1. By three points which do not lie in a line; 2. By two intersecting lines; 3. By a line and a point which does not lie in it. 
1. By three planes which do not pass through a line; 2. By two intersecting lines 3. By a plane and a line which does not lie in it. 
II. A line is determined—  
1. By two points;  2. By two planes. 
It will be observed that not only are planes determined by points, but also points by planes; that therefore the planes may be considered as elements, like points; and also that in any one of the above statements we may interchange the words point and plane, and we obtain again a correct statement, provided that these statements themselves are true. As they stand, we ought, in several cases, to add “if they are not parallel,” or some such words, parallel lines and planes being evidently left altogether out of consideration. To correct this we have to reconsider the theory of parallels.
Fig. 1. 
§ 2. Parallels. Point at Infinity.—Let us take in a plane a line p (fig. 1), a point S not in this line, and a line q drawn through S. Then this line q will meet the line p in a point A. If we turn the line q about S towards q′, its point of intersection with p will move along p towards B, passing, on continued turning, to a greater and greater distance, until it is moved out of our reach. If we turn q still farther, its continuation will meet p, but now at the other side of A. The point of intersection has disappeared to the right and reappeared to the left. There is one intermediate position where q is parallel to p—that is where it does not cut p. In every other position it cuts p in some finite point. If, on the other hand, we move the point A to an infinite distance in p, then the line q which passes through A will be a line which does not cut p at any finite point. Thus we are led to say: Every line through S which joins it to any point at an infinite distance in p is parallel to p. But by Euclid’s 12th axiom there is but one line parallel to p through S. The difficulty in which we are thus involved is due to the fact that we try to reason about infinity as if we, with our finite capabilities, could comprehend the infinite. To overcome this difficulty, we may say that all points at infinity in a line appear to us as one, and may be replaced by a single “ideal” point.
We may therefore now give the following definitions and axiom:—
Definition.—Lines which meet at infinity are called parallel.
Axiom.—All points at an infinite distance in a line may be considered as one single point.
Definition.—This ideal point is called the point at infinity in the line.
The axiom is equivalent to Euclid’s Axiom 12, for it follows from either that through any point only one line may be drawn parallel to a given line.
This point at infinity in a line is reached whether we move a point in the one or in the opposite direction of a line to infinity. A line thus appears closed by this point, and we speak as if we could move a point along the line from one position A to another B in two ways, either through the point at infinity or through finite points only.
It must never be forgotten that this point at infinity is ideal; in fact, the whole notion of “infinity” is only a mathematical conception, and owes its introduction (as a method of research) to the working generalizations which it permits.
§ 3. Line and Plane at Infinity.—Having arrived at the notion of replacing all points at infinity in a line by one ideal point, there is no difficulty in replacing all points at infinity in a plane by one ideal line.
To make this clear, let us suppose that a line p, which cuts two fixed lines a and b in the points A and B, moves parallel to itself to a greater and greater distance. It will at last cut both a and b at their points at infinity, so that a line which joins the two points at infinity in two intersecting lines lies altogether at infinity. Every other line in the plane will meet it therefore at infinity, and thus it contains all points at infinity in the plane.
All points at infinity in a plane lie in a line, which is called the line at infinity in the plane.
It follows that parallel planes must be considered as planes having a common line at infinity, for any other plane cuts them in parallel lines which have a point at infinity in common.
If we next take two intersecting planes, then the point at infinity in their line of intersection lies in both planes, so that their lines at infinity meet. Hence every line at infinity meets every other line at infinity, and they are therefore all in one plane.
All points at infinity in space may be considered as lying in one ideal plane, which is called the plane at infinity.
§ 4. Parallelism.—We have now the following definitions:—
Parallel lines are lines which meet at infinity;
Parallel planes are planes which meet at infinity;
A line is parallel to a plane if it meets it at infinity.
Theorems like this—Lines (or planes) which are parallel to a third are parallel to each other—follow at once.
This view of parallels leads therefore to no contradiction of Euclid’s Elements.
As immediate consequences we get the propositions:—
Every line meets a plane in one point, or it lies in it;
Every plane meets every other plane in a line;
Any two lines in the same plane meet.
§ 5. Aggregates of Geometrical Elements.—We have called points, lines and planes the elements of geometrical figures. We also say that an element of one kind contains one of the other if it lies in it or passes through it.
All the elements of one kind which are contained in one or two elements of a different kind form aggregates which have to be enumerated. They are the following:—
I. Of one dimension.
1. The row, or range, of points formed by all points in a line, which is called its base.
2. The flat pencil formed by all the lines through a point in a plane. Its base is the point in the plane.
3. The axial pencil formed by all planes through a line which is called its base or axis.
II. Of two dimensions.
1. The field of points and lines—that is, a plane with all its points and all its lines.
2. The pencil of lines and planes—that is, a point in space with all lines and all planes through it.
III. Of three dimensions.
The space of points—that is, all points in space.
The space of planes—that is, all planes in space.
IV. Of four dimensions.
The space of lines, or all lines in space.
§ 6. Meaning of “Dimensions.”—The word dimension in the above needs explanation. If in a plane we take a row p and a pencil with centre Q, then through every point in p one line in the pencil will pass, and every ray in Q will cut p in one point, so that we are entitled to say a row contains as many points as a flat pencil lines, and, we may add, as an axial pencil planes, because an axial pencil is cut by a plane in a flat pencil.
The number of elements in the row, in the flat pencil, and in the axial pencil is, of course, infinite and indefinite too, but the same in all. This number may be denoted by ∞. Then a plane contains ∞^{2} points and as many lines. To see this, take a flat pencil in a plane. It contains ∞ lines, and each line contains ∞ points, whilst each point in the plane lies on one of these lines. Similarly, in a plane each line cuts a fixed line in a point. But this line is cut at each point by ∞ lines and contains ∞ points; hence there are ∞^{2} lines in a plane.
A pencil in space contains as many lines as a plane contains points and as many planes as a plane contains lines, for any plane cuts the pencil in a field of points and lines. Hence a pencil contains ∞^{2} lines and ∞^{2} planes. The field and the pencil are of two dimensions.
To count the number of points in space we observe that each point lies on some line in a pencil. But the pencil contains ∞^{2} lines, and each line ∞ points; hence space contains ∞^{3} points. Each plane cuts any fixed plane in a line. But a plane contains ∞^{2} lines, and through each pass ∞ planes; therefore space contains ∞^{3} planes.
Hence space contains as many planes as points, but it contains an infinite number of times more lines than points or planes. To count them, notice that every line cuts a fixed plane in one point. But ∞^{2} lines pass through each point, and there are ∞^{2} points in the plane. Hence there are ∞^{4} lines in space. The space of points and planes is of three dimensions, but the space of lines is of four dimensions.
A field of points or lines contains an infinite number of rows and flat pencils; a pencil contains an infinite number of flat pencils and of axial pencils; space contains a triple infinite number of pencils and of fields, ∞^{4} rows and axial pencils and ∞^{5} flat pencils—or, in other words, each point is a centre of ∞^{2} flat pencils.
§ 7. The above enumeration allows a classification of figures. Figures in a row consist of groups of points only, and figures in the flat or axial pencil consist of groups of lines or planes. In the plane we may draw polygons; and in the pencil or in the point, solid angles, and so on.
We may also distinguish the different measurements. We have—

Segments of a Line
§ 8. Any two points A and B in space determine on the line through them a finite part, which may be considered as being described by a point moving from A to B. This we shall denote by AB, and distinguish it from BA, which is supposed as being described by a point moving from B to A, and hence in a direction or in a “sense” opposite to AB. Such a finite line, which has a definite sense, we shall call a “segment,” so that AB and BA denote different segments, which are said to be equal in length but of opposite sense. The one sense is often called positive and the other negative.
In introducing the word “sense” for direction in a line, we have the word direction reserved for direction of the line itself, so that different lines have different directions, unless they be parallel, whilst in each line we have a positive and negative sense.
We may also say, with Clifford, that AB denotes the “step” of going from A to B.
Fig. 2. 
§ 9. If we have three points A, B, C in a line (fig. 2), the step AB will bring us from A to B, and the step BC from B to C. Hence both steps are equivalent to the one step AC. This is expressed by saying that AC is the “sum” of AB and BC; in symbols—
where account is to be taken of the sense.
This equation is true whatever be the position of the three points on the line. As a special case we have
AB + BA = 0,  (1) 
and similarly
AB + BC + CA = 0,  (2) 
which again is true for any three points in a line.
We further write
where − denotes negative sense.
We can then, just as in algebra, change subtraction of segments into addition by changing the sense, so that AB − CB is the same as AB + (−CB) or AB + BC. A figure will at once show the truth of this. The sense is, in fact, in every respect equivalent to the “sign” of a number in algebra.
§ 10. Of the many formulae which exist between points in a line we shall have to use only one more, which connects the segments between any four points A, B, C, D in a line. We have
or multiplying these by AD, BD, CD respectively, we get
BC · AD = BD · AD + DC · AD = BD · AD − CD · AD 
It will be seen that the sum of the righthand sides vanishes, hence that
BC · AD + CA · BD + AB · CD = 0  (3) 
for any four points on a line.
Fig. 3. 
§ 11. If C is any point in the line AB, then we say that C divides the segment AB in the ratio AC/CB, account being taken of the sense of the two segments AC and CB. If C lies between A and B the ratio is positive, as AC and CB have the same sense. But if C lies without the segment AB, i.e. if C divides AB externally, then the ratio is negative. To see how the value of this ratio changes with C, we will move C along the whole line (fig. 3), whilst A and B remain fixed. If C lies at the point A, then AC = 0, hence the ratio AC : CB vanishes. As C moves towards B, AC increases and CB decreases, so that our ratio increases. At the middle point M of AB it assumes the value +1, and then increases till it reaches an infinitely large value, when C arrives at B. On passing beyond B the ratio becomes negative. If C is at P we have AC = AP = AB + BP, hence
AC  =  AB  +  BP  = −  AB  − 1. 
CB  PB  PB  BP 
In the last expression the ratio AB : BP is positive, has its greatest value ∞ when C coincides with B, and vanishes when BC becomes infinite. Hence, as C moves from B to the right to the point at infinity, the ratio AC : CB varies from −∞ to −1.
If, on the other hand, C is to the left of A, say at Q, we have AC = AQ = AB + BQ = AB − QB, hence ACCB = ABQB − 1.
Here AB < QB, hence the ratio AB : QB is positive and always less than one, so that the whole is negative and < 1. If C is at the point at infinity it is −1, and then increases as C moves to the right, till for C at A we get the ratio = 0. Hence—
“As C moves along the line from an infinite distance to the left to an infinite distance at the right, the ratio always increases; it starts with the value −1, reaches 0 at A, +1 at M, ∞ at B, now changes sign to −∞, and increases till at an infinite distance it reaches again the value −1. It assumes therefore all possible values from −∞ to +∞, and each value only once, so that not only does every position of C determine a definite value of the ratio AC : CB, but also, conversely, to every positive or negative value of this ratio belongs one single point in the line AB.
[Relations between segments of lines are interesting as showing an application of algebra to geometry. The genesis of such relations from algebraic identities is very simple. For example, if a, b, c, x be any four quantities, then
a  +  b  +  c  =  x  ; 
(a − b)(a − c)(x − a)  (b − c)(b − a)(x − b)  (c − a)(c − b)(x − c)  (x − a)(x − b)(x − c) 
this may be proved, cumbrously, by multiplying up, or, simply, by decomposing the righthand member of the identity into partial fractions. Now take a line ABCDX, and let AB = a, AC = b, AD = c, AX = x. Then obviously (a − b) = AB − AC = −BC, paying regard to signs; (a − c) = AB − AD = DB, and so on. Substituting these values in the identity we obtain the following relation connecting the segments formed by five points on a line:—
AB  +  AC  +  AD  =  AX  . 
BC · BD · BX  CD · CB · CX  DB · DC · DX  BX · CX · DX 
Conversely, if a metrical relation be given, its validity may be tested by reducing to an algebraic equation, which is an identity if the relation be true. For example, if ABCDX be five collinear points, prove
AD · AX  +  BD · BX  +  CD · CX  = 1. 
AB · AC  BC · BA  CA · CB 
Clearing of fractions by multiplying throughout by AB · BC · CA, we have to prove
Take A as origin and let AB = a, AC = b, AD = c, AX = x. Substituting for the segments in terms of a, b, c, x, we obtain on simplification
An alternative method of testing a relation is illustrated in the following example:— If A, B, C, D, E, F be six collinear points, then
AE · AF  +  BE · BF  +  CE · CF  +  DE · DF  = 0. 
AB · AC · AD  BC · BD · BA  CD · CA · CB  DA · DB · DC 
Clearing of fractions by multiplying throughout by AB · BC · CD · DA, and reducing to a common origin O (calling OA = a, OB = b, &c.), an equation containing the second and lower powers of OA ( = a), &c., is obtained. Calling OA = x, it is found that x = b, x = c, x = d are solutions. Hence the quadratic has three roots; consequently it is an identity.
The relations connecting five points which we have instanced above may be readily deduced from the sixpoint relation; the first by taking D at infinity, and the second by taking F at infinity, and then making the obvious permutations of the points.]
Projection and Crossratios
§ 12. If we join a point A to a point S, then the point where the line SA cuts a fixed plane π is called the projection of A on the plane π from S as centre of projection. If we have two planes π and π′ and a point S, we may project every point A in π to the other plane. If A′ is the projection of A, then A is also the projection of A′, so that the relations are reciprocal. To every figure in π we get as its projection a corresponding figure in π′.
We shall determine such properties of figures as remain true for the projection, and which are called projective properties. For this purpose it will be sufficient to consider at first only constructions in one plane.
Fig. 4.  Fig. 5. 
Let us suppose we have given in a plane two lines p and p′ and a centre S (fig. 4); we may then project the points in p from S to p′. Let A′, B′ ... be the projections of A, B ..., the point at infinity in p which we shall denote by I will be projected into a finite point I′ in p′, viz. into the point where the parallel to p through S cuts p′. Similarly one point J in p will be projected into the point J′ at infinity in p′. This point J is of course the point where the parallel to p′ through S cuts p. We thus see that every point in p is projected into a single point in p′.
Fig. 5 shows that a segment AB will be projected into a segment A′B′ which is not equal to it, at least not as a rule; and also that the ratio AC : CB is not equal to the ratio A′C′ : C′B′ formed by the projections. These ratios will become equal only if p and p′ are parallel, for in this case the triangle SAB is similar to the triangle SA′B′. Between three points in a line and their projections there exists therefore in general no relation. But between four points a relation does exist.
§ 13. Let A, B, C, D be four points in p, A′, B′, C, D′ their projections in p′, then the ratio of the two ratios AC : CB and AD : DB into which C and D divide the segment AB is equal to the corresponding expression between A′, B′, C′, D′. In symbols we have
AC  :  AD  =  A′C′  :  A′D′  . 
CB  DB  C′B′  D′B′ 
This is easily proved by aid of similar triangles.
Fig. 6. 
Through the points A and B on p draw parallels to p′, which cut the projecting rays in C_{2}, D_{2}, B_{2} and A_{1}, C_{1}, D_{1}, as indicated in fig. 6. The two triangles ACC_{2} and BCC_{1} will be similar, as will also be the triangles ADD_{2} and BDD_{1}.
The proof is left to the reader.
This result is of fundamental importance.
The expression AC/CB : AD/DB has been called by Chasles the “anharmonic ratio of the four points A, B, C, D.” Professor Clifford proposed the shorter name of “crossratio.” We shall adopt the latter. We have then the
Fundamental Theorem.—The crossratio of four points in a line is equal to the crossratio of their projections on any other line which lies in the same plane with it.
§ 14. Before we draw conclusions from this result, we must investigate the meaning of a crossratio somewhat more fully.
If four points A, B, C, D are given, and we wish to form their crossratio, we have first to divide them into two groups of two, the points in each group being taken in a definite order. Thus, let A, B be the first, C, D the second pair, A and C being the first points in each pair. The crossratio is then the ratio AC : CB divided by AD : DB. This will be denoted by (AB, CD), so that
(AB, CD) =  AC  :  AD  . 
CB  DB 
This is easily remembered. In order to write it out, make first the two lines for the fractions, and put above and below these the letters A and B in their places, thus, A/*B : A/*B; and then fill up, crosswise, the first by C and the other by D.
§ 15. If we take the points in a different order, the value of the crossratio will change. We can do this in twentyfour different ways by forming all permutations of the letters. But of these twentyfour crossratios groups of four are equal, so that there are really only six different ones, and these six are reciprocals in pairs.
We have the following rules:—
I. If in a crossratio the two groups be interchanged, its value remains unaltered, i.e.
II. If in a crossratio the two points belonging to one of the two groups be interchanged, the crossratio changes into its reciprocal, i.e.
From I. and II. we see that eight crossratios are associated with (AB, CD).
III. If in a crossratio the two middle letters be interchanged, the crossratio α changes into its complement 1 − α, i.e. (AB, CD) = 1 − (AC, BD).
[§ 16. If λ = (AB, CD), μ = (AC, DB), ν = (AD, BC), then λ, μ, ν and their reciprocals 1/λ, 1/μ, 1/ν are the values of the total number of twentyfour crossratios. Moreover, λ, μ, ν are connected by the relations
− λμν = 1;
this proposition may be proved by substituting for λ, μ, ν and reducing to a common origin. There are therefore four equations between three unknowns; hence if one crossratio be given, the remaining twentythree are determinate. Moreover, two of the quantities λ, μ, ν are positive, and the remaining one negative.
The following scheme shows the twentyfour crossratios expressed in terms of λ, μ, ν.]
(AB, CD) (BA, DC) (CD, AB) (DC, BA)  λ  1 − μ  1/(1 − ν)  (AC, DB) (BD, CA) (CA, BD) (DB, AC)  1/(1 − λ)  1/μ  (ν − 1)/ν  
(AB, DC) (BA, CD) (CD, BA) (DC, AB)  1/λ  1/(1 − μ)  1 − ν  (AD, BC) (BC, AD) (CB, DA) (DA, CB)  (λ − 1)/λ  μ/(μ − 1)  ν  
(AC, BD) (BD, AC) (CA, DB) (DB, CA)  1 − λ  μ  ν/(ν − 1)  (AD, CB) (BC, DA) (CB, AD) (DA, BC)  λ/(λ − 1)  (μ − 1)/μ  1/ν 
§ 17. If one of the points of which a crossratio is formed is the point at infinity in the line, the crossratio changes into a simple ratio. It is convenient to let the point at infinity occupy the last place in the symbolic expression for the crossratio. Thus if I is a point at infinity, we have (AB, CI) = −AC/CB, because AI : IB = −1.
Every common ratio of three points in a line may thus be expressed as a crossratio, by adding the point at infinity to the group of points.
Harmonic Ranges
§ 18. If the points have special positions, the crossratios may have such a value that, of the six different ones, two and two become equal. If the first two shall be equal, we get λ = 1/λ, or λ^{2} = 1, λ = ±1.
If we take λ = +1, we have (AB, CD) = 1, or AC/CB = AD/DB; that is, the points C and D coincide, provided that A and B are different.
If we take λ = −1, so that (AB, CD) = −1, we have AC/CB = −AD/DB. Hence C and D divide AB internally and externally in the same ratio.
The four points are in this case said to be harmonic points, and C and D are said to be harmonic conjugates with regard to A and B.
But we have also (CD, AB) = −1, so that A and B are harmonic conjugates with regard to C and D.
The principal property of harmonic points is that their crossratio remains unaltered if we interchange the two points belonging to one pair, viz.
For four harmonic points the six crossratios become equal two and two:
λ = −1, 1 − λ = 2,  λ  = 12,  1  = −1,  1  = 12,  λ − 1  = 2. 
λ − 1  λ  1 − λ  λ 
Hence if we get four points whose crossratio is 2 or 12, then they are harmonic, but not arranged so that conjugates are paired. If this is the case the crossratio = −1.
§ 19. If we equate any two of the above six values of the crossratios, we get either λ = 1, 0, ∞, or λ = −1, 2, 12, or else λ becomes a root of the equation λ^{2} − λ + 1 = 0, that is, an imaginary cube root of −1. In this case the six values become three and three equal, so that only two different values remain. This case, though important in the theory of cubic curves, is for our purposes of no interest, whilst harmonic points are allimportant.
§ 20. From the definition of harmonic points, and by aid of § 11, the following properties are easily deduced.
If C and D are harmonic conjugates with regard to A and B, then one of them lies in, the other without AB; it is impossible to move from A to B without passing either through C or through D; the one blocks the finite way, the other the way through infinity. This is expressed by saying A and B are “separated” by C and D.
For every position of C there will be one and only one point D which is its harmonic conjugate with regard to any point pair A, B.
If A and B are different points, and if C coincides with A or B, D does. But if A and B coincide, one of the points C or D, lying between them, coincides with them, and the other may be anywhere in the line. It follows that, “if of four harmonic conjugates two coincide, then a third coincides with them, and the fourth may be any point in the line.”
If C is the middle point between A and B, then D is the point at infinity; for AC : CB = +1, hence AD : DB must be equal to −1. The harmonic conjugate of the point at infinity in a line with regard to two points A, B is the middle point of AB.
This important property gives a first example how metric properties are connected with projective ones.
[§ 21. Harmonic properties of the complete quadrilateral and quadrangle. Fig. 7.  Fig. 8. 
A figure formed by four lines in a plane is called a complete quadrilateral, or, shorter, a fourside. The four sides meet in six points, named the “vertices,” which may be joined by three lines (other than the sides), named the “diagonals” or “harmonic lines.” The diagonals enclose the “harmonic triangle of the quadrilateral.” In fig. 7, A′B′C′, B′AC, C′AB, CBA′ are the sides, A, A′, B, B′, C, C′ the vertices, AA′, BB′, CC′ the harmonic lines, and αβγ the harmonic triangle of the quadrilateral. A figure formed by four coplanar points is named a complete quadrangle, or, shorter, a fourpoint. The four points may be joined by six lines, named the “sides,” which intersect in three other points, termed the “diagonal or harmonic points.” The harmonic points are the vertices of the “harmonic triangle of the complete quadrangle.” In fig. 8, AA′, BB′ are the points, AA′, BB′, A′B′, B′A, AB, BA′ are the sides, L, M, N are the diagonal points, and LMN is the harmonic triangle of the quadrangle.
The harmonic property of the complete quadrilateral is: Any diagonal or harmonic line is harmonically divided by the other two; and of a complete quadrangle: The angle at any harmonic point is divided harmonically by the joins to the other harmonic points. To prove the first theorem, we have to prove (AA′, βγ), (BB′, γα), (CC′, βα) are harmonic. Consider the crossratio (CC′, αβ). Then projecting from A on BB′ we have A(CC′, αβ) = A(B′B, αγ). Projecting from A′ on BB′, A′(CC′, αβ) = A′(BB′, αγ). Hence (B′B, αγ) = (BB′, αγ), i.e. the crossratio (BB′, αγ) equals that of its reciprocal; hence the range is harmonic.
The second theorem states that the pencils L(BA, NM), M(B′A, LN), N(BA, LM) are harmonic. Deferring the subject of harmonic pencils to the next section, it will suffice to state here that any transversal intersects an harmonic pencil in an harmonic range. Consider the pencil L(BA, NM), then it is sufficient to prove (BA′, NM′) is harmonic. This follows from the previous theorem by considering A′B as a diagonal of the quadrilateral ALB′M.]
This property of the complete quadrilateral allows the solution of the problem:
To construct the harmonic conjugate D to a point C with regard to two given points A and B.
Through A draw any two lines, and through C one cutting the former two in G and H. Join these points to B, cutting the former two lines in E and F. The point D where EF cuts AB will be the harmonic conjugate required.
This remarkable construction requires nothing but the drawing of lines, and is therefore independent of measurement. In a similar manner the harmonic conjugate of the line VA for two lines VC, VD is constructed with the aid of the property of the complete quadrangle.
§ 22. Harmonic Pencils.—The theory of crossratios may be extended from points in a row to lines in a flat pencil and to planes in an axial pencil. We have seen (§ 13) that if the lines which join four points A, B, C, D to any point S be cut by any other line in A′, B′, C′, D′, then (AB, CD) = (A′B′, C′D′). In other words, four lines in a flat pencil are cut by every other line in four points whose crossratio is constant.
Definition.—By the crossratio of four rays in a flat pencil is meant the crossratio of the four points in which the rays are cut by any line. If a, b, c, d be the lines, then this crossratio is denoted by (ab, cd).
Definition.—By the crossratio of four planes in an axial pencil is understood the crossratio of the four points in which any line cuts the planes, or, what is the same thing, the crossratio of the four rays in which any plane cuts the four planes.
In order that this definition may have a meaning, it has to be proved that all lines cut the pencil in points which have the same crossratio. This is seen at once for two intersecting lines, as their plane cuts the axial pencil in a flat pencil, which is itself cut by the two lines. The crossratio of the four points on one line is therefore equal to that on the other, and equal to that of the four rays in the flat pencil.
If two nonintersecting lines p and q cut the four planes in A, B, C, D and A′, B′, C′, D′, draw a line r to meet both p and q, and let this line cut the planes in A″, B″, C″, D″. Then (AB, CD) = (A′B′, C′D′), for each is equal to (A″B″, C″D″).
§ 23. We may now also extend the notion of harmonic elements, viz.
Definition.—Four rays in a flat pencil and four planes in an axial pencil are said to be harmonic if their crossratio equals 1, that is, if they are cut by a line in four harmonic points.
If we understand by a “median line” of a triangle a line which joins a vertex to the middle point of the opposite side, and by a “median line” of a parallelogram a line joining middle points of opposite sides, we get as special cases of the last theorem:
The diagonals and median lines of a parallelogram form an harmonic pencil; and
At a vertex of any triangle, the two sides, the median line, and the line parallel to the base form an harmonic pencil.
Taking the parallelogram a rectangle, or the triangle isosceles, we get:
Any two lines and the bisections of their angles form an harmonic pencil. Or:
In an harmonic pencil, if two conjugate rays are perpendicular, then the other two are equally inclined to them; and, conversely, if one ray bisects the angle between conjugate rays, it is perpendicular to its conjugate.
This connects perpendicularity and bisection of angles with projective properties.
§ 24. We add a few theorems and problems which are easily proved or solved by aid of harmonics.
An harmonic pencil is cut by a line parallel to one of its rays in three equidistant points.
Through a given point to draw a line such that the segment determined on it by a given angle is bisected at that point.
Having given two parallel lines, to bisect on either any given segment without using a pair of compasses.
Having given in a line a segment and its middle point, to draw through any given point in the plane a line parallel to the given line.
To draw a line which joins a given point to the intersection of two given lines which meet off the drawing paper (by aid of § 21).
Correspondence. Homographic and Perspective Ranges
§ 25. Two rows, p and p′, which are one the projection of the other (as in fig. 5), stand in a definite relation to each other, characterized by the following properties.
1. To each point in either corresponds one point in the other; that is, those points are said to correspond which are projections of one another.
2. The crossratio of any four points in one equals that of the corresponding points in the other.
3. The lines joining corresponding points all pass through the same point.
If we suppose corresponding points marked, and the rows brought into any other position, then the lines joining corresponding points will no longer meet in a common point, and hence the third of the above properties will not hold any longer; but we have still a correspondence between the points in the two rows possessing the first two properties. Such a correspondence has been called a oneone correspondence, whilst the two rows between which such correspondence has been established are said to be projective or homographic. Two rows which are each the projection of the other are therefore projective. We shall presently see, also, that any two projective rows may always be placed in such a position that one appears as the projection of the other. If they are in such a position the rows are said to be in perspective position, or simply to be in perspective.
§ 26. The notion of a oneone correspondence between rows may be extended to flat and axial pencils, viz. a flat pencil will be said to be projective to a flat pencil if to each ray in the first corresponds one ray in the second, and if the crossratio of four rays in one equals that of the corresponding rays in the second.
Similarly an axial pencil may be projective to an axial pencil. But a flat pencil may also be projective to an axial pencil, or either pencil may be projective to a row. The definition is the same in each case: there is a oneone correspondence between the elements, and four elements have the same crossratio as the corresponding ones.
§ 27. There is also in each case a special position which is called perspective, viz.
1. Two projective rows are perspective if they lie in the same plane, and if the one row is a projection of the other.
2. Two projective flat pencils are perspective—(1) if they lie in the same plane, and have a row as a common section; (2) if they lie in the same pencil (in space), and are both sections of the same axial pencil; (3) if they are in space and have a row as common section, or are both sections of the same axial pencil, one of the conditions involving the other.
3. Two projective axial pencils, if their axes meet, and if they have a flat pencil as a common section.
4. A row and a projective flat pencil, if the row is a section of the pencil, each point lying in its corresponding line.
5. A row and a projective axial pencil, if the row is a section of the pencil, each point lying in its corresponding line.
6. A flat and a projective axial pencil, if the former is a section of the other, each ray lying in its corresponding plane.
That in each case the correspondence established by the position indicated is such as has been called projective follows at once from the definition. It is not so evident that the perspective position may always be obtained. We shall show in § 30 this for the first three cases. First, however, we shall give a few theorems which relate to the general correspondence, not to the perspective position.
§ 28. Two rows or pencils, flat or axial, which are projective to a third are projective to each other; this follows at once from the definitions.
§ 29. If two rows, or two pencils, either flat or axial, or a row and a pencil, be projective, we may assume to any three elements in the one the three corresponding elements in the other, and then the correspondence is uniquely determined.
For if in two projective rows we assume that the points A, B, C in the first correspond to the given points A′, B′, C′ in the second, then to any fourth point D in the first will correspond a point D′ in the second, so that
(AB, CD) = (A′B′, C′D′).
But there is only one point, D′, which makes the crossratio (A′B′, C′D′) equal to the given number (AB, CD).
The same reasoning holds in the other cases.
§ 30. If two rows are perspective, then the lines joining corresponding points all meet in a point, the centre of projection; and the point in which the two bases of the rows intersect as a point in the first row coincides with its corresponding point in the second.
This follows from the definition. The converse also holds, viz.
If two projective rows have such a position that one point in the one coincides with its corresponding point in the other, then they are perspective, that is, the lines joining corresponding points all pass through a common point, and form a flat pencil.
For let A, B, C, D ... be points in the one, and A′, B′, C′, D′ ... the corresponding points in the other row, and let A be made to coincide with its corresponding point A′. Let S be the point where the lines BB′ and CC′ meet, and let us join S to the point D in the first row. This line will cut the second row in a point D″, so that A, B, C, D are projected from S into the points A, B′, C′, D″. The crossratio (AB, CD) is therefore equal to (AB′, C′D″), and by hypothesis it is equal to (A′B′, C′D′). Hence (A′B′, C′D″) = (A′B′, C&primeprime;D′), that is, D″ is the same point as D′.
§ 31. If two projected flat pencils in the same plane are in perspective, then the intersections of corresponding lines form a row, and the line joining the two centres as a line in the first pencil corresponds to the same line as a line in the second. And conversely,
If two projective pencils in the same plane, but with different centres, have one line in the one coincident with its corresponding line in the other, then the two pencils are perspective, that is, the intersection of corresponding lines lie in a line.
The proof is the same as in § 30.
§ 32. If two projective flat pencils in the same point (pencil in space), but not in the same plane, are perspective, then the planes joining corresponding rays all pass through a line (they form an axial pencil), and the line common to the two pencils (in which their planes intersect) corresponds to itself. And conversely:—
If two flat pencils which have a common centre, but do not lie in a common plane, are placed so that one ray in the one coincides with its corresponding ray in the other, then they are perspective, that is, the planes joining corresponding lines all pass through a line.
§ 33. If two projective axial pencils are perspective, then the intersection of corresponding planes lie in a plane, and the plane common to the two pencils (in which the two axes lie) corresponds to itself. And conversely:—
If two projective axial pencils are placed in such a position that a plane in the one coincides with its corresponding plane, then the two pencils are perspective, that is, corresponding planes meet in lines which lie in a plane.
The proof again is the same as in § 30.
§ 34. These theorems relating to perspective position become illusory if the projective rows of pencils have a common base. We then have:—
In two projective rows on the same line—and also in two projective and concentric flat pencils in the same plane, or in two projective axial pencils with a common axis—every element in the one coincides with its corresponding element in the other as soon as three elements in the one coincide with their corresponding elements in the other.
Proof (in case of two rows).—Between four elements A, B, C, D and their corresponding elements A′, B′, C′, D′ exists the relation (ABCD) = (A′B′C′D′). If now A′, B′, C′ coincide respectively with A, B, C, we get (AB, CD) = (AB, CD′), hence D and D′ coincide.
The last theorem may also be stated thus:—
In two projective rows or pencils, which have a common base but are not identical, not more than two elements in the one can coincide with their corresponding elements in the other.
Thus two projective rows on the same line cannot have more than two pairs of coincident points unless every point coincides with its corresponding point.
Fig. 9. 
Fig. 10. 
Fig. 11. 
It is easy to construct two projective rows on the same line, which have two pairs of corresponding points coincident. Let the points A, B, C as points belonging to the one row correspond to A, B, and C′ as points in the second. Then A and B coincide with their corresponding points, but C does not. It is, however, not necessary that two such rows have twice a point coincident with its corresponding point; it is possible that this happens only once or not at all. Of this we shall see examples later.
§ 35. If two projective rows or pencils are in perspective position, we know at once which element in one corresponds to any given element in the other. If p and q (fig. 9) are two projective rows, so that K corresponds to itself, and if we know that to A and B in p correspond A′ and B′ in q, then the point S, where AA′ meets BB′, is the centre of projection, and hence, in order to find the point C′ corresponding to C, we have only to join C to S; the point C′, where this line cuts q, is the point required.
If two flat pencils, S_{1} and S_{2}, in a plane are perspective (fig. 10), we need only to know two pairs, a, a′ and b, b′, of corresponding rays in order to find the axis s of projection. This being known, a ray c′ in S_{2}, corresponding to a given ray c in S_{1}, is found by joining S_{2} to the point where c cuts the axis s.
A similar construction holds in the other cases of perspective figures.
On this depends the solution of the following general problem.
§ 36. Three pairs of corresponding elements in two projective rows or pencils being given, to determine for any element in one the corresponding element in the other.
We solve this in the two cases of two projective rows and of two projective flat pencils in a plane.
Problem I.—Let A, B, C be three points in a row s, A′, B′, C′ the corresponding points in a projective row s′, both being in a plane; it is required to find for any point D in s the corresponding point D′ in s′.
Problem II.—Let a, b, c be three rays in a pencil S, a′, b′, c′ the corresponding rays in a projective pencil S′, both being in the same plane; it is required to find for any ray d in S the corresponding ray d′ in S′.
The solution is made to depend on the construction of an auxiliary row or pencil which is perspective to both the given ones. This is found as follows:—
Solution of Problem I.—On the line joining two corresponding points, say AA′ (fig. 11), take any two points, S and S′, as centres of auxiliary pencils. Join the intersection B_{1} of SB and S′B′ to the intersection C_{1} of SC and S′C′ by the line s_{1}. Then a row on s_{1} will be perspective to s with S as centre of projection, and to s′ with S′ as centre. To find now the point D′ on s′ corresponding to a point D on s we have only to determine the point D_{1}, where the line SD cuts s_{1}, and to draw S′D_{1}; the point where this line cuts s′ will be the required point D′.
Proof.—The rows s and s′ are both perspective to the row s_{1}, hence they are projective to one another. To A, B, C, D on s correspond A_{1}, B_{1}, C_{1}, D_{1} on s_{1}, and to these correspond A′, B′, C′, D′ on s′; so that D and D′ are
corresponding points as required. Fig. 12. 
Fig. 13. 
Solution of Problem II.—Through the intersection A of two corresponding rays a and a′ (fig. 12), take two lines, s and s′, as bases of auxiliary rows. Let S_{1} be the point where the line b_{1}, which joins B and B′, cuts the line c_{1}, which joins C and C′. Then a pencil S_{1} will be perspective to S with s as axis of projection. To find the ray d′ in S′ corresponding to a given ray d in S, cut d by s at D; project this point from S_{1} to D′ on s′ and join D′ to S′. This will be the required ray.
Proof.—That the pencil S_{1} is perspective to S and also to S′ follows from construction. To the lines a_{1}, b_{1}, c_{1}, d_{1} in S_{1} correspond the lines a, b, c, d in S and the lines a′, b′, c′, d′ in S′, so that d and d′ are corresponding rays.
In the first solution the two centres, S, S′, are any two points on a line joining any two corresponding points, so that the solution of the problem allows of a great many different constructions. But whatever construction be used, the point D′, corresponding to D, must be always the same, according to the theorem in § 29. This gives rise to a number of theorems, into which, however, we shall not enter. The same remarks hold for the second problem.
§ 37. Homological Triangles.—As a further application of the theorems about perspective rows and pencils we shall prove the following important theorem.
Theorem.—If ABC and A′B′C′ (fig. 13) be two triangles, such that the lines AA′, BB′, CC′ meet in a point S, then the intersections of BC and B′C′, of CA and C′A′, and of AB and A′B′ will lie in a line. Such triangles are said to be homological, or in perspective. The triangles are “coaxial” in virtue of the property that the meets of corresponding sides are collinear and copolar, since the lines joining corresponding vertices are concurrent.
Proof.—Let a, b, c denote the lines AA′, BB′, CC′, which meet at S. Then these may be taken as bases of projective rows, so that A, A′, S on a correspond to B, B′, S on b, and to C, C′, S on c. As the point S is common to all, any two of these rows will be perspective.
If  S_{1} be the centre of projection of rows  b and c, 
S_{2} ” ” ”  c and a,  
S_{3} ” ” ”  a and b, 
and if the line S_{1}S_{2} cuts a in A_{1}, and b in B_{1}, and c in C_{1}, then A_{1}, B_{1} will be corresponding points in a and b, both corresponding to C_{1} in c. But a and b are perspective, therefore the line A_{1}B_{1}, that is S_{1}S_{2}, joining corresponding points must pass through the centre of projection S_{3} of a and b. In other words, S_{1}, S_{2}, S_{3} lie in a line. This is Desargues’ celebrated theorem if we state it thus:—
Theorem of Desargues.—If each of two triangles has one vertex on each of three concurrent lines, then the intersections of corresponding sides lie in a line, those sides being called corresponding which are opposite to vertices on the same line.
The converse theorem holds also, viz.
Theorem.—If the sides of one triangle meet those of another in three points which lie in a line, then the vertices lie on three lines which meet in a point.
The proof is almost the same as before.
§ 38. Metrical Relations between Projective Rows.—Every row contains one point which is distinguished from all others, viz. the point at infinity. In two projective rows, to the point I at infinity in one corresponds a point I′ in the other, and to the point J′ at infinity in the second corresponds a point J in the first. The points I′ and J are in general finite. If now A and B are any two points in the one, A′, B′ the corresponding points in the other row, then
or
But, by § 17,
therefore the last equation changes into
that is to say—
Theorem.—The product of the distances of any two corresponding points in two projective rows from the points which correspond to the points at infinity in the other is constant, viz. AJ · A′I′ = k. Steiner has called this number k the Power of the correspondence.
[The relation AJ · A′I′ = k shows that if J, I′ be given then the point A′ corresponding to a specified point A is readily found; hence A, A′ generate homographic ranges of which I and J′ correspond to the points at infinity on the ranges. If we take any two origins O, O′, on the ranges and reduce the expression AJ · A′I′ = k to its algebraic equivalent, we derive an equation of the form αxx′ + βx + γx′ + δ = 0. Conversely, if a relation of this nature holds, then points corresponding to solutions in x, x′ form homographic ranges.]
§ 39. Similar Rows.—If the points at infinity in two projective rows correspond so that I′ and J are at infinity, this result loses its meaning. But if A, B, C be any three points in one, A′, B′, C′ the corresponding ones on the other row, we have
which reduces to
that is, corresponding segments are proportional. Conversely, if corresponding segments are proportional, then to the point at infinity in one corresponds the point at infinity in the other. If we call such rows similar, we may state the result thus—
Fig. 14. 
Theorem.—Two projective rows are similar if to the point at infinity in one corresponds the point at infinity in the other, and conversely, if two rows are similar then they are projective, and the points at infinity are corresponding points.
From this the wellknown propositions follow:—
Two lines are cut proportionally (in similar rows) by a series of parallels. The rows are perspective, with centre of projection at infinity.
If two similar rows are placed parallel, then the lines joining homologous points pass through a common point.
§ 40. If two flat pencils be projective, then there exists in either, one single pair of lines at right angles to one another, such that the corresponding lines in the other pencil are again at right angles.
To prove this, we place the pencils in perspective position (fig. 14) by making one ray coincident with its corresponding ray. Corresponding rays meet then on a line p. And now we draw the circle which has its centre O on p, and which passes through the centres S and S′ of the two pencils. This circle cuts p in two points H and K. The two pairs of rays, h, k, and h′, k′, joining these points to S and S′ will be pairs of corresponding rays at right angles. The construction gives in general but one circle, but if the line p is the perpendicular bisector of SS′, there exists an infinite number, and to every right angle in the one pencil corresponds a right angle in the other.
Principle of Duality
§ 41. It has been stated in § 1 that not only points, but also planes and lines, are taken as elements out of which figures are built up. We shall now see that the construction of one figure which possesses certain properties gives rise in many cases to the construction of another figure, by replacing, according to definite rules, elements of one kind by those of another. The new figure thus obtained will then possess properties which may be stated as soon as those of the original figure are known.
We obtain thus a principle, known as the principle of duality or of reciprocity, which enables us to construct to any figure not containing any measurement in its construction a reciprocal figure, as it is called, and to deduce from any theorem a reciprocal theorem, for which no further proof is needed.
It is convenient to print reciprocal propositions on opposite sides of a page broken into two columns, and this plan will occasionally be adopted.
We begin by repeating in this form a few of our former statements:—
Two points determine a line.  Two planes determine a line. 
Three points which are not in a line determine a plane.  Three planes which do not pass through a line determine a point. 
A line and a point without it determine a plane.  A line and a plane not through it determine a point. 
Two lines in a plane determine a point.  Two lines through a point determine a plane. 
These propositions show that it will be possible, when any figure is given, to construct a second figure by taking planes instead of points, and points instead of planes, but lines where we had lines.
For instance, if in the first figure we take a plane and three points in it, we have to take in the second figure a point and three planes through it. The three points in the first, together with the three lines joining them two and two, form a triangle; the three planes in the second and their three lines of intersection form a trihedral angle. A triangle and a trihedral angle are therefore reciprocal figures.
Similarly, to any figure in a plane consisting of points and lines will correspond a figure consisting of planes and lines passing through a point S, and hence belonging to the pencil which has S as centre.
The figure reciprocal to four points in space which do not lie in a plane will consist of four planes which do not meet in a point. In this case each figure forms a tetrahedron.
§ 42. As other examples we have the following:—
To a row  is reciprocal  an axial pencil, 
to a flat pencil  ”  a flat pencil, 
to a field of points and lines  ”  a pencil of planes and lines, 
to the space of points  ”  the space of planes. 
For the row consists of a line and all the points in it, reciprocal to it therefore will be a line with all planes through it, that is, an axial pencil; and so for the other cases.
This correspondence of reciprocity breaks down, however, if we take figures which contain measurement in their construction. For instance, there is no figure reciprocal to two planes at right angles, because there is no segment in a row which has a magnitude as definite as a right angle.
We add a few examples of reciprocal propositions which are easily proved.
Theorem.—If A, B, C, D are any four points in space, and if the lines AB and CD meet, then all four points lie in a plane, hence also AC and BD, as well as AD and BC, meet.  Theorem.—If α, β, γ, δ are four planes in space, and if the lines αβ and γδ meet, then all four planes lie in a point (pencil), hence also αγ and βδ, as well as αδ and βγ, meet. 
Theorem.—If of any number of lines every one meets every other, whilst all do not
lie in a point, then all lie in a plane.  lie in a plane, then all lie in a point (pencil). 
§ 43. Reciprocal figures as explained lie both in space of three dimensions. If the one is confined to a plane (is formed of elements which lie in a plane), then the reciprocal figure is confined to a pencil (is formed of elements which pass through a point).
But there is also a more special principle of duality, according to which figures are reciprocal which lie both in a plane or both in a pencil. In the plane we take points and lines as reciprocal elements, for they have this fundamental property in common, that two elements of one kind determine one of the other. In the pencil, on the other hand, lines and planes have to be taken as reciprocal, and here it holds again that two lines or planes determine one plane or line.
Thus, to one plane figure we can construct one reciprocal figure in the plane, and to each one reciprocal figure in a pencil. We mention a few of these. At first we explain a few names:—
A figure consisting of n points in a plane will be called an npoint.  A figure consisting of n lines in a plane will be called an nside. 
A figure consisting of n planes in a pencil will be called an nflat.  A figure consisting of n lines in a pencil will be called an nedge. 
It will be understood that an nside is different from a polygon of n sides. The latter has sides of finite length and n vertices, the former has sides all of infinite extension, and every point where two of the sides meet will be a vertex. A similar difference exists between a solid angle and an nedge or an nflat. We notice particularly—
A fourpoint has six sides, of which two and two are opposite, and three diagonal points, which are intersections of opposite sides.  A fourside has six vertices, of which two and two are opposite, and three diagonals, which join opposite vertices. 
A fourflat has six edges, of which two and two are opposite, and three diagonal planes, which pass through opposite edges.  A fouredge has six faces, of which two and two are opposite, and three diagonal edges, which are intersections of opposite faces. 
A fourside is usually called a complete quadrilateral, and a fourpoint a complete quadrangle. The above notation, however, seems better adapted for the statement of reciprocal propositions.
§ 44.
If a point moves in a plane it describes a plane curve.  If a line moves in a plane it envelopes a plane curve (fig. 15). 
If a plane moves in a pencil it envelopes a cone.  If a line moves in a pencil it describes a cone. 
Fig. 15. 
A curve thus appears as generated either by points, and then we call it a “locus,” or by lines, and then we call it an “envelope.” In the same manner a cone, which means here a surface, appears either as the locus of lines passing through a fixed point, the “vertex” of the cone, or as the envelope of planes passing through the same point.
To a surface as locus of points corresponds, in the same manner, a surface as envelope of planes; and to a curve in space as locus of points corresponds a developable surface as envelope of planes.
It will be seen from the above that we may, by aid of the principle of duality, construct for every figure a reciprocal figure, and that to any property of the one a reciprocal property of the other will exist, as long as we consider only properties which depend upon nothing but the positions and intersections of the different elements and not upon measurement.
For such propositions it will therefore be unnecessary to prove more than one of two reciprocal theorems.
Generation of Curves and Cones of Second Order or Second Class
§ 45. Conics.—If we have two projective pencils in a plane, corresponding rays will meet, and their point of intersection will constitute some locus which we have to investigate. Reciprocally, if two projective rows in a plane are given, then the lines which join corresponding points will envelope some curve. We prove first:—
Theorem.—If two projective flat pencils lie in a plane, but are neither in perspective nor concentric, then the locus of intersections of corresponding rays is a curve of the second order, that is, no line contains more than two points of the locus.  Theorem.—If two projective rows lie in a plane, but are neither in perspective nor on a common base, then the envelope of lines joining corresponding points is a curve of the second class, that is, through no point pass more than two of the enveloping lines. 
Proof.—We draw any line t. This cuts each of the pencils in a row, so that we have on t two rows, and these are projective because the pencils are projective. If corresponding rays of the two pencils meet on the line t, their intersection will be a point in the one row which coincides with its corresponding point in the other. But two projective rows on the same base cannot have more than two points of one coincident with their corresponding points in the other (§ 34).  Proof.—We take any point T and join it to all points in each row. This gives two concentric pencils, which are projective because the rows are projective. If a line joining corresponding points in the two rows passes through T, it will be a line in the one pencil which coincides with its corresponding line in the other. But two projective concentric flat pencils in the same plane cannot have more than two lines of one coincident with their corresponding line in the other (§ 34). 
It will be seen that the proofs are reciprocal, so that the one may be copied from the other by simply interchanging the words point and line, locus and envelope, row and pencil, and so on. We shall therefore in future prove seldom more than one of two reciprocal theorems, and often state one theorem only, the reader being recommended to go through the reciprocal proof by himself, and to supply the reciprocal theorems when not given.
§ 46. We state the theorems in the pencil reciprocal to the last, without proving them:—
Theorem.—If two projective flat pencils are concentric, but are neither perspective nor coplanar, then the envelope of the planes joining corresponding rays is a cone of the second class; that is, no line through the common centre contains more than two of the enveloping planes.  Theorem.—If two projective axial pencils lie in the same pencil (their axes meet in a point), but are neither perspective nor coaxial, then the locus of lines joining corresponding planes is a cone of the second order; that is, no plane in the pencil contains more than two of these lines. 
§ 47. Of theorems about cones of second order and cones of second class we shall state only very few. We point out, however, the following connexion between the curves and cones under consideration:
The lines which join any point in space to the points on a curve of the second order form a cone of the second order.  Every plane section of a cone of the second order is a curve of the second order. 
The planes which join any point in space to the lines enveloping a curve of the second class envelope themselves a cone of the second class.  Every plane section of a cone of the second class is a curve of the second class. 
By its aid, or by the principle of duality, it will be easy to obtain theorems about them from the theorems about the curves.
We prove the first. A curve of the second order is generated by two projective pencils. These pencils, when joined to the point in space, give rise to two projective axial pencils, which generate the cone in question as the locus of the lines where corresponding planes
meet. §48.
Theorem.—The curve of second order which is generated by two projective flat pencils passes through the centres of the two pencils.  Theorem.—The envelope of second class which is generated by two projective rows contains the bases of these rows as enveloping lines or tangents. 
Proof.—If S and S′ are the two pencils, then to the ray SS′ or p′ in the pencil S′ corresponds in the pencil S a ray p, which is different from p′, for the pencils are not perspective. But p and p′ meet at S, so that S is a point on the curve, and similarly S′.  Proof.—If s and s′ are the two rows, then to the point ss′ or P′ as a point in s′ corresponds in s a point P, which is not coincident with P′, for the rows are not perspective. But P and P′ are joined by s, so that s is one of the enveloping lines, and similarly s′. 
It follows that every line in one of the two pencils cuts the curve in two points, viz. once at the centre S of the pencil, and once where it cuts its corresponding ray in the other pencil. These two points, however, coincide, if the line is cut by its corresponding line at S itself. The line p in S, which corresponds to the line SS′ in S′, is therefore the only line through S which has but one point in common with the curve, or which cuts the curve in two coincident points. Such a line is called a tangent to the curve, touching the latter at the point S, which is called the “point of contact.”
In the same manner we get in the reciprocal investigation the result that through every point in one of the rows, say in s, two tangents may be drawn to the curve, the one being s, the other the line joining the point to its corresponding point in s′. There is, however, one point P in s for which these two lines coincide. Such a point in one of the tangents is called the “point of contact” of the tangent. We thus get—
Theorem.—To the line joining the centres of the projective pencils as a line in one pencil corresponds in the other the tangent at its centre.  Theorem.—To the point of intersection of the bases of two projective rows as a point in one row corresponds in the other the point of contact of its base. 
§ 49. Two projective pencils are determined if three pairs of corresponding lines are given. Hence if a_{1}, b_{1}, c_{1} are three lines in a pencil S_{1}, and a_{2}, b_{2}, c_{2} the corresponding lines in a projective pencil S_{2}, the correspondence and therefore the curve of the second order generated by the points of intersection of corresponding rays is determined. Of this curve we know the two centres S_{1} and S_{2}, and the three points a_{1}a_{2}, b_{1}b_{2}, c_{1}c_{2}, hence five points in all. This and the reciprocal considerations enable us to solve the following two problems:
Problem.—To construct a curve of the second order, of which five points S_{1}, S_{2}, A, B, C are given.  Problem.—To construct a curve of the second class, of which five tangents u_{1}, u_{2}, a, b, c are given. 
In order to solve the lefthand problem, we take two of the given points, say S_{1} and S_{2}, as centres of pencils. These we make projective by taking the rays a_{1}, b_{1}, c_{1}, which join S_{1} to A, B, C respectively, as corresponding to the rays a_{2}, b_{2}, c_{2}, which join S_{2} to A, B, C respectively, so that three rays meet their corresponding rays at the given points A, B, C. This determines the correspondence of the pencils which will generate a curve of the second order passing through A, B, C and through the centres S_{1} and S_{2}, hence through the five given points. To find more points on the curve we have to construct for any ray in S_{1} the corresponding ray in S_{2}. This has been done in § 36. But we repeat the construction in order to deduce further properties from it. We also solve the righthand problem. Here we select two, viz. u_{1}, u_{2} of the five given lines, u_{1}, u_{2}, a, b, c, as bases of two rows, and the points A_{1}, B_{1}, C_{1} where a, b, c cut u_{1} as corresponding to the points A_{2}, B_{2}, C_{2} where a, b, c cut u_{2}.
We get then the following solutions of the two problems:
Solution.—Through the point A draw any two lines, u_{1} and u_{2} (fig. 16), the first u_{1} to cut the pencil S_{1} in a row AB_{1}C_{1}, the other u_{2} to cut the pencil S_{2} in a row AB_{2}C_{2}. These two rows will be perspective, as the point A corresponds to itself, and the centre of projection will be the point S, where the lines B_{1}B_{2} and C_{1}C_{2} meet. To find now for any ray d_{1} in S_{1} its corresponding ray d_{2} in S_{2}, we determine the point D_{1} where d_{1} cuts u_{1}, project this point from S to D_{2} on u_{2} and join S_{2} to D_{2}. This will be the required ray d_{2} which cuts d_{1} at some point D on the curve.  Solution.—In the line a take any two points S_{1} and S_{2} as centres of pencils (fig. 17), the first S_{1} (A_{1}B_{1}C_{1}) to project the row u_{1}, the other S_{2} (A_{2}B_{2}C_{2}) to project the row u_{2}. These two pencils will be perspective, the line S_{1}A_{1} being the same as the corresponding line S_{2}A_{2}, and the axis of projection will be the line u, which joins the intersection B of S_{1}B_{1} and S_{2}B_{2} to the intersection C of S_{1}C_{1} and S_{2}C_{2}. To find now for any point D_{1} in u_{1} the corresponding point D_{2} in u_{2}, we draw S_{1}D_{1} and project the point D where this line cuts u from S_{2} to u_{2}. This will give the required point D_{2}, and the line d joining D_{1} to D_{2} will be a new tangent to the curve. 
§ 50. These constructions prove, when rightly interpreted, very important properties of the curves in question.
Fig. 16. 
If in fig. 16 we draw in the pencil S_{1} the ray k_{1} which passes through the auxiliary centre S, it will be found that the corresponding ray k_{2} cuts it on u_{2}. Hence—
Theorem.—In the above construction the bases of the auxiliary rows u_{1} and u_{2} cut the curve where they cut the rays S_{2}S and S_{1}S respectively.  Theorem.—In the above construction (fig. 17) the tangents to the curve from the centres of the auxiliary pencils S_{1} and S_{2} are the lines which pass through u_{2}u and u_{1}u respectively. 
As A is any given point on the curve, and u_{1} any line through it, we have solved the problems:
Problem.—To find the second point in which any line through a known point on the curve cuts the curve.  Problem.—To find the second tangent which can be drawn from any point in a given tangent to the curve. 
If we determine in S_{1} (fig. 16) the ray corresponding to the ray S_{2}S_{1} in S_{2}, we get the tangent at S_{1}. Similarly, we can determine the point of contact of the tangents u_{1} or u_{2} in fig. 17.
Fig. 17. 
Fig. 18. 
§ 51. If five points are given, of which not three are in a line, then we can, as has just been shown, always draw a curve of the second order through them; we select two of the points as centres of projective pencils, and then one such curve is determined. It will be presently shown that we get always the same curve if two other points are taken as centres of pencils, that therefore five points determine one curve of the second order, and reciprocally, that five tangents determine one curve of the second class. Six points taken at random will therefore not lie on a curve of the second order. In order that this may be the case a certain condition has to be satisfied, and this condition is easily obtained from the construction in § 49, fig. 16. If we consider the conic determined by the five points A, S_{1}, S_{2}, K, L, then the point D will be on the curve if, and only if, the points on D_{1}, S, D_{2} be in a line.
This may be stated differently if we take AKS_{1}DS_{2}L (figs. 16 and 18) as a hexagon inscribed in the conic, then AK and DS_{2} will be opposite sides, so will be KS_{1} and S_{2}L, as well as S_{1}D and LA. The first two meet in D_{2}, the others in S and D_{1} respectively. We may therefore state the required condition, together with the reciprocal one, as follows:—
Pascal’s Theorem.—If a hexagon be inscribed in a curve of the second order, then the intersections of opposite sides are three points in a line.  Brianchon’s Theorem.—If a hexagon be circumscribed about a curve of the second class, then the lines joining opposite vertices are three lines meeting in a point. 
These celebrated theorems, which are known by the names of their discoverers, are perhaps the most fruitful in the whole theory of conics. Before we go over to their applications we have to show that we obtain the same curve if we take, instead of S_{1}, S_{2}, any two other points on the curve as centres of projective pencils.
§ 52. We know that the curve depends only upon the correspondence between the pencils S_{1} and S_{2}, and not upon the special construction used for finding new points on the curve. The point A (fig. 16 or 18), through which the two auxiliary rows u_{1}, u_{2} were drawn, may therefore be changed to any other point on the curve. Let us now suppose the curve drawn, and keep the points S_{1}, S_{2}, K, L and D, and hence also the point S fixed, whilst we move A along the curve. Then the line AL will describe a pencil about L as centre, and the point D_{1} a row on S_{1}D perspective to the pencil L. At the same time AK describes a pencil about K and D_{2} a row perspective to it on S_{2}D. But by Pascal’s theorem D_{1} and D_{2} will always lie in a line with S, so that the rows described by D_{1} and D_{2} are perspective. It follows that the pencils K and L will themselves be projective, corresponding rays meeting on the curve. This proves that we get the same curve whatever pair of the five given points we take as centres of projective pencils. Hence—
Only one curve of the second order can be drawn which passes through five given points.  Only one curve of the second class can be drawn which touches five given lines. 
We have seen that if on a curve of the second order two points coincide at A, the line joining them becomes the tangent at A. If, therefore, a point on the curve and its tangent are given, this will be equivalent to having given two points on the curve. Similarly, if on the curve of second class a tangent and its point of contact are given, this will be equivalent to two given tangents.
We may therefore extend the last theorem:
Only one curve of the second order can be drawn, of which four points and the tangent at one of them, or three points and the tangents at two of them, are given.  Only one curve of the second class can be drawn, of which four tangents and the point of contact at one of them, or three tangents and the points of contact at two of them, are given. 
§ 53. At the same time it has been proved:
If all points on a curve of the second order be joined to any two of them, then the two pencils thus formed are projective, those rays being corresponding which meet on the curve. Hence—  All tangents to a curve of second class are cut by any two of them in projective rows, those being corresponding points which lie on the same tangent. Hence— 
The crossratio of four rays joining a point S on a curve of second order to four fixed points A, B, C, D in the curve is independent of the position of S, and is called the crossratio of the four points A, B, C, D.  The crossratio of the four points in which any tangent u is cut by four fixed tangents a, b, c, d is independent of the position of u, and is called the crossratio of the four tangents a, b, c, d. 
If this crossratio equals −1 the four points are said to be four harmonic points.  If this crossratio equals −1 the four tangents are said to be four harmonic tangents. 
We have seen that a curve of second order, as generated by projective pencils, has at the centre of each pencil one tangent; and further, that any point on the curve may be taken as centre of such pencil. Hence—
A curve of second order has at every point one tangent.  A curve of second class has on every tangent a point of contact. 
§ 54. We return to Pascal’s and Brianchon’s theorems and their applications, and shall, as before, state the results both for curves of the second order and curves of the second class, but prove them only for the former.
Pascal’s theorem may be used when five points are given to find more points on the curve, viz. it enables us to find the point where any line through one of the given points cuts the curve again. It is convenient, in making use of Pascal’s theorem, to number the points, to indicate the order in which they are to be taken in forming a hexagon, which, by the way, may be done in 60 different ways. It will be seen that 1 2 (leaving out 3) 4 5 are opposite sides, so are 2 3 and (leaving out 4) 5 6, and also 3 4 and (leaving out 5) 6 1.
If the points 1 2 3 4 5 are given, and we want a 6th point on a line drawn through 1, we know all the sides of the hexagon with the exception of 5 6, and this is found by Pascal’s theorem.
If this line should happen to pass through 1, then 6 and 1 coincide, or the line 6 1 is the tangent at 1. And always if two consecutive vertices of the hexagon approach nearer and nearer, then the side joining them will ultimately become a tangent.
We may therefore consider a pentagon inscribed in a curve of second order and the tangent at one of its vertices as a hexagon, and thus get the theorem:
Every pentagon inscribed in a curve of second order has the property that the intersections of two pairs of nonconsecutive sides lie in a line with the point where the fifth side cuts the tangent at the opposite vertex.  Every pentagon circumscribed about a curve of the second class has the property that the lines which join two pairs of nonconsecutive vertices meet on that line which joins the fifth vertex to the point of contact of the opposite side. 
This enables us also to solve the following problems.
Given five points on a curve of second order to construct the tangent at any one of them.  Given five tangents to a curve of second class to construct the point of contact of any one of them. 
Fig. 19. 
If two pairs of adjacent vertices coincide, the hexagon becomes a quadrilateral, with tangents at two vertices. These we take to be opposite, and get the following theorems:
If a quadrilateral be inscribed in a curve of second order, the intersections of opposite sides, and also the intersections of the tangents at opposite vertices, lie in a line (fig. 19).  If a quadrilateral be circumscribed about a curve of second class, the lines joining opposite vertices, and also the lines joining points of contact of opposite sides, meet in a point. 
Fig. 20. 
If we consider the hexagon made up of a triangle and the tangents at its vertices, we get—
If a triangle is inscribed in a curve of the second order, the points in which the sides are cut by the tangents at the opposite vertices meet in a point.  If a triangle be circumscribed about a curve of second class, the lines which join the vertices to the points of contact of the opposite sides meet in a point (fig. 20). 
§ 55. Of these theorems, those about the quadrilateral give rise to a number of others. Four points A, B, C, D may in three different ways be formed into a quadrilateral, for we may take them in the order ABCD, or ACBD, or ACDB, so that either of the points B, C, D may be taken as the vertex opposite to A. Accordingly we may apply the theorem in three different ways.
Let A, B, C, D be four points on a curve of second order (fig. 21), and let us take them as forming a quadrilateral by taking the points in the order ABCD, so that A, C and also B, D are pairs of opposite vertices. Then P, Q will be the points where opposite sides meet, and E, F the intersections of tangents at opposite vertices. The four points P, Q, E, F lie therefore in a line. The quadrilateral ACBD gives us in the same way the four points Q, R, G, H in a line, and the quadrilateral ABDC a line containing the four points R, P, I, K. These three lines form a triangle PQR.
The relation between the points and lines in this figure may be expressed more clearly if we consider ABCD as a fourpoint inscribed in a conic, and the tangents at these points as a fourside circumscribed about it,—viz. it will be seen that P, Q, R are the diagonal points of the fourpoint ABCD, whilst the sides of the triangle PQR are the diagonals of the circumscribing fourside. Hence the theorem—
Any fourpoint on a curve of the second order and the fourside formed by the tangents at these points stand in this relation that the diagonal points of the fourpoint lie in the diagonals of the fourside. And conversely,
If a fourpoint and a circumscribed fourside stand in the above relation, then a curve of the second order may be described which passes through the four points and touches there the four sides of these figures.
Fig. 21. 
That the last part of the theorem is true follows from the fact that the four points A, B, C, D and the line a, as tangent at A, determine a curve of the second order, and the tangents to this curve at the other points B, C, D are given by the construction which leads to fig. 21.
The theorem reciprocal to the last is—
Any fourside circumscribed about a curve of second class and the fourpoint formed by the points of contact stand in this relation that the diagonals of the fourside pass through the diagonal points of the fourpoint. And conversely,
If a fourside and an inscribed fourpoint stand in the above relation, then a curve of the second class may be described which touches the sides of the fourside at the points of the fourpoint.
§ 56. The fourpoint and the fourside in the two reciprocal theorems are alike. Hence if we have a fourpoint ABCD and a fourside abcd related in the manner described, then not only may a curve of the second order be drawn, but also a curve of the second class, which both touch the lines a, b, c, d at the points A, B, C, D.
The curve of second order is already more than determined by the points A, B, C and the tangents a, b, c at A, B and C. The point D may therefore be any point on this curve, and d any tangent to the curve. On the other hand the curve of the second class is more than determined by the three tangents a, b, c and their points of contact A, B, C, so that d is any tangent to this curve. It follows that every tangent to the curve of second order is a tangent of a curve of the second class having the same point of contact. In other words, the curve of second order is a curve of second class, and vice versa. Hence the important theorems—
Every curve of second order is a curve of second class.  Every curve of second class is a curve of second order. 
The curves of second order and of second class, having thus been proved to be identical, shall henceforth be called by the common name of Conics.
For these curves hold, therefore, all properties which have been proved for curves of second order or of second class. We may therefore now state Pascal’s and Brianchon’s theorem thus—
Pascal’s Theorem.—If a hexagon be inscribed in a conic, then the intersections of opposite sides lie in a line.
Brianchon’s Theorem.—If a hexagon be circumscribed about a conic, then the diagonals forming opposite centres meet in a point.
§ 57. If we suppose in fig. 21 that the point D together with the tangent d moves along the curve, whilst A, B, C and their tangents a, b, c remain fixed, then the ray DA will describe a pencil about A, the point Q a projective row on the fixed line BC, the point F the row b, and the ray EF a pencil about E. But EF passes always through Q. Hence the pencil described by AD is projective to the pencil described by EF, and therefore to the row described by F on b. At the same time the line BD describes a pencil about B projective to that described by AD (§ 53). Therefore the pencil BD and the row F on b are projective. Hence—
If on a conic a point A be taken and the tangent a at this point, then the crossratio of the four rays which join A to any four points on the curve is equal to the crossratio of the points in which the tangents at these points cut the tangent at A.
§ 58. There are theorems about cones of second order and second class in a pencil which are reciprocal to the above, according to § 43. We mention only a few of the more important ones.
The locus of intersections of corresponding planes in two projective axial pencils whose axes meet is a cone of the second order.
The envelope of planes which join corresponding lines in two projective flat pencils, not in the same plane, is a cone of the second class.
Cones of second order and cones of second class are identical.
Every plane cuts a cone of the second order in a conic.
A cone of second order is uniquely determined by five of its edges or by five of its tangent planes, or by four edges and the tangent plane at one of them, &c. &c.
Pascal’s Theorem.—If a solid angle of six faces be inscribed in a cone of the second order, then the intersections of opposite faces are three lines in a plane.
Brianchon’s Theorem.—If a solid angle of six edges be circumscribed about a cone of the second order, then the planes through opposite edges meet in a line.
Each of the other theorems about conics may be stated for cones of the second order.
Fig. 22. 
§ 59. Projective Definitions of the Conics.—We now consider the shape of the conics. We know that any line in the plane of the conic, and hence that the line at infinity, either has no point in common with the curve, or one (counting for two coincident points) or two distinct points. If the line at infinity has no point on the curve the latter is altogether finite, and is called an Ellipse (fig. 21). If the line at infinity has only one point in common with the conic, the latter extends to infinity, and has the line at infinity a tangent. It is called a Parabola (fig. 22). If, lastly, the line at infinity cuts the curve in two points, it consists of two separate parts which each extend in two branches to the points at infinity where they meet. The curve is in this case called an Hyperbola (see fig. 20). The tangents at the two points at infinity are finite because the line at infinity is not a tangent. They are called Asymptotes. The branches of the hyperbola approach these lines indefinitely as a point on the curves moves to infinity.
§ 60. That the circle belongs to the curves of the second order is seen at once if we state in a slightly different form the theorem that in a circle all angles at the circumference standing upon the same arc are equal. If two points S_{1}, S_{2} on a circle be joined to any other two points A and B on the circle, then the angle included by the rays S_{1}A and S_{1}B is equal to that between the rays S_{2}A and S_{2}B, so that as A moves along the circumference the rays S_{1}A and S_{2}A describe equal and therefore projective pencils. The circle can thus be generated by two projective pencils, and is a curve of the second order.
If we join a point in space to all points on a circle, we get a (circular) cone of the second order (§ 43). Every plane section of this cone is a conic. This conic will be an ellipse, a parabola, or an hyperbola, according as the line at infinity in the plane has no, one or two points in common with the conic in which the plane at infinity cuts the cone. It follows that our curves of second order may be obtained as sections of a circular cone, and that they are identical with the “Conic Sections” of the Greek mathematicians.
§ 61. Any two tangents to a parabola are cut by all others in projective rows; but the line at infinity being one of the tangents, the points at infinity on the rows are corresponding points, and the rows therefore similar. Hence the theorem—
The tangents to a parabola cut each other proportionally.
Pole and Polar
§ 62. We return once again to fig. 21, which we obtained in § 55.
If a fourside be circumscribed about and a fourpoint inscribed in a conic, so that the vertices of the second are the points of contact of the sides of the first, then the triangle formed by the diagonals of the first is the same as that formed by the diagonal points of the other.
Such a triangle will be called a polartriangle of the conic, so that PQR in fig. 21 is a polartriangle. It has the property that on the side p opposite P meet the tangents at A and B, and also those at C and D. From the harmonic properties of fourpoints and foursides it follows further that the points L, M, where it cuts the lines AB and CD, are harmonic conjugates with regard to AB and CD respectively.
If the point P is given, and we draw a line through it, cutting the conic in A and B, then the point Q harmonic conjugate to P with regard to AB, and the point H where the tangents at A and B meet, are determined. But they lie both on p, and therefore this line is determined. If we now draw a second line through P, cutting the conic in C and D, then the point M harmonic conjugate to P with regard to CD, and the point G where the tangents at C and D meet, must also lie on p. As the first line through P already determines p, the second may be any line through P. Now every two lines through P determine a fourpoint ABCD on the conic, and therefore a polartriangle which has one vertex at P and its opposite side at p. This result, together with its reciprocal, gives the theorems—
All polartriangles which have one vertex in common have also the opposite side in common.
All polartriangles which have one side in common have also the opposite vertex in common.
§ 63. To any point P in the plane of, but not on, a conic corresponds thus one line p as the side opposite to P in all polartriangles which have one vertex at P, and reciprocally to every line p corresponds one point P as the vertex opposite to p in all triangles which have p. as one side.
We call the line p the polar of P, and the point P the pole of the line p with regard to the conic.
If a point lies on the conic, we call the tangent at that point its polar; and reciprocally we call the point of contact the pole of tangent.
§ 64. From these definitions and former results follow—
The polar of any point P not on the conic is a line p, which has the following properties:—  The pole of any line p not a tangent to the conic is a point P, which has the following properties:— 
1. On every line through P which cuts the conic, the polar of P contains the harmonic conjugate of P with regard to those points on the conic.  1. Of all lines through a point on p from which two tangents may be drawn to the conic, the pole P contains the line which is harmonic conjugate to p, with regard to the two tangents. 
2. If tangents can be drawn from P, their points of contact lie on p.  2. If p cuts the conic, the tangents at the intersections meet at P. 
3. Tangents drawn at the points where any line through P cuts the conic meet on p; and conversely,  3. The point of contact of tangents drawn from any point on p to the conic lie in a line with P; and conversely, 
4. If from any point on p, tangents be drawn, their points of contact will lie in a line with P.  4. Tangents drawn at points where any line through P cuts the conic meet on p. 
5. Any fourpoint on the conic which has one diagonal point at P has the other two lying on p.  5. Any fourside circumscribed about a conic which has one diagonal on p has the other two meeting at P. 
The truth of 2 follows from 1. If T be a point where p cuts the conic, then one of the points where PT cuts the conic, and which are harmonic conjugates with regard to PT, coincides with T; hence the other does—that is, PT touches the curve at T.
That 4 is true follows thus: If we draw from a point H on the polar one tangent a to the conic, join its point of contact A to the pole P, determine the second point of intersection B of this line with the conic, and draw the tangent at B, it will pass through H, and will therefore be the second tangent which may be drawn from H to the curve.
§ 65. The second property of the polar or pole gives rise to the theorem—
From a point in the plane of a conic, two, one or no tangents may be drawn to the conic, as its polar has two, one, or no points in common with the curve.  A line in the plane of a conic has two, one or no points in common with the conic, according as two, one or no tangents can be drawn from its pole to the conic. 
Of any point in the plane of a conic we say that it was without, on or within the curve according as two, one or no tangents to the curve pass through it. The points on the conic separate those within the conic from those without. That this is true for a circle is known from elementary geometry. That it also holds for other conics follows from the fact that every conic may be considered as the projection of a circle, which will be proved later on.
The fifth property of pole and polar stated in § 64 shows how to find the polar of any point and the pole of any line by aid of the straightedge only. Practically it is often convenient to draw three secants through the pole, and to determine only one of the diagonal points for two of the fourpoints formed by pairs of these lines and the conic (fig. 22).
These constructions also solve the problem—
From a point without a conic, to draw the two tangents to the conic by aid of the straightedge only.
For we need only draw the polar of the point in order to find the points of contact.
§ 66. The property of a polartriangle may now be stated thus—
In a polartriangle each side is the polar of the opposite vertex, and each vertex is the pole of the opposite side.
Fig. 23. 
If P is one vertex of a polartriangle, then the other vertices, Q and R, lie on the polar p of P. One of these vertices we may choose arbitrarily. For if from any point Q on the polar a secant be drawn cutting the conic in A and D (fig. 23), and if the lines joining these points to P cut the conic again at B and C, then the line BC will pass through Q. Hence P and Q are two of the vertices on the polartriangle which is determined by the fourpoint ABCD. The third vertex R lies also on the line p. It follows, therefore, also—
If Q is a point on the polar of P, then P is a point on the polar of Q; and reciprocally,
If q is a line through the pole of p, then p is a line through the pole of q.
This is a very important theorem. It may also be stated thus—
If a point moves along a line describing a row, its polar turns about the pole of the line describing a pencil.
This pencil is projective to the row, so that the crossratio of four poles in a row equals the crossratio of its four polars, which pass through the pole of the row.
To prove the last part, let us suppose that P, A and B in fig. 23 remain fixed, whilst Q moves along the polar p of P. This will make CD turn about P and move R along p, whilst QD and RD describe projective pencils about A and B. Hence Q and R describe projective rows, and hence PR, which is the polar of Q, describes a pencil projective to either.
§ 67. Two points, of which one, and therefore each, lies on the polar of the other, are said to be conjugate with regard to the conic; and two lines, of which one, and therefore each, passes through the pole of the other, are said to be conjugate with regard to the conic. Hence all points conjugate to a point P lie on the polar of P; all lines conjugate to a line p pass through the pole of p.
If the line joining two conjugate poles cuts the conic, then the poles are harmonic conjugates with regard to the points of intersection; hence one lies within the other without the conic, and all points conjugate to a point within a conic lie without it.
Of a polartriangle any two vertices are conjugate poles, any two sides conjugate lines. If, therefore, one side cuts a conic, then one of the two vertices which lie on this side is within and the other without the conic. The vertex opposite this side lies also without, for it is the pole of a line which cuts the curve. In this case therefore one vertex lies within, the other two without. If, on the other hand, we begin with a side which does not cut the conic, then its pole lies within and the other vertices without. Hence—
Every polartriangle has one and only one vertex within the conic.
We add, without a proof, the theorem—
The four points in which a conic is cut by two conjugate polars are four harmonic points in the conic.
§ 68. If two conics intersect in four points (they cannot have more points in common, § 52), there exists one and only one fourpoint which is inscribed in both, and therefore one polartriangle common to both.
Theorem.—Two conics which intersect in four points have always one and only one common polartriangle; and reciprocally,
Two conics which have four common tangents have always one and only one common polartriangle.
Diameters and Axes of Conics
§ 69. Diameters.—The theorems about the harmonic properties of poles and polars contain, as special cases, a number of important metrical properties of conics. These are obtained if either the pole or the polar is moved to infinity,—it being remembered that the harmonic conjugate to a point at infinity, with regard to two points A, B, is the middle point of the segment AB. The most important properties are stated in the following theorems:—
The middle points of parallel chords of a conic lie in a line—viz. on the polar to the point at infinity on the parallel chords.
This line is called a diameter.
The polar of every point at infinity is a diameter.
The tangents at the end points of a diameter are parallel, and are parallel to the chords bisected by the diameter.
All diameters pass through a common point, the pole of the line at infinity.
All diameters of a parabola are parallel, the pole to the line at infinity being the point where the curve touches the line at infinity.
In case of the ellipse and hyperbola, the pole to the line at infinity is a finite point called the centre of the curve.
A centre of a conic bisects every chord through it.
The centre of an ellipse is within the curve, for the line at infinity does not cut the ellipse.
The centre of an hyperbola is without the curve, because the line at infinity cuts the curve. Hence also—
From the centre of an hyperbola two tangents can be drawn to the curve which have their point of contact at infinity. These are called Asymptotes (§ 59).
To construct a diameter of a conic, draw two parallel chords and join their middle points.
To find the centre of a conic, draw two diameters; their intersection will be the centre.
§ 70. Conjugate Diameters.—A polartriangle with one vertex at the centre will have the opposite side at infinity. The other two sides pass through the centre, and are called conjugate diameters, each being the polar of the point at infinity on the other.
Of two conjugate diameters each bisects the chords parallel to the other, and if one cuts the curve, the tangents at its ends are parallel to the other diameter.
Further—
Every parallelogram inscribed in a conic has its sides parallel to two conjugate diameters; and
Every parallelogram circumscribed about a conic has as diagonals two conjugate diameters.
This will be seen by considering the parallelogram in the first case as an inscribed fourpoint, in the other as a circumscribed fourside, and determining in each case the corresponding polartriangle. The first may also be enunciated thus—
The lines which join any point on an ellipse or an hyperbola to the ends of a diameter are parallel to two conjugate diameters.
§ 71. If every diameter is perpendicular to its conjugate the conic is a circle.
For the lines which join the ends of a diameter to any point on the curve include a right angle.
A conic which has more than one pair of conjugate diameters at right angles to each other is a circle.
Fig. 24. 
Let AA′ and BB′ (fig. 24) be one pair of conjugate diameters at right angles to each other, CC and DD′ a second pair. If we draw through the end point A of one diameter a chord AP parallel to DD′, and join P to A′, then PA and PA′ are, according to § 70, parallel to two conjugate diameters. But PA is parallel to DD′, hence PA′ is parallel to CC, and therefore PA and PA′ are perpendicular. If we further draw the tangents to the conic at A and A′, these will be perpendicular to AA′, they being parallel to the conjugate diameter BB′. We know thus five points on the conic, viz. the points A and A′ with their tangents, and the point P. Through these a circle may be drawn having AA′ as diameter; and as through five points one conic only can be drawn, this circle must coincide with the given conic.
§ 72. Axes.—Conjugate diameters perpendicular to each other are called axes, and the points where they cut the curve vertices of the conic.
In a circle every diameter is an axis, every point on it is a vertex; and any two lines at right angles to each other may be taken as a pair of axes of any circle which has its centre at their intersection.
Fig. 25. 
If we describe on a diameter AB of an ellipse or hyperbola a circle concentric to the conic, it will cut the latter in A and B (fig. 25). Each of the semicircles in which it is divided by AB will be partly within, partly without the curve, and must cut the latter therefore again in a point. The circle and the conic have thus four points A, B, C, D, and therefore one polartriangle, in common (§ 68). Of this the centre is one vertex, for the line at infinity is the polar to this point, both with regard to the circle and the other conic. The other two sides are conjugate diameters of both, hence perpendicular to each other. This gives—
An ellipse as well as an hyperbola has one pair of axes.
This reasoning shows at the same time how to construct the axis of an ellipse or of an hyperbola.
A parabola has one axis, if we define an axis as a diameter perpendicular to the chords which it bisects. It is easily constructed. The line which bisects any two parallel chords is a diameter. Chords perpendicular to it will be bisected by a parallel diameter, and this is the axis.
§ 73. The first part of the righthand theorem in § 64 may be stated thus: any two conjugate lines through a point P without a conic are harmonic conjugates with regard to the two tangents that may be drawn from P to the conic.
If we take instead of P the centre C of an hyperbola, then the conjugate lines become conjugate diameters, and the tangents asymptotes. Hence—
Any two conjugate diameters of an hyperbola are harmonic conjugates with regard to the asymptotes.
As the axes are conjugate diameters at right angles to one another, it follows (§ 23)—
The axes of an hyperbola bisect the angles between the asymptotes.
Fig. 26. 
Let O be the centre of the hyperbola (fig. 26), t any secant which cuts the hyperbola in C, D and the asymptotes in E, F, then the line OM which bisects the chord CD is a diameter conjugate to the diameter OK which is parallel to the secant t, so that OK and OM are harmonic with regard to the asymptotes. The point M therefore bisects EF. But by construction M bisects CD. It follows that DF = EC, and ED = CF; or
On any secant of an hyperbola the segments between the curve and the asymptotes are equal.
If the chord is changed into a tangent, this gives—
The segment between the asymptotes on any tangent to an hyperbola is bisected by the point of contact.
The first part allows a simple solution of the problem to find any number of points on an hyperbola, of which the asymptotes and one point are given. This is equivalent to three points and the tangents at two of them. This construction requires measurement.
§ 74. For the parabola, too, follow some metrical properties. A diameter PM (fig. 27) bisects every chord conjugate to it, and the pole P of such a chord BC lies on the diameter. But a diameter cuts the parabola once at infinity. Hence—
The segment PM which joins the middle point M of a chord of a parabola to the pole P of the chord is bisected by the parabola at A.
Fig. 27. 
§ 75. Two asymptotes and any two tangents to an hyperbola may be considered as a quadrilateral circumscribed about the hyperbola. But in such a quadrilateral the intersections of the diagonals and the points of contact of opposite sides lie in a line (§ 54). If therefore DEFG (fig. 28) is such a quadrilateral, then the diagonals DF and GE will meet on the line which joins the points of contact of the asymptotes, that is, on the line at infinity; hence they are parallel. From this the following theorem is a simple deduction:
All triangles formed by a tangent and the asymptotes of an hyperbola are equal in area.
If we draw at a point P (fig. 28) on an hyperbola a tangent, the part HK between the asymptotes is bisected at P. The parallelogram PQOQ′ formed by the asymptotes and lines parallel to them through P will be half the triangle OHK, and will therefore be constant. If we now take the asymptotes OX and OY as oblique axes of coordinates, the lines OQ and QP will be the coordinates of P, and will satisfy the equation xy = const. = a^{2}.
Fig. 28. 
For the asymptotes as axes of coordinates the equation of the hyperbola is xy = const.
Involution
Fig. 29. 
§ 76. If we have two projective rows, ABC on u and A′B′C′ on u′, and place their bases on the same line, then each point in this line counts twice, once as a point in the row u and once as a point in the row u ′. In fig. 29 we denote the points as points in the one row by letters above the line A, B, C ..., and as points in the second row by A′, B′, C′ ... below the line. Let now A and B′ be the same point, then to A will correspond a point A′ in the second, and to B′ a point B in the first row. In general these points A′ and B will be different. It may, however, happen that they coincide. Then the correspondence is a peculiar one, as the following theorem shows:
If two projective rows lie on the same base, and if it happens that to one point in the base the same point corresponds, whether we consider the point as belonging to the first or to the second row, then the same will happen for every point in the base—that is to say, to every point in the line corresponds the same point in the first as in the second row.
Fig. 30. 
In order to determine the correspondence, we may assume three pairs of corresponding points in two projective rows. Let then A′, B′, C′, in fig. 30, correspond to A, B, C, so that A and B′, and also B and A′, denote the same point. Let us further denote the point C′ when considered as a point in the first row by D; then it is to be proved that the point D′, which corresponds to D, is the same point as C. We know that the crossratio of four points is equal to that of the corresponding row. Hence
but replacing the dashed letters by those undashed ones which denote the same points, the second crossratio equals (BA, DD′), which, according to § 15, equals (AB, D′D); so that the equation becomes
This requires that C and D′ coincide.
§ 77. Two projective rows on the same base, which have the above property, that to every point, whether it be considered as a point in the one or in the other row, corresponds the same point, are said to be in involution, or to form an involution of points on the line.
We mention, but without proving it, that any two projective rows may be placed so as to form an involution.
An involution may be said to consist of a row of pairs of points, to every point A corresponding a point A′, and to A′ again the point A. These points are said to be conjugate, or, better, one point is termed the “mate” of the other.
From the definition, according to which an involution may be considered as made up of two projective rows, follow at once the following important properties:
1. The crossratio of four points equals that of the four conjugate points.
2. If we call a point which coincides with its mate a “focus” or “double point” of the involution, we may say: An involution has either two foci, or one, or none, and is called respectively a hyperbolic, parabolic or elliptic involution (§ 34).
3. In an hyperbolic involution any two conjugate points are harmonic conjugates with regard to the two foci.
For if A, A′ be two conjugate points, F_{1}, F_{2} the two foci, then to the points F_{1}, F_{2}, A, A′ in the one row correspond the points F_{1}, F_{2}, A′, A in the other, each focus corresponding to itself. Hence (F_{1}F_{2}, AA′) = (F_{1}F_{2}, A′A)—that is, we may interchange the two points AA′ without altering the value of the crossratio, which is the characteristic property of harmonic conjugates (§ 18).
4. The point conjugate to the point at infinity is called the “centre” of the involution. Every involution has a centre, unless the point at infinity be a focus, in which case we may say that the centre is at infinity.
In an hyperbolic involution the centre is the middle point between the foci.
5. The product of the distances of two conjugate points A, A′ from the centre O is constant: OA · OA′ = c.
For let A, A′ and B, B′ be two pairs of conjugate points, the centre, I the point at infinity, then
or
In order to determine the distances of the foci from the centre, we write F for A and A′ and get
Hence if c is positive OF is real, and has two values, equal and opposite. The involution is hyperbolic.
If c = 0, OF = 0, and the two foci both coincide with the centre. If c is negative, √c becomes imaginary, and there are no foci. Hence we may write—
In an hyperbolic involution,  OA · OA′ = k^{2}, 
In a parabolic involution,  OA · OA′ = 0, 
In an elliptic involution,  OA · OA′ = −k^{2}. 
From these expressions it follows that conjugate points A, A′ in an hyperbolic involution lie on the same side of the centre, and in an elliptic involution on opposite sides of the centre, and that in a parabolic involution one coincides with the centre.
In the first case, for instance, OA · OA′ is positive; hence OA and OA′ have the same sign.
It also follows that two segments, AA′ and BB′, between pairs of conjugate points have the following positions: in an hyperbolic involution they lie either one altogether within or altogether without each other; in a parabolic involution they have one point in common; and in an elliptic involution they overlap, each being partly within and partly without the other.
Proof.—We have OA . OA′ = OB · OB′ = k^{2} in case of an hyperbolic involution. Let A and B be the points in each pair which are nearer to the centre O. If now A, A′ and B, B′ lie on the same side of O, and if B is nearer to O than A, so that OB < OA, then OB′ > OA′; hence B′ lies farther away from O than A′, or the segment AA′ lies within BB′. And so on for the other cases.
6. An involution is determined—

7. The condition that A, B, C and A′, B′, C′ may form an involution may be written in one of the forms—
or
or
for each expresses that in the two projective rows in which A, B, C and A′, B′, C′ are conjugate points two conjugate elements may be interchanged.
8. Any three pairs. A, A′, B, B′, C, C′, of conjugate points are connected by the relations:
AB′ · BC′ · CA′  =  AB′ · BC · C′A′  =  AB · B′C′ · CA′  =  AB · B′C · C′A′  = −1. 
A′B · B′C · C′A  A′B · B′C′ · CA  A′B′ · BC · C′A  A′B′ · BC′ · CA 
These relations readily follow by working out the relations in (7) (above).
§ 78. Involution of a quadrangle.—The sides of any fourpoint are cut by any line in six points in involution, opposite sides being cut in conjugate points.
Let A_{1}B_{1}C_{1}D_{1} (fig. 31) be the fourpoint. If its sides be cut by the line p in the points A, A′, B, B′, C, C′, if further, C_{1}D_{1} cuts the line A_{1}B_{1} in C_{2}, and if we project the row A_{1}B_{1}C_{2}C to p once from D_{1} and once from C_{1}, we get (A′B′, C′C) = (BA, C′C).
Interchanging in the last crossratio the letters in each pair we get (A′B′, C′C) = (AB, CC′). Hence by § 77 (7) the points are in involution.
The theorem may also be stated thus:
The three points in which any line cuts the sides of a triangle and the projections, from any point in the plane, of the vertices of the triangle on to the same line are six points in involution.
Fig. 31. 
Or again—
The projections from any point on to any line of the six vertices of a fourside are six points in involution, the projections of opposite vertices being conjugate points.
This property gives a simple means to construct, by aid of the straight edge only, in an involution of which two pairs of conjugate points are given, to any point its conjugate.
§ 79. Pencils in Involution.—The theory of involution may at once be extended from the row to the flat and the axial pencil—viz. we say that there is an involution in a flat or in an axial pencil if any line cuts the pencil in an involution of points. An involution in a pencil consists of pairs of conjugate rays or planes; it has two, one or no focal rays (double lines) or planes, but nothing corresponding to a centre.
An involution in a flat pencil contains always one, and in general only one, pair of conjugate rays which are perpendicular to one another. For in two projective flat pencils exist always two corresponding right angles (§ 40).
Each involution in an axial pencil contains in the same manner one pair of conjugate planes at right angles to one another.
As a rule, there exists but one pair of conjugate lines or planes at right angles to each other. But it is possible that there are more, and then there is an infinite number of such pairs. An involution in a flat pencil, in which every ray is perpendicular to its conjugate ray, is said to be circular. That such involution is possible is easily seen thus: if in two concentric flat pencils each ray on one is made to correspond to that ray on the other which is perpendicular to it, then the two pencils are projective, for if we turn the one pencil through a right angle each ray in one coincides with its corresponding ray in the other. But these two projective pencils are in involution.
A circular involution has no focal rays, because no ray in a pencil coincides with the ray perpendicular to it.
§ 80. Every elliptical involution in a row may be considered as a section of a circular involution.
In an elliptical involution any two segments AA′ and BB′ lie partly within and partly without each other (fig. 32). Hence two circles described on AA′ and BB′ as diameters will intersect in two points E and E′. The line EE′ cuts the base of the involution at a point O, which has the property that OA . OA′ = OB · OB′, for each is equal to OE . OE′. The point O is therefore the centre of the involution. If we wish to construct to any point C the conjugate point C′, we may draw the circle through CEE′. This will cut the base in the required point C′ for OC · OC′ = OA · OA′. But EC and EC′ are at right angles. Hence the involution which is obtained by joining E or E′ to the points in the given involution is circular. This may also be expressed thus:
Fig. 32. 
Every elliptical involution has the property that there are two definite points in the plane from which any two conjugate points are seen under a right angle.
At the same time the following problem has been solved:
To determine the centre and also the point corresponding to any given point in an elliptical involution of which two pairs of conjugate points are given.
§ 81. Involution Range on a Conic.—By the aid of § 53, the points on a conic may be made to correspond to those on a line, so that the row of points on the conic is projective to a row of points on a line. We may also have two projective rows on the same conic, and these will be in involution as soon as one point on the conic has the same point corresponding to it all the same to whatever row it belongs. An involution of points on a conic will have the property (as follows from its definition, and from § 53) that the lines which join conjugate points of the involution to any point on the conic are conjugate lines of an involution in a pencil, and that a fixed tangent is cut by the tangents at conjugate points on the conic in points which are again conjugate points of an involution on the fixed tangent. For such involution on a conic the following theorem holds:
The lines which join corresponding points in an involution on a conic all pass through a fixed point; and reciprocally, the points of intersection of conjugate lines in an involution among tangents to a conic lie on a line.
Fig. 33 
We prove the first part only. The involution is determined by two pairs of conjugate points, say by A, A′ and B, B′ (fig. 33). Let AA′ and BB′ meet in P. If we join the points in involution to any point on the conic, and the conjugate points to another point on the conic, we obtain two projective pencils. We take A and A′ as centres of these pencils, so that the pencils A(A′BB′) and A′(AB′B) are projective, and in perspective position, because AA′ corresponds to A′A. Hence corresponding rays meet in a line, of which two points are found by joining AB′ to A′B and AB to A′B′. It follows that the axis of perspective is the polar of the point P, where AA′ and BB′ meet. If we now wish to construct to any other point C on the conic the corresponding point C′, we join C to A′ and the point where this line cuts p to A. The latter line cuts the conic again in C′. But we know from the theory of pole and polar that the line CC′ passes through P. The point of concurrence is called the “pole of the involution,” and the line of collinearity of the meets is called the “axis of the involution.”
Involution Determined by a Conic on a Line.—Foci
§ 82. The polars, with regard to a conic, of points in a row p form a pencil P projective to the row (§ 66). This pencil cuts the base of the row p in a projective row.
If A is a point in the given row, A′ the point where the polar of A cuts p, then A and A′ will be corresponding points. If we take A′ a point in the first row, then the polar of A′ will pass through A, so that A corresponds to A′—in other words, the rows are in involution. The conjugate points in this involution are conjugate points with regard to the conic. Conjugate points coincide only if the polar of a point A passes through A—that is, if A lies on the conic. Hence—
A conic determines on every line in its plane an involution, in which those points are conjugate which are also conjugate with regard to the conic.
If the line cuts the conic the involution is hyperbolic, the points of intersection being the foci.
If the line touches the conic the involution is parabolic, the two foci coinciding at the point of contact.
If the line does not cut the conic the involution is elliptic, having no foci.
If, on the other hand, we take a point P in the plane of a conic, we get to each line a through P one conjugate line which joins P to the pole of a. These pairs of conjugate lines through P form an involution in the pencil at P. The focal rays of this involution are the tangents drawn from P to the conic. This gives the theorem reciprocal to the last, viz:—
A conic determines in every pencil in its plane an involution, corresponding lines being conjugate lines with regard to the conic.
If the point is without the conic the involution is hyperbolic, the tangents from the points being the focal rays.
If the point lies on the conic the involution is parabolic, the tangent at the point counting for coincident focal rays.
If the point is within the conic the involution is elliptic, having no focal rays.
It will further be seen that the involution determined by a conic on any line p is a section of the involution, which is determined by the conic at the pole P of p.
§ 83. Foci.—The centre of a pencil in which the conic determines a circular involution is called a “focus” of the conic.
In other words, a focus is such a point that every line through it is perpendicular to its conjugate line. The polar to a focus is called a directrix of the conic.
From the definition it follows that every focus lies on an axis, for the line joining a focus to the centre of the conic is a diameter to which the conjugate lines are perpendicular; and every line joining two foci is an axis, for the perpendiculars to this line through the foci are conjugate to it. These conjugate lines pass through the pole of the line, the pole lies therefore at infinity, and the line is a diameter, hence by the last property an axis.
It follows that all foci lie on one axis, for no line joining a point in one axis to a point in the other can be an axis.
As the conic determines in the pencil which has its centre at a focus a circular involution, no tangents can be drawn from the focus to the conic. Hence each focus lies within a conic; and a directrix does not cut the conic.
Further properties are found by the following considerations:
§ 84. Through a point P one line p can be drawn, which is with regard to a given conic conjugate to a given line q, viz. that line which joins the point P to the pole of the line q. If the line q is made to describe a pencil about a point Q, then the line p will describe a pencil about P. These two pencils will be projective, for the line p passes through the pole of q, and whilst q describes the pencil Q, its pole describes a projective row, and this row is perspective to the pencil P.
We now take the point P on an axis of the conic, draw any line p through it, and from the pole of p draw a perpendicular q to p. Let q cut the axis in Q. Then, in the pencils of conjugate lines, which have their centres at P and Q, the lines p and q are conjugate lines at right angles to one another. Besides, to the axis as a ray in either pencil will correspond in the other the perpendicular to the axis (§ 72). The conic generated by the intersection of corresponding lines in the two pencils is therefore the circle on PQ as diameter, so that every line in P is perpendicular to its corresponding line in Q.
To every point P on an axis of a conic corresponds thus a point Q, such that conjugate lines through P and Q are perpendicular.
We shall show that these pointpairs P, Q form an involution. To do this let us move P along the axis, and with it the line p, keeping the latter parallel to itself. Then P describes a row, p a perspective pencil (of parallels), and the pole of p a projective row. At the same time the line q describes a pencil of parallels perpendicular to p, and perspective to the row formed by the pole of p. The point Q, therefore, where q cuts the axis, describes a row projective to the row of points P. The two points P and Q describe thus two projective rows on the axis; and not only does P as a point in the first row correspond to Q, but also Q as a point in the first corresponds to P. The two rows therefore form an involution. The centre of this involution, it is easily seen, is the centre of the conic.
A focus of this involution has the property that any two conjugate lines through it are perpendicular; hence, it is a focus to the conic.
Such involution exists on each axis. But only one of these can have foci, because all foci lie on the same axis. The involution on one of the axes is elliptic, and appears (§ 80) therefore as the section of two circular involutions in two pencils whose centres lie in the other axis. These centres are foci, hence the one axis contains two foci, the other axis none; or every central conic has two foci which lie on one axis equidistant from the centre.
The axis which contains the foci is called the principal axis; in case of an hyperbola it is the axis which cuts the curve, because the foci lie within the conic.
In case of the parabola there is but one axis. The involution on this axis has its centre at infinity. One focus is therefore at infinity, the one focus only is finite. A parabola has only one focus.
Fig. 34. 
§ 85. If through any point P (fig. 34) on a conic the tangent PT and the normal PN (i.e. the perpendicular to the tangent through the point of contact) be drawn, these will be conjugate lines with regard to the conic, and at right angles to each other. They will therefore cut the principal axis in two points, which are conjugate in the involution considered in § 84; hence they are harmonic conjugates with regard to the foci. If therefore the two foci F_{1} and F_{2} be joined to P, these lines will be harmonic with regard to the tangent and normal. As the latter are perpendicular, they will bisect the angles between the other pair. Hence—
The lines joining any point on a conic to the two foci are equally inclined to the tangent and normal at that point.
In case of the parabola this becomes—
The line joining any point on a parabola to the focus and the diameter through the point, are equally inclined to the tangent and normal at that point.
From the definition of a focus it follows that—
The segment of a tangent between the directrix and the point of contact is seen from the focus belonging to the directrix under a right angle, because the lines joining the focus to the ends of this segment are conjugate with regard to the conic, and therefore perpendicular.
With equal ease the following theorem is proved:
The two lines which join the points of contact of two tangents each to one focus, but not both to the same, are seen from the intersection of the tangents under equal angles.
§ 86. Other focal properties of a conic are obtained by the following considerations:
Fig. 35. 
Let F (fig. 35) be a focus to a conic, f the corresponding directrix, A and B the points of contact of two tangents meeting at T, and P the point where the line AB cuts the directrix. Then TF will be the polar of P (because polars of F and T meet at P). Hence TF and PF are conjugate lines through a focus, and therefore perpendicular. They are further harmonic conjugates with regard to FA and FB (§§ 64 and 13), so that they bisect the angles formed by these lines. This by the way proves—
The segments between the point of intersection of two tangents to a conic and their points of contact are seen from a focus under equal angles.
If we next draw through A and B lines parallel to TF, then the points A_{1}, B_{1} where these cut the directrix will be harmonic conjugates with regard to P and the point where FT cuts the directrix. The lines FT and FP bisect therefore also the angles between FA_{1} and FB_{1}. From this it follows easily that the triangles FAA_{1} and FBB_{1} are equiangular, and therefore similar, so that FA : AA_{1} = FB : BB_{1}.
The triangles AA_{1}A_{2} and BB_{1}B_{2} formed by drawing perpendiculars from A and B to the directrix are also similar, so that AA_{1} : AA_{2} = BB_{1} : BB_{2}. This, combined with the above proportion, gives FA : AA_{2} = FB : BB_{2}. Hence the theorem:
The ratio of the distances of any point on a conic from a focus and the corresponding directrix is constant.
To determine this ratio we consider its value for a vertex on the principal axis. In an ellipse the focus lies between the two vertices on this axis, hence the focus is nearer to a vertex than to the corresponding directrix. Similarly, in an hyperbola a vertex is nearer to the directrix than to the focus. In a parabola the vertex lies halfway between directrix and focus.
It follows in an ellipse the ratio between the distance of a point from the focus to that from the directrix is less than unity, in the parabola it equals unity, and in the hyperbola it is greater than unity.
It is here the same which focus we take, because the two foci lie symmetrical to the axis of the conic. If now P is any point on the conic having the distances r_{1} and r_{2} from the foci and the distances d_{1} and d_{2} from the corresponding directrices, then r_{1}/d_{1} = r_{2}/d_{2} = e, where e is constant. Hence also r_{1} ± r_{2}d_{1} ± d_{2} = e.
In the ellipse, which lies between the directrices, d_{1} + d_{2} is constant, therefore also r_{1} + r_{2}. In the hyperbola on the other hand d_{1} − d_{2} is constant, equal to the distance between the directrices, therefore in this case r_{1} − r_{2} is constant.
If we call the distances of a point on a conic from the focus its focal distances we have the theorem:
In an ellipse the sum of the focal distances is constant; and in an hyperbola the difference of the focal distances is constant.
This constant sum or difference equals in both cases the length of the principal axis.
Pencil of Conics
§ 87. Through four points A, B, C, D in a plane, of which no three lie in a line, an infinite number of conics may be drawn, viz. through these four points and any fifth one single conic. This system of conics is called a pencil of conics. Similarly, all conics touching four fixed lines form a system such that any fifth tangent determines one and only one conic. We have here the theorems:
The pairs of points in which any line is cut by a system of conics through four fixed points are in involution.  The pairs of tangents which can be drawn from a point to a system of conics touching four fixed lines are in involution. 
Fig. 36. 
We prove the first theorem only. Let ABCD (fig. 36) be the fourpoint, then any line t will cut two opposite sides AC, BD in the points E, E′, the pair AD, BC in points F, F′, and any conic of the system in M, N, and we have A(CD, MN) = B(CD, MN).
If we cut these pencils by t we get
or
But this is, according to § 77 (7), the condition that M, N are corresponding points in the involution determined by the point pairs E, E′, F, F′ in which the line t cuts pairs of opposite sides of the fourpoint ABCD. This involution is independent of the particular conic chosen.
§ 88. There follow several important theorems:
Through four points two, one, or no conics may be drawn which touch any given line, according as the involution determined by the given fourpoint on the line has real, coincident or imaginary foci.
Two, one, or no conics may be drawn which touch four given lines and pass through a given point, according as the involution determined by the given fourside at the point has real, coincident or imaginary focal rays.
For the conic through four points which touches a given line has its point of contact at a focus of the involution determined by the fourpoint on the line.
As a special case we get, by taking the line at infinity:
Through four points of which none is at infinity either two or no parabolas may be drawn.
The problem of drawing a conic through four points and touching a given line is solved by determining the points of contact on the line, that is, by determining the foci of the involution in which the line cuts the sides of the fourpoint. The corresponding remark holds for the problem of drawing the conics which touch four lines and pass through a given point.
Ruled Quadric Surfaces
§ 89. We have considered hitherto projective rows which lie in the same plane, in which case lines joining corresponding points envelop a conic. We shall now consider projective rows whose bases do not meet. In this case, corresponding points will be joined by lines which do not lie in a plane, but on some surface, which like every surface generated by lines is called a ruled surface. This surface clearly contains the bases of the two rows.
If the points in either row be joined to the base of the other, we obtain two axial pencils which are also projective, those planes being corresponding which pass through corresponding points in the given rows. If A′, A be two corresponding points, α, α′ the planes in the axial pencils passing through them, then AA′ will be the line of intersection of the corresponding planes α, α′ and also the line joining corresponding points in the rows.
If we cut the whole figure by a plane this will cut the axial pencils in two projective flat pencils, and the curve of the second order generated by these will be the curve in which the plane cuts the surface. Hence
The locus of lines joining corresponding points in two projective rows which do not lie in the same plane is a surface which contains the bases of the rows, and which can also be generated by the lines of intersection of corresponding planes in two projective axial pencils. This surface is cut by every plane in a curve of the second order, hence either in a conic or in a linepair. No line which does not lie altogether on the surface can have more than two points in common with the surface, which is therefore said to be of the second order or is called a ruled quadric surface.
That no line which does not lie on the surface can cut the surface in more than two points is seen at once if a plane be drawn through the line, for this will cut the surface in a conic. It follows also that a line which contains more than two points of the surface lies altogether on the surface.
§ 90. Through any point in space one line can always be drawn cutting two given lines which do not themselves meet.
If therefore three lines in space be given of which no two meet, then through every point in either one line may be drawn cutting the other two.
If a line moves so that it always cuts three given lines of which no two meet, then it generates a ruled quadric surface.
Let a, b, c be the given lines, and p, q, r . . . lines cutting them in the points A, A′, A″ . . .; B, B′, B″ . . .; C, C′, C″ . . . respectively; then the planes through a containing p, q, r, and the planes through b containing the same lines, may be taken as corresponding planes in two axial pencils which are projective, because both pencils cut the line c in the same row, C, C′, C″ . . .; the surface can therefore be generated by projective axial pencils.
Of the lines p, q, r . . . no two can meet, for otherwise the lines a, b, c which cut them would also lie in their plane. There is a single infinite number of them, for one passes through each point of a. These lines are said to form a set of lines on the surface.
If now three of the lines p, q, r be taken, then every line d cutting them will have three points in common with the surface, and will therefore lie altogether on it. This gives rise to a second set of lines on the surface. From what has been said the theorem follows:
A ruled quadric surface contains two sets of straight lines. Every line of one set cuts every line of the other, but no two lines of the same set meet.
Any two lines of the same set may be taken as bases of two projective rows, or of two projective pencils which generate the surface. They are cut by the lines of the other set in two projective rows.
The plane at infinity like every other plane cuts the surface either in a conic proper or in a linepair. In the first case the surface is called an Hyperboloid of one sheet, in the second an Hyperbolic Paraboloid.
The latter may be generated by a line cutting three lines of which one lies at infinity, that is, cutting two lines and remaining parallel to a given plane.
Quadric Surfaces
§ 91. The conics, the cones of the second order, and the ruled quadric surfaces complete the figures which can be generated by projective rows or flat and axial pencils, that is, by those aggregates of elements which are of one dimension (§§ 5, 6). We shall now consider the simpler figures which are generated by aggregates of two dimensions. The space at our disposal will not, however, allow us to do more than indicate a few of the results.
§ 92. We establish a correspondence between the lines and planes in pencils in space, or reciprocally between the points and lines in two or more planes, but consider principally pencils.
In two pencils we may either make planes correspond to planes and lines to lines, or else planes to lines and lines to planes. If hereby the condition be satisfied that to a flat, or axial, pencil corresponds in the first case a projective flat, or axial, pencil, and in the second a projective axial, or flat, pencil, the pencils are said to be projective in the first case and reciprocal in the second.
For instance, two pencils which join two points S_{1} and S_{2} to the different points and lines in a given plane π are projective (and in perspective position), if those lines and planes be taken as corresponding which meet the plane π in the same point or in the same line. In this case every plane through both centres S_{1} and S_{2} of the two pencils will correspond to itself. If these pencils are brought into any other position they will be projective (but not perspective).
The correspondence between two projective pencils is uniquely determined, if to four rays (or planes) in the one the corresponding rays (or planes) in the other are given, provided that no three rays of either set lie in a plane.
Let a, b, c, d be four rays in the one, a′, b′, c′, d′ the corresponding rays in the other pencil. We shall show that we can find for every ray e in the first a single corresponding ray e′ in the second. To the axial pencil a (b, c, d . . .) formed by the planes which join a to b, c, d . . ., respectively corresponds the axial pencil a′ (b′, c′, d′ . . . ), and this correspondence is determined. Hence, the plane a′e′ which corresponds to the plane ae is determined. Similarly the plane b′e′ may be found and both together determine the ray e′.
Similarly the correspondence between two reciprocal pencils is determined if for four rays in the one the corresponding planes in the other are given.
§ 93. We may now combine—
1. Two reciprocal pencils.
Each ray cuts its corresponding plane in a point, the locus of these points is a quadric surface.
2. Two projective pencils.
Each plane cuts its corresponding plane in a line, but a ray as a rule does not cut its corresponding ray. The locus of points where a ray cuts its corresponding ray is a twisted cubic. The lines where a plane cuts its corresponding plane are secants.
3. Three projective pencils.
The locus of intersection of corresponding planes is a cubic surface.
Of these we consider only the first two cases.
§ 94. If two pencils are reciprocal, then to a plane in either corresponds a line in the other, to a flat pencil an axial pencil, and so on. Every line cuts its corresponding plane in a point. If S_{1} and S_{2} be the centres of the two pencils, and P be a point where a line a_{1} in the first cuts its corresponding plane α_{2}, then the line b_{2} in the pencil S_{2} which passes through P will meet its corresponding plane β_{1} in P. For b_{2} is a line in the plane α_{2}. The corresponding plane β_{1} must therefore pass through the line a_{1}, hence through P.
The points in which the lines in S_{1} cut the planes corresponding to them in S_{2} are therefore the same as the points in which the lines in S_{2} cut the planes corresponding to them in S_{1}.
The locus of these points is a surface which is cut by a plane in a conic or in a linepair and by a line in not more than two points unless it lies altogether on the surface. The surface itself is therefore called a quadric surface, or a surface of the second order.
To prove this we consider any line p in space.
The flat pencil in S_{1} which lies in the plane drawn through p and the corresponding axial pencil in S_{2} determine on p two projective rows, and those points in these which coincide with their corresponding points lie on the surface. But there exist only two, or one, or no such points, unless every point coincides with its corresponding point. In the latter case the line lies altogether on the surface.
This proves also that a plane cuts the surface in a curve of the second order, as no line can have more than two points in common with it. To show that this is a curve of the same kind as those considered before, we have to show that it can be generated by projective flat pencils. We prove first that this is true for any plane through the centre of one of the pencils, and afterwards that every point on the surface may be taken as the centre of such pencil. Let then α_{1} be a plane through S_{1}. To the flat pencil in S_{1} which it contains corresponds in S_{2} a projective axial pencil with axis a_{2} and this cuts α_{1} in a second flat pencil. These two flat pencils in α_{1} are projective, and, in general, neither concentric nor perspective. They generate therefore a conic. But if the line a_{2} passes through S_{1} the pencils will have S_{1} as common centre, and may therefore have two, or one, or no lines united with their corresponding lines. The section of the surface by the plane α_{1} will be accordingly a linepair or a single line, or else the plane α_{1} will have only the point S_{1} in common with the surface.
Every line l_{1} through S_{1} cuts the surface in two points, viz. first in S_{1} and then at the point where it cuts its corresponding plane. If now the corresponding plane passes through S_{1}, as in the case just considered, then the two points where l_{1} cuts the surface coincide at S_{1}, and the line is called a tangent to the surface with S_{1} as point of contact. Hence if l_{1} be a tangent, it lies in that plane τ_{1} which corresponds to the line S_{2}S_{1} as a line in the pencil S_{2}. The section of this plane has just been considered. It follows that—
All tangents to quadric surface at the centre of one of the reciprocal pencils lie in a plane which is called the tangent plane to the surface at that point as point of contact.
To the line joining the centres of the two pencils as a line in one corresponds in the other the tangent plane at its centre.
The tangent plane to a quadric surface either cuts the surface in two lines, or it has only a single line, or else only a single point in common with the surface.
In the first case the point of contact is said to be hyperbolic, in the second parabolic, in the third elliptic.
§ 95. It remains to be proved that every point S on the surface may be taken as centre of one of the pencils which generate the surface. Let S be any point on the surface Φ′ generated by the reciprocal pencils S_{1} and S_{2}. We have to establish a reciprocal correspondence between the pencils S and S_{1}, so that the surface generated by them is identical with Φ. To do this we draw two planes α_{1} and β_{1} through S_{1}, cutting the surface Φ in two conics which we also denote by α_{1} and β_{1}. These conics meet at S_{1}, and at some other point T where the line of intersection of α_{1} and β_{1} cuts the surface.
In the pencil S we draw some plane σ which passes through T, but not through S_{1} or S_{2}. It will cut the two conics first at T, and therefore each at some other point which we call A and B respectively. These we join to S by lines a and b, and now establish the required correspondence between the pencils S_{1} and S as follows:—To S_{1}T shall correspond the plane σ, to the plane α_{1} the line a, and to β_{1} the line b, hence to the flat pencil in α_{1} the axial pencil a. These pencils are made projective by aid of the conic in α_{1}.
In the same manner the flat pencil in β_{1} is made projective to the axial pencil b by aid of the conic in β_{1}, corresponding elements being those which meet on the conic. This determines the correspondence, for we know for more than four rays in S_{1} the corresponding planes in S. The two pencils S and S_{1} thus made reciprocal generate a quadric surface Φ′, which passes through the point S and through the two conics α_{1} and β_{1}.
The two surfaces Φ and Φ′ have therefore the points S and S_{1} and the conics α_{1} and β_{1} in common. To show that they are identical, we draw a plane through S and S_{2}, cutting each of the conics α_{1} and β_{1} in two points, which will always be possible. This plane cuts Φ and Φ′ in two conics which have the point S and the points where it cuts α_{1} and β_{1} in common, that is five points in all. The conics therefore coincide.
This proves that all those points P on Φ′ lie on Φ which have the property that the plane SS_{2}P cuts the conics α_{1}, β_{1} in two points each. If the plane SS_{2}P has not this property, then we draw a plane SS_{1}P. This cuts each surface in a conic, and these conics have in common the points S, S_{1}, one point on each of the conics α_{1}, β_{1}, and one point on one of the conics through S and S_{2} which lie on both surfaces, hence five points. They are therefore coincident, and our theorem is proved.
§ 96. The following propositions follow:—
A quadric surface has at every point a tangent plane.
Every plane section of a quadric surface is a conic or a linepair.
Every line which has three points in common with a quadric surface lies on the surface.
Every conic which has five points in common with a quadric surface lies on the surface.
Through two conics which lie in different planes, but have two points in common, and through one external point always one quadric surface may be drawn.
§ 97. Every plane which cuts a quadric surface in a linepair is a tangent plane. For every line in this plane through the centre of the linepair (the point of intersection of the two lines) cuts the surface in two coincident points and is therefore a tangent to the surface, the centre of the linepair being the point of contact.
If a quadric surface contains a line, then every plane through this line cuts the surface in a linepair (or in two coincident lines). For this plane cannot cut the surface in a conic. Hence:—
If a quadric surface contains one line p then it contains an infinite number of lines, and through every point Q on the surface, one line q can be drawn which cuts p. For the plane through the point Q and the line p cuts the surface in a linepair which must pass through Q and of which p is one line.
No two such lines q on the surface can meet. For as both meet p their plane would contain p and therefore cut the surface in a triangle.
Every line which cuts three lines q will be on the surface; for it has three points in common with it.
Hence the quadric surfaces which contain lines are the same as the ruled quadric surfaces considered in §§ 8993, but with one important exception. In the last investigation we have left out of consideration the possibility of a plane having only one line (two coincident lines) in common with a quadric surface.
§ 98. To investigate this case we suppose first that there is one point A on the surface through which two different lines a, b can be drawn, which lie altogether on the surface.
If P is any other point on the surface which lies neither on a nor b, then the plane through P and a will cut the surface in a second line a′ which passes through P and which cuts a. Similarly there is a line b′ through P which cuts b. These two lines a′ and b′ may coincide, but then they must coincide with PA.
If this happens for one point P, it happens for every other point Q. For if two different lines could be drawn through Q, then by the same reasoning the line PQ would be altogether on the surface, hence two lines would be drawn through P against the assumption. From this follows:—
If there is one point on a quadric surface through which one, but only one, line can be drawn on the surface, then through every point one line can be drawn, and all these lines meet in a point. The surface is a cone of the second order.
If through one point on a quadric surface, two, and only two, lines can be drawn on the surface, then through every point two lines may be drawn, and the surface is ruled quadric surface.
If through one point on a quadric surface no line on the surface can be drawn, then the surface contains no lines.
Using the definitions at the end of § 95, we may also say:—
On a quadric surface the points are all hyperbolic, or all parabolic, or all elliptic.
As an example of a quadric surface with elliptical points, we mention the sphere which may be generated by two reciprocal pencils, where to each line in one corresponds the plane perpendicular to it in the other.
§ 99. Poles and Polar Planes.—The theory of poles and polars with regard to a conic is easily extended to quadric surfaces.
Let P be a point in space not on the surface, which we suppose not to be a cone. On every line through P which cuts the surface in two points we determine the harmonic conjugate Q of P with regard to the points of intersection. Through one of these lines we draw two planes α and β. The locus of the points Q in α is a line a, the polar of P with regard to the conic in which α cuts the surface. Similarly the locus of points Q in β is a line b. This cuts a, because the line of intersection of α and β contains but one point Q. The locus of all points Q therefore is a plane. This plane is called the polar plane of the point P, with regard to the quadric surface. If P lies on the surface we take the tangent plane of P as its polar.
The following propositions hold:—
1. Every point has a polar plane, which is constructed by drawing the polars of the point with regard to the conics in which two planes through the point cut the surface.
2. If Q is a point in the polar of P, then P is a point in the polar of Q, because this is true with regard to the conic in which a plane through PQ cuts the surface.
3. Every plane is the polar plane of one point, which is called the Pole of the plane.
The pole to a plane is found by constructing the polar planes of three points in the plane. Their intersection will be the pole.
4. The points in which the polar plane of P cuts the surface are points of contact of tangents drawn from P to the surface, as is easily seen. Hence:—
5. The tangents drawn from a point P to a quadric surface form a cone of the second order, for the polar plane of P cuts it in a conic.
6. If the pole describes a line a, its polar plane will turn about another line a′, as follows from 2. These lines a and a′ are said to be conjugate with regard to the surface.
§ 100. The pole of the line at infinity is called the centre of the surface. If it lies at the infinity, the plane at infinity is a tangent plane, and the surface is called a paraboloid.
The polar plane to any point at infinity passes through the centre, and is called a diametrical plane.
A line through the centre is called a diameter. It is bisected at the centre. The line conjugate to it lies at infinity.
If a point moves along a diameter its polar plane turns about the conjugate line at infinity; that is, it moves parallel to itself, its centre moving on the first line.
The middle points of parallel chords lie in a plane, viz. in the polar plane of the point at infinity through which the chords are drawn.
The centres of parallel sections lie in a diameter which is a line conjugate to the line at infinity in which the planes meet.
Twisted Cubics
§ 101. If two pencils with centres S_{1} and S_{2} are made projective, then to a ray in one corresponds a ray in the other, to a plane a plane, to a flat or axial pencil a projective flat or axial pencil, and so on.
There is a double infinite number of lines in a pencil. We shall see that a single infinite number of lines in one pencil meets its corresponding ray, and that the points of intersection form a curve in space.
Of the double infinite number of planes in the pencils each will meet its corresponding plane. This gives a system of a double infinite number of lines in space. We know (§ 5) that there is a quadruple infinite number of lines in space. From among these we may select those which satisfy one or more given conditions. The systems of lines thus obtained were first systematically investigated and classified by Plücker, in his Geometrie des Raumes. He uses the following names:—
A treble infinite number of lines, that is, all lines which satisfy one condition, are said to form a complex of lines; e.g. all lines cutting a given line, or all lines touching a surface.
A double infinite number of lines, that is, all lines which satisfy two conditions, or which are common to two complexes, are said to form a congruence of lines; e.g. all lines in a plane, or all lines cutting two curves, or all lines cutting a given curve twice.
A single infinite number of lines, that is, all lines which satisfy three conditions, or which belong to three complexes, form a ruled surface; e.g. one set of lines on a ruled quadric surface, or developable surfaces which are formed by the tangents to a curve.
It follows that all lines in which corresponding planes in two projective pencils meet form a congruence. We shall see this congruence consists of all lines which cut a twisted cubic twice, or of all secants to a twisted cubic.
§ 102. Let l_{1} be the line S_{1}S_{2} as a line in the pencil S_{1}. To it corresponds a line l_{2} in S_{2}. At each of the centres two corresponding lines meet. The two axial pencils with l_{1} and l_{2} as axes are projective, and, as, their axes meet at S_{2}, the intersections of corresponding planes form a cone of the second order (§ 58), with S_{2} as centre. If π_{1} and π_{2} be corresponding planes, then their intersection will be a line p_{2} which passes through S_{2}. Corresponding to it in S_{1} will be a line p_{1} which lies in the plane π_{1}, and which therefore meets p_{2} at some point P. Conversely, if p_{2} be any line in S_{2} which meets its corresponding line p_{1} at a point P, then to the plane l_{2}p_{2} will correspond the plane l_{1}p_{1}, that is, the plane S_{1}S_{2}P. These planes intersect in p_{2}, so that p_{2} is a line on the quadric cone generated by the axial pencils l_{1} and l_{2}. Hence:—
All lines in one pencil which meet their corresponding lines in the other form a cone of the second order which has its centre at the centre of the first pencil, and passes through the centre of the second.
From this follows that the points in which corresponding rays meet lie on two cones of the second order which have the ray joining their centres in common, and form therefore, together with the line S_{1}S_{2} or l_{1}, the intersection of these cones. Any plane cuts each of the cones in a conic. These two conics have necessarily that point in common in which it cuts the line l_{1}, and therefore besides either one or three other points. It follows that the curve is of the third order as a plane may cut it in three, but not in more than three, points. Hence:—
The locus of points in which corresponding lines on two projective pencils meet is a curve of the third order or a “twisted cubic” k, which passes through the centres of the pencils, and which appears as the intersection of two cones of the second order, which have one line in common.
A line belonging to the congruence determined by the pencils is a secant of the cubic; it has two, or one, or no points in common with this cubic, and is called accordingly a secant proper, a tangent, or a secant improper of the cubic. A secant improper may be considered, to use the language of coordinate geometry, as a secant with imaginary points of intersection.
§ 103. If a_{1} and a_{2} be any two corresponding lines in the two pencils, then corresponding planes in the axial pencils having a_{1} and a_{2} as axes generate a ruled quadric surface. If P be any point on the cubic k, and if p_{1}, p_{2} be the corresponding rays in S_{1} and S_{2} which meet at P, then to the plane a_{1}p_{1} in S_{1} corresponds a_{2}p_{2} in S_{2}. These therefore meet in a line through P.
This may be stated thus:—
Those secants of the cubic which cut a ray a_{1}, drawn through the centre S_{1} of one pencil, form a ruled quadric surface which passes through both centres, and which contains the twisted cubic k. Of such surfaces an infinite number exists. Every ray through S_{1} or S_{2} which is not a secant determines one of them.
If, however, the rays a_{1} and a_{2} are secants meeting at A, then the ruled quadric surface becomes a cone of the second order, having A as centre. Or all lines of the congruence which pass through a point on the twisted cubic k form a cone of the second order. In other words, the projection of a twisted cubic from any point in the curve on to any plane is a conic.
If a_{1} is not a secant, but made to pass through any point Q in space, the ruled quadric surface determined by a_{1} will pass through Q. There will therefore be one line of the congruence passing through Q, and only one. For if two such lines pass through Q, then the lines S_{1}Q and S_{2}Q will be corresponding lines; hence Q will be a point on the cubic k, and an infinite number of secants will pass through it. Hence:—
Through every point in space not on the twisted cubic one and only one secant to the cubic can be drawn.
§ 104. The fact that all the secants through a point on the cubic form a quadric cone shows that the centres of the projective pencils generating the cubic are not distinguished from any other points on the cubic. If we take any two points S, S′ on the cubic, and draw the secants through each of them, we obtain two quadric cones, which have the line SS′ in common, and which intersect besides along the cubic. If we make these two pencils having S and S′ as centres projective by taking four rays on the one cone as corresponding to the four rays on the other which meet the first on the cubic, the correspondence is determined. These two pencils will generate a cubic, and the two cones of secants having S and S′ as centres will be identical with the above cones, for each has five rays in common with one of the first, viz. the line SS′ and the four lines determined for the correspondence; therefore these two cones intersect in the original cubic. This gives the theorem:—
On a twisted cubic any two points may be taken as centres of projective pencils which generate the cubic, corresponding planes being those which meet on the same secant.
Of the two projective pencils at S and S′ we may keep the first fixed, and move the centre of the other along the curve. The pencils will hereby remain projective, and a plane α in S will be cut by its corresponding plane α′ always in the same secant a. Whilst S′ moves along the curve the plane α′ will turn about a, describing an axial pencil.
Authorities.—In this article we have given a purely geometrical theory of conics, cones of the second order, quadric surfaces, &c. In doing so we have followed, to a great extent, Reye’s Geometrie der Lage, and to this excellent work those readers are referred who wish for a more exhaustive treatment of the subject. Other works especially valuable as showing the development of the subject are: Monge, Géométrie descriptive: Carnot, Géométrie de position (1803), containing a theory of transversals; Poncelet’s great work Traité des propriétés projectives des figures (1822); Möbins, Barycentrischer Calcul (1826); Steiner, Abhängigkeit geometrischer Gestalten (1832), containing the first full discussion of the projective relations between rows, pencils, &c.; Von Staudt, Geometrie der Lage (1847) and Beiträge zur Geometrie der Lage (1856–1860), in which a system of geometry is built up from the beginning without any reference to number, so that ultimately a number itself gets a geometrical definition, and in which imaginary elements are systematically introduced into pure geometry; Chasles, Aperçu historique (1837), in which the author gives a brilliant account of the progress of modern geometrical methods, pointing out the advantages of the different purely geometrical methods as compared with the analytical ones, but without taking as much account of the German as of the French authors; Id., Rapport sur les progrès de la géométrie (1870), a continuation of the Aperçu; Id., Traité de géométrie supérieure (1852); Cremona, Introduzione ad una teoria geometrica delle curve piane (1862) and its continuation Preliminari di una teoria geometrica delle superficie (German translations by Curtze). As more elementary books, we mention: Cremona, Elements of Projective Geometry, translated from the Italian by C. Leudesdorf (2nd ed., 1894); J. W. Russell, Pure Geometry (2nd ed., 1905). (O. H.)
III. Descriptive Geometry
This branch of geometry is concerned with the methods for representing solids and other figures in three dimensions by drawings in one plane. The most important method is that which was invented by Monge towards the end of the 18th century. It is based on parallel projections to a plane by rays perpendicular to the plane. Such a projection is called orthographic (see Projection, § 18). If the plane is horizontal the projection is called the plan of the figure, and if the plane is vertical the elevation. In Monge’s method a figure is represented by its plan and elevation. It is therefore often called drawing in plan and elevation, and sometimes simply orthographic projection.
§ 1. We suppose then that we have two planes, one horizontal,
the other vertical, and these we call the planes of plan and of elevation
respectively, or the horizontal and the vertical plane, and
denote them by the letters π_{1} and π_{2}. Their line of intersection is
called the axis, and will be denoted by xy.
If the surface of the drawing paper is taken as the plane of the plan, then the vertical plane will be the plane perpendicular to it through the axis xy. To bring this also into the plane of the drawing paper we turn it about the axis till it coincides with the horizontal plane. This process of turning one plane down till it coincides with another is called rabatting one to the other. Of course there is no necessity to have one of the two planes horizontal, but even when this is not the case it is convenient to retain the above names.
Fig. 37.  Fig. 38. 
The whole arrangement will be better understood by referring to fig. 37. A point A in space is there projected by the perpendicular AA_{1} and AA_{2} to the planes π_{1} and π_{2} so that A_{1} and A_{2} are the horizontal and vertical projections of A.
If we remember that a line is perpendicular to a plane that is perpendicular to every line in the plane if only it is perpendicular to any two intersecting lines in the plane, we see that the axis which is perpendicular both to AA_{1} and to AA_{2} is also perpendicular to A_{1}A_{0} and to A_{2}A_{0} because these four lines are all in the same plane. Hence, if the plane π_{2} be turned about the axis till it coincides with the plane π_{1}, then A_{2}A_{0} will be the continuation of A_{1}A_{0}. This position of the planes is represented in fig. 38, in which the line A_{1}A_{2} is perpendicular to the axis x.
Conversely any two points A_{1}, A_{2} in a line perpendicular to the axis will be the projections of some point in space when the plane π_{2} is turned about the axis till it is perpendicular to the plane π_{1}, because in this position the two perpendiculars to the planes π_{1} and π_{2} through the points A_{1} and A_{2} will be in a plane and therefore meet at some point A.
Representation of Points.—We have thus the following method of representing in a single plane the position of points in space:—we take in the plane a line xy as the axis, and then any pair of points A_{1}, A_{2} in the plane on a line perpendicular to the axis represent a point A in space. If the line A_{1}A_{2} cuts the axis at A_{0}, and if at A_{1} a perpendicular be erected to the plane, then the point A will be in it at a height A_{1}A = A_{0}A_{2} above the plane. This gives the position of the point A relative to the plane π_{1}. In the same way, if in a perpendicular to π_{2} through A_{2} a point A be taken such that A_{2}A = A_{0}A_{1}, then this will give the point A relative to the plane π_{2}.
Fig. 39. 
§ 2. The two planes π_{1}, π_{2} in their original position divide space into four parts. These are called the four quadrants. We suppose that the plane π_{2} is turned as indicated in fig. 37, so that the point P comes to Q and R to S, then the quadrant in which the point A lies is called the first, and we say that in the first quadrant a point lies above the horizontal and in front of the vertical plane. Now we go round the axis in the sense in which the plane π_{2} is turned and come in succession to the second, third and fourth quadrant. In the second a point lies above the plane of the plan and behind the plane of elevation, and so on. In fig. 39, which represents a side view of the planes in fig. 37 the quadrants are marked, and in each a point with its projection is taken. Fig. 38 shows how these are represented when the plane π_{2} is turned down. We see that
A point lies in the first quadrant if the plan lies below, the elevation above the axis; in the second if plan and elevation both lie above; in the third if the plan lies above, the elevation below; in the fourth if plan and elevation both lie below the axis.
If a point lies in the horizontal plane, its elevation lies in the axis and the plan coincides with the point itself. If a point lies in the vertical plane, its plan lies in the axis and the elevation coincides with the point itself. If a point lies in the axis, both its plan and elevation lie in the axis and coincide with it.
Of each of these propositions, which will easily be seen to be true, the converse holds also.
§ 3. Representation of a Plane.—As we are thus enabled to represent points in a plane, we can represent any finite figure by representing its separate points. It is, however, not possible to represent a plane in this way, for the projections of its points completely cover the planes π_{1} and π_{2}, and no plane would appear different from any other. But any plane α cuts each of the planes π_{1}, π_{2} in a line. These are called the traces of the plane. They cut each other in the axis at the point where the latter cuts the plane α.
A plane is determined by its two traces, which are two lines that meet on the axis, and, conversely, any two lines which meet on the axis determine a plane.
If the plane is parallel to the axis its traces are parallel to the axis. Of these one may be at infinity; then the plane will cut one of the planes of projection at infinity and will be parallel to it. Thus a plane parallel to the horizontal plane of the plan has only one finite trace, viz. that with the plane of elevation.
Fig. 40. 
If the plane passes through the axis both its traces coincide with the axis. This is the only case in which the representation of the plane by its two traces fails. A third plane of projection is therefore introduced, which is best taken perpendicular to the other two. We call it simply the third plane and denote it by π_{3}. As it is perpendicular to π_{1}, it may be taken as the plane of elevation, its line of intersection γ with π_{1} being the axis, and be turned down to coincide with π_{1}. This is represented in fig. 40. OC is the axis xy whilst OA and OB are the traces of the third plane. They lie in one line γ. The plane is rabatted about γ to the horizontal plane. A plane α through the axis xy will then show in it a trace α_{3}. In fig. 40 the lines OC and OP will thus be the traces of a plane through the axis xy, which makes an angle POQ with the horizontal plane.
We can also find the trace which any other plane makes with π_{3}. In rabatting the plane π_{3} its trace OB with the plane π_{2} will come to the position OD. Hence a plane β having the traces CA and CB will have with the third plane the trace β_{3}, or AD if OD = OB.
It also follows immediately that—
If a plane α is perpendicular to the horizontal plane, then every point in it has its horizontal projection in the horizontal trace of the plane, as all the rays projecting these points lie in the plane itself.
Any plane which is perpendicular to the horizontal plane has its vertical trace perpendicular to the axis.
Any plane which is perpendicular to the vertical plane has its horizontal trace perpendicular to the axis and the vertical projections of all points in the plane lie in this trace.
§ 4. Representation of a Line.—A line is determined either by two points in it or by two planes through it. We get accordingly two representations of it either by projections or by traces.
First.—A line a is represented by its projections a_{1} and a_{2} on the two planes π_{1} and π_{2}. These may be any two lines, for, bringing the planes π_{1}, π_{2} into their original position, the planes through these lines perpendicular to π_{1} and π_{2} respectively will intersect in some line a which has a_{1}, a_{2} as its projections.
Secondly.—A line a is represented by its traces—that is, by the points in which it cuts the two planes π_{1}, π_{2}. Any two points may be taken as the traces of a line in space, for it is determined when the planes are in their original position as the line joining the two traces. This representation becomes undetermined if the two traces coincide in the axis. In this case we again use a third plane, or else the projections of the line.
The fact that there are different methods of representing points and planes, and hence two methods of representing lines, suggests the principle of duality (section ii., Projective Geometry, § 41). It is worth while to keep this in mind. It is also worth remembering that traces of planes or lines always lie in the planes or lines which they represent. Projections do not as a rule do this excepting when the point or line projected lies in one of the planes of projection.
Having now shown how to represent points, planes and lines, we have to state the conditions which must hold in order that these elements may lie one in the other, or else that the figure formed by them may possess certain metrical properties. It will be found that the former are very much simpler than the latter.
Before we do this, however, we shall explain the notation used; for it is of great importance to have a systematic notation. We shall denote points in space by capitals A, B, C; planes in space by Greek letters α, β, γ; lines in space by small letters a, b, c; horizontal projections by suffixes 1, like A_{1}, a_{1}; vertical projections by suffixes 2, like A_{2}, a_{2}; traces by single and double dashes α′ α″, a′, a″. Hence P_{1} will be the horizontal projection of a point P in space; a line a will have the projections a_{1}, a_{2} and the traces a′ and a″; a plane α has the traces α′ and α″.
§ 5. If a point lies in a line, the projections of the point lie in the projections of the line.
If a line lies in a plane, the traces of the line lie in the traces of the plane.
These propositions follow at once from the definitions of the projections and of the traces.
If a point lies in two lines its projections must lie in the projections of both. Hence
If two lines, given by their projections, intersect, the intersection of their planes and the intersection of their elevations must lie in a line perpendicular to the axis, because they must be the projections of the point common to the two lines.
Similarly—If two lines given by their traces lie in the same plane or intersect, then the lines joining their horizontal and vertical traces respectively must meet on the axis, because they must be the traces of the plane through them.
§ 6. To find the projections of a line which joins two points A, B given by their projections A_{1}, A_{2} and B_{1}, B_{2}, we join A_{1}, B_{1} and A_{2}, B_{2}; these will be the projections required. For example, the traces of a line are two points in the line whose projections are known or at all events easily found. They are the traces themselves and the feet of the perpendiculars from them to the axis.
Fig. 41. 
Hence if a′ a″ (fig. 41) are the traces of a line a, and if the perpendiculars from them cut the axis in P and Q respectively, then the line a′Q will be the horizontal and a″P the vertical projection of the line.
Conversely, if the projections a_{1}, a_{2} of a line are given, and if these cut the axis in Q and P respectively, then the perpendiculars Pa′ and Qa″ to the axis drawn through these points cut the projections a_{1} and a_{2} in the traces a′ and a″.
To find the line of intersection of two planes, we observe that this line lies in both planes; its traces must therefore lie in the traces of both. Hence the points where the horizontal traces of the given planes meet will be the horizontal, and the point where the vertical traces meet the vertical trace of the line required.
§ 7. To decide whether a point A, given by its projections, lies in a plane α, given by its traces, we draw a line p by joining A to some point in the plane α and determine its traces. If these lie in the traces of the plane, then the line, and therefore the point A, lies in the plane; otherwise not. This is conveniently done by joining A_{1} to some point p′ in the trace α′; this gives p_{1}; and the point where the perpendicular from p′ to the axis cuts the latter we join to A_{2}; this gives p_{2}. If the vertical trace of this line lies in the vertical trace of the plane, then, and then only, does the line p, and with it the point A, lie in the plane α.
§ 8. Parallel planes have parallel traces, because parallel planes are cut by any plane, hence also by π_{1} and by π_{2}, in parallel lines.
Parallel lines have parallel projections, because points at infinity are projected to infinity.
If a line is parallel to a plane, then lines through the traces of the line and parallel to the traces of the plane must meet on the axis, because these lines are the traces of a plane parallel to the given plane.
§ 9. To draw a plane through two intersecting lines or through two parallel lines, we determine the traces of the lines; the lines joining their horizontal and vertical traces respectively will be the horizontal and vertical traces of the plane. They will meet, at a finite point or at infinity, on the axis if the lines do intersect.
To draw a plane through a line and a point without the line, we join the given point to any point in the line and determine the plane through this and the given line.
To draw a plane through three points which are not in a line, we draw two of the lines which each join two of the given points and draw the plane through them. If the traces of all three lines AB, BC, CA be found, these must lie in two lines which meet on the axis.
§ 10. We have in the last example got more points, or can easily get more points, than are necessary for the determination of the figure required—in this case the traces of the plane. This will happen in a great many constructions and is of considerable importance. It may happen that some of the points or lines obtained are not convenient in the actual construction. The horizontal traces of the lines AB and AC may, for instance, fall very near together, in which case the line joining them is not well defined. Or, one or both of them may fall beyond the drawing paper, so that they are practically nonexistent for the construction. In this case the traces of the line BC may be used. Or, if the vertical traces of AB and AC are both in convenient position, so that the vertical trace of the required plane is found and one of the horizontal traces is got, then we may join the latter to the point where the vertical trace cuts the axis.
The draughtsman must remember that the lines which he draws are not mathematical lines without thickness, and therefore every drawing is affected by some errors. It is therefore very desirable to be able constantly to check the latter. Such checks always present themselves when the same result can be obtained by different constructions, or when, as in the above case, some lines must meet on the axis, or if three points must lie in a line. A careful draughtsman will always avail himself of these checks.
§ 11. To draw a plane through a given point parallel to a given plane α, we draw through the point two lines which are parallel to the plane α, and determine the plane through them; or, as we know that the traces of the required plane are parallel to those of the given one (§ 8), we need only draw one line l through the point parallel to the plane and find one of its traces, say the vertical trace l″; a line through this parallel to the vertical trace of α will be the vertical trace β″ of the required plane β, and a line parallel to the horizontal trace of α meeting β″ on the axis will be the horizontal trace β′.
Fig. 42. 
Let A_{1} A_{2} (fig. 42) be the given point, α′ α″ the given plane, a line l_{1} through A_{1}, parallel to α′ and a horizontal line l_{2} through A_{2} will be the projections of a line l through A parallel to the plane, because the horizontal plane through this line will cut the plane α in a line c which has its horizontal projection c_{1} parallel to α′.
§ 12. We now come to the metrical properties of figures.
A line is perpendicular to a plane if the projections of the line are perpendicular to the traces of the plane. We prove it for the horizontal projection. If a line p is perpendicular to a plane α, every plane through p is perpendicular to α; hence also the vertical plane which projects the line p to p_{1}. As this plane is perpendicular both to the horizontal plane and to the plane α, it is also perpendicular to their intersection—that is, to the horizontal trace of α. It follows that every line in this projecting plane, therefore also p_{1}, the plan of p, is perpendicular to the horizontal trace of α.
To draw a plane through a given point A perpendicular to a given line p, we first draw through some point O in the axis lines γ′, γ″ perpendicular respectively to the projections p_{1} and p_{2} of the given line. These will be the traces of a plane γ which is perpendicular to the given line. We next draw through the given point A a plane parallel to the plane γ; this will be the plane required.
Other metrical properties depend on the determination of the real size or shape of a figure.
In general the projection of a figure differs both in size and shape from the figure itself. But figures in a plane parallel to a plane of projection will be identical with their projections, and will thus be given in their true dimensions. In other cases there is the problem, constantly recurring, either to find the true shape and size of a plane figure when plan and elevation are given, or, conversely, to find the latter from the known true shape of the figure itself. To do this, the plane is turned about one of its traces till it is laid down into that plane of projection to which the trace belongs. This is technically called rabatting the plane respectively into the plane of the plan or the elevation. As there is no difference in the treatment of the two cases, we shall consider only the case of rabatting a plane α into the plane of the plan. The plan of the figure is a parallel (orthographic) projection of the figure itself. The results of parallel projection (see Projection, §§ 17 and 18) may therefore now be used. The trace α′ will hereby take the place of what formerly was called the axis of projection. Hence we see that corresponding points in the plan and in the rabatted plane are joined by lines which are perpendicular to the trace α′ and that corresponding lines meet on this trace. We also see that the correspondence is completely determined if we know for one point or one line in the plan the corresponding point or line in the rabatted plane.
Before, however, we treat of this we consider some special cases.
§ 13. To determine the distance between two points A, B given by their projections A_{1}, B_{1} and A_{2}, B_{2}, or, in other words, to determine the true length of a line the plan and elevation of which are given.
Fig. 43. 
Fig. 44. 
Solution.—The two points A, B in space lie vertically above their plans A_{1}, B_{1} (fig. 43) and A_{1}A = A_{0}A_{2}, B_{1}B = B_{0}B_{2}. The four points A, B, A_{1}, B_{1} therefore form a plane quadrilateral on the base A_{1}B_{1} and having right angles at the base. This plane we rabatt about A_{1}B_{1} by drawing A_{1}A and B_{1}B perpendicular to A_{1}B_{1} and making A_{1}A = A_{0}A_{2}, B_{1}B = B_{0}B_{2}. Then AB will give the length required.
The construction might have been performed in the elevation by making A_{2}A = A_{0}A_{1} and B_{2}B = B_{0}B_{1} on lines perpendicular to A_{2}B_{2}. Of course AB must have the same length in both cases.
This figure may be turned into a model. Cut the paper along A_{1}A, AB and BB_{1}, and fold the piece A_{1}ABB_{1} over along A_{1}B_{1} till it stands upright at right angles to the horizontal plane. The points A, B will then be in their true position in space relative to π_{1}. Similarly if B_{2}BAA_{2} be cut out and turned along A_{2}B_{2} through a right angle we shall get AB in its true position relative to the plane π_{2}. Lastly we fold the whole plane of the paper along the axis x till the plane π_{2} is at right angles to π_{1}. In this position the two sets of points AB will coincide if the drawing has been accurate.
Models of this kind can be made in many cases and their construction cannot be too highly recommended in order to realize orthographic projection.
§ 14. To find the angle between two given lines a, b of which the projections a_{1}, b_{1} and a_{2}, b_{2} are given.
Solution.—Let a_{1}, b_{1} (fig. 44) meet in P_{1}, a_{2}, b_{2} in T, then if the line P_{1}T is not perpendicular to the axis the two lines will not meet. In this case we draw a line parallel to b to meet the line a. This is easiest done by drawing first the line P_{1}P_{2} perpendicular to the axis to meet a_{2} in P_{2}, and then drawing through P_{2} a line c_{2} parallel to b_{2}; then b_{1}, c_{2} will be the projections of a line c which is parallel to b and meets a in P. The plane α which these two lines determine we rabatt to the plan. We determine the traces a′ and c′ of the lines a and c; then a′c′ is the trace α′ of their plane. On rabatting the point P comes to a point S on the line P_{1}Q perpendicular to a′c′, so that QS = QP. But QP is the hypotenuse of a triangle PP_{1}Q with a right angle P_{1}. This we construct by making QR = P_{0}P_{2}; then P_{1}R = PQ. The lines a′S and c′S will therefore include angles equal to those made by the given lines. It is to be remembered that two lines include two angles which are supplementary. Which of these is to be taken in any special case depends upon the circumstances.
To determine the angle between a line and a plane, we draw through any point in the line a perpendicular to the plane (§ 12) and determine the angle between it and the given line. The complement of this angle is the required one.
To determine the angle between two planes, we draw through any point two lines perpendicular to the two planes and determine the angle between the latter as above.
In special cases it is simpler to determine at once the angle between the two planes by taking a plane section perpendicular to the intersection of the two planes and rabatt this. This is especially the case if one of the planes is the horizontal or vertical plane of projection.
Thus in fig. 45 the angle P_{1}QR is the angle which the plane α makes with the horizontal plane.
§ 15. We return to the general case of rabatting a plane α of which the traces α′ α″ are given.
Fig. 45. 
Here it will be convenient to determine first the position which the trace α″—which is a line in α—assumes when rabatted. Points in this line coincide with their elevations. Hence it is given in its true dimension, and we can measure off along it the true distance between two points in it. If therefore (fig. 45) P is any point in α″ originally coincident with its elevation P_{2}, and if O is the point where α″ cuts the axis xy, so that O is also in α′, then the point P will after rabatting the plane assume such a position that OP = OP_{2}. At the same time the plan is an orthographic projection of the plane α. Hence the line joining P to the plan P_{1} will after rabatting be perpendicular to α′. But P_{1} is known; it is the foot of the perpendicular from P_{2} to the axis xy. We draw therefore, to find P, from P_{1} a perpendicular P_{1}Q to α′ and find on it a point P such that OP = OP_{2}. Then the line OP will be the position of α″ when rabatted. This line corresponds therefore to the plan of α″—that is, to the axis xy, corresponding points on these lines being those which lie on a perpendicular to α′.
We have thus one pair of corresponding lines and can now find for any point B_{1} in the plan the corresponding point B in the rabatted plane. We draw a line through B_{1}, say B_{1}P_{1}, cutting α′ in C. To it corresponds the line CP, and the point where this is cut by the projecting ray through B_{1}, perpendicular to α′, is the required point B.
Similarly any figure in the rabatted plane can be found when the plan is known; but this is usually found in a different manner without any reference to the general theory of parallel projection. As this method and the reasoning employed for it have their peculiar advantages, we give it also.
Supposing the planes π_{1} and π_{2} to be in their positions in space perpendicular to each other, we take a section of the whole figure by a plane perpendicular to the trace α′ about which we are going to rabatt the plane α. Let this section pass through the point Q in α′. Its traces will then be the lines QP_{1} and P_{1}P_{2} (fig. 9). These will be at right angles, and will therefore, together with the section QP_{2} of the plane α, form a rightangled triangle QP_{1}P_{2} with the right angle at P_{1}, and having the sides P_{1}Q and P_{1}P_{2} which both are given in their true lengths. This triangle we rabatt about its base P_{1}Q, making P_{1}R = P_{1}P_{2}. The line QR will then give the true length of the line QP in space. If now the plane α be turned about α′ the point P will describe a circle about Q as centre with radius QP = QR, in a plane perpendicular to the trace α′. Hence when the plane α has been rabatted into the horizontal plane the point P will lie in the perpendicular P_{1}Q to α′, so that QP = QR.
If A_{1} is the plan of a point A in the plane α, and if A_{1} lies in QP_{1}, then the point A will lie vertically above A_{1} in the line QP. On turning down the triangle QP_{1}P_{2}, the point A will come to A_{0}, the line A_{1}A_{0} being perpendicular to QP_{1}. Hence A will be a point in QP such that QA = QA_{0}.
If B_{1} is the plan of another point, but such that A_{1}B_{1} is parallel to α′, then the corresponding line AB will also be parallel to α′. Hence, if through A a line AB be drawn parallel to α′, and B_{1}B perpendicular to α′, then their intersection gives the point B. Thus of any point given in plan the real position in the plane α, when rabatted, can be found by this second method. This is the one most generally given in books on geometrical drawing. The first method explained is, however, in most cases preferable as it gives the draughtsman a greater variety of constructions. It requires a somewhat greater amount of theoretical knowledge.
If instead of our knowing the plan of a figure the latter is itself given, then the process of finding the plan is the reverse of the above and needs little explanation. We give an example.
§ 16. It is required to draw the plan and elevation of a polygon of which the real shape and position in a given plane α are known.
Fig. 46. 
We first rabatt the plane α (fig. 46) as before so that P_{1} comes to P, hence OP_{1} to OP. Let the given polygon in α be the figure ABCDE. We project, not the vertices, but the sides. To project the line AB, we produce it to cut α′ in F and OP in G, and draw GG_{1} perpendicular to α′; then G_{1} corresponds to G, therefore FG_{1} to FG. In the same manner we might project all the other sides, at least those which cut OF and OP in convenient points. It will be best, however, first to produce all the sides to cut OP and α′ and then to draw all the projecting rays through A, B, C ... perpendicular to α′, and in the same direction the lines G, G_{1}, &c. By drawing FG we get the points A_{1}, B_{1} on the projecting ray through A and B. We then join B to the point M where BC produced meets the trace α′. This gives C_{1}. So we go on till we have found E_{1}. The line A_{1} E_{1} must then meet AE in α′, and this gives a check. If one of the sides cuts α′ or OP beyond the drawing paper this method fails, but then we may easily find the projection of some other line, say of a diagonal, or directly the projection of a point, by the former methods. The diagonals may also serve to check the drawing, for two corresponding diagonals must meet in the trace α′.
Having got the plan we easily find the elevation. The elevation of G is above G_{1} in α″, and that of F is at F_{2} in the axis. This gives the elevation F_{2}G_{2} of FG and in it we get A_{2}B_{2} in the verticals through A_{1} and B_{1}. As a check we have OG = OG_{2}. Similarly the elevation of the other sides and vertices are found.
§ 17. We proceed to give some applications of the above principles to the representation of solids and of the solution of problems connected with them.
Of a pyramid are given its base, the length of the perpendicular from the vertex to the base, and the point where this perpendicular cuts the base; it is required first to develop the whole surface of the pyramid into one plane, and second to determine its section by a plane which cuts the plane of the base in a given line and makes a given angle with it.
1. As the planes of projection are not given we can take them as we like, and we select them in such a manner that the solution becomes as simple as possible. We take the plane of the base as the horizontal plane and the vertical plane perpendicular to the plane of the section. Let then (fig. 47) ABCD be the base of the pyramid, V_{1} the plan of the vertex, then the elevations of A, B, C, D will be in the axis at A_{2}, B_{2}, C_{2}, D_{2}, and the vertex at some point V_{2} above V_{1} at a known distance from the axis. The lines V_{1}A, V_{1}B, &c., will be the plans and the lines V_{2}A_{2}, V_{2}B_{2}, &c., the elevations of the edges of the pyramid, of which thus plan and elevation are known.
We develop the surface into the plane of the base by turning each lateral face about its lower edge into the horizontal plane by the method used in § 14. If one face has been turned down, say ABV to ABP, then the point Q to which the vertex of the next face BCV comes can be got more simply by finding on the line V_{1}Q perpendicular to BC the point Q such that BQ = BP, for these lines represent the same edge BV of the pyramid. Next R is found by making CR = CQ, and so on till we have got the last vertex—in this case S. The fact that AS must equal AP gives a convenient check.
2. The plane α whose section we have to determine has its horizontal trace given perpendicular to the axis, and its vertical trace makes the given angle with the axis. This determines it. To find the section of the pyramid by this plane there are two methods applicable: we find the sections of the plane either with the faces or with the edges of the pyramid. We use the latter.
As the plane α is perpendicular to the vertical plane, the trace α″ contains the projection of every figure in it; the points E_{2}, F_{2}, G_{2}, H_{2} where this trace cuts the elevations of the edges will therefore be the elevations of the points where the edges cut α. From these we find the plans E_{1}, F_{1}, G_{1}, H_{1}, and by joining them the plan of the section. If from E_{1}, F_{1} lines be drawn perpendicular to AB, these will determine the points E, F on the developed face in which the plane α cuts it; hence also the line EF. Similarly on the other faces. Of course BF must be the same length on BP and on BQ. If the plane α be rabatted to the plan, we get the real shape of the section as shown in the figure in EFGH. This is done easily by making F_{0}F = OF_{2}, &c. If the figure representing the development of the pyramid, or better a copy of it, is cut out, and if the lateral faces be bent along the lines AB, BC, &c., we get a model of the pyramid with the section marked on its faces. This may be placed on its plan ABCD and the plane of elevation bent about the axis x. The pyramid stands then in front of its elevations. If next the plane α with a hole cut out representing the true section be bent along the trace α′ till its edge coincides with α″, the edges of the hole ought to coincide with the lines EF, FG, &c., on the faces.
§ 18. Polyhedra like the pyramid in § 17 are represented by the projections of their edges and vertices. But solids bounded by curved surfaces, or surfaces themselves, cannot be thus represented.
For a surface we may use, as in case of the plane, its traces—that is, the curves in which it cuts the planes of projection. We may also project points and curves on the surface. A ray cuts the surface generally in more than one point; hence it will happen that some of the rays touch the surface, if two of these points coincide. The points of contact of these rays will form some curve on the surface, and this will appear from the centre of projection as the boundary of the surface or of part of the surface. The outlines of all surfaces of solids which we see about us are formed by the points at which rays through our eye touch the surface. The projections of these contours are therefore best adapted to give an idea of the shape of a surface.
Fig. 47. 
Thus the tangents drawn from any finite centre to a sphere form a right circular cone, and this will be cut by any plane in a conic. It is often called the projection of a sphere, but it is better called the contourline of the sphere, as it is the boundary of the projections of all points on the sphere.
If the centre is at infinity the tangent cone becomes a right circular cylinder touching the sphere along a great circle, and if the projection is, as in our case, orthographic, then the section of this cone by a plane of projection will be a circle equal to the great circle of the sphere. We get such a circle in the plan and another in the elevation, their centres being plan and elevation of the centre of the sphere.
Similarly the rays touching a cone of the second order will lie in two planes which pass through the vertex of the cone, the contourline of the projection of the cone consists therefore of two lines meeting in the projection of the vertex. These may, however, be invisible if no real tangent rays can be drawn from the centre of projection; and this happens when the ray projecting the centre of the vertex lies within the cone. In this case the traces of the cone are of importance. Thus in representing a cone of revolution with a vertical axis we get in the plan a circular trace of the surface whose centre is the plan of the vertex of the cone, and in the elevation the contour, consisting of a pair of lines intersecting in the elevation of the vertex of the cone. The circle in the plan and the pair of lines in the elevation do not determine the surface, for an infinite number of surfaces might be conceived which pass through the circular trace and touch two planes through the contour lines in the vertical plane. The surface becomes only completely defined if we write down to the figure that it shall represent a cone. The same holds for all surfaces. Even a plane is fully represented by its traces only under the silent understanding that the traces are those of a plane.
§ 19. Some of the simpler problems connected with the representation of surfaces are the determination of plane sections and of the curves of intersection of two such surfaces. The former is constantly used in nearly all problems concerning surfaces. Its solution depends of course on the nature of the surface.
To determine the curve of intersection of two surfaces, we take a plane and determine its section with each of the two surfaces, rabatting this plane if necessary. This gives two curves which lie in the same plane and whose intersections will give us points on both surfaces. It must here be remembered that two curves in space do not necessarily intersect, hence that the points in which their projections intersect are not necessarily the projections of points common to the two curves. This will, however, be the case if the two curves lie in a common plane. By taking then a number of plane sections of the surfaces we can get as many points on their curve of intersection as we like. These planes have, of course, to be selected in such a way that the sections are curves as simple as the case permits of, and such that they can be easily and accurately drawn. Thus when possible the sections should be straight lines or circles. This not only saves time in drawing but determines all points on the sections, and therefore also the points where the two curves meet, with equal accuracy.
§ 20. We give a few examples how these sections have to be selected. A cone is cut by every plane through the vertex in lines, and if it is a cone of revolution by planes perpendicular to the axis in circles.
A cylinder is cut by every plane parallel to the axis in lines, and if it is a cylinder of revolution by planes perpendicular to the axis in circles.
A sphere is cut by every plane in a circle.
Hence in case of two cones situated anywhere in space we take sections through both vertices. These will cut both cones in lines. Similarly in case of two cylinders we may take sections parallel to the axis of both. In case of a sphere and a cone of revolution with vertical axis, horizontal sections will cut both surfaces in circles whose plans are circles and whose elevations are lines, whilst vertical sections through the vertex of the cone cut the latter in lines and the sphere in circles. To avoid drawing the projections of these circles, which would in general be ellipses, we rabatt the plane and then draw the circles in their real shape. And so on in other cases.
Special attention should in all cases be paid to those points in which the tangents to the projection of the curve of intersection are parallel or perpendicular to the axis x, or where these projections touch the contour of one of the surfaces. (O. H.)
1. In the name geometry there is a lasting record that the science had its origin in the knowledge that two distances may be compared by measurement, and in the idea that measurement must be effectual in the dissociation of different directions as well as in the comparison of distances in the same direction. The distance from an observer’s eye of an object seen would be specified as soon as it was ascertained that a rod, straight to the eye and of length taken as known, could be given the direction of the line of vision, and had to be moved along it a certain number of times through lengths equal to its own in order to reach the object from the eye. Moreover, if a field had for two of its boundaries lines straight to the eye, one running from south to north and the other from west to east, the position of a point in the field would be specified if the rod, when directed west, had to be shifted from the point one observed number of times westward to meet the former boundary, and also, when directed south, had to be shifted another observed number of times southward to meet the latter. Comparison by measurement, the beginning of geometry, involved counting, the basis of arithmetic; and the science of number was marked out from the first as of geometrical importance.
But the arithmetic of the ancients was inadequate as a science of number. Though a length might be recognized as known when measurement certified that it was so many times a standard length, it was not every length which could be thus specified in terms of the same standard length, even by an arithmetic enriched with the notion of fractional number. The idea of possible incommensurability of lengths was introduced into Europe by Pythagoras; and the corresponding idea of irrationality of number was absent from a crude arithmetic, while there were great practical difficulties in the way of its introduction. Hence perhaps it arose that, till comparatively modern times, appeal to arithmetical aid in geometrical reasoning was in all possible ways restrained. Geometry figured rather as the helper of the more difficult science of arithmetic.
2. It was reserved for algebra to remove the disabilities of arithmetic, and to restore the earliest ideas of the landmeasurer to the position of controlling ideas in geometrical investigation. This unified science of pure number made comparatively little headway in the hands of the ancients, but began to receive due attention shortly after the revival of learning. It expresses whole classes of arithmetical facts in single statements, gives to arithmetical laws the form of equations involving symbols which may mean any known or sought numbers, and provides processes which enable us to analyse the information given by an equation and derive from that equation other equations, which express laws that are in effect consequences or causes of a law started from, but differ greatly from it in form. Above all, for present purposes, it deals not only with integral and fractional number, but with number regarded as capable of continuous growth, just as distance is capable of continuous growth. The difficulty of the arithmetical expression of irrational number, a difficulty considered by the modern school of analysts to have been at length surmounted (see Function), is not vital to it. It can call the ratio of the diagonal of a square to a side, for instance, or that of the circumference of a circle to a diameter, a number, and let a or x denote that number, just as properly as it may allow either letter to denote any rational number which may be greater or less than the ratio in question by a difference less than any minute one we choose to assign.
Counting only, and not the counting of objects, is of the essence of arithmetic, and of algebra. But it is lawful to count objects, and in particular to count equal lengths by measure. The widened idea is that even when a or x is an irrational number we may speak of a or x unit lengths by measure. We may give concrete interpretation to an algebraical equation by allowing its terms all to mean numbers of times the same unit length, or the same unit area, or &c. and in any equation lawfully derived from the first by algebraical processes we may do the same. Descartes in his Géométrie (1637) was the first to systematize the application of this principle to the inherent first notions of geometry; and the methods which he instituted have become the most potent methods of all in geometrical research. It is hardly too much to say that, when known facts as to a geometrical figure have once been expressed in algebraical terms, all strictly consequential facts as to the figure can be deduced by almost mechanical processes. Some may well be unexpected consequences; and in obtaining those of which there has been suggestion beforehand the often bewildering labour of constant attention to the figure is obviated. These are the methods of what is now called analytical, or sometimes algebraical, geometry.
3. The modern use of the term “analytical” in geometry has obscured, but not made obsolete, an earlier use, one as old as Plato. There is nothing algebraical in this analysis, as distinguished from synthesis, of the Greeks, and of the expositors of pure geometry. It has reference to an order of ideas in demonstration, or, more frequently, in discovering means to effect the geometrical construction of a figure with an assigned special property. We have to suppose hypothetically that the construction has been performed, drawing a rough figure which exhibits it as nearly as is practicable. We then analyse or critically examine the figure, treated as correct, and ascertain other properties which it can only possess in association with the one in question. Presently one of these properties will often be found which is of such a character that the construction of a figure possessing it is simple. The means of effecting synthetically a construction such as was desired is thus brought to light by what Plato called analysis. Or again, being asked to prove a theorem A, we ascertain that it must be true if another theorem B is, that B must be if C is, and so on, thus eventually finding that the theorem A is the consequence, through a chain of intermediaries, of a theorem Z of which the establishment is easy. This geometrical analysis is not the subject of the present article; but in the reasoning from form to form of an equation or system of equations, with the object of basing the algebraical proof of a geometrical fact on other facts of a more obvious character, the same logic is utilized, and the name “analytical geometry” is thus in part explained.
4. In algebra real positive number was alone at first dealt with, and in geometry actual signless distance. But in algebra it became of importance to say that every equation of the first degree has a root, and the notion of negative number was introduced. The negative unit had to be defined as what can be added to the positive unit and produce the sum zero. The corresponding notion was readily at hand in geometry, where it was clear that a unit distance can be measured to the left or down from the farther end of a unit distance already measured to the right or up from a point O, with the result of reaching O again. Thus, to give full interpretation in geometry to the algebraically negative, it was only necessary to associate distinctness of sign with oppositeness of direction. Later it was discovered that algebraical reasoning would be much facilitated, and that conclusions as to the real would retain all their soundness, if a pair of imaginary units ±√−1 of what might be called number were allowed to be contemplated, the pair being defined, though not separately, by the two properties of having the real sum 0 and the real product 1. Only in these two real combinations do they enter in conclusions as to the real. An advantage gained was that every quadratic equation, and not some quadratics only, could be spoken of as having two roots. These admissions of new units into algebra were final, as it admitted of proof that all equations of degrees higher than two have the full numbers of roots possible for their respective degrees in any case, and that every root has a value included in the form a + b √−1, with a, b, real. The corresponding enrichment could be given to geometry, with corresponding advantages and the same absence of danger, and this was done. On a line of measurement of distance we contemplate as existing, not only an infinite continuum of points at real distances from an origin of measurement O, but a doubly infinite continuum of points, all but the singly infinite continuum of real ones imaginary, and imaginary in conjugate pairs, a conjugate pair being at imaginary distances from O, which have a real arithmetic and a real geometric mean. To geometry enriched with this conception all algebra has its application.
5. Actual geometry is one, two or threedimensional, i.e. lineal, plane or solid. In onedimensional geometry positions and measurements in a single line only are admitted. Now descriptive constructions for points in a line are impossible without going out of the line. It has therefore been held that there is a sense in which no science of geometry strictly confined to one dimension exists. But an algebra of one variable can be applied to the study of distances along a line measured from a chosen point on it, so that the idea of construction as distinct from measurement is not essential to a onedimensional geometry aided by algebra. In geometry of two dimensions, the flat of the landmeasurer, the passage from one point O to any other point, can be effected by two successive marches, one east or west and one north or south, and, as will be seen, an algebra of two variables suffices for geometrical exploitation. In geometry of three dimensions, that of space, any point can be reached from a chosen one by three marches, one east or west, one north or south, and one up or down; and we shall see that an algebra of three variables is all that is necessary. With three dimensions actual geometry stops; but algebra can supply any number of variables. Four or more variables have been used in ways analogous to those in which one, two and three variables are used for the purposes of one, two and threedimensional geometry, and the results have been expressed in quasigeometrical language on the supposition that a higher space can be conceived of, though not realized, in which four independent directions exist, such that no succession of marches along three of them can effect the same displacement of a point as a march along the fourth; and similarly for higher numbers than four. Thus analytical, though not actual, geometries exist for four and more dimensions. They are in fact algebras furnished with nomenclature of a geometrical cast, suggested by convenient forms of expression which actual geometry has, in return for benefits received, conferred on algebras of one, two and three variables.
We will confine ourselves to the dimensions of actual geometry, and will devote no space to the onedimensional, except incidentally as existing within the twodimensional. The analytical method will now be explained for the cases of two and three dimensions in succession. The form of it originated by Descartes, and thence known as Cartesian, will alone be considered in much detail.
Fig. 48.  Fig. 49. 
6. Coordinates.—It is assumed that the points, lines and figures considered lie in one and the same plane, which plane therefore need not be in any way referred to. In the plane a point O, and two lines x′Ox, y ′Oy, intersecting in O, are taken once for all, and regarded as fixed. O is called the origin, and x′Ox, y ′Oy the axes of x and y respectively. Other positions in the plane are specified in relation to this fixed origin and these fixed axes. From any point P we suppose PM drawn parallel to the axis of y to meet the axis of x in M, and may also suppose PN drawn parallel to the axis of x to meet the axis of y in N, so that OMPN is a parallelogram. The position of P is determined when we know OM (= NP) and MP (= ON). If OM is x times the unit of a scale of measurement chosen at pleasure, and MP is y times the unit, so that x and y have numerical values, we call x and y the (Cartesian) coordinates of P. To distinguish them we often speak of y as the ordinate, and of x as the abscissa.
It is necessary to attend to signs; x has one sign or the other according as the point P is on one side or the other of the axis of y, and y one sign or the other according as P is on one side or the other of the axis of x. Using the letters N, E, S, W, as in a map, and considering the plane as divided into four quadrants by the axes, the signs are usually taken to be:
x  y  For quadrant 
+  +  N E 
+  −  S E 
−  +  N W 
−  −  S W 
A point is referred to as the point (a, b), when its coordinates are x = a, y = b. A point may be fixed, or it may be variable, i.e. be regarded for the time being as free to move in the plane. The coordinates (x, y) of a variable point are algebraic variables, and are said to be “current coordinates.”
The axes of x and y are usually (as in fig. 48) taken at right angles to one another, and we then speak of them as rectangular axes, and of x and y as “rectangular coordinates” of a point P; OMPN is then a rectangle. Sometimes, however, it is convenient to use axes which are oblique to one another, so that (as in fig. 49) the angle xOy between their positive directions is some known angle ω distinct from a right angle, and OMPN is always an oblique parallelogram with given angles; and we then speak of x and y as “oblique coordinates.” The coordinates are as a rule taken to be rectangular in what follows.
7. Equations and loci. If (x, y) is the point P, and if we are given that x = 0, we are told that, in fig. 48 or fig. 49, the point M lies at O, whatever value y may have, i.e. we are told the one fact that P lies on the axis of y. Conversely, if P lies anywhere on the axis of y, we have always OM = 0, i.e. x = 0. Thus the equation x = 0 is one satisfied by the coordinates (x, y) of every point in the axis of y, and not by those of any other point. We say that x = 0 is the equation of the axis of y, and that the axis of y is the locus represented by the equation x = 0. Similarly y = 0 is the equation of the axis of x. An equation x = a, where a is a constant, expresses that P lies on a parallel to the axis of y through a point M on the axis of x such that OM = a. Every line parallel to the axis of y has an equation of this form. Similarly, every line parallel to the axis of x has an equation of the form y = b, where b is some definite constant.
These are simple cases of the fact that a single equation in the current coordinates of a variable point (x, y) imposes one limitation on the freedom of that point to vary. The coordinates of a point taken at random in the plane will, as a rule, not satisfy the equation, but infinitely many points, and in most cases infinitely many real ones, have coordinates which do satisfy it, and these points are exactly those which lie upon some locus of one dimension, a straight line or more frequently a curve, which is said to be represented by the equation. Take, for instance, the equation y = mx, where m is a given constant. It is satisfied by the coordinates of every point P, which is such that, in fig. 48, the distance MP, with its proper sign, is m times the distance OM, with its proper sign, i.e. by the coordinates of every point in the straight line through O which we arrive at by making a line, originally coincident with x′Ox, revolve about O in the direction opposite to that of the hands of a watch through an angle of which m is the tangent, and by those of no other points. That line is the locus which it represents. Take, more generally, the equation y = φ(x), where φ(x) is any given nonambiguous function of x. Choosing any point M on x′Ox in fig. 1, and giving to x the value of the numerical measure of OM, the equation determines a single corresponding y, and so determines a single point P on the line through M parallel to y ′Oy. This is one point whose coordinates satisfy the equation. Now let M move from the extreme left to the extreme right of the line x′Ox, regarded as extended both ways as far as we like, i.e. let x take all real values from −∞ to ∞. With every value goes a point P, as above, on the parallel to y ′Oy through the corresponding M; and we thus find that there is a path from the extreme left to the extreme right of the figure, all points P along which are distinguished from other points by the exceptional property of satisfying the equation by their coordinates. This path is a locus; and the equation y = φ(x) represents it. More generally still, take an equation ƒ(x, y) = 0 which involves both x and y under a functional form. Any particular value given to x in it produces from it an equation for the determination of a value or values of y, which go with that value of x in specifying a point or points (x, y), of which the coordinates satisfy the equation ƒ(x, y) = 0. Here again, as x takes all values, the point or points describe a path or paths, which constitute a locus represented by the equation. Except when y enters to the first degree only in ƒ(x, y), it is not to be expected that all the values of y, determined as going with a chosen value of x, will be necessarily real; indeed it is not uncommon for all to be imaginary for some ranges of values of x. The locus may largely consist of continua of imaginary points; but the real parts of it constitute a real curve or real curves. Note that we have to allow x to admit of all imaginary, as well as of all real, values, in order to obtain all imaginary parts of the locus.
A locus or curve may be algebraically specified in another way; viz. we may be given two equations x = ƒ(θ), y = F(θ), which express the coordinates of any point of it as two functions of the same variable parameter θ to which all values are open. As θ takes all values in turn, the point (x, y) traverses the curve.
It is a good exercise to trace a number of curves, taken as defined by the equations which represent them. This, in simple cases, can be done approximately by plotting the values of y given by the equation of a curve as going with a considerable number of values of x, and connecting the various points (x, y) thus obtained. But methods exist for diminishing the labour of this tentative process.
Fig. 50. 
Another problem, which will be more attended to here, is that of determining the equations of curves of known interest, taken as defined by geometrical properties. It is not a matter for surprise that the curves which have been most and longest studied geometrically are among those represented by equations of the simplest character.
8. The Straight Line.—This is the simplest type of locus. Also the simplest type of equation in x and y is Ax + By + C = 0, one of the first degree. Here the coefficients A, B, C are constants. They are, like the current coordinates, x, y, numerical. But, in giving interpretation to such an equation, we must of course refer to numbers Ax, By, C of unit magnitudes of the same kind, of units of counting for instance, or unit lengths or unit squares. It will now be seen that every straight line has an equation of the first degree, and that every equation of the first degree represents a straight line.
It has been seen (§ 7) that lines parallel to the axes have equations of the first degree, free from one of the variables. Take now a straight line ABC inclined to both axes. Let it make a given angle α with the positive direction of the axis of x, i.e. in fig. 50 let this be the angle through which Ax must be revolved counterclockwise about A in order to be made coincident with the line. Let C, of coordinates (h, k), be a fixed point on the line, and P(x, y) any other point upon it. Draw the ordinates CD, PM of C and P, and let the parallel to the axis of x through C meet PM, produced if necessary, in R. The rightangled triangle CRP tells us that, with the signs appropriate to their directions attached to CR and RP,
and this gives that
an equation of the first degree satisfied by x and y. No point not on the line satisfies the same equation; for the line from C to any point off the line would make with CR some angle β different from α, and the point in question would satisfy an equation y − k = tan β(x − h), which is inconsistent with the above equation.
The equation of the line may also be written y = mx + b, where m = tan α, and b = k − h tan α. Here b is the value obtained for y from the equation when 0 is put for x, i.e. it is the numerical measure, with proper sign, of OB, the intercept made by the line on the axis of y, measured from the origin. For different straight lines, m and b may have any constant values we like.
Now the general equation of the first degree Ax + By + C = 0 may be written y = −(A/B)x − C/B, unless B = 0, in which case it represents a line parallel to the axis of y; and −A/B, −C/B are values which can be given to m and b, so that every equation of the first degree represents a straight line. It is important to notice that the general equation, which in appearance contains three constants A, B, C, in effect depends on two only, the ratios of two of them to the third. In virtue of this last remark, we see that two distinct conditions suffice to determine a straight line. For instance, it is easy from the above to see that
x  +  y  = 1 
a  b 
is the equation of a straight line determined by the two conditions that it makes intercepts OA, OB on the two axes, of which a and b are the numerical measures with proper signs: note that in fig. 50 a is negative. Again,
y − y_{1} =  y_{2} − y_{1}  (x − x_{1}), 
x_{2} − x_{1} 
i.e.
represents the line determined by the data that it passes through two given points (x_{1}, y_{1}) and (x_{2}, y_{2}). To prove this find m in the equation y − y_{1} = m(x − x_{1}) of a line through (x_{1}, y_{1}), from the condition that (x_{2}, y_{2}) lies on the line.
In this paragraph the coordinates have been assumed rectangular. Had they been oblique, the doctrine of similar triangles would have given the same results, except that in the forms of equation y − k = m(x − h), y = mx + b, we should not have had m = tan α.
9. The Circle.—It is easy to write down the equation of a given circle. Let (h, k) be its given centre C, and ρ the numerical measure of its given radius. Take P (x, y) any point on its circumference, and construct the triangle CRP, in fig. 50 as above. The fact that this is rightangled tells us that
and this at once gives the equation
A point not upon the circumference of the particular circle is at some distance from (h, k) different from ρ, and satisfies an equation inconsistent with this one; which accordingly represents the circumference, or, as we say, the circle.
The equation is of the form
Conversely every equation of this form represents a circle: we have only to take −A, −B, A^{2} + B^{2} − C for h, k, ρ^{2} respectively, to obtain its centre and radius. But this statement must appear too unrestricted. Ought we not to require A^{2} + B^{2} − C to be positive? Certainly, if by circle we are only to mean the visible round circumference of the geometrical definition. Yet, analytically, we contemplate altogether imaginary circles, for which ρ^{2} is negative, and circles, for which ρ = 0, with all their reality condensed into their centres. Even when ρ^{2} is positive, so that a visible round circumference exists, we do not regard this as constituting the whole of the circle. Giving to x any value whatever in (x − h)^{2} + (y − k)^{2} = ρ^{2}, we obtain two values of y, real, coincident or imaginary, each of which goes with the abscissa x as the ordinate of a point, real or imaginary, on what is represented by the equation of the circle.
The doctrine of the imaginary on a circle, and in geometry generally, is of purely algebraical inception; but it has been in its entirety accepted by modern pure geometers, and signal success has attended the efforts of those who, like K. G. C. von Staudt, have striven to base its conclusions on principles not at all algebraical in form, though of course cognate to those adopted in introducing the imaginary into algebra.
A circle with its centre at the origin has an equation x^{2} + y^{2} = ρ^{2}.
In oblique coordinates the general equation of a circle is x^{2} + 2xy cos ω + y^{2} + 2Ax + 2By + C = 0.
10. The conic sections are the next simplest loci; and it will be seen later that they are the loci represented by equations of the second degree. Circles are particular cases of conic sections; and they have just been seen to have for their equations a particular class of equations of the second degree. Another particular class of such equations is that included in the form (Ax + By + C)(A′x + B′y + C′) = 0, which represents two straight lines, because the product on the left vanishes if, and only if, one of the two factors does, i.e. if, and only if, (x, y) lies on one or other of two straight lines. The condition that ax^{2} + 2hxy + by^{2} + 2gx + 2fy + c = 0, which is often written (a, b, c, f, g, h)(x, y, I)^{2} = 0, takes this form is abc + 2fgh − af^{2} − bg^{2} − ch^{2} = 0. Note that the two lines may, in particular cases, be parallel or coincident.
Any equation like F_{1}(x, y) F_{2}(x, y) ... F_{n}(x, y) = 0, of which the lefthand side breaks up into factors, represents all the loci separately represented by F_{1}(x, y) = 0, F_{2}(x, y) = 0, ... F_{n}(x, y) = 0. In particular an equation of degree n which is free from x represents n straight lines parallel to the axis of x, and one of degree n which is homogeneous in x and y, i.e. one which upon division by x^{n}, becomes an equation in the ratio y/x, represents n straight lines through the origin.
Curves represented by equations of the third degree are called cubic curves. The general equation of this degree will be written (*)(x, y, I)^{3} = 0.
Fig. 51. 
11. Descriptive Geometry.—A geometrical proposition is either descriptive or metrical: in the former case the statement of it is independent of the idea of magnitude (length, inclination, &c.), and in the latter it has reference to this idea. The method of coordinates seems to be by its inception essentially metrical. Yet in dealing by this method with descriptive propositions we are eminently free from metrical considerations, because of our power to use general equations, and to avoid all assumption that measurements implied are any particular measurements.
12. It is worth while to illustrate this by the instance of the wellknown theorem of the radical centre of three circles. The theorem is that, given any three circles A, B, C (fig. 51), the common chords αα′, ββ′, γγ′ of the three pairs of circles meet in a point.
The geometrical proof is metrical throughout:—
Take O the point of intersection of αα′, ββ′, and joining this with γ′, suppose that γ′O does not pass through γ, but that it meets the circles A, B in two distinct points γ_{2}, γ_{1} respectively. We have then the known metrical property of intersecting chords of a circle; viz. in circle C, where αα′, ββ′, are chords meeting at a point O,
where, as well as in what immediately follows, Oα, &c. denote, of course, lengths or distances.
Similarly in circle A,
and in circle B,
Consequently Oγ_{1}·Oγ′ = Oγ_{2}·Oγ′, that is, Oγ_{1} = Oγ_{2}, or the points γ_{1} and γ_{2} coincide; that is, they each coincide with γ.
We contrast this with the analytical method:—
Here it only requires to be known that an equation Ax + By + C = 0 represents a line, and an equation x^{2} + y^{2} + Ax + By + C = 0 represents a circle. A, B, C have, in the two cases respectively, metrical significations; but these we are not concerned with. Using S to denote the function x^{2} + y^{2} + Ax + By + C, the equation of a circle is S = o. Let the equation of any other circle be S′, = x^{2} + y^{2} + A′x + B′y + C′ = 0; the equation SS′ = 0 is a linear equation (S − S′ is in fact = (A − A′)x + (B − B′)y + CC), and it thus represents a line; this equation is satisfied by the coordinates of each of the points of intersection of the two circles (for at each of these points S = 0 and S′ = 0, therefore also S − S′ = 0); hence the equation S − S′ = 0 is that of the line joining the two points of intersection of the two circles, or say it is the equation of the common chord of the two circles. Considering then a third circle S″, = x^{2} + y^{2} + A″x + B″y + C″ = 0, the equations of the common chords are S − S′ = 0, S − S″ = 0, S′ − S″ = 0 (each of these a linear equation); at the intersection of the first and second of these lines S = S′ and S = S″, therefore also S′ = S″, or the equation of the third line is satisfied by the coordinates of the point in question; that is, the three chords intersect in a point O, the coordinates of which are determined by the equations S = S′ = S″.
It further appears that if the two circles S = 0, S′ = 0 do not intersect in any real points, they must be regarded as intersecting in two imaginary points, such that the line joining them is the real line represented by the equation S − S′ = 0; or that two circles, whether their intersections be real or imaginary, have always a real common chord (or radical axis), and that for any three circles the common chords intersect in a point (of course real) which is the radical centre. And by this very theorem, given two circles with imaginary intersections, we can, by drawing circles which meet each of them in real points, construct the radical axis of the firstmentioned two circles.
13. The principle employed in showing that the equation of the common chord of two circles is S − S′ = 0 is one of very extensive application, and some more illustrations of it may be given.
Suppose S = 0, S′ = 0 are lines (that is, let S, S′ now denote linear functions Ax + By + C, A′x + B′y + C′), then S − kS′ = 0 (k an arbitrary constant) is the equation of any line passing through the point of intersection of the two given lines. Such a line may be made to pass through any given point, say the point (x_{0}, y_{0}); if S_{0}, S′_{0} are what S, S′ respectively become on writing for (x, y) the values (x_{0}, y_{0}), then the value of k is k = S_{0} ÷ S′_{0}. The equation in fact is SS′_{0} − S_{0}S′ = 0; and starting from this equation we at once verify it a posteriori; the equation is a linear equation satisfied by the values of (x, y) which make S = 0, S′ = 0; and satisfied also by the values (x_{0}, y_{0}); and it is thus the equation of the line in question.
If, as before, S = 0, S′ = 0 represent circles, then (k being arbitrary) S − kS′ = 0 is the equation of any circle passing through the two points of intersection of the two circles; and to make this pass through a given point (x_{0}, y_{0}) we have again k = S_{0} ÷ S′_{0}. In the particular case k = 1, the circle becomes the common chord (more accurately it becomes the common chord together with the line infinity; see § 23 below).
If S denote the general quadric function,
then the equation S = 0 represents a conic; assuming this, then, if S′ = 0 represents another conic, the equation S − kS′ = 0 represents any conic through the four points of intersection of the two conics.
Fig. 52. 
14. The object still being to illustrate the mode of working with coordinates for descriptive purposes, we consider the theorem of the polar of a point in regard to a circle. Given a circle and a point O (fig. 52), we draw through O any two lines meeting the circle in the points A, A′ and B, B′ respectively, and then taking Q as the intersection of the lines AB′ and A′B, the theorem is that the locus of the point Q is a right line depending only upon O and the circle, but independent of the particular lines OAA′ and OBB′.
Taking O as the origin, and for the axes any two lines through O at right angles to each other, the equation of the circle will be
and if the equation of the line OAA′ is taken to be y = mx, then the points A, A′ are found as the intersections of the straight line with the circle; or to determine x we have
If (x_{1}, y_{1}) are the coordinates of A, and (x_{2}, y_{2}) of A′, then the roots of this equation are x_{1}, x_{2}, whence easily
1  +  1  = −2  A + Bm  . 
x_{1}  x_{2}  C 
And similarly, if the equation of the line OBB′ is taken to be y = m′x_{1} and the coordinates of B, B′ to be (x_{3}, y_{3}) and (x_{4}, y_{4}) respectively, then
1  +  1  = −2  A + Bm′  . 
x_{3}  x_{4}  C′ 
We have then by § 8
x (y_{1} − y_{4}) − y (x_{1} − x_{4}) + x_{1}y_{4} − x_{4}y_{1} = 0, x (y_{2} − y_{3}) − y (x_{2} − x_{3}) + x_{2}y_{3} − x_{3}y_{2} = 0, 
as the equations of the lines AB′ and A′B respectively. Reducing by means of the relations y_{1} − mx_{1} = 0, y_{2} − mx_{2} = 0, y_{3} − m′x_{3} = 0, y_{4} − m′x_{4} = 0, the two equations become
x (mx_{1} − m′x_{4}) − y (x_{1} − x_{4}) + (m′ − m) x_{1}x_{4} = 0, x (mx_{2} − m′x_{3}) − y (x_{2} − x_{3}) + (m′ − m) x_{2}x_{3} = 0, 
and if we divide the first of these equations by x_{1}x_{4}, and the second by x_{2}x_{3} and then add, we obtain
x { m (  1  +  1  ) − m′ (  1  +  1  ) } − y {  1  +  1  − (  1  +  1  ) } + 2m′ − 2m = 0, 
x_{3}  x_{4}  x_{1}  x_{2}  x_{3}  x_{4}  x_{1}  x_{2} 
or, what is the same thing,
(  1  +  1  ) (y − m′x) − (  1  +  1  ) (y − mx) + 2m′ − 2m = 0, 
x_{1}  x_{2}  x_{3}  x_{4} 
which by what precedes is the equation of a line through the point Q. Substituting herein for 1/x_{1} + 1/x_{2}, 1/x_{3} + 1/x_{4} their foregoing values, the equation becomes
that is,
or finally it is Ax + By + C = 0, showing that the point Q lies in a line the position of which is independent of the particular lines OAA′, OBB′ used in the construction. It is proper to notice that there is no correspondence to each other of the points A, A′ and B, B′; the grouping might as well have been A, A′ and B′, B; and it thence appears that the line Ax + By + C = 0 just obtained is in fact the line joining the point Q with the point R which is the intersection of AB and A′B′.
15. In § 8 it has been seen that two conditions determine the equation of a straight line, because in Ax + By + C = 0 one of the coefficients may be divided out, leaving only two parameters to be determined. Similarly five conditions instead of six determine an equation of the second degree (a, b, c, f, g, h)(x, y, 1)^{2} = 0, and nine instead of ten determine a cubic (*)(x, y, 1)^{3} = 0. It thus appears that a cubic can be made to pass through 9 given points, and that the cubic so passing through 9 given points is completely determined. There is, however, a remarkable exception. Considering two given cubic curves S = 0, S′ = 0, these intersect in 9 points, and through these 9 points we have the whole series of cubics S − kS′ = 0, where k is an arbitrary constant: k may be determined so that the cubic shall pass through a given tenth point (k = S_{0} ÷ S′_{0}, if the coordinates are (x_{0}, y_{0}), and S_{0}, S′_{0} denote the corresponding values of S, S′). The resulting curve SS′_{0} − S′S_{0} = 0 may be regarded as the cubic determined by the conditions of passing through 8 of the 9 points and through the given point (x_{0}, y_{0}); and from the equation it thence appears that the curve passes through the remaining one of the 9 points. In other words, we thus have the theorem, any cubic curve which passes through 8 of the 9 intersections of two given cubic curves passes through the 9th intersection.
The applications of this theorem are very numerous; for instance, we derive from it Pascal’s theorem of the inscribed hexagon. Consider a hexagon inscribed in a conic. The three alternate sides constitute a cubic, and the other three alternate sides another cubic. The cubics intersect in 9 points, being the 6 vertices of the hexagon, and the 3 Pascalian points, or intersections of the pairs of opposite sides of the hexagon. Drawing a line through two of the Pascalian points, the conic and this line constitute a cubic passing through 8 of the 9 points of intersection, and it therefore passes through the remaining point of intersection—that is, the third Pascalian point; and since obviously this does not lie on the conic, it must lie on the line—that is, we have the theorem that the three Pascalian points (or points of intersection of the pairs of opposite sides) lie on a line.
16. Metrical Theory resumed. Projections and Perpendiculars.—It is a metrical fact of fundamental importance, already used in § 8, that, if a finite line PQ be projected on any other line OO′ by perpendiculars PP′, QQ′ to OO′, the length of the projection P′Q′ is equal to that of PQ multiplied by the cosine of the acute angle between the two lines. Also the algebraical sum of the projections of the sides of any closed polygon upon any line is zero, because as a point goes round the polygon, from any vertex A to A again, the point which is its projection on the line passes from A′ the projection of A to A′ again, i.e. traverses equal distances along the line in positive and negative senses. If we consider the polygon as consisting of two broken lines, each extending from the same initial to the same terminal point, the sum of the projections of the lines which compose the one is equal, in sign and magnitude, to the sum of the projections of the lines composing the other. Observe that the projection on a line of a length perpendicular to the line is zero.
Let us hence find the equation of a straight line such that the perpendicular OD on it from the origin is of length ρ taken as positive, and is inclined to the axis of x at an angle xOD = α, measured counterclockwise from Ox. Take any point P(x, y) on the line, and construct OM and MP as in fig. 48. The sum of the projections of OM and MP on OD is OD itself; and this gives the equation of the line
Observe that cos α and sin α here are the sin α and −cos α, or the −sin α and cos α of § 8 according to circumstances.
We can write down an expression for the perpendicular distance from this line of any point (x′, y ′) which does not lie upon it. If the parallel through (x′, y ′) to the line meet OD in E, we have x′ cos α + y ′ sin α = OE, and the perpendicular distance required is OD − OE, i.e. ρ − x′ cos α − y ′ sin α; it is the perpendicular distance taken positively or negatively according as (x′, y ′) lies on the same side of the line as the origin or not.
The general equation Ax + By + C = 0 may be given the form x cos α + y sin α − ρ = 0 by dividing it by √(A^{2} + B^{3}). Thus (Ax′ + By ′ + C) ÷ √(A^{2} + B^{2}) is in absolute value the perpendicular distance of (x′, y ′) from the line Ax + By + C = 0. Remember, however, that there is an essential ambiguity of sign attached to a square root. The expression found gives the distance taken positively when (x′, y ′) is on the origin side of the line, if the sign of C is given to √(A^{2} + B^{2}).
17. Transformation of Coordinates.—We often need to adopt new axes of reference in place of old ones; and the above principle of projections readily expresses the old coordinates of any point in terms of the new.
Fig. 53. 
Suppose, for instance, that we want to take for new origin the point O′ of old coordinates OA = h, AO′ = k, and for new axes of X and Y lines through O′ obtained by rotating parallels to the old axes of x and y through an angle θ counterclockwise. Construct (fig. 53) the old and new coordinates of any point P. Expressing that the projections, first on the old axis of x and secondly on the old axis of y, of OP are equal to the sums of the projections, on those axes respectively, of the parts of the broken line OO′M′P, we obtain:
and
Be careful to observe that these formulae do not apply to every conceivable change of reference from one set of rectangular axes to another. It might have been required to take O′X, O′Y′ for the positive directions of the new axes, so that the change of directions of the axes could not be effected by rotation. We must then write −Y for Y in the above.
Were the new axes oblique, making angles α, β respectively with the old axis of x, and so inclined at the angle β − α, the same method would give the formulae
18. The Conic Sections.—The conics, as they are now called, were at first defined as curves of intersection of planes and a cone; but Apollonius substituted a definition free from reference to space of three dimensions. This, in effect, is that a conic is the locus of a point the distance of which from a given point, called the focus, has a given ratio to its distance from a given line, called the directrix (see Conic Section). If e : 1 is the ratio, e is called the eccentricity. The distances are considered signless.
Take (h, k) for the focus, and x cos α + y sin α − p = 0 for the directrix. The absolute values of √{(x − h)^{2} + (y − k)^{2}} and p − x cos α − y sin α are to have the ratio e : 1; and this gives
as the general equation, in rectangular coordinates, of a conic.
It is of the second degree, and is the general equation of that degree. If, in fact, we multiply it by an unknown λ, we can, by solving six simultaneous equations in the six unknowns λ, h, k, e, p, α, so choose values for these as to make the coefficients in the equation equal to those in any equation of the second degree which may be given. There is no failure of this statement in the special case when the given equation represents two straight lines, as in § 10, but there is speciality: if the two lines intersect, the intersection and either bisector of the angle between them are a focus and directrix; if they are united in one line, any point on the line and a perpendicular to it through the point are: if they are parallel, the case is a limiting one in which e and h^{2} + k^{2} have become infinite while e^{−2}(h^{2} + k^{2}) remains finite. In the case (§ 9) of an equation such as represents a circle there is another instance of proceeding to a limit: e has to become 0, while ep remains finite: moreover α is indeterminate. The centre of a circle is its focus, and its directrix has gone to infinity, having no special direction. This last fact illustrates the necessity, which is also forced on plane geometry by threedimensional considerations, of treating all points at infinity in a plane as lying on a single straight line.
Sometimes, in reducing an equation to the above focus and directrix form, we find for h, k, e, p, tan α, or some of them, only imaginary values, as quadratic equations have to be solved; and we have in fact to contemplate the existence of entirely imaginary conics. For instance, no real values of x and y satisfy x^{2} + 2y^{2} + 3 = 0. Even when the locus represented is real, we obtain, as a rule, four sets of values of h, k, e, p, of which two sets are imaginary; a real conic has, besides two real foci and corresponding directrices, two others that are imaginary.
In oblique as well as rectangular coordinates equations of the second degree represent conics.
19. The three Species of Conics.—A real conic, which does not degenerate into straight lines, is called an ellipse, parabola or hyperbola according as e <, = , or > 1. To trace the three forms it is best so to choose the axes of reference as to simplify their equations.
In the case of a parabola, let 2c be the distance between the given focus and directrix, and take axes referred to which these are the point (c, 0) and the line x = − c. The equation becomes (x − c)^{2} + y^{2} = (x + c)^{2}, i.e. y^{2} = 4cx.
In the other cases, take a such that a(e ~ e^{−1}) is the distance of focus from directrix, and so choose axes that these are (ae, 0) and x = ae^{−1}, thus getting the equation(x − ae)^{2} + y^{2} = e^{2}(x − ae^{−1})^{2}, i.e. (1 − e^{2})x^{2} + y^{2} = a^{2}(1 − e^{2}). When e < 1, i.e. in the case of an ellipse, this may be written x^{2}/a^{2} + y^{2}/b^{2} = 1, where b^{2} = a^{2}(1 − e^{2}); and when e > 1, i.e. in the case of an hyperbola, x^{2}/a^{2} − y^{2}/b^{2} = 1, where b^{2} = a^{2}(e^{2} − 1). The axes thus chosen for the ellipse and hyperbola are called the principal axes.
In figs. 54, 55, 56 in order, conics of the three species, thus referred, are depicted.
Fig. 54  Fig. 55 
Fig. 56. 
The oblique straight lines in fig. 56 are the asymptotes x/a = ±y/b of the hyperbola, lines to which the curve tends with unlimited closeness as it goes to infinity. The hyperbola would have an equation of the form xy = c if referred to its asymptotes as axes, the coordinates being then oblique, unless a = b, in which case the hyperbola is called rectangular. An ellipse has two imaginary asymptotes. In particular a circle x^{2} + y^{2} = a^{2}, a particular ellipse, has for asymptotes the imaginary lines x = ±y √−1. These run from the centre to the socalled circular points at infinity.
20. Tangents and Curvature.—Let (x′, y ′) and (x′ + h, y ′ + k) be two neighbouring points P, P′ on a curve. The equation of the line on which both lie is h(y − y ′) = k(x − x′). Now keep P fixed, and let P′ move towards coincidence with it along the curve. The connecting line will tend towards a limiting position, to which it can never attain as long as P and P′ are distinct. The line which occupies this limiting position is the tangent at P. Now if we subtract the equation of the curve, with (x′, y ′) for the coordinates in it, from the like equation in (x′ + h, y ′ + k), we obtain a relation in h and k, which will, as a rule, be of the form 0 = Ah + Bk + terms of higher degrees in h and k, where A, B and the other coefficients involve x′ and y ′. This gives k/h = −A/B + terms which tend to vanish as h and k do, so that −A : B is the limiting value tended to by k : h. Hence the equation of the tangent is B(y − y ′) + A(x − x′) = 0.
The normal at (x′, y ′) is the line through it at right angles to the tangent, and its equation is A(y − y ′) − B(x − x′) = 0.
In the case of the conic (a, b, c, f, g, h) (x, y, 1)^{2} = 0 we find that A/B = (ax′ + hy ′ + g)/(hx′ + by ′ + f).
We can obtain the coordinates of Q, the intersection of the normals QP, QP′ at (x′, y ′) and (x′ + h, y ′ + k), and then, using the limiting value of k : h, deduce those of its limiting position as P′ moves up to P. This is the centre of curvature of the curve at P (x′, y ′), and is so called because it is the centre of the circle of closest contact with the curve at that point. That it is so follows from the facts that the closest circle is the limit tended to by the circle which touches the curve at P and passes through P′, and that the arc from P to P′ of this circle lies between the circles of centre Q and radii QP, QP′, which circles tend, not to different limits as P′ moves up to P, but to one. The distance from P to the centre of curvature is the radius of curvature.
21. Differential Plane Geometry.—The language and notation of the differential calculus are very useful in the study of tangents and curvature. Denoting by (ξ, η) the current coordinates, we find, as above, that the tangent at a point (x, y) of a curve is η − y = (ξ − x)dy/dx, where dy/dx is found from the equation of the curve. If this be ƒ(x, y) = 0 the tangent is (ξ − x) (∂f/∂x) + (η − y) (∂f/∂y) = 0. If ρ and (α, β) are the radius and centre of curvature at (x, y), we find that q(α − x) = −p(1 + p^{2}), q(β − y) = 1 + p^{2}, q^{2}ρ^{2} = (1 + p^{2})^{3}, where p, q denote dy/dx, d^{2}y/dx^{2} respectively. (See Infinitesimal Calculus.)
In any given case we can, at all events in theory, eliminate x, y between the above equations for α − x and β − y, and the equation of the curve. The resulting equation in (α, β) represents the locus of the centre of curvature. This is the evolute of the curve.
22. Polar Coordinates.—In plane geometry the distance of any point P from a fixed origin (or pole) O, and the inclination xOP of OP to a fixed line Ox, determine the point: r, the numerical measure of OP, the radius vector, and θ, the circular measure of xOP, the inclination, are called polar coordinates of P. The formulae x = r cos θ, y = r sin θ connect Cartesian and polar coordinates, and make transition from either system to the other easy. In polar coordinates the equations of a circle through O, and of a conic with O as focus, take the simple forms r = 2a cos (θ − α), r{1 − e cos (θ − α)} = l. The use of polar coordinates is very convenient in discussing curves which have properties of symmetry akin to that of a regular polygon, such curves for instance as r = a cos m θ, with m integral, and also the curves called spirals, which have equations giving r as functions of θ itself, and not merely of sin θ and cos θ. In the geometry of motion under central forces the advantage of working with polar coordinates is great.
23. Trilinear and Areal Coordinates.—Consider a fixed triangle ABC, and regard its sides as produced without limit. Denote, as in trigonometry, by a, b, c the positive numbers of units of a chosen scale contained in the lengths BC, CA, AB, by A, B, C the angles, and by Δ the area, of the triangle. We might, as in § 6, take CA, CB as axes of x and y, inclined at an angle C. Any point P (x, y) in the plane is at perpendicular distances y sin C and x sin C from CA and CB. Call these β and α respectively. The signs of β and α are those of y and x, i.e. β is positive or negative according as P lies on the same side of CA as B does or the opposite, and similarly for α. An equation in (x, y) of any degree may, upon replacing in it x and y by α cosec C and β cosec C, be written as one of the same degree in (α, β). Now let γ be the perpendicular distance of P from the third side AB, taken as positive or negative as P is on the C side of AB or not. The geometry of the figure tells us that aα + bβ + cγ = 2Δ. By means of this relation in α, β, γ we can give an equation considered countless other forms, involving two or all of α, β, γ. In particular we may make it homogeneous in α, β, γ: to do this we have only to multiply the terms of every degree less than the highest present in the equation by a power of (aα + bβ + cγ)/2Δ just sufficient to raise them, in each case, to the highest degree.
We call (α, β, γ) trilinear coordinates, and an equation in them the trilinear equation of the locus represented. Trilinear equations are, as a rule, dealt with in their homogeneous forms. An advantage thus gained is that we need not mean by (α, β, γ) the actual measures of the perpendicular distances, but any properly signed numbers which have the same ratio two and two as these distances.
In place of α, β, γ it is lawful to use, as coordinates specifying the position of a point in the plane of a triangle of reference ABC, any given multiples of these. For instance, we may use x = aα/2Δ, y = bβ/2Δ, z = cγ/2Δ, the properly signed ratios of the triangular areas PBC, PCA, PAB to the triangular area ABC. These are called the areal coordinates of P. In areal coordinates the relation which enables us to make any equation homogeneous takes the simple form x + y + z = 1; and, as before, we need mean by x, y, z, in a homogeneous equation, only signed numbers in the right ratios.
Straight lines and conics are represented in trilinear and in areal, because in Cartesian, coordinates by equations of the first and second degrees respectively, and these degrees are preserved when the equations are made homogeneous. What must be said about points infinitely far off in order to make universal the statement, to which there is no exception as long as finite distances alone are considered, that every homogeneous equation of the first degree represents a straight line? Let the point of areal coordinates (x′, y ′, z′) move infinitely far off, and mean by x, y, z finite quantities in the ratios which x′, y ′, z′ tend to assume as they become infinite. The relation x′ + y ′ + z′ = 1 gives that the limiting state of things tended to is expressed by x + y + z = 0. This particular equation of the first degree is satisfied by no point at a finite distance; but we see the propriety of saying that it has to be taken as satisfied by all the points conceived of as actually at infinity. Accordingly the special property of these points is expressed by saying that they lie on a special straight line, of which the areal equation is x + y + z = 0. In trilinear coordinates this line at infinity has for equation aα + bβ + cγ = 0.
On the one special line at infinity parallel lines are treated as meeting. There are on it two special (imaginary) points, the circular points at infinity of § 19, through which all circles pass in the same sense. In fact if S = O be one circle, in areal coordinates, S + (x + y + z)(lx + my + nz) = 0 may, by proper choice of l, m, n, be made any other; since the added terms are once lx + my + nz, and have the generality of any expression like a′x + b′y + c′ in Cartesian coordinates. Now these two circles intersect in the two points where either meets x + y + z = 0 as well as in two points on the radical axis lx + my + nz = 0.
24. Let us consider the perpendicular distance of a point (α′, β′, γ′) from a line lα + mβ + nγ. We can take rectangular axes of Cartesian coordinates (for clearness as to equalities of angle it is best to choose an origin inside ABC), and refer to them, by putting expressions p − x cos θ − y sin θ, &c., for α &c.; we can then apply § 16 to get the perpendicular distance; and finally revert to the trilinear notation. The result is to find that the required distance is
where {l, m, n}^{2} = l^{2} + m^{2} + n^{2} − 2mn cos A − 2nl cos B − 2lm cos C.
In areal coordinates the perpendicular distance from (x′, y ′, z′) to lx + my + nz = 0 is 2Δ(lx′ + my ′ + nz′)/{al, bm, cn}. In both cases the coordinates are of course actual values.
Now let ξ, η, ζ be the perpendiculars on the line from the vertices A, B, C, i.e. the points (1, 0, 0), (0, 1, 0), (0, 0, 1), with signs in accord with a convention that oppositeness of sign implies distinction between one side of the line and the other. Three applications of the result above give
and we thus have the important fact that ξx′ + ηy ′ + ζz′ is the perpendicular distance between a point of areal coordinates (x′y ′z′) and a line on which the perpendiculars from A, B, C are ξ, η, ζ respectively. We have also that ξx + ηy + ζz = 0 is the areal equation of the line on which the perpendiculars are ξ, η, ζ; and, by equating the two expressions for the perpendiculars from (x′, y ′, z′) on the line, that in all cases {aξ, bη, cζ}^{2} = 4Δ^{2}.
25. Linecoordinates. Duality.—A quite different order of ideas may be followed in applying analysis to geometry. The notion of a straight line specified may precede that of a point, and points may be dealt with as the intersections of lines. The specification of a line may be by means of coordinates, and that of a point by an equation, satisfied by the coordinates of lines which pass through it. Systems of linecoordinates will here be only briefly considered. Every such system is allied to some system of pointcoordinates; and space will be saved by giving prominence to this fact, and not recommencing ab initio.
Suppose that any particular system of pointcoordinates, in which lx + my + nz = 0 may represent any straight line, is before us: notice that not only are trilinear and areal coordinates such systems, but Cartesian coordinates also, since we may write x/z, y/z for the Cartesian x, y, and multiply through by z. The line is exactly assigned if l, m, n, or their mutual ratios, are known. Call (l, m, n) the coordinates of the line. Now keep x, y, z constant, and let the coordinates of the line vary, but always so as to satisfy the equation. This equation, which we now write xl + ym + zn = 0, is satisfied by the coordinates of every line through a certain fixed point, and by those of no other line; it is the equation of that point in the linecoordinates l, m, n.
Linecoordinates are also called tangential coordinates. A curve is the envelope of lines which touch it, as well as the locus of points which lie on it. A homogeneous equation of degree above the first in l, m, n is a relation connecting the coordinates of every line which touches some curve, and represents that curve, regarded as an envelope. For instance, the condition that the line of coordinates (l, m, n), i.e. the line of which the allied pointcoordinate equation is lx + my + nz = 0, may touch a conic (a, b, c, f, g, h) (x, y, z)^{2} = 0, is readily found to be of the form (A, B, C, F, G, H) (l, m, n)^{2} = 0, i.e. to be of the second degree in the linecoordinates. It is not hard to show that the general equation of the second degree in l, m, n thus represents a conic; but the degenerate conics of linecoordinates are not linepairs, as in pointcoordinates, but pointpairs.
The degree of the pointcoordinate equation of a curve is the order of the curve, the number of points in which it cuts a straight line. That of the linecoordinate equation is its class, the number of tangents to it from a point. The order and class of a curve are generally different when either exceeds two.
26. The system of linecoordinates allied to the areal system of pointcoordinates has special interest.
The l, m, n of this system are the perpendiculars ξ, η, ζ of § 24; and x′ξ + y ′η + z′ζ = 0 is the equation of the point of areal coordinates (x′, y ′, z′), i.e. is a relation which the perpendiculars from the vertices of the triangle of reference on every line through the point, but no other line, satisfy. Notice that a nonhomogeneous equation of the first degree in ξ, η, ζ does not, as a homogeneous one does, represent a point, but a circle. In fact x′ξ + y ′η + z′ζ = R expresses the constancy of the perpendicular distance of the fixed point x′ξ + y ′η + z′ζ = 0 from the variable line (ξ, η, ζ), i.e. the fact that (ξ, η, ζ) touches a circle with the fixed point for centre. The relation in any ξ, η, ζ which enables us to make an equation homogeneous is not linear, as in pointcoordinates, but quadratic, viz. it is the relation {aξ, bη, cζ}^{2} = 4Δ^{2} of § 24. Accordingly the homogeneous equation of the above circle is
Every circle has an equation of this form in the present system of linecoordinates. Notice that the equation of any circle is satisfied by those coordinates of lines which satisfy both x′ξ + y ′η + z′ζ = 0, the equation of its centre, and {aξ, bη, cζ}^{2} = 0. This last equation, of which the lefthand side satisfies the condition for breaking up into two factors, represents the two imaginary circular points at infinity, through which all circles and their asymptotes pass.
There is strict duality in descriptive geometry between pointlinelocus and linepointenvelope theorems. But in metrical geometry duality is encumbered by the fact that there is in a plane one special line only, associated with distance, while of special points, associated with direction, there are two: moreover the line is real, and the points both imaginary.
II. Solid Analytical Geometry.
27. Any point in space may be specified by three coordinates. We consider three fixed planes of reference, and generally, as in all that follows, three which are at right angles two and two. They intersect, two and two, in lines x′Ox, y ′Oy, z′Oz, called the axes of x, y, z respectively, and divide all space into eight parts called octants. If from any point P in space we draw PN parallel to zOz′ to meet the plane xOy in N, and then from N draw NM parallel to yOy ′ to meet x′Ox in M, the coordinates (x, y, z) of P are the numerical measures of OM, MN, NP; in the case of rectangular coordinates these are the perpendicular distances of P from the three planes of reference. The sign of each coordinate is positive or negative as P lies on one side or the other of the corresponding plane. In the octant delineated the signs are taken all positive.
Fig. 57.  Fig. 58. 
In fig. 57 the delineation is on a plane of the paper taken parallel to the plane zOx, the points of a solid figure being projected on that plane by parallels to some chosen line through O in the positive octant. Sometimes it is clearer to delineate, as in fig. 58, by projection parallel to that line in the octant which is equally inclined to Ox, Oy, Oz upon a plane of the paper perpendicular to it. It is possible by parallel projection to delineate equal scales along Ox, Oy, Oz by scales having any ratios we like along lines in a plane having any mutual inclinations we like.
Fig. 59. 
For the delineation of a surface of simple form it frequently suffices to delineate the sections by the coordinate planes; and, in particular, when the surface has symmetry about each coordinate plane, to delineate the quartersections belonging to a single octant. Thus fig. 59 conveniently represents an octant of the wave surface, which cuts each coordinate plane in a circle and an ellipse. Or we may delineate a series of contour lines, i.e. sections by planes parallel to xOy, or some other chosen plane; of course other sections may be indicated too for greater clearness. For the delineation of a curve a good method is to represent, as above, a series of points P thereof, each accompanied by its ordinate PN, which serves to refer it to the plane of xy. The employment of stereographic projection is also interesting.
28. In plane geometry, reckoning the line as a curve of the first order, we have only the point and the curve. In solid geometry, reckoning a line as a curve of the first order, and the plane as a surface of the first order, we have the point, the curve and the surface; but the increase of complexity is far greater than would hence at first sight appear. In plane geometry a curve is considered in connexion with lines (its tangents); but in solid geometry the curve is considered in connexion with lines and planes (its tangents and osculating planes), and the surface also in connexion with lines and planes (its tangent lines and tangent planes); there are surfaces arising out of the line—cones, skew surfaces, developables, doubly and triply infinite systems of lines, and whole classes of theories which have nothing analogous to them in plane geometry: it is thus a very small part indeed of the subject which can be even referred to in the present article.
In the case of a surface we have between the coordinates (x, y, z) a single, or say a onefold relation, which can be represented by a single relation ƒ(x, y, z) = 0; or we may consider the coordinates expressed each of them as a given function of two variable parameters p, q; the form z = ƒ(x, y) is a particular case of each of these modes of representation; in other words, we have in the first mode ƒ(x, y, z) = z − ƒ(x, y), and in the second mode x = p, y = q for the expression of two of the coordinates in terms of the parameters.
In the case of a curve we have between the coordinates (x, y, z) a twofold relation: two equations ƒ(x, y, z) = 0, φ(x, y, z) = 0 give such a relation; i.e. the curve is here considered as the intersection of two surfaces (but the curve is not always the complete intersection of two surfaces, and there are hence difficulties); or, again, the coordinates may be given each of them as a function of a single variable parameter. The form y = φ(x), z = ψ(x), where two of the coordinates are given in terms of the third, is a particular case of each of these modes of representation.
29. The remarks under plane geometry as to descriptive and metrical propositions, and as to the nonmetrical character of the method of coordinates when used for the proof of a descriptive proposition, apply also to solid geometry; and they might be illustrated in like manner by the instance of the theorem of the radical centre of four spheres. The proof is obtained from the consideration that S and S′ being each of them a function of the form x^{2} + y^{2} + z^{2} + ax + by + cz + d, the difference S−S′ is a mere linear function of the coordinates, and consequently that S−S′ = 0 is the equation of the plane containing the circle of intersection of the two spheres S = 0 and S′ = 0.
Fig. 60. 
30. Metrical Theory.—The foundation in solid geometry of the metrical theory is in fact the beforementioned theorem that if a finite right line PQ be projected upon any other line OO′ by lines perpendicular to OO′, then the length of the projection P′Q′ is equal to the length of PQ into the cosine of its inclination to P′Q′—or (in the form in which it is now convenient to state the theorem) the perpendicular distance P′Q′ of two parallel planes is equal to the inclined distance PQ into the cosine of the inclination. The principle of § 16, that the algebraical sum of the projections of the sides of any closed polygon on any line is zero, or that the two sets of sides of the polygon which connect a vertex A and a vertex B have the same sum of projections on the line, in sign and magnitude, as we pass from A to B, is applicable when the sides do not all lie in one plane.
31. Consider the skew quadrilateral QMNP, the sides QM, MN, NP being respectively parallel to the three rectangular axes Ox, Oy, Oz; let the lengths of these sides be ξ, η, ζ, and that of the side QP be = ρ; and let the cosines of the inclinations (or say the cosineinclinations) of ρ to the three axes be α, β, γ; then projecting successively on the three sides and on QP we have
and
whence ρ^{2} = ξ^{2} + η^{2} + ζ^{2}, which is the relation between a distance ρ and its projections ξ, η, ζ upon three rectangular axes. And from the same equations we obtain α^{2} + β^{2} + γ^{2} = 1, which is a relation connecting the cosineinclinations of a line to three rectangular axes.
Suppose we have through Q any other line QT, and let the cosineinclinations of this to the axes be α′, β′, γ′, and δ be its cosineinclination to QP; also let ρ be the length of the projection of QP upon QT; then projecting on QT we have
And in the last equation substituting for ξ, η, ζ their values ρα, ρβ, ργ we find
which is an expression for the mutual cosineinclination of two lines, the cosineinclinations of which to the axes are α, β, γ and α′, β′, γ′ respectively. We have of course α^{2} + β^{2} + γ^{2} = 1 and α′^{2} + β′^{2} + γ′^{2} = 1; and hence also
= (βγ′ − β′γ)^{2} + (γα′ − γ′α)^{2} + (αβ′ − α′β)^{2};
so that the sine of the inclination can only be expressed as a square root. These formulae are the foundation of spherical trigonometry.
32. Straight Lines, Planes and Spheres.—The foregoing formulae give at once the equations of these loci.
For first, taking Q to be a fixed point, coordinates (a, b, c), and the cosineinclinations (α, β, γ) to be constant, then P will be a point in the line through Q in the direction thus determined; or, taking (x, y, z) for its coordinates, these will be the current coordinates of a point in the line. The values of ξ, η, ζ then are x − a, y − b, z − c, and we thus have
x − a  =  y − b  =  z − c  (= ρ), 
α  β  γ 
which (omitting the last equation, = ρ) are the equations of the line through the point (a, b, c), the cosineinclinations to the axes being α, β, γ, and these quantities being connected by the relation α^{2} + β^{2} + γ^{2} = 1. This equation may be omitted, and then α, β, γ, instead of being equal, will only be proportional, to the cosineinclinations.
Using the last equation, and writing
these are expressions for the current coordinates in terms of a parameter ρ, which is in fact the distance from the fixed point (a, b, c).
It is easy to see that, if the coordinates (x, y, z) are connected by any two linear equations, these equations can always be brought into the foregoing form, and hence that the two linear equations represent a line.
Secondly, taking for greater simplicity the point Q to be coincident with the origin, and α′, β′, γ′, p to be constant, then p is the perpendicular distance of a plane from the origin, and α′, β′, γ′ are the cosineinclinations of this distance to the axes (α′^{2} + β′^{2} + γ′^{2} = 1). P is any point in this plane, and taking its coordinates to be (x, y, z) then (ξ, η, ζ) are = (x, y, z), and the foregoing equation p = α′ξ + β′η + γ′ζ becomes
which is the equation of the plane in question.
If, more generally, Q is not coincident with the origin, then, taking its coordinates to be (a, b, c), and writing p_{1} instead of p, the equation is
and we thence have p_{1} = p − (aα′ + bβ′ + cγ′), which is an expression for the perpendicular distance of the point (a, b, c) from the plane in question.
It is obvious that any linear equation Ax + By + Cz + D = O between the coordinates can always be brought into the foregoing form, and hence that such an equation represents a plane.
Thirdly, supposing Q to be a fixed point, coordinates (a, b, c), and the distance QP = ρ, to be constant, say this is = d, then, as before, the values of ξ, η, ζ are x − a, y − b, z − c, and the equation ξ^{2} + η^{2} + ζ^{2} = ρ^{2} becomes
which is the equation of the sphere, coordinates of the centre = (a, b, c), and radius = d.
A quadric equation wherein the terms of the second order are x^{2} + y^{2} + z^{2}, viz. an equation
can always, it is clear, be brought into the foregoing form; and it thus appears that this is the equation of a sphere, coordinates of the centre −12A, −12B, −12C, and squared radius = 14(A^{2} + B^{2} + C^{2}) − D.
33. Cylinders, Cones, ruled Surfaces.—If the two equations of a straight line involve a parameter to which any value may be given, we have a singly infinite system of lines. They cover a surface, and the equation of the surface is obtained by eliminating the parameter between the two equations.
If the lines all pass through a given point, then the surface is a cone; and, in particular, if the lines are all parallel to a given line, then the surface is a cylinder.
Beginning with this last case, suppose the lines are parallel to the line x = mz, y = nz, the equations of a line of the system are x = mz + a, y = nz + b,—where a, b are supposed to be functions of the variable parameter, or, what is the same thing, there is between them a relation ƒ(a, b) = 0: we have a = x − mz, b = y − nz, and the result of the elimination of the parameter therefore is ƒ(x − mz, y − nz) = 0, which is thus the general equation of the cylinder the generating lines whereof are parallel to the line x = mz, y = nz. The equation of the section by the plane z = 0 is ƒ(x, y) = 0, and conversely if the cylinder be determined by means of its curve of intersection with the plane z = 0, then, taking the equation of this curve to be ƒ(x, y) = 0, the equation of the cylinder is ƒ(x − mz, y − nz) = 0. Thus, if the curve of intersection be the circle (x − α)^{2} + (y − β)^{2} = γ^{2}, we have (x − mz − α)^{2} + (y − nz − β)^{2} = γ^{2} as the equation of an oblique cylinder on this base, and thus also (x − α)^{2} + (y − β)^{2} = γ^{2} as the equation of the right cylinder.
If the lines all pass through a given point (a, b, c), then the equations of a line are x − a = α(z − c), y − b = β(z − c), where α, β are functions of the variable parameter, or, what is the same thing, there exists between them an equation ƒ(α, β) = 0; the elimination of the parameter gives, therefore, ƒ(x − az − c, y − bz − c) = 0; and this equation, or, what is the same thing, any homogeneous equation ƒ(x − a, y − b, z − c) = 0, or, taking ƒ to be a rational and integral function of the order n, say (*) (x − a, y − b, z − c)^{n} = 0, is the general equation of the cone having the point (a, b, c) for its vertex. Taking the vertex to be at the origin, the equation is (*) (x, y, z)^{n} = 0; and, in particular, (*) (x, y, z)^{2} = 0 is the equation of a cone of the second order, or quadricone, having the origin for its vertex.
34. In the general case of a singly infinite system of lines, the locus is a ruled surface (or regulus). Now, when a line is changing its position in space, it may be looked upon as in a state of turning about some point in itself, while that point is, as a rule, in a state of moving out of the plane in which the turning takes place. If instantaneously it is only in a state of turning, it is usual, though not strictly accurate, to say that it intersects its consecutive position. A regulus such that consecutive lines on it do not intersect, in this sense, is called a skew surface, or scroll; one on which they do is called a developable surface or torse.
Suppose, for instance, that the equations of a line (depending on the variable parameter θ) are xa + yc = θ (1 + yb), xa − zc = 1θ(1 − yb); then, eliminating θ we have x^{2}a^{2} − z^{2}c^{2} = 1 − y^{2}b^{2}, or say, x^{2}a^{2} + y^{2}b^{2} − z^{2}c^{2} = 1, the equation of a quadric surface, afterwards called the hyperboloid of one sheet; this surface is consequently a scroll. It is to be remarked that we have upon the surface a second singly infinite series of lines; the equations of a line of this second system (depending on the variable parameter φ) are
x  +  z  = φ ( 1 −  y  ),  x  −  z  =  1  ( 1 +  y  ). 
a  c  b  a  c  φ  b 
It is easily shown that any line of the one system intersects every line of the other system.
Considering any curve (of double curvature) whatever, the tangent lines of the curve form a singly infinite system of lines, each line intersecting the consecutive line of the system,—that is, they form a developable, or torse; the curve and torse are thus inseparably connected together, forming a single geometrical figure. An osculating plane of the curve (see § 38 below) is a tangent plane of the torse all along a generating line.
35. Transformation of Coordinates.—There is no difficulty in changing the origin, and it is for brevity assumed that the origin remains unaltered. We have, then, two sets of rectangular axes, Ox, Oy, Oz, and Ox_{1}, Oy_{1}, Ozx_{1}, the mutual cosineinclinations being shown by the diagram—
x  y  z  
x_{1}  α  β  γ 
y_{1}  α′  β′  γ′ 
z_{1}  α″  β″  γ″ 
that is, α, β, γ are the cosineinclinations of Ox_{1} to Ox, Oy, Oz; α′, β′, γ′ those of Oy_{1}, &c.
And this diagram gives also the linear expressions of the coordinates (x_{1}, y_{1}, z_{1}) or (x, y, z) of either set in terms of those of the other set; we thus have
x_{1} = α x + β y + γ z,  x = αx_{1} + α′y_{1} + α″z_{1}, 
y_{1} = α′x + β′y + γ′z,  y = βx_{1} + β′y_{1} + β″z_{1}, 
z_{1} = α″x + β″y + γ″z,  z = γx_{1} + γ′y_{1} + γ″z_{1}, 
which are obtained by projection, as above explained. Each of these equations is, in fact, nothing else than the beforementioned equation p = α′ξ + β′η + γ′ζ, adapted to the problem in hand.
But we have to consider the relations between the nine coefficients. By what precedes, or by the consideration that we must have identically x^{2} + y^{2} + z^{2} = x_{1}^{2} + y_{1}^{2} + z_{1}^{2}, it appears that these satisfy the relations—
α^{2}  + β^{2}  + γ^{2}  = 1,  α^{2} +  α′^{2}  + α″^{2}  = 1, 
α′^{2}  + β′^{2}  + γ′^{2}  = 1,  β^{2}  + β′^{2}  + β″^{2}  = 1, 
α″^{2}  + β″^{2}  + γ″^{2}  = 1,  γ^{2}  + γ′^{2}  + γ″^{2}  = 1, 
α′a″  + β′β″  + γ′γ″  = 0,  βγ  +β′γ′  + β″γ″  = 0, 
α″α  + β″β  + γ″γ  = 0,  γα  + γ′α′  + γ″α″  = 0, 
αα′  + ββ′  + γγ′  = 0,  αβ  +α′β′  + α″β″  = 0, 
either set of six equations being implied in the other set.
It follows that the square of the determinant
α,  β,  γ 
α′,  β′,  γ′ 
α″,  β″,  γ″ 
is = 1; and hence that the determinant itself is = ±1. The distinction of the two cases is an important one: if the determinant is = + 1, then the axes Ox_{1}, Oy_{1}, Oz_{1} are such that they can by a rotation about O be brought to coincide with Ox, Oy, Oz respectively; if it is = −1, then they cannot. But in the latter case, by measuring x_{1}, y_{1}, z_{1} in the opposite directions we change the signs of all the coefficients and so make the determinant to be = + 1; hence the former case need alone be considered, and it is accordingly assumed that the determinant is = +1. This being so, it is found that we have the equality α = β′γ″ − β″γ′, and eight like ones, obtained from this by cyclical interchanges of the letters α, β, γ, and of unaccented, singly and doubly accented letters.
36. The nine cosineinclinations above are, as has been seen, connected by six equations. It ought then to be possible to express them all in terms of three parameters. An elegant means of doing this has been given by Rodrigues, who has shown that the tabular expression of the formulae of transformation may be written
x  y  z  
x_{1}  1 + λ^{2} − μ^{2} − ν^{2}  2(λμ − ν)  2(νλ + μ) 
y_{1}  2(λμ + ν)  1 − λ^{2} + μ^{2} − ν^{2}  2(μν + λ) 
z_{1}  2(νλ − μ)  2(μν + λ)  1 − λ^{2} − μ^{2} + ν^{2} 
÷ (1 + λ^{2} + μ^{2} + ν^{2}), 
the meaning being that the coefficients in the transformation are fractions, with numerators expressed as in the table, and the common denominator.
37. The Species of Quadric Surfaces.—Surfaces represented by equations of the second degree are called quadric surfaces. Quadric surfaces are either proper or special. The special ones arise when the coefficients in the general equation are limited to satisfy certain special equations; they comprise (1) planepairs, including in particular one plane twice repeated, and (2) cones, including in particular cylinders; there is but one form of cone, but cylinders may be elliptic, parabolic or hyperbolic.
A discussion of the general equation of the second degree shows that the proper quadric surfaces are of five kinds, represented respectively, when referred to the most convenient axes of reference, by equations of the five types (a and b positive):
(1)  z = x^{2}2a + y^{2}2b, elliptic paraboloid. 
(2)  z = x^{2}2a − y^{2}2b, hyperbolic paraboloid. 
(3)  x^{2}a^{2} + y^{2}b^{2} + z^{2}c^{2} = 1, ellipsoid. 
(4)  x^{2}a^{2} + y^{2}b^{2} − z^{2}c^{2} = 1, hyperboloid of one sheet. 
(5)  x^{2}a^{2} + y^{2}b^{2} − z^{2}c^{2} = −1, hyperboloid of two sheets. 
Fig. 61. 
It is at once seen that these are distinct surfaces; and the equations also show very readily the general form and mode of generation of the several surfaces.
In the elliptic paraboloid (fig. 61) the sections by the planes of zx and zy are the parabolas
z =  x^{2}  , z =  y^{2}  , 
2a  2b 
having the common axes Oz; and the section by any plane z = γ parallel to that of xy is the ellipse
γ =  x^{2}  +  y^{2}  ; 
2a  2b 
so that the surface is generated by a variable ellipse moving parallel to itself along the parabolas as directrices.
Fig. 62.  Fig. 63. 
Fig. 64. 
In the hyperbolic paraboloid (figs. 62 and 63) the sections by the planes of zx, zy are the parabolas z = x^{2}2a, z = − y^{2}2b, having the opposite axes Oz, Oz′, and the section by a plane z = γ parallel to that of xy is the hyperbola γ = x^{2}2a − y^{2}2b, which has its transverse axis parallel to Ox or Oy according as γ is positive or negative. The surface is thus generated by a variable hyperbola moving parallel to itself along the parabolas as directrices. The form is best seen from fig. 63, which represents the sections by planes parallel to the plane of xy, or say the contour lines; the continuous lines are the sections above the plane of xy, and the dotted lines the sections below this plane. The form is, in fact, that of a saddle.
In the ellipsoid (fig. 64) the sections by the planes of zx, zy, and xy are each of them an ellipse, and the section by any parallel plane is also an ellipse. The surface may be considered as generated by an ellipse moving parallel to itself along two ellipses as directrices.
In the hyperboloid of one sheet (fig. 65), the sections by the planes of zx, zy are the hyperbolas
x^{2}  −  z^{2}  = 1,  y^{2}  −  z^{2}  = 1, 
c^{2}  c^{2}  b^{2}  c^{2} 
having a common conjugate axis zOz′; the section by the plane of x, y, and that by any parallel plane, is an ellipse; and the surface may be considered as generated by a variable ellipse moving parallel to itself along the two hyperbolas as directrices. If we imagine two equal and parallel circular disks, their points connected by strings of equal lengths, so that these are the generators of a right circular cylinder, and if we turn one of the disks about its centre through an angle in its plane, the strings in their new positions will be one system of generators of a hyperboloid of one sheet, for which a = b; and if we turn it through the same angle in the opposite direction, we get in like manner the generators of the other system; there will be the same general configuration when a ≠ b. The hyperbolic paraboloid is also covered by two systems of rectilinear generators as a method like that used in § 34 establishes without difficulty. The figures should be studied to see how they can lie.
Fig. 65.  Fig. 66. 
In the hyperboloid of two sheets (fig. 66) the sections by the planes of zx and zy are the hyperbolas
z^{2}  −  x^{2}  = 1,  z^{2}  −  y^{2}  = 1, 
c^{2}  a^{2}  c^{2}  b^{2} 
having a common transverse axis along z′Oz; the section by any plane z = ±γ parallel to that of xy is the ellipse
x^{2}  +  y^{2}  =  γ^{2}  − 1, 
a^{2}  b^{2}  c^{2} 
provided γ^{2} > c^{2}, and the surface, consisting of two distinct portions or sheets, may be considered as generated by a variable ellipse moving parallel to itself along the hyperbolas as directrices.
38. Differential Geometry of Curves.—For convenience consider the coordinates (x, y, z) of a point on a curve in space to be given as functions of a variable parameter θ, which may in particular be one of themselves. Use the notation x′, x″ for dx/dθ, d^{2}x/dθ^{2}, and similarly as to y and z. Only a few formulae will be given. Call the current coordinates (ξ, η, ζ).
The tangent at (x, y, z) is the line tended to as a limit by the connector of (x, y, z) and a neighbouring point of the curve when the latter moves up to the former: its equations are
The osculating plane at (x, y, z) is the plane tended to as a limit by that through (x, y, z) and two neighbouring points of the curve as these, remaining distinct, both move up to (x, y, z): its one equation is
The normal plane is the plane through (x, y, z) at right angles to the tangent line, i.e. the plane
It cuts the osculating plane in a line called the principal normal. Every line through (x, y, z) in the normal plane is a normal. The normal perpendicular to the osculating plane is called the binormal. A tangent, principal normal, and binormal are a convenient set of rectangular axes to use as those of reference, when the nature of a curve near a point on it is to be discussed.
Through (x, y, z) and three neighbouring points, all on the curve, passes a single sphere; and as the three points all move up to (x, y, z) continuing distinct, the sphere tends to a limiting size and position. The limit tended to is the sphere of closest contact with the curve at (x, y, z); its centre and radius are called the centre and radius of spherical curvature. It cuts the osculating plane in a circle, called the circle of absolute curvature; and the centre and radius of this circle are the centre and radius of absolute curvature. The centre of absolute curvature is the limiting position of the point where the principal normal at (x, y, z) is cut by the normal plane at a neighbouring point, as that point moves up to (x, y, z).
39. Differential Geometry of Surfaces.—Let (x, y, z) be any chosen point on a surface ƒ(x, y, z) = 0. As a second point of the surface moves up to (x, y, z), its connector with (x, y, z) tends to a limiting position, a tangent line to the surface at (x, y, z). All these tangent lines at (x, y, z), obtained by approaching (x, y, z) from different directions on a surface, lie in one plane
∂ƒ  (ξ − x) +  ∂ƒ  (η − y) +  ∂ƒ  (ζ − z) = 0. 
∂x  ∂y  ∂z 
This plane is called the tangent plane at (x, y, z). One line through (x, y, z) is at right angles to the tangent plane. This is the normal
(ξ − x) /  ∂ƒ  = (η − y) /  ∂ƒ  = (ζ − z) /  ∂ƒ  . 
∂x  ∂y  ∂z 
The tangent plane is cut by the surface in a curve, real or imaginary, with a node or double point at (x, y, z). Two of the tangent lines touch this curve at the node. They are called the “chief tangents” (Haupttangenten) at (x, y, z); they have closer contact with the surface than any other tangents.
In the case of a quadric surface the curve of intersection of a tangent and the surface is of the second order and has a node, it must therefore consist of two straight lines. Consequently a quadric surface is covered by two sets of straight lines, a pair through every point on it; these are imaginary for the ellipsoid, hyperboloid of two sheets, and elliptic paraboloid.
A surface of any order is covered by two singly infinite systems of curves, a pair through every point, the tangents to which are all chief tangents at their respective points of contact. These are called chieftangent curves; on a quadric surface they are the above straight lines.
40. The tangents at a point of a surface which bisect the angles between the chief tangents are called the principal tangents at the point. They are at right angles, and together with the normal constitute a convenient set of rectangular axes to which to refer the surface when its properties near the point are under discussion. At a special point which is such that the chief tangents there run to the circular points at infinity in the tangent plane, the principal tangents are indeterminate; such a special point is called an umbilic of the surface.
There are two singly infinite systems of curves on a surface, a pair cutting one another at right angles through every point upon it, all tangents to which are principal tangents of the surface at their respective points of contact. These are called lines of curvature, because of a property next to be mentioned.
As a point Q moves in an arbitrary direction on a surface from coincidence with a chosen point P, the normal at it, as a rule, at once fails to meet the normal at P; but, if it takes the direction of a line of curvature through P, this is instantaneously not the case. We have thus on the normal two centres of curvature, and the distances of these from the point on the surface are the two principal radii of curvature of the surface at that point; these are also the radii of curvature of the sections of the surface by planes through the normal and the two principal tangents respectively; or say they are the radii of curvature of the normal sections through the two principal tangents respectively. Take at the point the axis of z in the direction of the normal, and those of x and y in the directions of the principal tangents respectively, then, if the radii of curvature be a, b (the signs being such that the coordinates of the two centres of curvature are z = a and z = b respectively), the surface has in the neighbourhood of the point the form of the paraboloid
z =  x^{2}  +  y^{2}  , 
2a  2b 
and the chieftangents are determined by the equation 0 = x^{2}2a + y^{2}2b. The two centres of curvature may be on the same side of the point or on opposite sides; in the former case a and b have the same sign, the paraboloid is elliptic, and the chieftangents are imaginary; in the latter case a and b have opposite signs, the paraboloid is hyperbolic, and the chieftangents are real.
The normal sections of the surface and the paraboloid by the same plane have the same radius of curvature; and it thence readily follows that the radius of curvature of a normal section of the surface by a plane inclined at an angle θ to that of zx is given by the equation
1  =  cos^{2}θ  +  sin^{2}θ  . 
ρ  a  b 
The section in question is that by a plane through the normal and a line in the tangent plane inclined at an angle θ to the principal tangent along the axis of x. To complete the theory, consider the section by a plane having the same trace upon the tangent plane, but inclined to the normal at an angle φ; then it is shown without difficulty (Meunier’s theorem) that the radius of curvature of this inclined section of the surface is = ρ cos φ.
Authorities.—The above article is largely based on that by Arthur Cayley in the 9th edition of this work. Of early and important recent publications on analytical geometry, special mention is to be made of R. Descartes, Géométrie (Leyden, 1637); John Wallis, Tractatus de sectionibus conicis nova methodo expositis (1655, Opera mathematica, i., Oxford, 1695); de l’Hospital, Traité analytique des sections coniques (Paris, 1720); Leonhard Euler, Introductio in analysin infinitorum, ii. (Lausanne, 1748); Gaspard Monge, “Application d’algèbre à la géométrie” (Journ. École Polytech., 1801); Julius Plücker, Analytischgeometrische Entwickelungen, 3 Bde. (Essen, 1828–1831); System der analytischen Geometrie (Berlin, 1835); G. Salmon, A Treatise on Conic Sections (Dublin, 1848; 6th ed., London, 1879); Ch. Briot and J. Bouquet, Leçons de géométrie analytique (Paris, 1851; 16th ed., 1897); M. Chasles, Traité de géométrie supérieure (Paris, 1852); Wilhelm Fiedler, Analytische Geometrie der Kegelschnitte nach G. Salmon frei bearbeitet (Leipzig, 5te Aufl., 1887–1888); N. M. Ferrers, An Elementary Treatise on Trilinear Coordinates (London, 1861); Otto Hesse, Vorlesungen aus der analytischen Geometrie (Leipzig, 1865, 1881); W. A. Whitworth, Trilinear Coordinates and other Methods of Modern Analytical Geometry (Cambridge, 1866); J. Booth, A Treatise on Some New Geometrical Methods (London, i., 1873; ii., 1877); A. ClebschF. Lindemann, Vorlesungen über Geometrie, Bd. i. (Leipzig, 1876, 2te Aufl., 1891); R. Baltser, Analytische Geometrie (Leipzig, 1882); Charlotte A. Scott, Modern Methods of Analytical Geometry (London, 1894); G. Salmon, A Treatise on the Analytical Geometry of three Dimensions (Dublin, 1862; 4th ed., 1882); SalmonFiedler, Analytische Geometrie des Raumes (Leipzig, 1863; 4te Aufl., 1898); P. Frost, Solid Geometry (London, 3rd ed., 1886; 1st ed., Frost and J. Wolstenholme). See also E. Pascal, Repertorio di matematiche superiori, II. Geometria (Milan, 1900), and articles now appearing in the Encyklopädie der mathematischen Wissenschaften, Bd. iii. 1, 2. (E. B. El.)
V. Line Geometry
Line geometry is the name applied to those geometrical investigations in which the straight line replaces the point as element. Just as ordinary geometry deals primarily with points and systems of points, this theory deals in the first instance with straight lines and systems of straight lines. In two dimensions there is no necessity for a special line geometry, inasmuch as the straight line and the point are interchangeable by the principle of duality; but in three dimensions the straight line is its own reciprocal, and for the better discussion of systems of lines we require some new apparatus, e.g., a system of coordinates applicable to straight lines rather than to points. The essential features of the subject are most easily elucidated by analytical methods: we shall therefore begin with the notion of line coordinates, and in order to emphasize the merits of the system of coordinates ultimately adopted, we first notice a system without these advantages, but often useful in special investigations.
In ordinary Cartesian coordinates the two equations of a straight line may be reduced to the form y = rx + s, z = tx + u, and r, s, t, u may be regarded as the four coordinates of the line. These coordinates lack symmetry: moreover, in changing from one base of reference to another the transformation is not linear, so that the degree of an equation is deprived of real significance. For purposes of the general theory we employ homogeneous coordinates; if x_{1}y_{1}z_{1}w_{1} and x_{2}y_{2}z_{2}w_{2} are two points on the line, it is easily verified that the six determinants of the array
x_{1}y_{1}z_{1}w_{1} 
x_{2}y_{2}z_{2}w_{2} 
are in the same ratios for all pointpairs on the line, and further, that when the point coordinates undergo a linear transformation so also do these six determinants. We therefore adopt these six determinants for the coordinates of the line, and express them by the symbols l, λ, m, μ, n, ν where l = x_{1}w_{2} − x_{2}w_{1}, λ = y_{1}z_{2} − y_{2}z_{1}, &c. There is the further advantage that if a_{1}b_{1}c_{1}d_{1} and a_{2}b_{2}c_{2}d_{2} be two planes through the line, the six determinants
a_{1}b_{1}c_{1}d_{1} 
a_{2}b_{2}c_{2}d_{2} 
are in the same ratios as the foregoing, so that except as regards a factor of proportionality we have λ = b_{1}c_{2} − b_{2}c_{1}, l = c_{1}d_{2} − c_{2}d_{1}, &c. The identical relation lλ + mμ + nν = 0 reduces the number of independent constants in the six coordinates to four, for we are only concerned with their mutual ratios; and the quadratic character of this relation marks an essential difference between point geometry and line geometry. The condition of intersection of two lines is
where the accented letters refer to the second line. If the coordinates are Cartesian and l, m, n are direction cosines, the quantity on the left is the mutual moment of the two lines.
Since a line depends on four constants, there are three distinct types of configurations arising in line geometry—those containing a triplyinfinite, a doublyinfinite and a singlyinfinite number of lines; they are called Complexes, Congruences, and Ruled Surfaces or Skews respectively. A Complex is thus a system of lines satisfying one condition—that is, the coordinates are connected by a single relation; and the degree of the complex is the degree of this equation supposing it to be algebraic. The lines of a complex of the nth degree which pass through any point lie on a cone of the nth degree, those which lie in any plane envelop a curve of the nth class and there are n lines of the complex in any plane pencil; the last statement combines the former two, for it shows that the cone is of the nth degree and the curve is of the nth class. To find the lines common to four complexes of degrees n_{1}, n_{2}, n_{3}, n_{4}, we have to solve five equations, viz. the four complex equations together with the quadratic equation connecting the line coordinates, therefore the number of common lines is 2n_{1}n_{2}n_{3}n_{4}. As an example of complexes we have the lines meeting a twisted curve of the nth degree, which form a complex of the nth degree.
A Congruence is the set of lines satisfying two conditions: thus a finite number m of the lines pass through any point, and a finite number n lie in any plane; these numbers are called the degree and class respectively, and the congruence is symbolically written (m, n).
The simplest example of a congruence is the system of lines constituted by all those that pass through m points and those that lie in n planes; through any other point there pass m of these lines, and in any other plane there lie n, therefore the congruence is of degree m and class n. It has been shown by G. H. Halphen that the number of lines common to two congruences is mm′ + nn′, which may be verified by taking one of them to be of this simple type. The lines meeting two fixed lines form the general (1, 1) congruence; and the chords of a twisted cubic form the general type of a (1, 3) congruence; Halphen’s result shows that two twisted cubics have in general ten common chords. As regards the analytical treatment, the difficulty is of the same nature as that arising in the theory of curves in space, for a congruence is not in general the complete intersection of two complexes.
A Ruled Surface, Regulus or Skew is a configuration of lines which satisfy three conditions, and therefore depend on only one parameter. Such lines all lie on a surface, for we cannot draw one through an arbitrary point; only one line passes through a point of the surface; the simplest example, that of a quadric surface, is really two skews on the same surface.
The degree of a ruled surface qua line geometry is the number of its generating lines contained in a linear complex. Now the number which meets a given line is the degree of the surface qua point geometry, and as the lines meeting a given line form a particular case of linear complex, it follows that the degree is the same from whichever point of view we regard it. The lines common to three complexes of degrees, n_{1}n_{2}n_{3}, form a ruled surface of degree 2n_{1}n_{2}n_{3}; but not every ruled surface is the complete intersection of three complexes.
In the case of a complex of the first degree (or linear complex) the lines through a fixed point lie in a plane called the polar plane or nulplane of that point, and those lying in a fixed plane pass through a point called the nulpoint or pole of the plane. If the nulplane of A pass through B, then the Linear complex. nulplane of B will pass through A; the nulplanes of all points on one line l_{1} pass through another line l_{2}. The relation between l_{1} and l_{2} is reciprocal; any line of the complex that meets one will also meet the other, and every line meeting both belongs to the complex. They are called conjugate or polar lines with respect to the complex. On these principles can be founded a theory of reciprocation with respect to a linear complex.
This may be aptly illustrated by an elegant example due to A. Voss. Since a twisted cubic can be made to satisfy twelve conditions, it might be supposed that a finite number could be drawn to touch four given lines, but this is not the case. For, suppose one such can be drawn, then its reciprocal with respect to any linear complex containing the four lines is a curve of the third class, i.e. another twisted cubic, touching the same four lines, which are unaltered in the process of reciprocation; as there is an infinite number of complexes containing the four lines, there is an infinite number of cubics touching the four lines, and the problem is poristic.
The following are some geometrical constructions relating to the unique linear complex that can be drawn to contain five arbitrary lines:
To construct the nulplane of any point O, we observe that the two lines which meet any four of the given five are conjugate lines of the complex, and the line drawn through O to meet them is therefore a ray of the complex; similarly, by choosing another four we can find another ray through O: these rays lie in the nulplane, and there is clearly a result involved that the five lines so obtained all lie in one plane. A reciprocal construction will enable us to find the nulpoint of any plane. Proceeding now to the metrical properties and the statical and dynamical applications, we remark that there is just one line such that the nulplane of any point on it is perpendicular to it. This is called the central axis; if d be the shortest distance, θ the angle between it and a ray of the complex, then d tan θ = p, where p is a constant called the pitch or parameter. Any system of forces can be reduced to a force R along a certain line, and a couple G perpendicular to that line; the lines of nulmoment for the system form a linear complex of which the given line is the central axis and the quotient G/R is the pitch. Any motion of a rigid body can be reduced to a screw motion about a certain line, i.e. to an angular velocity ω about that line combined with a linear velocity u along the line. The plane drawn through any point perpendicular to the direction of its motion is its nulplane with respect to a linear complex having this line for central axis, and the quotient u/ω for pitch (cf. Sir R. S. Ball, Theory of Screws).
The following are some properties of a configuration of two linear complexes:
The lines common to the twocomplexes also belong to an infinite number of linear complexes, of which two reduce to single straight lines. These two lines are conjugate lines with respect to each of the complexes, but they may coincide, and then some simple modifications are required. The locus of the central axis of this system of complexes is a surface of the third degree called the cylindroid, which plays a leading part in the theory of screws as developed synthetically by Ball. Since a linear complex has an invariant of the second degree in its coefficients, it follows that two linear complexes have a lineolinear invariant. This invariant is fundamental: if the complexes be both straight lines, its vanishing is the condition of their intersection as given above; if only one of them be a straight line, its vanishing is the condition that this line should belong to the other complex. When it vanishes for any two complexes they are said to be in involution or apolar; the nulpoints P, Q of any plane then divide harmonically the points in which the plane meets the common conjugate lines, and each complex is its own reciprocal with respect to the other. As regards a configuration of these linear complexes, the common lines from one system of generators of a quadric, and the doubly infinite system of complexes containing the common lines, include an infinite number of straight lines which form the other system of generators of the same quadric.
If the equation of a linear complex is Al + Bm + Cn + Dλ + Eμ + Fν = 0, then for a line not belonging to the complex we may regard the expression on the lefthand side as a multiple of the moment of the line with respect to the complex, the word moment being used in the statical sense; and we infer General line coordinates. that when the coordinates are replaced by linear functions of themselves the new coordinates are multiples of the moments of the line with respect to six fixed complexes. The essential features of this coordinate system are the same as those of the original one, viz. there are six coordinates connected by a quadratic equation, but this relation has in general a different form. By suitable choice of the six fundamental complexes, as they may be called, this connecting relation may be brought into other simple forms of which we mention two: (i.) When the six are mutually in involution it can be reduced to x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2} + x_{5}^{2} + x_{6}^{2} = 0; (ii.) When the first four are in involution and the other two are the lines common to the first four it is x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2} − 2x_{5}x_{6} = 0. These generalized coordinates might be explained without reference to actual magnitude, just as homogeneous point coordinates can be; the essential remark is that the equation of any coordinate to zero represents a linear complex, a point of view which includes our original system, for the equation of a coordinate to zero represents all the lines meeting an edge of the fundamental tetrahedron.
The system of coordinates referred to six complexes mutually in involution was introduced by Felix Klein, and in many cases is more useful than that derived directly from point coordinates; e.g. in the discussion of quadratic complexes: by means of it Klein has developed an analogy between line geometry and the geometry of spheres as treated by G. Darboux and others. In fact, in that geometry a point is represented by five coordinates, connected by a relation of the same type as the one just mentioned when the five fundamental spheres are mutually at right angles and the equation of a sphere is of the first degree. Extending this to four dimensions of space, we obtain an exact analogue of line geometry, in which (i.) a point corresponds to a line; (ii.) a linear complex to a hypersphere; (iii.) two linear complexes in involution to two orthogonal hyperspheres; (iv.) a linear complex and two conjugate lines to a hypersphere and two inverse points. Many results may be obtained by this principle, and more still are suggested by trying to extend the properties of circles to spheres in three and four dimensions. Thus the elementary theorem, that, given four lines, the circles circumscribed to the four triangles formed by them are concurrent, may be extended to six hyperplanes in four dimensions; and then we can derive a result in line geometry by translating the inverse of this theorem. Again, just as there is an infinite number of spheres touching a surface at a given point, two of them having contact of a closer nature, so there is an infinite number of linear complexes touching a nonlinear complex at a given line, and three of these have contact of a closer nature (cf. Klein, Math. Ann. v.).
Sophus Lie has pointed out a different analogy with sphere geometry. Suppose, in fact, that the equation of a sphere of radius r is
so that r^{2} = a^{2} + b^{2} + c^{2} − d; then introducing the quantity e to make this equation homogeneous, we may regard the sphere as given by the six coordinates a, b, c, d, e, r connected by the equation a^{2} + b^{2} + c^{2} − r^{2} − de = 0, and it is easy to see that two spheres touch, if the polar form 2aa_{1} + 2bb_{1} + 2cc_{1} − 2rr_{1} − de_{1} − d_{1}e vanishes. Comparing this with the equation x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2} − 2x_{5}x_{6} = 0 given above, it appears that this sphere geometry and line geometry are identical, for we may write a = x_{1}, b = x_{2}, c = x_{3}, r = x_{4}δ − 1, d = x_{5}, e = 12x_{6}; but it is to be noticed that a sphere is really replaced by two lines whose coordinates only differ in the sign of x_{4}, so that they are polar lines with respect to the complex x_{4} = 0. Two spheres which touch correspond to two lines which intersect, or more accurately to two pairs of lines (p, p′) and (q, q′), of which the pairs (p, q) and (p′, q′) both intersect. By this means the problem of describing a sphere to touch four given spheres is reduced to that of drawing a pair of lines (t, t′) (of which t intersects one line of the four pairs (pp′), (qq′), (rr′), (ss′), and t′ intersects the remaining four). We may, however, ignore the accented letters in translating theorems, for a configuration of lines and its polar with respect to a linear complex have the same projective properties. In Lie’s transformation a linear complex corresponds to the totality of spheres cutting a given sphere at a given angle. A most remarkable result is that lines of curvature in the sphere geometry become asymptotic lines in the line geometry.
Some of the principles of line geometry may be brought into clearer light by admitting the ideas of space of four and five dimensions.
Thus, regarding the coordinates of a line as homogeneous coordinates in five dimensions, we may say that line geometry is equivalent to geometry on a quadric surface in five dimensions. A linear complex is represented by a hyperplane section; and if two such complexes are in involution, the corresponding hyperplanes are conjugate with respect to the fundamental quadric. By projecting this quadric stereographically into space of four dimensions we obtain Klein’s analogy. In the same way geometry in a linear complex is equivalent to geometry on a quadric in four dimensions; when two lines intersect the representative points are on the same generator of this quadric. Stereographic projection, therefore, converts a curve in a linear complex, i.e. one whose tangents all belong to the complex, into one whose tangents intersect a fixed conic: when this conic is the imaginary circle at infinity the curve is what Lie calls a minimal curve. Curves in a linear complex have been extensively studied. The osculating plane at any point of such a curve is the nulplane of the point with respect to the complex, and points of superosculation always coincide in pairs at the points of contact of stationary tangents. When a point of such a curve is given, the osculating plane is determined, hence all the curves through a given point with the same tangent have the same torsion.
The lines through a given point that belong to a complex of the nth degree lie on a cone of the nth degree: if this cone has a double line the point is said to be a singular point. Similarly, a plane is said to be singular when the envelope of the Nonlinear complexes. lines in it has a double tangent. It is very remarkable that the same surface is the locus of the singular points and the envelope of the singular planes: this surface is called the singular surface, and both its degree and class are in general 2n(n − 1)^{2}, which is equal to four for the quadratic complex.
The singular lines of a complex F = 0 are the lines common to F and the complex
δF  δF  +  δF  δF  +  δF  δF  = 0.  
δl  δλ  δm  δμ  δn  δν 
As already mentioned, at each line l of a complex there is an infinite number of tangent linear complexes, and they all contain the lines adjacent to l. If now l be a singular line, these complexes all reduce to straight lines which form a plane pencil containing the line l. Suppose the vertex of the pencil is A, its plane a, and one of its lines ξ, then l′ being a complex line near l, meets ξ, or more accurately the mutual moment of l′, and is of the second order of small quantities. If P be a point on l, a line through P quite near l in the plane a will meet ξ and is therefore a line of the complex; hence the complexcones of all points on l touch a and the complexcurves of all planes through l touch l at A. It follows that l is a double line of the complexcone of A, and a double tangent of the complexcurve of a. Conversely, a double line of a cone or curve is a singular line, and a singular line clearly touches the curves of all planes through it in the same point. Suppose now that the consecutive line l′ is also a singular line, A′ being the allied singular point, a′ the singular plane and ξ′ any line of the pencil (A′, a′) so that ξ′ is a tangent line at l′ to the complex: the mutual moments of the pairs l′, ξ and l, ξ are each of the second order; hence the plane a′ meets the lines l and ξ′ in two points very near A. This being true for all singular planes, near a the point of contact of a with its envelope is in A, i.e. the locus of singular points is the same as the envelope of singular planes. Further, when a line touches a complex it touches the singular surface, for it belongs to a plane pencil like (Aa), and thus in Klein’s analogy the analogue of a focus of a hypersurface being a bitangent line of the complex is also a bitangent line of the singular surface. The theory of cosingular complexes is thus brought into line with that of confocal surfaces in four dimensions, and guided by these principles the existence of cosingular quadratic complexes can easily be established, the analysis required being almost the same as that invented for confocal cyclides by Darboux and others. Of cosingular complexes of higher degree nothing is known.
Following J. Plücker, we give an account of the lines of a quadratic complex that meet a given line.
The cones whose vertices are on the given line all pass through eight fixed points and envelop a surface of the fourth degree; the conics whose planes contain the given line all lie on a surface of the fourth class and touch eight fixed planes. It is easy to see by elementary geometry that these two surfaces are identical. Further, the given line contains four singular points A_{1}, A_{2}, A_{3}, A_{4}, and the planes into which their cones degenerate are the eight common tangent planes mentioned above; similarly, there are four singular planes, a_{1}, a_{2}, a_{3}, a_{4}, through the line, and the eight points into which their conics degenerate are the eight common points above. The locus of the pole of the line with respect to all the conics in planes through it is a straight line called the polar line of the given one; and through this line passes the polar plane of the given line with respect to each of the cones. The name polar is applied in the ordinary analytical sense; any line has an infinite number of polar complexes with respect to the given complex, for the equation of the latter can be written in an infinite number of ways; one of these polars is a straight line, and is the polar line already introduced. The surface on which lie all the conics through a line l is called the Plücker surface of that line: from the known properties of (2, 2) correspondences it can be shown that the Plücker surface of l cuts l_{1} in a range of the same cross ratio as that of the range in which the Plücker surface of l_{1} cuts l. Applying this to the case in which l_{1} is the polar of l, we find that the cross ratios of (A_{1}, A_{2}, A_{3}, A_{4}) and (a_{1}, a_{2}, a_{3}, a_{4}) are equal. The identity of the locus of the A′s with the envelope of the a′s follows at once; moreover, a line meets the singular surface in four points having the same cross ratio as that of the four tangent planes drawn through the line to touch the surface. The Plücker surface has eight nodes, eight singular tangent planes, and is a double line. The relation between a line and its polar line is not a reciprocal one with respect to the complex; but W. Stahl has pointed out that the relation is reciprocal as far as the singular surface is concerned.
To facilitate the discussion of the general quadratic complex we introduce Klein’s canonical form. We have, in fact, to deal with two quadratic equations in six variables; and by suitable linear transformations these can be reduced to the formQuadratic complexes.
a_{1}x_{1}^{2}  + a_{2}x_{2}^{2}  + a_{3}x_{3}^{2}  + a_{4}x_{4}^{2}  + a_{5}x_{5}^{2}  + a_{6}x_{6}^{2}  = 0 
x_{1}^{2}  + x_{2}^{2}  + x_{3}^{2}  + x_{4}^{2}  + x_{5}^{2}  + x_{6}^{2}  = 0 
subject to certain exceptions, which will be mentioned later.
Taking the first equation to be that of the complex, we remark that both equations are unaltered by changing the sign of any coordinate; the geometrical meaning of this is, that the quadratic complex is its own reciprocal with respect to each of the six fundamental complexes, for changing the sign of a coordinate is equivalent to taking the polar of a line with respect to the corresponding fundamental complex. It is easy to establish the existence of six systems of bitangent linear complexes, for the complex l_{1}x_{1} + l_{2}x_{2} + l_{3}x_{3} + l_{4}x_{4} + l_{5}x_{5} + l_{6}x_{6} = 0 is a bitangent when
l_{1} = 0, and  l_{2}^{2}  +  l_{3}^{2}  +  l_{4}^{2}  +  l_{5}^{2}  +  l_{6}^{2}  = 0, 
a_{2} − a_{1}  a_{3} − a_{1}  a_{4} − a_{1}  a_{5} − a_{1}  a_{6} − a_{1} 
and its lines of contact are conjugate lines with respect to the first fundamental complex. We therefore infer the existence of six systems of bitangent lines of the complex, of which the first is given by
x_{1} = 0,  x_{2}^{2}  +  x_{3}^{2}  +  x_{4}^{2}  +  x_{5}^{2}  +  x_{6}^{2}  = 0, 
a_{2} − a_{1}  a_{3} − a_{1}  a_{4} − a_{1}  a_{5} − a_{1}  a_{6} − a_{1} 
Each of these lines is a bitangent of the singular surface, which is therefore completely determined as being the focal surface of the (2, 2) congruence above. It is thence easy to verify that the two complexes Σax^{2} = 0 and Σbx^{2} = 0 are cosingular if b_{r} = a_{r }λ + μ/a_{r}ν + ρ.
The singular surface of the general quadratic complex is the famous quartic, with sixteen nodes and sixteen singular tangent planes, first discovered by E. E. Kümmer.
We cannot give a full account of its properties here, but we deduce at once from the above that its bitangents break up into six (2, 2) congruences, and the six linear complexes containing these are mutually in involution. The nodes of the singular surface are points whose complex cones are coincident planes, and the complex conic in a singular tangent plane consists of two coincident points. This configuration of sixteen points and planes has many interesting properties; thus each plane contains six points which lie on a conic, while through each point there pass six planes which touch a quadric cone. In many respects the Kümmer quartic plays a part in three dimensions analogous to the general quartic curve in two; it further gives a natural representation of certain relations between hyperelliptic functions (cf. R. W. H. T. Hudson, Kümmer’s Quartic, 1905).
As might be expected from the magnitude of a form in six variables, the number of projectivally distinct varieties of quadratic complexes is very great; and in fact Adolf Weiler, by whom the question was first systematically studied on lines indicated by Klein, enumerated no fewer than fortynine different Classification of quadratic complexes. types. But the principle of the classification is so important, and withal so simple, that we give a brief sketch which indicates its essential features.
We have practically to study the intersection of two quadrics F and F′ in six variables, and to classify the different cases arising we make use of the results of Karl Weierstrass on the equivalence conditions of two pairs of quadratics. As far as at present required, they are as follows: Suppose that the factorized form of the determinantal equation Disct (F + λF′) = 0 is
where the root α occurs s_{1} + s_{2} + s_{3} . . . times in the determinant, s_{2} + s_{3} . . . times in every first minor, s_{3} + . . . times in every second minor, and so on; the meaning of each exponent is then perfectly definite. Every factor of the type (λ − α)^{s} is called an elementartheil (elementary divisor) of the determinant, and the condition of equivalence of two pairs of quadratics is simply that their determinants have the same elementary divisors. We write the pair of forms symbolically thus [(s_{1}s_{2} . . .), (t_{1}t_{2} . . .), . . .], letters in the inner brackets referring to the same factor. Returning now to the two quadratics representing the complex, the sum of the exponents will be six, and two complexes are put in the same class if they have the same symbolical expression; i.e. the actual values of the roots of the determinantal equation need not be the same for both, but their manner of occurrence, as far as here indicated, must be identical in the two. The enumeration of all possible cases is thus reduced to a simple question in combinatorial analysis, and the actual study of any particular case is much facilitated by a useful rule of Klein’s for writing down in a simple form two quadratics belonging to a given class—one of which, of course, represents the equation connecting line coordinates, and the other the equation of the complex. The general complex is naturally [111111]; the complex of tangents to a quadric is [(111), (111)] and that of lines meeting a conic is [(222)]. Full information will be found in Weiler’s memoir, Math. Ann. vol. vii.
The detailed study of each variety of complex opens up a vast subject; we only mention two special cases, the harmonic complex and the tetrahedral complex.
The harmonic complex, first studied by Battaglini, is generated in an infinite number of ways by the lines cutting two quadrics harmonically. Taking the most general case, and referring the quadrics to their common selfconjugate tetrahedron, we can find its equation in a simple form, and verify that this complex really depends only on seventeen constants, so that it is not the most general quadratic complex. It belongs to the general type in so far as it is discussed above, but the roots of the determinant are in involution. The singular surface is the “tetrahedroid” discussed by Cayley. As a particular case, from a metrical point of view, we have L. F. Painvin’s complex generated by the lines of intersection of perpendicular tangent planes of a quadric, the singular surface now being Fresnel’s wave surface. The tetrahedral or Reye complex is the simplest and best known of proper quadratic complexes. It is generated by the lines which cut the faces of a tetrahedron in a constant cross ratio, and therefore by those subtending the same cross ratio at the four vertices. The singular surface is made up of the faces or the vertices of the fundamental tetrahedron, and each edge of this tetrahedron is a double line of the complex. The complex was first discussed by K. T. Reye as the assemblage of lines joining corresponding points in a homographic transformation of space, and this point of view leads to many important and elegant properties. A (metrically) particular case of great interest is the complex generated by the normals to a family of confocal quadrics, and for many investigations it is convenient to deal with this complex referred to the principal axes. For example, Lie has developed the theory of curves in a Reye complex (i.e. curves whose tangents belong to the complex) as solutions of a differential equation of the form (b − c)xdydz + (c − a)ydzdx + (a − b)zdxdy = 0, and we can simplify this equation by a logarithmic transformation. Many theorems connecting complexes with differential equations have been given by Lie and his school. A line complex, in fact, corresponds to a Mongian equation having ∞^{3} line integrals.
As the coordinates of a line belonging to a congruence are functions of two independent parameters, the theory of congruences is analogous to that of surfaces, and we may regard it as a fundamental inquiry to find the simplest form of surface into which Congruences. a given congruence can be transformed. Most of those whose properties have been extensively discussed can be represented on a plane by a birational transformation. But in addition to the difficulties of the theory of algebraic surfaces, a subject still in its infancy, the theory of congruences has other difficulties in that a congruence is seldom completely represented, even by two equations.
A fundamental theorem is that the lines of a congruence are in general bitangents of a surface; in fact, since the condition of intersection of two consecutive straight lines is ldλ + dmdμ + dndν = 0, a line l of the congruence meets two adjacent lines, say l_{1} and l_{2}. Suppose l, l_{1} lie in the plane pencil (A_{1}a_{1}) and l, l_{2} in the plane pencil (A_{2}a_{2}), then the locus of the A′s is the same as the envelope of the a′s, but a_{2} is the tangent plane at A_{1} and a_{1} at A_{2}. This surface is called the focal surface of the congruence, and to it all the lines l are bitangent. The distinctive property of the points A is that two of the congruence lines through them coincide, and in like manner the planes a each contain two coincident lines. The focal surface consists of two sheets, but one or both may degenerate into curves; thus, for example, the normals to a surface are bitangents of the surface of centres, and in the case of Dupin’s cyclide this surface degenerates into two conics.
In the discussion of congruences it soon becomes necessary to introduce another number r, called the rank, which expresses the number of plane pencils each of which contains an arbitrary line and two lines of the congruence. The order of the focal surface is 2m(n − 1) − 2r, and its class is m(m − 1) − 2r. Our knowledge of congruences is almost exclusively confined to those in which either m or n does not exceed two. We give a brief account of those of the second order without singular lines, those of order unity not being especially interesting. A congruence generally has singular points through which an infinite number of lines pass; a singular point is said to be of order r when the lines through it lie on a cone of the rth degree. By means of formulae connecting the number of singular points and their orders with the class m of quadratic congruence Kümmer proved that the class cannot exceed seven. The focal surface is of degree four and class 2m; this kind of quartic surface has been extensively studied by Kümmer, Cayley, Rohn and others. The varieties (2, 2), (2, 3), (2, 4), (2, 5) all belong to at least one Reye complex; and so also does the most important class of (2, 6) congruences which includes all the above as special cases. The congruence (2, 2) belongs to a linear complex and forty different Reye complexes; as above remarked, the singular surface is Kümmer’s sixteennodal quartic, and the same surface is focal for six different congruences of this variety. The theory of (2, 2) congruences is completely analogous to that of the surfaces called cyclides in three dimensions. Further particulars regarding quadratic congruences will be found in Kümmer’s memoir of 1866, and the second volume of Sturm’s treatise. The properties of quadratic congruences having singular lines, i.e. degenerate focal surfaces, are not so interesting as those of the above class; they have been discussed by Kümmer, Sturm and others.
Since a ruled surface contains only ∞¹ elements, this theory is practically the same as that of curves. If a linear complex contains more than n generators of a ruled surface of the nth degree, it contains all the generators, hence for n = 2 there are Ruled surfaces. three linearly independent complexes, containing all the generators, and this is a wellknown property of quadric surfaces. In ruled cubics the generators all meet two lines which may or may not coincide; these two cases correspond to the two main classes of cubics discussed by Cayley and Cremona. As regards ruled quartics, the generators must lie in one and may lie in two linear complexes. The first class is equivalent to a quartic in four dimensions and is always rational, but the latter class has to be subdivided into the elliptic and the rational, just like twisted quartic curves. A quintic skew may not lie in a linear complex, and then it is unicursal, while of sextics we have two classes not in a linear complex, viz. the elliptic variety, having thirtysix places where a linear complex contains six consecutive generators, and the rational, having six such places.
The general theory of skews in two linear complexes is identical with that of curves on a quadric in three dimensions and is known. But for skews lying in only one linear complex there are difficulties; the curve now lies in four dimensions, and we represent it in three by stereographic projection as a curve meeting a given plane in n points on a conic. To find the maximum deficiency for a given degree would probably be difficult, but as far as degree eight the spacecurve theory of Halphen and Nöther can be translated into line geometry at once. When the skew does not lie in a linear complex at all the theory is more difficult still, and the general theory clearly cannot advance until further progress is made in the study of twisted curves.
References.—The earliest works of a general nature are Plücker, Neue Geometrie des Raumes (Leipzig, 1868); and Kümmer, “Über die algebraischen Strahlensysteme,” Berlin Academy (1866). Systematic development on purely synthetic lines will be found in the three volumes of Sturm, Liniengeometrie (Leipzig, 1892, 1893, 1896); vol. i. deals with the linear and Reye complexes, vols. ii. and iii. with quadratic congruences and complexes respectively. For a highly suggestive review by Gino Loria see Bulletin des sciences mathématiques (1893, 1897). A shorter treatise, giving a very interesting account of Klein’s coordinates, is the work of Koenigs, La Géométrie réglée et ses applications (Paris, 1898). English treatises are C. M. Jessop, Treatise on the Line Complex (1903); R. W. H. T. Hudson, Kümmer’s Quartic (1905). Many references to memoirs on line geometry will be found in Hagen, Synopsis der höheren Mathematik, ii. (Berlin, 1894); Loria, Il passato ed il presente delle principali teorie geometriche (Milan, 1897); a clear résumé of the principal results is contained in the very elegant volume of Pascal, Repertorio di mathematiche superiori, ii. (Milan, 1900). Another treatise dealing extensively with line geometry is Lie, Geometrie der Berührungstransformationen (Leipzig, 1896). Many memoirs on the subject have appeared in the Mathematische Annalen; a full list of these will be found in the index to the first fifty volumes, p. 115. Perhaps the two memoirs which have left most impression on the subsequent development of the subject are Klein, “Zur Theorie der Liniencomplexe des ersten und zweiten Grades,” Math. Ann. ii.; and Lie, “Über Complexe, insbesondere Linien und Kugelcomplexe,” Math. Ann. v. (J. H. Gr.)
VI. NonEuclidean Geometry
The various metrical geometries are concerned with the properties of the various types of congruencegroups, which are defined in the study of the axioms of geometry and of their immediate consequences. But this point of view of the subject is the outcome of recent research, and historically the subject has a different origin. NonEuclidean geometry arose from the discussion, extending from the Greek period to the present day, of the various assumptions which are implicit in the traditional Euclidean system of geometry. In the course of these investigations it became evident that metrical geometries, each internally consistent but inconsistent in many respects with each other and with the Euclidean system, could be developed. A short historical sketch will explain this origin of the subject, and describe the famous and interesting progress of thought on the subject. But previously a description of the chief characteristic properties of elliptic and of hyperbolic geometries will be given, assuming the standpoint arrived at below under VII. Axioms of Geometry.
First assume the equation to the absolute (cf. loc. cit.) to be w^{2} − x^{2} − y^{2} − z^{2} = 0. The absolute is then real, and the geometry is hyberbolic.
The distance (d_{12}) between the two points (x_{1}, y_{1}, z_{1}, w_{1}) and (x_{2}, y_{2}, z_{2}, w_{2}) is given by
cosh (d_{12}/γ) = (w_{1}w_{2} − x_{1}x_{2} − y_{1}y_{2} − z_{1}z_{2}) / {(w_{1}^{2} − x_{1}^{2} − y_{1}^{2} − z_{1}^{2})
(w_{2}^{2} − x_{2}^{2} − y_{2}^{2} − z_{2}^{2})}^{1/2} 
(1) 
The only points to which the metrical geometry applies are those within the region enclosed by the quadric; the other points are “improper ideal points.” The angle (θ_{12}) between two planes, l_{1}x + m_{1}y + n_{1}z + r_{1}w = 0 and l_{2}x + m_{2}y + n_{2}z + r_{2}w = 0, is given by
cos θ_{12} = (l_{1}l_{2} + m_{1}m_{2} + n_{1}n_{2} − r_{1}r_{2}) / {(l_{1}^{2} + m_{1}^{2} + n_{1}^{2} − r_{1}^{2})
(l_{2}^{2} + m_{2}^{2} + n_{2}^{2} − r_{2}^{2})}^{1/2} 
(2) 
These planes only have a real angle of inclination if they possess a line of intersection within the actual space, i.e. if they intersect. Planes which do not intersect possess a shortest distance along a line which is perpendicular to both of them. If this shortest distance is δ_{12}, we have
cosh (δ_{12}/γ) = (l_{1}l_{2} + m_{1}m_{2} + n_{1}n_{2} − r_{1}r_{2}) / {(l_{1}^{2} + m_{1}^{2} + n_{1}^{2} − r_{1}^{2})
(l_{2}^{2} + m_{2}^{2} + n_{2}^{2} − r_{2}^{2})}^{1/2} 
(3) 
Fig. 67. 
Thus in the case of the two planes one and only one of the two, θ12 and δ_{12}, is real. The same considerations hold for coplanar straight lines (see VII. Axioms of Geometry). Let O (fig. 67) be the point (0, 0, 0, 1), OX the line y = 0, z = 0, OY the line z = 0, x = 0, and OZ the line x = 0, y = 0. These are the coordinate axes and are at right angles to each other. Let P be any point, and let ρ be the distance OP, θ the angle POZ, and φ the angle between the planes ZOX and ZOP. Then the coordinates of P can be taken to be
sin φ, sinh (ρ/γ) cos θ, cosh (ρ/γ).
If ABC is a triangle, and the sides and angles are named according to the usual convention, we have
sinh (a/γ) / sin A = sinh (b/γ) / sin B = sinh (c/γ) / sin C,  (4) 
and also
cosh (a/γ) = cosh (b/γ) cosh (c/γ) − sinh (b/γ) sinh (c/γ) cos A,  (5) 
Fig. 68. 
with two similar equations. The sum of the three angles of a triangle is always less than two right angles. The area of the triangle ABC is λ^{2}(π − A − B − C). If the base BC of a triangle is kept fixed and the vertex A moves in the fixed plane ABC so that the area ABC is constant, then the locus of A is a line of equal distance from BC. This locus is not a straight line. The whole theory of similarity is inapplicable; two triangles are either congruent, or their angles are not equal two by two. Thus the elements of a triangle are determined when its three angles are given. By keeping A and B and the line BC fixed, but by making C move off to infinity along BC, the lines BC and AC become parallel, and the sides a and b become infinite. Hence from equation (5) above, it follows that two parallel lines (cf. Section VII. Axioms of Geometry) must be considered as making a zero angle with each other. Also if B be a right angle, from the equation (5), remembering that, in the limit,
The angle A is called by N. I. Lobatchewsky the “angle of parallelism.”
The whole theory of lines and planes at right angles to each other is simply the theory of conjugate elements with respect to the absolute, where ideal lines and planes are introduced.
Thus if l and l′ be any two conjugate lines with respect to the absolute (of which one of the two must be improper, say l′), then any plane through l′ and containing proper points is perpendicular to l. Also if p is any plane containing proper points, and P is its pole, which is necessarily improper, then the lines through P are the normals to P. The equation of the sphere, centre (x_{1}, y_{1}, z_{1}, w_{1}) and radius ρ, is
.  (7). 
The equation of the surface of equal distance (σ) from the plane lx + my + nz + rw = 0 is
(l^{2} + m^{2} + n^{2} − r^{2}) (w^{2} − x^{2} − y^{2} − z^{2}) sinh^{2} (σ/γ) = (rw + lx + my + nz)^{2}  (8). 
A surface of equal distance is a sphere whose centre is improper; and both types of surface are included in the family
k^{2} (w^{2} − x^{2} − y^{2} − z^{2}) = (ax + by + cz + dw)^{2}  (9). 
But this family also includes a third type of surfaces, which can be looked on either as the limits of spheres whose centres have approached the absolute, or as the limits of surfaces of equal distance whose central planes have approached a position tangential to the absolute. These surfaces are called limitsurfaces. Thus (9) denotes a limitsurface, if d^{2} − a^{2} − b^{2} − c^{2} = 0. Two limitsurfaces only differ in position. Thus the two limitsurfaces which touch the plane YOZ at O, but have their concavities turned in opposite directions, have as their equations
The geodesic geometry of a sphere is elliptic, that of a surface of equal distance is hyperbolic, and that of a limitsurface is parabolic (i.e. Euclidean). The equation of the surface (cylinder) of equal distance (δ) from the line OX is
This is not a ruled surface. Hence in this geometry it is not possible for two straight lines to be at a constant distance from each other.
Secondly, let the equation of the absolute be x^{2} + y^{2} + z^{2} + w^{2} = 0. The absolute is now imaginary and the geometry is elliptic.
The distance (d_{12}) between the two points (x_{1}, y_{1}, z_{1}, w_{1}) and (x_{2}, y_{2}, z_{2}, w_{2}) is given by
cos (d_{12}/γ) = ± (x_{1}x_{2} + y_{1}y_{2} + z_{1}z_{2} + w_{1}w_{2}) /
{(x_{1}^{2} + y_{1}^{2} + z_{1}^{2} + w_{1}^{2}) (x_{2}^{2} + y_{2}^{2} + z_{2}^{2} + w_{2}^{2})}^{1/2} 
(10). 
Thus there are two distances between the points, and if one is d_{12}, the other is πγd_{12}. Every straight line returns into itself, forming a closed series. Thus there are two segments between any two points, together forming the whole line which contains them; one distance is associated with one segment, and the other distance with the other segment. The complete length of every straight line is πγ.
The angle between the two planes l_{1}x + m_{1}y + n_{1}z + r + _{1}w = 0 and l_{2}x + m_{2}y + n_{2}z + r_{2}w = 0 is
cos θ_{12} = (l_{1}l_{2} + m_{1}m_{2} + n_{1}n_{2} + r_{1}r_{2}) /
{(l_{1}^{2} + m_{1}^{2} + n_{1}^{2} +r_{1}^{2}) (l_{2}^{2} + m_{2}^{2} + n_{2}^{2} + r_{2}^{2})}^{1/2} 
(11). 
The polar plane with respect to the absolute of the point (x_{1}, y_{1}, z_{1}, w_{1}) is the real plane x_{1}x + y_{1}y + z_{1}z + w_{1}w = 0, and the pole of the plane l_{1}x + m_{1}y + n_{1}z + r_{1}w = 0 is the point (l_{1}, m_{1}, n_{1}, r_{1}). Thus (from equations 10 and 11) it follows that the angle between the polar planes of the points (x_{1}, ...) and (x_{2}, ...) is d_{12}/γ, and that the distance between the poles of the planes (l_{1}, ...) and (l_{2}, ...) is γθ_{12}. Thus there is complete reciprocity between points and planes in respect to all properties. This complete reign of the principle of duality is one of the great beauties of this geometry. The theory of lines and planes at right angles is simply the theory of conjugate elements with respect to the absolute. A tetrahedron selfconjugate with respect to the absolute has all its intersecting elements (edges and planes) at right angles. If l and l′ are two conjugate lines, the planes through one are the planes perpendicular to the other. If P is the pole of the plane p, the lines through P are the normals to the plane p. The distance from P to p is 12πγ. Thus every sphere is also a surface of equal distance from the polar of its centre, and conversely. A plane does not divide space; for the line joining any two points P and Q only cuts the plane once, in L say, then it is always possible to go from P to Q by the segment of the line PQ which does not contain L. But P and Q may be said to be separated by a plane p, if the point in which PQ cuts p lies on the shortest segment between P and Q. With this sense of “separation,” it is possible^{[2]} to find three points P, Q, R such that P and Q are separated by the plane p, but P and R are not separated by p, nor are Q and R.
Let A, B, C be any three noncollinear points, then four triangles are defined by these points. Thus if a, b, c and A, B, C are the elements of any one triangle, then the four triangles have as their elements:
(1)  a ,  b,  c,  A,  B,  C. 
(2)  a,  πγ − b,  πγ − c,  A,  π − B,  π − C. 
(3)  πγ − a,  b,  πγ − c,  π − A,  B,  π − C. 
(4)  πγ − a,  πγ − b,  c,  π − A,  π − B,  C. 
The formulae connecting the elements are
sin A/sin (a/γ) = sin B/sin (b/γ) = sin C/sin (c/γ),  (12). 
and
cos (a/γ) = cos (b/γ) cos (c/γ) + sin (b/γ) sin (c/γ) cos A,  (13). 
with two similar equations.
Two cases arise, namely (I.) according as one of the four triangles has as its sides the shortest segments between the angular points, or (II.) according as this is not the case. When case I. holds there is said to be a “principal triangle.”^{[3]} If all the figures considered lie within a sphere of radius 14πγ only case I. can hold, and the principal triangle is the triangle wholly within this sphere, also the peculiarities in respect to the separation of points by a plane cannot then arise. The sum of the three angles of a triangle ABC is always greater than two right angles, and the area of the triangle is γ^{2}(A + B + C − π). Thus as in hyperbolic geometry the theory of similarity does not hold, and the elements of a triangle are determined when its three angles are given. The coordinates of a point can be written in the form
where ρ, Φ and φ have the same meanings as in the corresponding formulae in hyperbolic geometry. Again, suppose a watch is laid on the plane OXY, face upwards with its centre at O, and the line 12 to 6 (as marked on dial) along the line YOY. Let the watch be continually pushed along the plane along the line OX, that is, in the direction 9 to 3. Then the line XOX being of finite length, the watch will return to O, but at its first return it will be found to be face downwards on the other side of the plane, with the line 12 to 6 reversed in direction along the line YOY. This peculiarity was first pointed out by Felix Klein. The theory of parallels as it exists in hyperbolic space has no application in elliptic geometry. But another property of Euclidean parallel lines holds in elliptic geometry, and by the use of it parallel lines are defined. For the equation of the surface (cylinder) of equal distance (δ) from the line XOX is
This is also the surface of equal distance, 12πγ−δ, from the line conjugate to XOX. Now from the form of the above equation this is a ruled surface, and through every point of it two generators pass. But these generators are lines of equal distance from XOX. Thus throughout every point of space two lines can be drawn which are lines of equal distance from a given line l. This property was discovered by W. K. Clifford. The two lines are called Clifford’s right and left parallels to l through the point. This property of parallelism is reciprocal, so that if m is a left parallel to l, then l is a left parallel to m. Note also that two parallel lines l and m are not coplanar. Many of those properties of Euclidean parallels, which do not hold for Lobatchewsky’s parallels in hyperbolic geometry, do hold for Clifford’s parallels in elliptic geometry. The geodesic geometry of spheres is elliptic, the geodesic geometry of surfaces of equal distance from lines (cylinders) is Euclidean, and surfaces of revolution can be found^{[4]} of which the geodesic geometry is hyperbolic. But it is to be noticed that the connectivity of these surfaces is different to that of a Euclidean plane. For instance there are only ∞^{2} congruence transformations of the cylindrical surfaces of equal distance into themselves, instead of the ∞^{3} for the ordinary plane. It would obviously be possible to state “axioms” which these geodesics satisfy, and thus to define independently, and not as loci, quasispaces of these peculiar types. The existence of such Euclidean quasigeometries was first pointed out by Clifford.^{[5]}
In both elliptic and hyperbolic geometry the spherical geometry, i.e. the relations between the angles formed by lines and planes passing through the same point, is the same as the “spherical trigonometry” in Euclidean geometry. The constant γ, which appears in the formulae both of hyperbolic and elliptic geometry, does not by its variation produce different types of geometry. There is only one type of elliptic geometry and one type of hyperbolic geometry; and the magnitude of the constant γ in each case simply depends upon the magnitude of the arbitrary unit of length in comparison with the natural unit of length which each particular instance of either geometry presents. The existence of a natural unit of length is a peculiarity common both to hyperbolic and elliptic geometries, and differentiates them from Euclidean geometry. It is the reason for the failure of the theory of similarity in them. If γ is very large, that is, if the natural unit is very large compared to the arbitrary unit, and if the lengths involved in the figures considered are not large compared to the arbitrary unit, then both the elliptic and hyperbolic geometries approximate to the Euclidean. For from formulae (4) and (5) and also from (12) and (13) we find, after retaining only the lowest powers of small quantities, as the formulae for any triangle ABC,
and
with two similar equations. Thus the geometries of small figures are in both types Euclidean.
History.—“In pulcherrimo Geometriae corpore,” wrote Sir Henry Savile in 1621, “duo sunt naevi, duae labes nec quod sciam plures, in quibus eluendis et emaculendis cum veterum tum recentiorum ... vigilavit industria.” These two blemishes are the theory of parallels and Theory of parallels before Gauss. the theory of proportion. The “industry of the moderns,” in both respects, has given rise to important branches of mathematics, while at the same time showing that Euclid is in these respects more free from blemish than had been previously credible. It was from endeavours to improve the theory of parallels that nonEuclidean geometry arose; and though it has now acquired a far wider scope, its historical origin remains instructive and interesting. Euclid’s “axiom of parallels” appears as Postulate V. to the first book of his Elements, and is stated thus, “And that, if a straight line falling on two straight lines make the angles, internal and on the same side, less than two right angles, the two straight lines, being produced indefinitely, meet on the side on which are the angles less than two right angles.” The original Greek is καὶ ἐὰν εἰς δύο εὐθείας εὐθεῖα ἐμπίπτουσα τὰς ἐντὸς καὶ ἐπὶ τὰ αὐτὰ μέρη γωνίας δύο ὀρθῶν ἐλάσσονας ποιῇ, ἐκβαλλομένας τὰς δύο εὐθείας ἐπ᾽ ἄπειρον συμπίπτειν, ἐφ᾽ ἃ μέρη εἰσὶν αἱ τῶν δύο ὀρθῶν ἐλάσσονες.
To Euclid’s successors this axiom had signally failed to appear selfevident, and had failed equally to appear indemonstrable. Without the use of the postulate its converse is proved in Euclid’s 28th proposition, and it was hoped that by further efforts the postulate itself could be also proved. The first step consisted in the discovery of equivalent axioms. Christoph Clavius in 1574 deduced the axiom from the assumption that a line whose points are all equidistant from a straight line is itself straight. John Wallis in 1663 showed that the postulate follows from the possibility of similar triangles on different scales. Girolamo Saccheri (1733) showed that it is sufficient to have a single triangle, the sum of whose angles is two right angles. Other equivalent forms may be obtained, but none shows any essential superiority to Euclid’s. Indeed plausibility, which is chiefly aimed at, becomes a positive demerit where it conceals a real assumption.
A new method, which, though it failed to lead to the desired goal, proved in the end immensely fruitful, was invented by Saccheri, in a work entitled Euclides ab omni naevo vindicatus (Milan, 1733). If the postulate of parallels Saccheri. is involved in Euclid’s other assumptions, contradictions must emerge when it is denied while the others are maintained. This led Saccheri to attempt a reductio ad absurdum, in which he mistakenly believed himself to have succeeded. What is interesting, however, is not his fallacious conclusion, but the nonEuclidean results which he obtains in the process. Saccheri distinguishes three hypotheses (corresponding to what are now known as Euclidean or parabolic, elliptic and hyperbolic geometry), and proves that some one of the three must be universally true. His three hypotheses are thus obtained: equal perpendiculars AC, BD are drawn from a straight line AB, and CD are joined. It is shown that the angles ACD, BDC are equal. The first hypothesis is that these are both right angles; the second, that they are both obtuse; and the third, that they are both acute. Many of the results afterwards obtained by Lobatchewsky and Bolyai are here developed. Saccheri fails to be the founder of nonEuclidean geometry only because he does not perceive the possible truth of his nonEuclidean hypotheses.
Some advance is made by Johann Heinrich Lambert in his Theorie der Parallellinien (written 1766; posthumously published 1786). Though he still believed in the necessary truth of Euclidean geometry, he confessed that, in Lambert. all his attempted proofs, something remained undemonstrated. He deals with the same three hypotheses as Saccheri, showing that the second holds on a sphere, while the third would hold on a sphere of purely imaginary radius. The second hypothesis he succeeds in condemning, since, like all who preceded Bernhard Riemann, he is unable to conceive of the straight line as finite and closed. But the third hypothesis, which is the same as Lobatchewsky’s, is not even professedly refuted.^{[6]}
NonEuclidean geometry proper begins with Karl Friedrich Gauss. The advance which he made was rather philosophical than mathematical: it was he (probably) who first recognized that the postulate of parallels is possibly false, and should be empirically tested by measuring Three periods of nonEuclidean geometry. the angles of large triangles. The history of nonEuclidean geometry has been aptly divided by Felix Klein into three very distinct periods. The first—which contains only Gauss, Lobatchewsky and Bolyai—is characterized by its synthetic method and by its close relation to Euclid. The attempt at indirect proof of the disputed postulate would seem to have been the source of these three men’s discoveries; but when the postulate had been denied, they found that the results, so far from showing contradictions, were just as selfconsistent as Euclid. They inferred that the postulate, if true at all, can only be proved by observations and measurements. Only one kind of nonEuclidean space is known to them, namely, that which is now called hyperbolic. The second period is analytical, and is characterized by a close relation to the theory of surfaces. It begins with Riemann’s inaugural dissertation, which regards space as a particular case of a manifold; but the characteristic standpoint of the period is chiefly emphasized by Eugenio Beltrami. The conception of measure of curvature is extended by Riemann from surfaces to spaces, and a new kind of space, finite but unbounded (corresponding to the second hypothesis of Saccheri and Lambert), is shown to be possible. As opposed to the second period, which is purely metrical, the third period is essentially projective in its method. It begins with Arthur Cayley, who showed that metrical properties are projective properties relative to a certain fundamental quadric, and that different geometries arise according as this quadric is real, imaginary or degenerate. Klein, to whom the development of Cayley’s work is due, showed further that there are two forms of Riemann’s space, called by him the elliptic and the spherical. Finally, it has been shown by Sophus Lie, that if figures are to be freely movable throughout all space in ∞^{6} ways, no other threedimensional spaces than the above four are possible.
Gauss published nothing on the theory of parallels, and it was not generally known until after his death that he had interested himself in that theory from a very early date. In 1799 he announces that Euclidean geometry would follow from the assumption that a triangle can be drawn Gauss. greater than any given triangle. Though unwilling to assume this, we find him in 1804 still hoping to prove the postulate of parallels. In 1830 he announces his conviction that geometry is not an a priori science; in the following year he explains that nonEuclidean geometry is free from contradictions, and that, in this system, the angles of a triangle diminish without limit when all the sides are increased. He also gives for the circumference of a circle of radius the formula , where is a constant depending upon the nature of the space. In 1832, in reply to the receipt of Bolyai’s Appendix, he gives an elegant proof that the amount by which the sum of the angles of a triangle falls short of two right angles is proportional to the area of the triangle. From these and a few other remarks it appears that Gauss possessed the foundations of hyperbolic geometry, which he was probably the first to regard as perhaps true. It is not known with certainty whether he influenced Lobatchewsky and Bolyai, but the evidence we possess is against such a view.^{[7]}
The first to publish a nonEuclidean geometry was Nicholas Lobatchewsky, professor of mathematics in the new university of Kazañ.^{[8]} In the place of the disputed postulate he puts the following: “All straight lines which, in a plane, radiate from a given point, can, with respect Lobatchewsky. to any other straight line in the same plane, be divided into two classes, the intersecting and the nonintersecting. The boundary line of the one and the other class is called parallel to the given line.” It follows that there are two parallels to the given line through any point, each meeting the line at infinity, like a Euclidean parallel. (Hence a line has two distinct points at infinity, and not one only as in ordinary geometry.) The two parallels to a line through a point make equal acute angles with the perpendicular to the line through the point. If p be the length of the perpendicular, either of these angles is denoted by Π(p). The determination of Π(p) is the chief problem (cf. equation (6) above); it appears finally that, with a suitable choice of the unit of length,
tan .
Before obtaining this result it is shown that spherical trigonometry is unchanged, and that the normals to a circle or a sphere still pass through its centre. When the radius of the circle or sphere becomes infinite all these normals become parallel, but the circle or sphere does not become a straight line or plane. It becomes what Lobatchewsky calls a limitline or limitsurface. The geometry on such a surface is shown to be Euclidean, limitlines replacing Euclidean straight lines. (It is, in fact, a surface of zero measure of curvature.) By the help of these propositions Lobatchewsky obtains the above value of Π(p), and thence the solution of triangles. He points out that his formulae result from those of spherical trigonometry by substituting ia, ib, ic, for the sides a, b, c.
John Bolyai, a Hungarian, obtained results closely corresponding to those of Lobatchewsky. These he published in an appendix to a work by his father, entitled Appendix Scientiam spatii absolute veram exhibens: a veritate aut falsitate Axiomatis XI. Euclidei (a priori haud unquam decidenda) independentem: Bolyai. adjecta ad casum falsitatis, quadratura circuli geometrica.^{[9]} This work was published in 1831, but its conception dates from 1823. It reveals a profounder appreciation of the importance of the new ideas, but otherwise differs little from Lobatchewsky’s. Both men point out that Euclidean geometry as a limiting case of their own more general system, that the geometry of very small spaces is always approximately Euclidean, that no a priori grounds exist for a decision, and that observation can only give an approximate answer. Bolyai gives also, as his title indicates, a geometrical construction, in hyperbolic space, for the quadrature of the circle, and shows that the area of the greatest possible triangle, which has all its sides parallel and all its angles zero, is πi^{2}, where i is what we should now call the spaceconstant.
The works of Lobatchewsky and Bolyai, though known and valued by Gauss, remained obscure and ineffective until, in 1866, they were translated into French by J. Hoüel. But at this time Riemann’s dissertation, Über die Hypothesen, welche der Geometrie zu Grunde liegen,^{[10]} was already about to be Riemann. published. In this work Riemann, without any knowledge of his predecessors in the same field, inaugurated a far more profound discussion, based on a far more general standpoint; and by its publication in 1867 the attention of mathematicians and philosophers was at last secured. (The dissertation dates from 1854, but owing to changes which Riemann wished to make in it, it remained unpublished until after his death.)
Riemann’s work contains two fundamental conceptions, that
of a manifold and that of the measure of curvature of a continuous
manifold possessed of what he calls flatness in the smallest parts.
By means of these conceptions space is made to appear
at the end of a gradual series of more and more specialized
Definition of
a manifold.
conceptions. Conceptions of magnitude, he explains,
are only possible where we have a general conception
capable of determination in various ways. The manifold consists
of all these various determinations, each of which is an element
of the manifold. The passage from one element to another may
be discrete or continuous; the manifold is called discrete or
continuous accordingly. Where it is discrete two portions of
it can be compared, as to magnitude, by counting; where
continuous, by measurement. But measurement demands
superposition, and consequently some magnitude independent
of its place in the manifold. In passing, in a continuous manifold,
from one element to another in a determinate way, we pass
through a series of intermediate terms, which form a onedimensional
manifold. If this whole manifold be similarly
caused to pass over into another, each of its elements passes
through a onedimensional manifold, and thus on the whole
a twodimensional manifold is generated. In this way we can
proceed to n dimensions. Conversely, a manifold of n dimensions
can be analysed into one of one dimension and one of (n − 1)
dimensions. By repetitions of this process the position of an
element may be at last determined by n magnitudes. We may
here stop to observe that the above conception of a manifold
is akin to that due to Hermann Grassmann in the first edition
(1847) of his Ausdehnungslehre.^{[11]}
Both concepts have been elaborated and superseded by the modern procedure in respect to the axioms of geometry, and by the conception of abstract geometry involved therein. Riemann proceeds to specialize the manifold by considerations as to measurement. If measurement is to Measure of curvature. be possible, some magnitude, we saw, must be independent of position; let us consider manifolds in which lengths of lines are such magnitudes, so that every line is measurable by every other. The coordinates of a point being x_{1}, x_{2}, ... x_{n}, let us confine ourselves to lines along which the ratios dx_{1} : dx_{2} : ... : dx_{n} alter continuously. Let us also assume that the element of length, ds, is unchanged (to the first order) when all its points undergo the same infinitesimal motion. Then if all the increments dx be altered in the same ratio, ds is also altered in this ratio. Hence ds is a homogeneous function of the first degree of the increments dx. Moreover, ds must be unchanged when all the dx change sign. The simplest possible case is, therefore, that in which ds is the square root of a quadratic function of the dx. This case includes space, and is alone considered in what follows. It is called the case of flatness in the smallest parts. Its further discussion depends upon the measure of curvature, the second of Riemann’s fundamental conceptions. This conception, derived from the theory of surfaces, is applied as follows. Any one of the shortest lines which issue from a given point (say the origin) is completely determined by the initial ratios of the dx. Two such lines, defined by dx and δx say, determine a pencil, or onedimensional series, of shortest lines, any one of which is defined by λdx + μδx, where the parameter λ : μ may have any value. This pencil generates a twodimensional series of points, which may be regarded as a surface, and for which we may apply Gauss’s formula for the measure of curvature at any point. Thus at every point of our manifold there is a measure of curvature corresponding to every such pencil; but all these can be found when n·n − 1/2 of them are known. If figures are to be freely movable, it is necessary and sufficient that the measure of curvature should be the same for all points and all directions at each point. Where this is the case, if α be the measure of curvature, the linear element can be put into the form
If α be positive, space is finite, though still unbounded, and every straight line is closed—a possibility first recognized by Riemann. It is pointed out that, since the possible values of a form a continuous series, observations cannot prove that our space is strictly Euclidean. It is also regarded as possible that, in the infinitesimal, the measure of curvature of our space should be variable.
There are four points in which this profound and epochmaking work is open to criticism or development—(1) the idea of a manifold requires more precise determination; (2) the introduction of coordinates is entirely unexplained and the requisite presuppositions are unanalysed; (3) the assumption that ds is the square root of a quadratic function of dx_{1}, dx_{2}, ... is arbitrary; (4) the idea of superposition, or congruence, is not adequately analysed. The modern solution of these difficulties is properly considered in connexion with the general subject of the axioms of geometry.
The publication of Riemann’s dissertation was closely followed by two works of Hermann von Helmholtz,^{[12]} again undertaken in ignorance of the work of predecessors. In these a Helmholtz. proof is attempted that ds must be a rational integral quadratic function of the increments of the coordinates. This proof has since been shown by Lie to stand in need of correction (see VII. Axioms of Geometry). Helmholtz’s remaining works on the subject^{[13]} are of almost exclusively philosophical interest. We shall return to them later.
The only other writer of importance in the second period is Eugenio Beltrami, by whom Riemann’s work was brought into connexion with that of Lobatchewsky and Bolyai. As he gave, by an elegant method, a convenient Beltrami. Euclidean interpretation of hyperbolic plane geometry, his results will be stated at some length^{[14]}. The Saggio shows that Lobatchewsky’s plane geometry holds in Euclidean geometry on surfaces of constant negative curvature, straight lines being replaced by geodesics. Such surfaces are capable of a conformal representation on a plane, by which geodesics are represented by straight lines. Hence if we take, as coordinates on the surface, the Cartesian coordinates of corresponding points on the plane, the geodesics must have linear equations.
Hence it follows that
where w^{2} = a^{2} − u^{2} − v^{2}, and −1/R^{2} is the measure of curvature of our surface (note that k = γ as used above). The angle between two geodesics u = const., v = const. is θ, where
Thus u = 0 is orthogonal to all geodesies v = const., and vice versa. In order that sin θ may be real, w^{2} must be positive; thus geodesics have no real intersection when the corresponding straight lines intersect outside the circle u^{2} + v^{2} = α^{2}. When they intersect on this circle, θ = 0. Thus Lobatchewsky’s parallels are represented by straight lines intersecting on the circle. Again, transforming to polar coordinates u = r cos μ, v = r sin μ, and calling ρ the geodesic distance of u, v from the origin, we have, for a geodesic through the origin,
Thus points on the surface corresponding to points in the plane on the limiting circle r = a, are all at an infinite distance from the origin. Again, considering r constant, the arc of a geodesic circle subtending an angle μ at the origin is
whence the circumference of a circle of radius ρ is 2πR sinh (ρ/R). Again, if α be the angle between any two geodesics
then
Thus α is imaginary when u, v is outside the limiting circle, and is zero when, and only when, u, v is on the limiting circle. All these results agree with those of Lobatchewsky and Bolyai. The maximum triangle, whose angles are all zero, is represented in the auxiliary plane by a triangle inscribed in the limiting circle. The angle of parallelism is also easily obtained. The perpendicular to v = 0 at a distance δ from the origin is u = a tanh (δ/R), and the parallel to this through the origin is u = v sinh (δ/R). Hence Π (δ), the angle which this parallel makes with v = 0, is given by
which is Lobatchewsky’s formula. We also obtain easily for the area of a triangle the formula R^{2}(π − A − B − C).
Beltrami’s treatment connects two curves which, in the earlier treatment, had no connexion. These are limitlines and curves of constant distance from a straight line. Both may be regarded as circles, the first having an infinite, the second an imaginary radius. The equation to a circle of radius ρ and centre u_{0}v_{0} is
This equation remains real when ρ is a pure imaginary, and remains finite when w_{0} = 0, provided ρ becomes infinite in such a way that w_{0} cosh (ρ/R) remains finite. In the latter case the equation represents a limitline. In the former case, by giving different values to C, we obtain concentric circles with the imaginary centre u_{0}v_{0}. One of these, obtained by putting C = 0, is the straight line a^{2} − uu_{0} − vv_{0} = 0. Hence the others are each throughout at a constant distance from this line. (It may be shown that all motions in a hyperbolic plane consist, in a general sense, of rotations; but three types must be distinguished according as the centre is real, imaginary or at infinity. All points describe, accordingly, one of the three types of circles.)
The above Euclidean interpretation fails for three or more dimensions. In the Teoria fondamentale, accordingly, where n dimensions are considered, Beltrami treats hyperbolic space in a purely analytical spirit. The paper shows that Lobatchewsky’s space of any number of dimensions has, in Riemann’s sense, a constant negative measure of curvature. Beltrami starts with the formula (analogous to that of the Saggio)
where
He shows that geodesics are represented by linear equations between x_{1}, x_{2}, ..., x_{n}, and that the geodesic distance ρ between two points x and x′ is given by
cosh  ρ  =  a^{2} − x_{1}x′_{1} − x_{2}x′_{2} − ... − x_{n}x′_{n} 
R  {(a^{2} − x_{1}^{2} − x_{2}^{2} − ... − x_{n}^{2}) (a^{2} − x′_{1}^{2} − x′_{2}^{2} − ... − x′_{n}^{2})}^{1⁄2} 
(a formula practically identical with Cayley’s, though obtained by a very different method). In order to show that the measure of curvature is constant, we make the substitutions
Also calling ρ the geodesic distance from the origin, we have
cos h (ρ/R) =  a  , sinh (ρ/R) =  r  . 
√(a^{2} − r^{2})  √(a^{2} − r^{2}) 
we obtain
ds^{2} = Σdz^{2} +  1  { (  R  sinh  ρ  )  ^{2}  − 1 } Σ (z_{i}dz_{k} − z_{k}dz_{i})^{2}. 
ρ^{2}  ρ  R 
Hence when ρ is small, we have approximately
ds^{2} = Σdz^{2} +13R^{2}Σ (z_{i}dz_{k} − z_{k}dz_{i})^{2}  (1). 
Considering a surface element through the origin, we may choose
our axes so that, for this element,
Thus
ds^{2} = dz^{2}1+dz^{2}2+13R^{2} (z_{1}dz_{2} − z_{2}dz_{1})^{2}  (2). 
Now the area of the triangle whose vertices are (0, 0), (z_{1}, z_{2}), (dz_{1}, dz_{2}) is 12(z_{1}, dz_{2} − z_{2}dz_{1}). Hence the quotient when the terms of the fourth order in (2) are divided by the square of this triangle is 4/3R^{2}; hence, returning to general axes, the same is the quotient when the terms of the fourth order in (1) are divided by the square of the triangle whose vertices are (0, 0, . . . 0), (z_{1}, z_{2}, z_{3}, . . . z_{n}), (dz_{1}, dz_{2}, dz_{3} . . . dz_{n}). But −34 of this quotient is defined by Riemann as the measure of curvature.^{[15]} Hence the measure of curvature is −1/R^{2}, i.e. is constant and negative. The properties of parallels, triangles, &c., are as in the Saggio. It is also shown that the analogues of limit surfaces have zero curvature; and that spheres of radius ρ have constant positive curvature 1/R^{2} sinh^{2} (ρ/R), so that spherical geometry may be regarded as contained in the pseudospherical (as Beltrami calls Lobatchewsky’s system).
The Saggio, as we saw, gives a Euclidean interpretation confined to two dimensions. But a consideration of the auxiliary plane suggests a different interpretation, which may be extended to any number of dimensions. If, instead of referring to the pseudosphere, we merely define Transition to the projective method. distance and angle, in the Euclidean plane, as those functions of the coordinates which gave us distance and angle on the pseudosphere, we find that the geometry of our plane has become Lobatchewsky’s. All the points of the limiting circle are now at infinity, and points beyond it are imaginary. If we give our circle an imaginary radius the geometry on the plane becomes elliptic. Replacing the circle by a sphere, we obtain an analogous representation for three dimensions. Instead of a circle or sphere we may take any conic or quadric. With this definition, if the fundamental quadric be Σ_{xx} = 0, and if Σ_{xx}′ be the polar form of Σ_{xx}, the distance ρ between x and x′ is given by the projective formula
That this formula is projective is rendered evident by observing that e^{−2iρ/k} is the anharmonic ratio of the range consisting of the two points and the intersections of the line joining them with the fundamental quadric. With this we are brought to the third or projective period. The method of this period is due to Cayley; its application to previous nonEuclidean geometry is due to Klein. The projective method contains a generalization of discoveries already made by Laguerre^{[16]} in 1853 as regards Euclidean geometry. The arbitrariness of this procedure of deriving metrical geometry from the properties of conics is removed by Lie’s theory of congruence. We then arrive at the stage of thought which finds its expression in the modern treatment of the axioms of geometry.
The projective method leads to a discrimination, first made by Klein,^{[17]} of two varieties of Riemann’s space; Klein calls these elliptic and spherical. They are also called the polar and antipodal forms of elliptic space. The latter names will here be used. The difference is strictly The two kinds of elliptic space. analogous to that between the diameters and the points of a sphere. In the polar form two straight lines in a plane always intersect in one and only one point; in the antipodal form they intersect always in two points, which are antipodes. According to the definition of geometry adopted in section VII. (Axioms of Geometry), the antipodal form is not to be termed “geometry,” since any pair of coplanar straight lines intersect each other in two points. It may be called a “quasigeometry.” Similarly in the antipodal form two diameters always determine a plane, but two points on a sphere do not determine a great circle when they are antipodes, and two great circles always intersect in two points. Again, a plane does not form a boundary among lines through a point: we can pass from any one such line to any other without passing through the plane. But a great circle does divide the surface of a sphere. So, in the polar form, a complete straight line does not divide a plane, and a plane does not divide space, and does not, like a Euclidean plane, have two sides.^{[18]} But, in the antipodal form, a plane is, in these respects, like a Euclidean plane.
It is explained in section VII. in what sense the metrical geometry of the material world can be considered to be determinate and not a matter of arbitrary choice. The scientific question as to the best available evidence concerning the nature of this geometry is one beset with difficulties of a peculiar kind. We are obstructed by the fact that all existing physical science assumes the Euclidean hypothesis. This hypothesis has been involved in all actual measurements of large distances, and in all the laws of astronomy and physics. The principle of simplicity would therefore lead us, in general, where an observation conflicted with one or more of those laws, to ascribe this anomaly, not to the falsity of Euclidean geometry, but to the falsity of the laws in question. This applies especially to astronomy. On the earth our means of measurement are many and direct, and so long as no great accuracy is sought they involve few scientific laws. Thus we acquire, from such direct measurements, a very high degree of probability that the spaceconstant, if not infinite, is yet large as compared with terrestrial distances. But astronomical distances and triangles can only be measured by means of the received laws of astronomy and optics, all of which have been established by assuming the truth of the Euclidean hypothesis. It therefore remains possible (until a detailed proof of the contrary is forthcoming) that a large but finite spaceconstant, with different laws of astronomy and optics, would have equally explained the phenomena. We cannot, therefore, accept the measurements of stellar parallaxes, &c., as conclusive evidence that the spaceconstant is large as compared with stellar distances. For the present, on grounds of simplicity, we may rightly adopt this view; but it must remain possible that, in view of some hitherto undiscovered discrepancy, a slight correction of the sort suggested might prove the simplest alternative. But conversely, a finite parallax for very distant stars, or a negative parallax for any star, could not be accepted as conclusive evidence that our geometry is nonEuclidean, unless it were shown—and this seems scarcely possible—that no modification of astronomy or optics could account for the phenomenon. Thus although we may admit a probability that the spaceconstant is large in comparison with stellar distances, a conclusive proof or disproof seems scarcely possible.
Finally, it is of interest to note that, though it is theoretically possible to prove, by scientific methods, that our geometry is nonEuclidean, it is wholly impossible to prove by such methods that it is accurately Euclidean. For the unavoidable errors of observation must always leave a slight margin in our measurements. A triangle might be found whose angles were certainly greater, or certainly less, than two right angles; but to prove them exactly equal to two right angles must always be beyond our powers. If, therefore, any man cherishes a hope of proving the exact truth of Euclid, such a hope must be based, not upon scientific, but upon philosophical considerations.
Bibliography.—The bibliography appended to section VII. should be consulted in this connexion. Also, in addition to the citations already made, the following works may be mentioned.
For Lobatchewsky’s writings, cf. Urkunden zur Geschichte der nichteuklidischen Geometrie, i., Nikolaj Iwanowitsch Lobatschefsky, by F. Engel and P. Stäckel (Leipzig, 1898). For John Bolyai’s Appendix, cf. Absolute Geometrie nach Johann Bolyai, by J. Frischauf (Leipzig, 1872), and also the new edition of his father’s large work, Tentamen . . ., published by the Mathematical Society of Budapest; the second volume contains the appendix. Cf. also J. Frischauf, Elemente der absoluten Geometrie (Leipzig, 1876); M. L. Gérard, Sur la géométrie nonEuclidienne (thesis for doctorate) (Paris, 1892); de Tilly, Essai sur les principes fondamentales de la géométrie et de la mécanique (Bordeaux, 1879); Sir R. S. Ball, “On the Theory of Content,” Trans. Roy. Irish Acad. vol. xxix. (1889); F. Lindemann, “Mechanik bei projectiver Maasbestimmung,” Math. Annal. vol. vii.; W. K. Clifford, “Preliminary Sketch of Biquaternions,” Proc. of Lond. Math. Soc. (1873), and Coll. Works; A. Buchheim, “On the Theory of Screws in Elliptic Space,” Proc. Lond. Math. Soc. vols. xv., xvi., xvii.; H. Cox, “On the Application of Quaternions and Grassmann’s Algebra to different Kinds of Uniform Space,” Trans. Camb. Phil. Soc. (1882); M. Dehn, “Die Legendarischen Sätze über die Winkelsumme im Dreieck,” Math. Ann. vol. 53 (1900), and “Über den Rauminhalt,” Math. Annal. vol. 55 (1902).
For expositions of the whole subject, cf. F. Klein, NichtEuklidische Geometrie (Göttingen, 1893); R. Bonola, La Geometria nonEuclidea (Bologna, 1906); P. Barbarin, La Géométrie nonEuclidienne (Paris, 1902); W. Killing, Die nichtEuklidischen Raumformen in analytischer Behandlung (Leipzig, 1885). The lastnamed work also deals with geometry of more than three dimensions; in this connexion cf. also G. Veronese, Fondamenti di geometria a più dimensioni ed a più specie di unità rettilinee . . . (Padua, 1891, German translation, Leipzig, 1894); G. Fontené, L’Hyperespace à (n − 1) dimensions (Paris, 1892); and A. N. Whitehead, loc. cit. Cf. also E. Study, “Über nichtEuklidische und Liniengeometrie,” Jahr. d. Deutsch. Math. Ver. vol. xv. (1906); W. Burnside, “On the Kinematics of nonEuclidean Space,” Proc. Lond. Math. Soc. vol. xxvi. (1894). A bibliography on the subject up to 1878 has been published by G. B. Halsted, Amer. Journ. of Math. vols. i. and ii.; and one up to 1900 by R. Bonola, Index operum ad geometriam absolutam spectantium . . . (1902, and Leipzig, 1903). (B. A. W. R.; A. N. W.)
VII. Axioms of Geometry
Until the discovery of the nonEuclidean geometries (Lobatchewsky, 1826 and 1829; J. Bolyai, 1832; B. Riemann, 1854),
geometry was universally considered as being exclusively
the science of existent space. (See section VI. NonEuclidean Geometry.) In respect to the Theories
of space.
science, as thus conceived, two controversies may be noticed.
First, there is the controversy respecting the absolute and
relational theories of space. According to the absolute theory,
which is the traditional view (held explicitly by Newton), space
has an existence, in some sense whatever it may be, independent
of the bodies which it contains. The bodies occupy space, and
it is not intrinsically unmeaning to say that any definite body
occupies this part of space, and not that part of space, without
reference to other bodies occupying space. According to the
relational theory of space, of which the chief exponent was
Leibnitz,^{[19]} space is nothing but a certain assemblage of the relations
between the various particular bodies in space. The idea of
space with no bodies in it is absurd. Accordingly there can be
no meaning in saying that a body is here and not there, apart
from a reference to the other bodies in the universe. Thus, on
this theory, absolute motion is intrinsically unmeaning. It is
admitted on all hands that in practice only relative motion is
directly measurable. Newton, however, maintains in the
Principia (scholium to the 8th definition) that it is indirectly
measurable by means of the effects of “centrifugal force” as
it occurs in the phenomena of rotation. This irrelevance of
absolute motion (if there be such a thing) to science has led to
the general adoption of the relational theory by modern men
of science. But no decisive argument for either view has at
present been elaborated.^{[20]} Kant’s view of space as being a form
of perception at first sight appears to cut across this controversy.
But he, saturated as he was with the spirit of the Newtonian
physics, must (at least in both editions of the Critique) be classed
with the upholders of the absolute theory. The form of perception
has a type of existence proper to itself independently
of the particular bodies which it contains. For example he
writes:^{[21]} “Space does not represent any quality of objects by
themselves, or objects in their relation to one another, i.e. space
does not represent any determination which is inherent in the
objects themselves, and would remain, even if all subjective
conditions of intuition were removed.”
The second controversy is that between the view that the axioms applicable to space are known only from experience, and the view that in some sense these axioms are given a priori. Both these views, thus broadly stated, Axioms. are capable of various subtle modifications, and a discussion of them would merge into a general treatise on epistemology. The cruder forms of the a priori view have been made quite untenable by the modern mathematical discoveries. Geometers now profess ignorance in many respects of the exact axioms which apply to existent space, and it seems unlikely that a profound study of the question should thus obliterate a priori intuitions.
Another question irrelevant to this article, but with some relevance to the above controversy, is that of the derivation of our perception of existent space from our various types of sensation. This is a question for psychology.^{[22]}
Definition of Abstract Geometry.—Existent space is the subject matter of only one of the applications of the modern science of abstract geometry, viewed as a branch of pure mathematics. Geometry has been defined^{[23]} as “the study of series of two or more dimensions.” It has also been defined^{[24]} as “the science of cross classification.” These definitions are founded upon the actual practice of mathematicians in respect to their use of the term “Geometry.” Either of them brings out the fact that geometry is not a science with a determinate subject matter. It is concerned with any subject matter to which the formal axioms may apply. Geometry is not peculiar in this respect. All branches of pure mathematics deal merely with types of relations. Thus the fundamental ideas of geometry (e.g. those of points and of straight lines) are not ideas of determinate entities, but of any entities for which the axioms are true. And a set of formal geometrical axioms cannot in themselves be true or false, since they are not determinate propositions, in that they do not refer to a determinate subject matter. The axioms are propositional functions.^{[25]} When a set of axioms is given, we can ask (1) whether they are consistent, (2) whether their “existence theorem” is proved, (3) whether they are independent. Axioms are consistent when the contradictory of any axiom cannot be deduced from the remaining axioms. Their existence theorem is the proof that they are true when the fundamental ideas are considered as denoting some determinate subject matter, so that the axioms are developed into determinate propositions. It follows from the logical law of contradiction that the proof of the existence theorem proves also the consistency of the axioms. This is the only method of proof of consistency. The axioms of a set are independent of each other when no axiom can be deduced from the remaining axioms of the set. The independence of a given axiom is proved by establishing the consistency of the remaining axioms of the set, together with the contradictory of the given axiom. The enumeration of the axioms is simply the enumeration of the hypotheses^{[26]} (with respect to the undetermined subject matter) of which some at least occur in each of the subsequent propositions.
Any science is called a “geometry” if it investigates the theory of the classification of a set of entities (the points) into classes (the straight lines), such that (1) there is one and only one class which contains any given pair of the entities, and (2) every such class contains more than two members. In the two geometries, important from their relevance to existent space, axioms which secure an order of the points on any line also occur. These geometries will be called “Projective Geometry” and “Descriptive Geometry.” In projective geometry any two straight lines in a plane intersect, and the straight lines are closed series which return into themselves, like the circumference of a circle. In descriptive geometry two straight lines in a plane do not necessarily intersect, and a straight line is an open series without beginning or end. Ordinary Euclidean geometry is a descriptive geometry; it becomes a projective geometry when the socalled “points at infinity” are added.
Projective Geometry.
Projective geometry may be developed from two undefined fundamental ideas, namely, that of a “point” and that of a “straight line.” These undetermined ideas take different specific meanings for the various specific subject matters to which projective geometry can be applied. The number of the axioms is always to some extent arbitrary, being dependent upon the verbal forms of statement which are adopted. They will be presented^{[27]} here as twelve in number, eight being “axioms of classification,” and four being “axioms of order.”
Axioms of Classification.—The eight axioms of classification are as follows:
1. Points form a class of entities with at least two members.
2. Any straight line is a class of points containing at least three members.
3. Any two distinct points lie in one and only one straight line.
4. There is at least one straight line which does not contain all the points.
5. If A, B, C are noncollinear points, and A′ is on the straight line BC, and B′ is on the straight line CA, then the straight lines AA′ and BB′ possess a point in common.
Definition.—If A, B, C are any three noncollinear points, the plane ABC is the class of points lying on the straight lines joining A with the various points on the straight line BC.
6. There is at least one plane which does not contain all the points.
7. There exists a plane α, and a point A not incident in α, such that any point lies in some straight line which contains both A and a point in α.
Definition.—Harm. (ABCD) symbolizes the following conjoint statements: (1) that the points A, B, C, D are collinear, and (2) that a quadrilateral can be found with one pair of opposite sides intersecting at A, with the other pair intersecting at C, and with its diagonals passing through B and D respectively. Then B and D are said to be “harmonic conjugates” with respect to A and C.
8. Harm. (ABCD) implies that B and D are distinct points.
In the above axioms 4 secures at least two dimensions, axiom 5 is the fundamental axiom of the plane, axiom 6 secures at least three dimensions, and axiom 7 secures at most three dimensions. From axioms 15 it can be proved that any two distinct points in a straight line determine that line, that any three noncollinear points in a plane determine that plane, that the straight line containing any two points in a plane lies wholly in that plane, and that any two straight lines in a plane intersect. From axioms 16 Desargue’s wellknown theorem on triangles in perspective can be proved.
The enunciation of this theorem is as follows: If ABC and A′B′C′ are two coplanar triangles such that the lines AA′, BB′, CC′ are concurrent, then the three points of intersection of BC and B′C′ of CA and C′A′, and of AB and A′B′ are collinear; and conversely if the three points of intersection are collinear, the three lines are concurrent. The proof which can be applied is the usual projective proof by which a third triangle A″B″C″ is constructed not coplanar with the other two, but in perspective with each of them.
It has been proved^{[28]} that Desargues’s theorem cannot be deduced from axioms 15, that is, if the geometry be confined to two dimensions. All the proofs proceed by the method of producing a specification of “points” and “straight lines” which satisfies axioms 15, and such that Desargues’s theorem does not hold.
It follows from axioms 15 that Harm. (ABCD) implies Harm. (ADCB) and Harm. (CBAD), and that, if A, B, C be any three distinct collinear points, there exists at least one point D such that Harm. (ABCD). But it requires Desargues’s theorem, and hence axiom 6, to prove that Harm. (ABCD) and Harm. (ABCD′) imply the identity of D and D′.
The necessity for axiom 8 has been proved by G. Fano,^{[29]} who has produced a three dimensional geometry of fifteen points, i.e. a method of cross classification of fifteen entities, in which each straight line contains three points, and each plane contains seven straight lines. In this geometry axiom 8 does not hold. Also from axioms 16 and 8 it follows that Harm. (ABCD) implies Harm. (BCDA).
Definitions.—When two plane figures can be derived from one another by a single projection, they are said to be in perspective. When two plane figures can be derived one from the other by a finite series of perspective relations between intermediate figures, they are said to be projectively related. Any property of a plane figure which necessarily also belongs to any projectively related figure, is called a projective property.
The following theorem, known from its importance as “the fundamental theorem of projective geometry,” cannot be proved^{[30]} from axioms 18. The enunciation is: “A projective correspondence between the points on two straight lines is completely determined when the correspondents of three distinct points on one line are determined on the other.” This theorem is equivalent^{[31]} (assuming axioms 18) to another theorem, known as Pappus’s Theorem, namely: “If l and l ′ are two distinct coplanar lines, and A, B, C are three distinct points on l, and A′, B′, C′ are three distinct points on l ′, then the three points of intersection of AA′ and B′C, of A′B and CC′, of BB′ and C′A, are collinear.” This theorem is obviously Pascal’s wellknown theorem respecting a hexagon inscribed in a conic, for the special case when the conic has degenerated into the two lines l and l ′. Another theorem also equivalent (assuming axioms 18) to the fundamental theorem is the following:^{[32]} If the three collinear pairs of points, A and A′, B and B′, C and C′, are such that the three pairs of opposite sides of a complete quadrangle pass respectively through them, i.e. one pair through A and A′ respectively, and so on, and if also the three sides of the quadrangle which pass through A, B, and C, are concurrent in one of the corners of the quadrangle, then another quadrangle can be found with the same relation to the three pairs of points, except that its three sides which pass through A, B, and C, are not concurrent.
Thus, if we choose to take any one of these three theorems as an axiom, all the theorems of projective geometry which do not require ordinal or metrical ideas for their enunciation can be proved. Also a conic can be defined as the locus of the points found by the usual construction, based upon Pascal’s theorem, for points on the conic through five given points. But it is unnecessary to assume here any one of the suggested axioms; for the fundamental theorem can be deduced from the axioms of order together with axioms 18.
Axioms of Order.—It is possible to define (cf. Pieri, loc. cit.) the property upon which the order of points on a straight line depends. But to secure that this property does in fact range the points in a serial order, some axioms are required. A straight line is to be a closed series; thus, when the points are in order, it requires two points on the line to divide it into two distinct complementary segments, which do not overlap, and together form the whole line. Accordingly the problem of the definition of order reduces itself to the definition of these two segments formed by any two points on the line; and the axioms are stated relatively to these segments.
Definition.—If A, B, C are three collinear points, the points on the segment ABC are defined to be those points such as X, for which there exist two points Y and Y′ with the property that Harm. (AYCY′) and Harm. (BYXY′) both hold. The supplementary segment ABC is defined to be the rest of the points on the line. This definition is elucidated by noticing that with our ordinary geometrical ideas, if B and X are any two points between A and C, then the two pairs of points, A and C, B and X, define an involution with real double points, namely, the Y and Y′ of the above definition. The property of belonging to a segment ABC is projective, since the harmonic relation is projective.
The first three axioms of order (cf. Pieri, loc. cit.) are:
9. If A, B, C are three distinct collinear points, the supplementary segment ABC is contained within the segment BCA.
10. If A, B, C are three distinct collinear points, the common part of the segments BCA and CAB is contained in the supplementary segment ABC.
11. If A, B, C are three distinct collinear points, and D lies In the segment ABC, then the segment ADC is contained within the segment ABC.
From these axioms all the usual properties of a closed order follow. It will be noticed that, if A, B, C are any three collinear points, C is necessarily traversed in passing from A to B by one route along the line, and is not traversed in passing from A to B along the other route. Thus there is no meaning, as referred to closed straight lines, in the simple statement that C lies between A and B. But there may be a relation of separation between two pairs of collinear points, such as A and C, and B and D. The couple B and D is said to separate A and C, if the four points are collinear and D lies in the segment complementary to the segment ABC. The property of the separation of pairs of points by pairs of points is projective. Also it can be proved that Harm. (ABCD) implies that B and D separate A and C.
Definitions.—A series of entities arranged in a serial order, open or closed, is said to be compact, if the series contains no immediately consecutive entities, so that in traversing the series from any one entity to any other entity it is necessary to pass through entities distinct from either. It was the merit of R. Dedekind and of G. Cantor explicitly to formulate another fundamental property of series. The Dedekind property^{[33]} as applied to an open series can be defined thus: An open series possesses the Dedekind property, if, however, it be divided into two mutually exclusive classes u and v, which (1) contain between them the whole series, and (2) are such that every member of u precedes in the serial order every member of v, there is always a member of the series, belonging to one of the two, u or v, which precedes every member of v (other than itself if it belong to v), and also succeeds every member of u (other than itself if it belong to u). Accordingly in an open series with the Dedekind property there is always a member of the series marking the junction of two classes such as u and v. An open series is continuous if it is compact and possesses the Dedekind property. A closed series can always be transformed into an open series by taking any arbitrary member as the first term and by taking one of the two ways round as the ascending order of the series. Thus the definitions of compactness and of the Dedekind property can be at once transferred to a closed series.
12. The last axiom of order is that there exists at least one straight line for which the point order possesses the Dedekind property.
It follows from axioms 112 by projection that the Dedekind property is true for all lines. Again the harmonic system ABC, where A, B, C are collinear points, is defined^{[34]} thus: take the harmonic conjugates A′, B′, C′ of each point with respect to the other two, again take the harmonic conjugates of each of the six points A, B, C, A′, B′, C′ with respect to each pair of the remaining five, and proceed in this way by an unending series of steps. The set of points thus obtained is called the harmonic system ABC. It can be proved that a harmonic system is compact, and that every segment of the line containing it possesses members of it. Furthermore, it is easy to prove that the fundamental theorem holds for harmonic systems, in the sense that, if A, B, C are three points on a line l, and A′, B′, C′ are three points on a line l′, and if by any two distinct series of projections A, B, C are projected into A′, B′, C′, then any point of the harmonic system ABC corresponds to the same point of the harmonic system A′B′C′ according to both the projective relations which are thus established between l and l′. It now follows immediately that the fundamental theorem must hold for all the points on the lines l and l′, since (as has been pointed out) harmonic systems are “everywhere dense” on their containing lines. Thus the fundamental theorem follows from the axioms of order.
A system of numerical coordinates can now be introduced, possessing the property that linear equations represent planes and straight lines. The outline of the argument by which this remarkable problem (in that “distance” is as yet undefined) is solved, will now be given. It is first proved that the points on any line can in a certain way be definitely associated with all the positive and negative real numbers, so as to form with them a oneone correspondence. The arbitrary elements in the establishment of this relation are the points on the line associated with 0, 1 and ∞.
This association^{[35]} is most easily effected by considering a class of projective relations of the line with itself, called by F. Schur (loc. cit.) prospectivities.
Fig. 69. 
Fig. 70. 
Fig. 71. 
Fig. 72. 
Let l (fig. 69) be the given line, m and n any two lines intersecting at U on l, S and S′ two points on n. Then a projective relation between l and itself is formed by projecting l from S on to m, and then by projecting m from S′ back on to l. All such projective relations, however m, n, S and S′ be varied, are called “prospectivities,” and U is the double point of the prospectivity. If a point O on l is related to A by a prospectivity, then all prospectivities, which (1) have the same double point U, and (2) relate O to A, give the same correspondent (Q, in figure) to any point P on the line l; in fact they are all the same prospectivity, however m, n, S, and S′ may have been varied subject to these conditions. Such a prospectivity will be denoted by (OAU^{2}).
The sum of two prospectivities, written (OAU^{2}) + (OBU^{2}), is defined to be that transformation of the line l into itself which is obtained by first applying the prospectivity (OAU^{2}) and then applying the prospectivity (OBU^{2}). Such a transformation, when the two summands have the same double point, is itself a prospectivity with that double point.
With this definition of addition it can be proved that prospectivities with the same double point satisfy all the axioms of magnitude. Accordingly they can be associated in a oneone correspondence with the positive and negative real numbers. Let E (fig. 70) be any point on l, distinct from O and U. Then the prospectivity (OEU^{2}) is associated with unity, the prospectivity (OOU^{2}) is associated with zero, and (OUU^{2}) with ∞. The prospectivities of the type (OPU^{2}), where P is any point on the segment OEU, correspond to the positive numbers; also if P′ is the harmonic conjugate of P with respect to O and U, the prospectivity (OP′U^{2}) is associated with the corresponding negative number. (The subjoined figure explains this relation of the positive and negative prospectivities.) Then any point P on l is associated with the same number as is the prospectivity (OPU^{2}).
It can be proved that the order of the numbers in algebraic order of magnitude agrees with the order on the line of the associated points. Let the numbers, assigned according to the preceding specification, be said to be associated with the points according to the “numerationsystem (OEU).” The introduction of a coordinate system for a plane is now managed as follows: Take any triangle OUV in the plane, and on the lines OU and OV establish the numeration systems (OE_{1}U) and (OE_{2}V), where E_{1} and E_{2} are arbitrarily chosen. Then (cf. fig. 71) if M and N are associated with the numbers x and y according to these systems, the coordinates of P are x and y. It then follows that the equation of a straight line is of the form ax + by + c = 0. Both coordinates of any point on the line UV are infinite. This can be avoided by introducing homogeneous coordinates X, Y, Z, where x = X/Z, and y = Y/Z, and Z = 0 is the equation of UV.
The procedure for three dimensions is similar. Let OUVW (fig. 72) be any tetrahedron, and associate points on OU, OV, OW with numbers according to the numeration systems (OE_{1}U), (OE_{2}V), and (OE_{3}W). Let the planes VWP, WUP, UVP cut OU, OV, OW in L, M, N respectively; and let x, y, z be the numbers associated with L, M, N respectively. Then P is the point (x, y, z). Also homogeneous coordinates can be introduced as before, thus avoiding the infinities on the plane UVW.
The cross ratio of a range of four collinear points can now be defined as a number characteristic of that range. Let the coordinates of any point P_{r} of the range P_{1} P_{2} P_{3} P_{4} be
λ_{r}a + μ_{r} + a′  ,  λ_{r}b + μ_{r}b′  ,  λ_{r}c + μ_{r}c′  , (r = 1, 2, 3, 4) 
λ_{r} + μ_{r}  λ_{r} + μ_{r}  λ_{r} + μ_{r} 
and let (λ_{r}μ_{s}) be written for λ_{r}μ_{s} −λ_{s}μ_{r}. Then the cross ratio {P_{1} P_{2} P_{3} P_{4}} is defined to be the number (λ_{1}μ_{2})(λ_{3}μ_{4}) / (λ_{1}μ_{4})(λ_{3}μ_{2}). The equality of the cross ratios of the ranges (P_{1} P_{2} P_{3} P_{4}) and (Q_{1} Q_{2} Q_{3} Q_{4}) is proved to be the necessary and sufficient condition for their mutual projectivity. The cross ratios of all harmonic ranges are then easily seen to be all equal to −1, by comparing with the range (OE_{1}UE′_{1}) on the axis of x.
Thus all the ordinary propositions of geometry in which distance and angular measure do not enter otherwise than in cross ratios can now be enunciated and proved. Accordingly the greater part of the analytical theory of conics and quadrics belongs to geometry at this stage The theory of distance will be considered after the principles of descriptive geometry have been developed.
Descriptive geometry is essentially the science of multiple order for open series. The first satisfactory system of axioms was given by M. Pasch.^{[36]} An improved version is due to G. Peano.^{[37]} Both these authors treat the idea of the class of points constituting the segment lying between two points as an undefined fundamental idea. Thus in fact there are in this system two fundamental ideas, namely, of points and of segments. It is then easy enough to define the prolongations of the segments, so as to form the complete straight lines. D. Hilbert’s^{[38]} formulation of the axioms is in this respect practically based on the same fundamental ideas. His work is justly famous for some of the mathematical investigations contained in it, but his exposition of the axioms is distinctly inferior to that of Peano. Descriptive geometry can also be considered^{[39]} as the science of a class of relations, each relation being a twotermed serial relation, as considered in the logic of relations, ranging the points between which it holds into a linear open order. Thus the relations are the straight lines, and the terms between which they hold are the points. But a combination of these two points of view yields^{[40]} the simplest statement of all. Descriptive geometry is then conceived as the investigation of an undefined fundamental relation between three terms (points); and when the relation holds between three points A, B, C, the points are said to be “in the [linear] order ABC.”
O. Veblen’s axioms and definitions, slightly modified, are as follows:—
1. If the points A, B, C are in the order ABC, they are in the order CBA.
2. If the points A, B, C are in the order ABC, they are not in the order BCA.
3. If the points A, B, C are in the order ABC, A is distinct from C.
4. If A and B are any two distinct points, there exists a point C such that A, B, C are in the order ABC.
Definition.—The line (A ± B) consists of A and B, and of all points X in one of the possible orders, ABX, AXB, XAB. The points X in the order AXB constitute the segment AB.
5. If points C and D (C ± D) lie on the line AB, then A lies on the line CD.
6. There exist three distinct points A, B, C not in any of the orders ABC, BCA, CAB.
Fig. 73. 
7. If three distinct points A, B, C (fig. 73) do not lie on the same line, and D and E are two distinct points in the orders BCD and CEA, then a point F exists in the order AFB, and such that D, E, F are collinear.
Definition.—If A, B, C are three noncollinear points, the plane ABC is the class of points which lie on any one of the lines joining any two of the points belonging to the boundary of the triangle ABC, the boundary being formed by the segments BC, CA and AB. The interior of the triangle ABC is formed by the points in segments such as PQ, where P and Q are points respectively on two of the segments BC, CA, AB.
8. There exists a plane ABC, which does not contain all the points.
Definition.—If A, B, C, D are four noncoplanar points, the space ABCD is the class of points which lie on any of the lines containing two points on the surface of the tetrahedron ABCD, the surface being formed by the interiors of the triangles ABC, BCD, DCA, DAB.
9. There exists a space ABCD which contains all the points.
10. The Dedekind property holds for the order of the points on any straight line.
It follows from axioms 19 that the points on any straight line are arranged in an open serial order. Also all the ordinary theorems respecting a point dividing a straight line into two parts, a straight line dividing a plane into two parts, and a plane dividing space into two parts, follow.
Again, in any plane α consider a line l and a point A (fig. 74).
Fig. 74. 
Let any point B divide l into two halflines l_{1} and l_{2}. Then it can be proved that the set of halflines, emanating from A and intersecting l_{1} (such as m), are bounded by two halflines, of which ABC is one. Let r be the other. Then it can be proved that r does not intersect l_{1}. Similarly for the halfline, such as n, intersecting l_{2}. Let s be its bounding halfline. Then two cases are possible. (1) The halflines r and s are collinear, and together form one complete line. In this case, there is one and only one line (viz. r + s) through A and lying in α which does not intersect l. This is the Euclidean case, and the assumption that this case holds is the Euclidean parallel axiom. But (2) the halflines r and s may not be collinear. In this case there will be an infinite number of lines, such as k for instance, containing A and lying in α, which do not intersect l. Then the lines through A in α are divided into two classes by reference to l, namely, the secant lines which intersect l, and the nonsecant lines which do not intersect l. The two boundary nonsecant lines, of which r and s are respectively halves, may be called the two parallels to l through A.
The perception of the possibility of case 2 constituted the startingpoint from which Lobatchewsky constructed the first explicit coherent theory of nonEuclidean geometry, and thus created a revolution in the philosophy of the subject. For many centuries the speculations of mathematicians on the foundations of geometry were almost confined to hopeless attempts to prove the “parallel axiom” without the introduction of some equivalent axiom.^{[41]}
Associated Projective and Descriptive Spaces.—A region of a projective space, such that one, and only one, of the two supplementary segments between any pair of points within it lies entirely within it, satisfies the above axioms (110) of descriptive geometry, where the points of the region are the descriptive points, and the portions of straight lines within the region are the descriptive lines. If the excluded part of the original projective space is a single plane, the Euclidean parallel axiom also holds, otherwise it does not hold for the descriptive space of the limited region. Again, conversely, starting from an original descriptive space an associated projective space can be constructed by means of the concept of ideal points.^{[42]} These are also called projective points, where it is understood that the simple points are the points of the original descriptive space. An ideal point is the class of straight lines which is composed of two coplanar lines a and b, together with the lines of intersection of all pairs of intersecting planes which respectively contain a and b, together with the lines of intersection with the plane ab of all planes containing any one of the lines (other than a or b) already specified as belonging to the ideal point. It is evident that, if the two original lines a and b intersect, the corresponding ideal point is nothing else than the whole class of lines which are concurrent at the point ab. But the essence of the definition is that an ideal point has an existence when the lines a and b do not intersect, so long as they are coplanar. An ideal point is termed proper, if the lines composing it intersect; otherwise it is improper.
A theorem essential to the whole theory is the following: if any two of the three lines a, b, c are coplanar, but the three lines are not all coplanar, and similarly for the lines a, b, d, then c and d are coplanar. It follows that any two lines belonging to an ideal point can be used as the pair of guiding lines in the definition. An ideal point is said to be coherent with a plane, if any of the lines composing it lie in the plane. An ideal line is the class of ideal points each of which is coherent with two given planes. If the planes intersect, the ideal line is termed proper, otherwise it is improper. It can be proved that any two planes, with which any two of the ideal points are both coherent, will serve as the guiding planes used in the definition. The ideal planes are defined as in projective geometry, and all the other definitions (for segments, order, &c.) of projective geometry are applied to the ideal elements. If an ideal plane contains some proper ideal points, it is called proper, otherwise it is improper. Every ideal plane contains some improper ideal points.
It can now be proved that all the axioms of projective geometry hold of the ideal elements as thus obtained; and also that the order of the ideal points as obtained by the projective method agrees with the order of the proper ideal points as obtained from that of the associated points of the descriptive geometry. Thus a projective space has been constructed out of the ideal elements, and the proper ideal elements correspond element by element with the associated descriptive elements. Thus the proper ideal elements form a region in the projective space within which the descriptive axioms hold. Accordingly, by substituting ideal elements, a descriptive space can always be considered as a region within a projective space. This is the justification for the ordinary use of the “points at infinity” in the ordinary Euclidean geometry; the reasoning has been transferred from the original descriptive space to the associated projective space of ideal elements; and with the Euclidean parallel axiom the improper ideal elements reduce to the ideal points on a single improper ideal plane, namely, the plane at infinity.^{[43]}
Congruence and Measurement.—The property of physical space which is expressed by the term “measurability” has now to be considered. This property has often been considered as essential to the very idea of space. For example, Kant writes,^{[44]} “Space is represented as an infinite given quantity.” This quantitative aspect of space arises from the measurability of distances, of angles, of surfaces and of volumes. These four types of quantity depend upon the two first among them as fundamental. The measurability of space is essentially connected with the idea of congruence, of which the simplest examples are to be found in the proofs of equality by the method of superposition, as used in elementary plane geometry. The mere concepts of “part” and of “whole” must of necessity be inadequate as the foundation of measurement, since we require the comparison as to quantity of regions of space which have no portions in common. The idea of congruence, as exemplified by the method of superposition in geometrical reasoning, appears to be founded upon that of the “rigid body,” which moves from one position to another with its internal spatial relations unchanged. But unless there is a previous concept of the metrical relations between the parts of the body, there can be no basis from which to deduce that they are unchanged.
It would therefore appear as if the idea of the congruence, or metrical equality, of two portions of space (as empirically suggested by the motion of rigid bodies) must be considered as a fundamental idea incapable of definition in terms of those geometrical concepts which have already been enumerated. This was in effect the point of view of Pasch.^{[45]} It has, however, been proved by Sophus Lie^{[46]} that congruence is capable of definition without recourse to a new fundamental idea. This he does by means of his theory of finite continuous groups (see Groups, Theory of), of which the definition is possible in terms of our established geometrical ideas, remembering that coordinates have already been introduced. The displacement of a rigid body is simply a mode of defining to the senses a oneone transformation of all space into itself. For at any point of space a particle may be conceived to be placed, and to be rigidly connected with the rigid body; and thus there is a definite correspondence of any point of space with the new point occupied by the associated particle after displacement. Again two successive displacements of a rigid body from position A to position B, and from position B to position C, are the same in effect as one displacement from A to C. But this is the characteristic “group” property. Thus the transformations of space into itself defined by displacements of rigid bodies form a group.
Call this group of transformations a congruencegroup. Now according to Lie a congruencegroup is defined by the following characteristics:—
1. A congruencegroup is a finite continuous group of oneone transformations, containing the identical transformation.
2. It is a subgroup of the general projective group, i.e. of the group of which any transformation converts planes into planes, and straight lines into straight lines.
3. An infinitesimal transformation can always be found satisfying the condition that, at least throughout a certain enclosed region, any definite line and any definite point on the line are latent, i.e. correspond to themselves.
4. No infinitesimal transformation of the group exists, such that, at least in the region for which (3) holds, a straight line, a point on it, and a plane through it, shall all be latent.
The property enunciated by conditions (3) and (4), taken together, is named by Lie “Free mobility in the infinitesimal.” Lie proves the following theorems for a projective space:—
1. If the above four conditions are only satisfied by a group throughout part of projective space, this part either (α) must be the region enclosed by a real closed quadric, or (β) must be the whole of the projective space with the exception of a single plane. In case (α) the corresponding congruence group is the continuous group for which the enclosing quadric is latent; and in case (β) an imaginary conic (with a real equation) lying in the latent plane is also latent, and the congruence group is the continuous group for which the plane and conic are latent.
2. If the above four conditions are satisfied by a group throughout the whole of projective space, the congruence group is the continuous group for which some imaginary quadric (with a real equation) is latent.
By a proper choice of nonhomogeneous coordinates the equation of any quadrics of the types considered, either in theorem 1 (α), or in theorem 2, can be written in the form 1 + c(x^{2} + y^{2} + z^{2}) = 0, where c is negative for a real closed quadric, and positive for an imaginary quadric. Then the general infinitesimal transformation is defined by the three equations:
dx/dt = u − ω_{3}y + ω_{2}z + cx (ux + vy + wz), 