(4. 5.); that is, the prism BNM is to the parallelopiped CD as the triangle AME to the parallelogram LG.

PROP. IX. THEOR.

Solid parallelopipeds are to one another in the ratio that is compounded of the ratios of the areas of their bases, and of their altitudes.

Let AF and GO be two solid parallelopipeds, of which the bases are the parallelograms AC and GK, and the altitudes, the perpendicu lars let fall on the planes of these bases from any point in the opposite planes EF and MO; the solid AF is to the solid GO in a ratio compounded of the ratios of the base AC to the base GK, and of the perpendicular on AC, to the perpendicular on GK.

Case 1. When the insisting lines are perpendicular to the bases AC and GK, or when the solids are upright.

In GM, one of the insisting lines of the solid GO, take GQ equal to AE, one of the insisting lines of the solid AF, and through Q let a plane pass parallel to the plane GK, meeting the other insisting lines

of the solid GO in the points R, S and T. It is evident that GS is a solid parallelopiped (def. 5. 3. Sup.), and that it has the same altitude with AF, viz. GQ or AE. Now the solid AF is to the solid GO in a ratio compounded of the ratios of the solid AF to the solid GS (def. 10. 5.), and of the solid GS to the solid GO; but the ratio of the solid AF to the solid GS, is the same with that of the base AC to the base GK (8. 3. Sup.), because their altitudes AE and GQ are equal; and the ratio of the solid GS to the solid GO, is the same with that of GQ to GM (3. 2. Sup.); therefore, the ratio which is compounded of the ratios of the solid AF to the solid GS, and of the solid GS to the solid GO, is the same with the ratio which is compounded of the ratios of the base AC to the base GK, and of the altitude AE to the altitude GM (F. 5.). But the ratio of the solid AF to the solid GO, is that which is compounded of the ratios of AF to GS, and of GS to GO; therefore,

the ratio of the solid AF to the solid GO is compounded of the ratios of the base AC to the base GK, and of the altitude AE to the altitude GM. Case 2. When the insisting lines are not perpendicular to the bases. Let the parallelograms AC and GK be the bases as before, and let AE and GM be the altitudes of two parallelopipeds Y and Z on these bases. Then, if the upright parallelopipeds AF and GO be constituted on the bases AC and GK, with the altitudes AE and GM, they will be equal to the parallelopipeds Y and Z (7.3. Sup.). Now, the solids AF and GO, by the first case, are in the ratio compounded of the ratios of the bases AČ and GK, and of the altitudes AE and GM; therefore also the solids Y and Z have to one another a ratio that is compounded of the same ratios, Therefore, &c. Q. E. D.

4

COR. 1. Hence, two straight lines may be found having the same ratio with the two parallelopipeds AF and GO. To AB, one of the sides of the parallelogram AC, apply the parallelogram BV equal to GK, having an angle equal to the angle BAD (44.1.); and as AE to GM, so let AV be to AX (12.6.), then AD is to AX as the solid AF to the solid GO. For the ratio of AD to AX is compounded of the ratios (def. 10. 5.) of AD to AV, and of AV to AX; but the ratio of AD to AV is the same with that of the parallelogram AC to the parallelogram BV (1. 6.) or GK; and the ratio of AV to AX is the same with that of AE to GM; therefore the ratio of AD to AX is compounded of the ratios of AC to GK, and of AE to GM (E. 5.). But the ratio of the solid AF to the solid GO is compounded of the same ratios; therefore, as AD to AX, so is the solid AF to the solid GO.

COR. 2. If AF and GO are two parallelopipeds, and if to AB, to the perpendicular from A upon DC, and to the altitude of the parallelopiped AF, the numbers L, M, N be proportional: and if to AB, to GH, to the perpendicular from G on LK, and to the altitude of the parallelopiped GO, the numbers L, l, m, n be proportional; the solid AF is to the solid GO as LxMxN to lxmxn.

For it may be proved, as in the 7th of the 1st of the Sup. that LX MXN is to lxmxn in the ratio compounded of the ratio of LxM to lxm, and of the ratio of N to n. Now the ratio of LxM to lxm is that of the area of the parallelogram AC to that of the parallelogram GK; and the ratio of N to n is the ratio of the altitudes of the parallelopipeds, by hypothesis, therefore, the ratio of LxMXN to lxmxn is compounded of the ratio of the areas of the bases, and of the ratio of the altitudes of the parallelopipeds AF and GO; and the ratio of the parallelopipeds themselves is shown, in this proposition, to be compounded of the same ratios; therefore it is the same with that of the product LX MXN to the product /xmxn.

COR. 3. Hence all prisms are to one another in the ratio compounded of the ratios of their bases, and of their altitudes. For every prism is equal to a parallelopiped of the same altitude with it, and of an equal base (2. Cor. 8. 3. Sup.).

PROP. X. THEOR.

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 alse

by hypothesis, AC to KM as KO to AE, KO has the same ratio to S that it has to AE (11. 5.); wherefore AE is equal to S (9. 5.). But the solid AG is to the solid KQ, in the ratio compounded of the ratios of AE to KO, and of AC to KM (9.3. Sup.), that is, in the ratio compounded of the ratios of AE to KO, and of KO to S. And the ratio of AE to S is also compounded of the same ratios (def. 10. 5.); 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. Therefore, Q. E. D.

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

PROP. XI. THEOR.

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 its triplicate of the ratio of AB to KL.

Because the solids are similar, the parallelograms AF, KB are similar (def. 2. 3. Sup.), as also the parallelograms AH, KR; therefore,

the ratios of AB to KL, of AE to KO, and of AD to KN are all equal (def. 1. 6.). But the ratio of the solid AG to the solid KQ is compounded of the ratios of AC to KM, and of AE to KO. Now, the ratio of AC to KM, because they are equiangular parallelograms, is compounded (23. 6.) 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, viz. the ratio of the solid AG to the solid KQ, is triplicate of any of them (def. 12. 5.); it is therefore triplicate of the ratio of AB to KL. Therefore, similar solid parallelopipeds, &c. Q. E. D.

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 nis triplicate of the ratio of AB to KL (def. 12. 5.), 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.

PROP. XII. THEOR.

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.

Let ABCD and EFGH be two pyramids, having equal bases BDC and FGH, and equal altitudes, viz. 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 be the triangles KLM, NOP; KLM and NOP are equal to one another.

Because the plane ABD cuts the parallel planes BDC, KLM, the common sections BD and KM are parallel (14. 2. Sup.). For the same reason, DC and ML are parallel. Since therefore KM and ML are parallel to BD an: DC, each to each, though not in the same plane with them, the angle KML is equal to the angle BDC (9.2. Sup.). 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 (16.2. Sup.), or QR: RA :: BK : KA; and AQ : AR :: AB : AK (18. 5.), 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: KL:: FG: NO. And, when four straight lines are proportionals, the similar figures described on them are also proportionals (22. 6); 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 (1.5.). Therefore, &c. Q. E. D.

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, in polygonal pyramids of equal bases and alii