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 non-intersecting 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 starting-point 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 V1 and V2. 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
