Suppl.

COR. 2. 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. Therefore similar solid parallelepipeds are to one another as the cubes on their homologous sides.

COR. 3. Hence similar prisms are to one another in the triplicate ratio, or in the ratio of the cubes, of their homologous sides. For a prism is equal to half of a parallelepiped of the 84. 3. Sup. same base and altitudes.

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 eacl: other.

Let ABCD, EFGH be two pyramids, having equal bases BDC, FGH, and equal altitudes, viz. the perpendiculars AQ, ES, drawn from A, E upon the planes BDC, FGH ; let

them be cut by planes parallel to BDC, FGH, at equal altitudes QR, ST above those planes, and let the sections be the triangles KLM, NOP; KLM and NOP are equal to each other.

Because the plane ABD cuts the parallel planes BDC, Book III. KLM, the common sections BD, KM are parallel. For

the same reason DC, ML are parallel. Since KM, ML are a 14.2. Sup. parallel to BD, DC, each to each, though not in the same plane with them, the angle KML is equal to the angle BDCb. 69. 2. Sup. In like manner the other angles of the triangles KLM, BCD may be proved to be equal; therefore the triangle are equiangular, and consequently similar. The same is rue of the triangles NOP, FGH.

Since the straight lines ARQ, AKB meet the parallel planes BDC, KML, they are cut by them proportionaly;

therefore

therefore

In the the same manner

QR: RA :: BK : ĮA,
AQ: AR:: AB : Ka.
ES: ET:: EF : LN.

Hence (because AQ-ES, and AR-ET)

AB: AK: EF EN.

Again, because the triangles ABC, AKL are smilar,

In the same manner
Therefore

c16.2. Sup.

d 18. 5.

e 11. 5.

AB: AK:: BC: KL.

EF:EN:: FC: NO.

BC: KL:: F: NO.

Therefore the triangle BCD is to the triang KLM as the triangle FGH to the triangle NOPf. But the riangles BCD, f 22. 6. FGH are equal; therefore the triangles KLM, NOP are equals. Therefore, if two triangular pyramids, &c. Q. E. D. g 14. 5.

COR. 1. 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 may be 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 altitudes, the sections parallel to the bases, and at equal distances from them, are equal to one another.

2 L

Suppl.

PROP. XIII. THEOR.

a 1. Cor 12. 3. Sup.

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 ABID be a pyramid, and Z* a given solid; a series of prisms having all the same altitude may be circumscribed about the Framid ABCD, so that their sum shall exceed ABCD by solid less than Z.

N

W

M

Let Z be qual to a prism standing on the same base BCD with the pyramid, 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 a many equal parts as there are units in m, and let these be CF, FG, GH, HA, tach 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 sections FPQ, GRS, HTU, which will be similar to one another, and to the base BCDa. From the point B draw, in the plane of the triangle ABC, the straight line BK parallel to CF, and meeting FP produced

K

B

2

H

R

G

m

E

F

D

C

in K; and, in like manner, from D draw DL parallel to CF, and meeting FQ in L; and join KL. It is plain that the so

* The solid Z is not represented in the figure of this, or the following propo

sition.

lid KBCDLF is a prism. By the same construction let the Book III. prisms PM, RO, TV be described. Let the straight line IP, which is in the plane of the triangle ABC, be produced till it b 11. Def. meet BC in h; and let the line MQ be produced till it meet 3. Sup. DC in g; and join hg; then hCgQFP is a prism, and is equal to the prism PM. In the same manner is described c 1. Cor. 8. 3. Sup. the prism mS, equal to the prism RO, and the prism qU equal to the prism TV. Therefore the sum of all the inscribed prisms, hQ, mS, qU is equal to the sum of the prisms PM, RO, 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 in 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 hypothesis, equal to the given solid Z; therefore the excess of the circumscribed above the inscribed prisms is less than the given solid Z. But the excess of the circumscribed prisms above the inscribed is greater than their excess above the pyramid ABCD, because ABCD is greater than the sum of the inscribed prisms. Much more, therefore, is the excess of the circumscribed prisms above the pyramid less than the solid Z. Therefore a series of prisms of the same altitude has been circumscribed about the pyramid ABCD exceeding it by a solid less than the given solid Z. Q. E. D.

PYRAMIDS which have equal bases and altitudes are equal to one another.

Let ABCD, EFGH be two pyramids that have equal bases BCD, FGH, and also equal altitudes, viz. the perpendiculars drawn from the vertices A and E upon the planes BCD, FGH; the pyramid ABCD is equal to the pyramid EFGH.

Suppl.

If they be not equal, let the pyramid EFGH exceed the pyramid ABCD by the solid Z. Then a series of prisms of the same altitude may be described about the pyramid a13.3. Sup. ABCD, which shall exceed it by a solid less than Za.

Let

these be the prisms that have for their bases the triangles BCD, NQL, ORI, PSM. Divide EH into the same number of equal parts into which AD is divided, viz. HT, TU,

UV, VE; and through the points T, U, V let the sections TZW, UKX, VJY be made parallel to the base FGH. The b 12.3.Sup. section NQL is equal to the section WZT, ORI to XKU,

ç 1. Cor. & 3. Sup.

and PSM to YJV; therefore the prisms which stand upon the equal sections are equale, that is, the prism which stands on the base BCD, and is between the planes BCD and NQL, is equal to the prism which stands on the base FGH, and is between the planes FGH and WZT; and so of the rest, because they have the same altitude. Wherefore the sum of all the prisms described about the pyramid ABCD is equal to the sum of all the prisms described about the pyramid EFGH. But the excess of the prisms described about the pyramid ABCD above the pyramid ABCD is less than Z; therefore the excess of the prisms described about the pyramid EFGH above the pyramid ABCD is also less than Z. But the excess of the pyramid EFGH above the pyramid ABCD is equal to Z, by hypothesis; therefore the pyramid EFGH exceeds the pyramid ABCD more than the