(II. 11) P: Q = (A·E : B·F, I: K), and R:S= (C•G: D.H, L: M); and the two component ratios of P: Q are equal respectively to those of R: S; therefore (Pl. Ge. V. E.) P: QR:S.

COR. 4.-A right rectangular parallelopiped is equal to the cube described on any unit of measure, multiplied by the product of the numbers denoting the number of times that it is contained in any three contiguous edges of the parallelopiped, or its length, breadth, and height. Let A, B, and C, be the three sides; and a, b, and c, the three numbers, whether terminate or interminate, that denote the number of times that a line M, taken as the unit of measure, is contained in these sides respectively; then, considering C and M as the respective altitudes of the parallelopiped A B C and the cube M3, and AB and M2 as their bases (II. 11), A BC: M3 (A B: M2, C: M). But (Pl. Ge. Qu. 10, and Ad. V. 7) (A B: M2, C: M) = (ab: 1, c:1) abc: 1 (Pl. Ge. Ad. V. 26). A B C : M3 — abc : 1, or A B·C=abc M3.

=

PROPOSITION XII. THEOREM.

Therefore

Solid parallelopipeds which have their bases and altitudes reciprocally proportional, are equal; and parallelopipeds which are equal, have their bases and altitudes reciprocally proportional.

Let AG and KQ be two solid parallelopipeds, of which the bases are AC and KM, and the altitudes AE and KO, and let AC be to KM as KO to AE, the solids AG and KQ are equal.

As the base AC to the base KM, so let the straight line KO be to the straight line S. Then, since AC is to KM as KO to S, and also by hypo- н thesis, AC to KM as KO to AE, KO has the same ratio to S that it has to AE (Pl. Ge. V. 11); P wherefore AE is equal to S. But the solid AG is to the solid KQ in the ratio compounded of the

Ꮐ

R

Q

F

C

N

M

K

ratios of AE to KO, and of AC to KM (II. 11), that is, in the ratio compounded of the ratios of AE to KO, and of

KO to S. And the ratio of AE to Sis also compounded of the same ratios (Pl. Ge. V. Def. 17); therefore, the solid AG has to the solid KQ the same ratio that AE has to S. But AE was proved to be equal to S, therefore AG is equal to KQ.

Again, if the solids AG and KQ be equal, the base AC is to the base KM as the altitude KO to the altitude AE. Take S, so that AC may be to KM as KO to S, and it will be shown, as was done above, that the solid AG is to the solid KQ as AE to S. Now, the solid AG is, by hypothesis, equal to the solid KQ; therefore, AE is equal to S; but, by construction, AC is to KM as KO is to S; therefore, AC is to KM as KO to AE.

COR.-In the same manner, may it be demonstrated that equal prisms have their bases and altitudes reciprocally proportional, and conversely.

PROPOSITION XIII. THEOREM.

Similar solid parallelopipeds are to one another in the triplicate ratio of their homologous sides.

Let AG, KQ, be two similar parallelopipeds, of which AB and KL are two homologous sides; the ratio of the solid AG to the solid KQ is triplicate of the ratio of AB to KL.

Because the solids are similar, the parallelograms AF, KP, are similar (II. Def. 8), as also the parallelograms AH, KR ; therefore, the ratios of H

Ꮐ

AB to KL, of AE to KO,

and of AD to KN, are all

equal (Pl. Ge. VI. Def. 9). But the ratio of the solid D AG to the solid KQ is compounded of the ratios

R

of AC to KM, and of AE to KO. Now, the ratio of AC to KM, because they are equiangular parallelograms, is compounded (Pl. Ge. VI. 23) of the ratios of AB to KL, and of AD to KN. Wherefore, the ratio of AG to KQ is compounded of the three ratios of AB to KL, AD to KN, and AE to KO; and these three ratios have already been proved to be equal; therefore, the ratio that is compounded

of them, namely, the ratio of the solid AG to the solid KQ, is triplicate of any of them (Pl. Ge. V. Def. 19); it is therefore triplicate of the ratio of AB to KL.

COR. 1.-If as AB to KL, so KL to m, and as KL to m, so is m to n, then AB is to n as the solid AG to the solid KQ. For the ratio of AB to n is triplicate of the ratio of AB to KL, and is therefore equal to that of the solid AG to the solid KQ.

COR. 2.-As cubes are similar solids, therefore the cube on AB is to the cube on KL in the triplicate ratio of AB to KL, that is, in the same ratio with the solid AG to the solid KQ. Similar solid parallelopipeds are therefore to one another as the cubes on their homologous sides.

COR. 3.-In the same manner, it is proved that similar prisms are to one another in the triplicate ratio, or in the ratio of the cubes, of their homologous sides.

PROPOSITION XIV. THEOREM.

If two triangular pyramids which have equal bases and altitudes be cut by planes that are parallel to the bases, and at equal distances from them, the sections are equal to one another.

E

Let ABCD and EFGH be two pyramids, having equal bases BDC and FGH, and equal altitudes, namely, the perpendiculars AQ and ES drawn from A and E upon the planes BDC and FGH; and let them be cut by planes parallel to BDC and FGH, and at equal altitudes QR and ST above those planes, and let the sections K

be the triangles KLM, NOP; KLM and NOP are equal to one another.

Because the plane ABD B

N

H

cuts the parallel planes BDC, KLM, the common sections BD and KM are parallel (I. 14). For the same reason, DC and ML are parallel. Since therefore KM and ML are parallel to BD and DC, each to each, though not in the

same plane with them, the angle KML is equal to the angle BDC (I. 9). In like manner, the other angles of these triangles are proved to be equal; therefore the triangles are equiangular, and consequently similar; and the same is true of the triangles NOP, FGH.

Now, since the straight lines ARQ, AKB, meet the parallel planes BDC and KML, they are cut by them proportionally (I. 16), or QR: RA=BK: KA; and AQ:AR AB: AK (Pl. Ge. V. 18), for the same reason, ES: ET: EF EN; therefore AB: AK=EF: EN, because AQ is equal to ES, and AR to ET. Again, because the triangles ABC, AKL, are similar,

:

AB: AK=BC: KL; and for the same reason,

EF: EN=FG: NO; therefore

=

BC: KLFG: NO. And, when four straight lines are proportionals, the similar figures described on them are also proportionals (Pl. Ge. VI. 22); therefore the triangle BCD is to the triangle KLM as the triangle FGH to the triangle NOP; but the triangles BDC, FGH, are equal; therefore the triangle KLM is also equal to the triangle NOP (Pl. Ge. V. 14).

COR. 1.-Because it has been shown that the triangle KLM is similar to the base BCD, therefore, any section of a triangular pyramid parallel to the base, is a triangle similar to the base. And in the same manner, it is shown that the sections parallel to the base of a polygonal pyramid are similar to the base.

COR. 2.-Hence also, the sections parallel to the bases of two polygonal pyramids, and at equal distances from the bases, are equal to one another.

PROPOSITION XV. THEOREM.

A series of prisms of the same altitude may be circumscribed about any pyramid, such that the sum of the prisms shall exceed the pyramid by a solid less than any given solid.

Let ABCD be a pyramid, and Z* a given solid; a series of prisms having all the same altitude, may be circumscribed

* The solid Z is not represented in the figure of this or the following proposition.

about the pyramid ABCD, so that their sum shall exceed ABCD by a solid less than Z..

K

B

I

N

W

C

V

H

M

F

Let Z be equal to a prism standing on the same base with the pyramid, namely, the triangle BCD, and having for its altitude the perpendicular drawn from a certain point E in the line AC upon the plane BCD. It is evident that CE, multiplied by a certain number m, will be greater than AC; divide CA into as many equal parts as there are units in m, and let these be CF, FG, GH, HA, each of which will be less than CE. Through each of the points F, G, H, let planes be made to pass parallel to the plane BCD, making with the sides of the pyramid the section FPQ, GRS, HTU, which will be all similar to one another, and to the base BCD (II. 14, Cor. 1). From the point B draw in the plane of the triangle ABC the straight line BK parallel to CF, meeting FP produced in K. In like manner, from D draw DL parallel to CF, meeting FQ in L. Join KL, and it is plain that the solid KBCDLF is a prism (II. Def. 4). By the same construction, let the prisms PM, RO, TV, be described. Also, let the straight line IP, which is in the plane of the triangle ABC, be produced till it meet BC in h; and let MQ be produced till it meet DC in g. Join hg; then hCgQFP is a prism, and is equal to the prism PM (II. 10, Cor. 1). In the same manner is described the prism mS equal to the prism RO, and the prism qU equal to the prism TV. The sum, therefore, of all the inscribed prisms hQ, mS, and qU, is equal to the sum of the prisms PM, RO, and TV, that is, to the sum of all the circumscribed prisms except the prism BL; wherefore, BL is the excess of the prisms circumscribed about the pyramid ABCD above the prisms inscribed within it. But the prism BL is less than the prism which has the triangle BCD for its base, and for its altitude the perpendicular from E upon the plane BCD; and the prism which has BCD for its base, and the perpendicular from E for its altitude, is by