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 acute-angled, right-angled or obtuse-angled.
§ 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 2n + 1 sides may be constructed if this number 2n + 1 be prime, and that no others except those with 2m (2n + 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 = 24 + 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,
