PROP. XXXIV.

THEOR.

Book XI.

THE

HE bafes and altitudes of equal folid parallelepi- see Npeds, are reciprocally proportional; and if the bafes and altitudes be reciprocally proportional, the folid parallelepids are equal.

Let AB, CD be equal folid parallelepipeds; their bases are reciprocally proportional to their altitudes; that is, as the bafe EH is to the bafe NP, fo is the altitude of the folid CD to the altitude of the folid AB.

First, Let the infifting ftraight lines AG, EF, CM, NX, OD, PR be at right angles to the bases.

EH to the base NP, so is CM to AG. If the bafe EH be equal to the bafe NP, then because the folid AB is likewife equal to the folid CD, CM fhall

LB, HK;
As the bafe

be equal to AG. Becaufe, H

10

if the bafes EH, NP be e

qual, but the altitudes

N

AG, CM be not equal,

neither shall the folid AB be equal to the folid CD. But the folids are equal, by the hypothefis. Therefore the altitude CM = is not unequal to the altitude AG; that is, they are equal. Wherefore as the bafe EH to the bafe NP, fo is CM to AG.

Next, Let the bafes EH, NP not be equal, but EH greater than the other: Since then the folid AB is equal to the folid CD, CM is therefore greater than AG: For, if it be not, neither alfo, in this cafe, would the folids AB, CD be equal, which, by the hypothefis, are equal. Make then CT equal to AG, and complete the folid parallelepiped CV H

of which the base is

NP, and altitude CT.

Because the folid AB

1

R D

T

is equal to the folid CD, therefore the folid AB is to the

a 7. 5. b 32. II.

Book XI. folid CV, as a the folid CD to the folid CV. But as the folid AB to the folid CV, fob is the base EH to the base NP; for the folids AB, CV are of the fame altitude; and as the folid CD to CV, fo is the base MP to the base PT, and fod is the ftraight line MC to CT; and CT is equal to AG. Therefore, as the bafe EH to the bafe NP, fo is MC to AG. Wherefore the bafes of the folid parallelepipeds AB, CD are reciprocally proportional to their altitudes.

c 25. II. d 1. 6.

e A. s.

f31. II.

Let now the bafes of the folid parallelepipeds AB, CD be reciprocally proportional to their altitudes; viz. as the bafe EH to the bafe NP, fo the altitude of the folid CD to the altitude of the folid AB; the folid AB is equal to the folid CD. Let the infifting lines be, as before, at right angles to H the bafes. Then, if the bafe EH be equal to the

bafe NP, fince EH is to

K

B

R

D

G

M

X

Р

N

NP, as the altitude of the folid CD is to the altitude of the folid AB, therefore the altitude of CD is equal to the altitude of AB. But folid parallelepipeds upon equal bafes, and of the fame altitude, are equal f to one another; therefore the folid AB is equal to the folid CD.

But let the bases EH, NP be unequal, and let EH be the greater of the two. Therefore, fince as the bafe EH to the base NP, fo is CM the altitude of the folid CD to AG the altitude of AB, CM is greater than AG. Again, Take CT equal to AG, and com

K

R D

B

M

EH is to the base NP, as MC to CT. But as the bafe EH is to NP, fo is the folid AB to the folid CV; for the folids AB, CV are of the fame altitude; and as MC to CT, fo is the bafe MP to the bafe

PI

241

PT, and the folid CD to the folid CV: And therefore as the Book XI. folid AB to the folid CV, fo is the folid CD to the folid CV; that is, each of the folids AB, CD has the fame ratio to the € 25. II. folid CV; and therefore the folid AB is equal to the folid CD.

с

Second general cafe. Let the infifting ftraight lines FE, BL, GA, KH; XN, DO, MC, RP not be at right angles to the bafes of the folids; and from the points F, B, K, G; X, D, R, M draw perpendiculars to the planes in which are the bafes EH, NP meeting those planes in the points S, Y, V, T; Q,I, U, Z; and complete the folids FV, XU, which are parallelepipeds, as was proved in the last part of prop. 31. of this book. In this cafe, likewife, if the folids AB, CD be equal, their bafes are reciprocally proportional to their altitudes, viz. the bafe EH to the bafe NP, as the altitude of the folid CD to the altitude of the folid AB. Because the folid AB is equal to the folid CD, and that the folid BT is equal to the 8 29. or 30. folid BA, for they are upon the fame bafe FK, and of the

8

II.

fame altitude; and that the folid DC is equal to the folid DZ, being upon the fame bafe XR, and of the fame altitude; therefore the folid BT is equal to the folid DZ: But the bafes are reciprocally proportional to the altitudes of equal folid parallelepipeds of which the infifting ftraight lines are at right angles to their bafes, as before was proved: Therefore as the bafe FK to the base XR, fo is the altitude of the folid DZ to the altitude of the folid BT: And the bafe FK is equal to the bafe EH, and the base XR to the base NP: Wherefore, as the bafe EH to the bafe NP, fo is the altitude of the folid DZ to the altitude of the folid BT: But the altitudes of the folids DZ, DC, as alfo of the folids BT, BA are the fame. Therefore, as the bafe EH to the base NP, fo is the altitude of the folid

e

Book XI. folid CD to the altitude of the folid AB; that is, the bafes of པ the folid parallelepipeds AB, CD are reciprocally proportional to their altitudes.

Next, Let the bafes of the folids AB, CD be reciprocally proportional to their altitudes, viz. the bafe EH to the bafe NP, as the altitude of the folid CD to the altitude of the folid AB; the folid AB is equal to the folid CD: The fame conftruction being made; because, as the base EH to the base NP, fo is the altitude of the folid CD to the altitude of the folid AB; and that the base EH is equal to the bafe FK; and NP to XR; therefore the base FK is to the base XR, as the altitude of the folid CD to the altitude of AB: But the alti

tudes of the folids AB, BT are the fame, as alfo of CD and DZ; therefore, as the bafe FK to the bafe XR, fo is the altitude of the folid DZ to the altitude of the folid BT: Wherefore the bafes of the folids BT, DZ are reciprocally proportional to their altitudes; and their infifting ftraight lines are at right angles to the bafes; wherefore, as was before proved, the 8 29. or 30. folid BT is equal to the folid DZ: But BT is equal to the folid BA, and DZ to the folid DC, because they are upon the fame bafes, and of the fame altitude. Therefore the folid AB is equal to the folid CD. Q. E. D.

11.

PROR

PROP. XXXV.

THEOR.

Book XI.

IF

F, from the vertices of two equal plane angles, there See N. be drawn two ftraight lines elevated above the planes in which the angles are, and containing equal angles with the fides of those angles, each to each; and if in the lines above the planes there be taken any points, and from them perpendiculars be drawn to the planes in which the first named angles are: And from the points in which they meet the planes, ftraight lines be drawn to the vertices of the angles first named; these straight lines hall contain equal angles with the ftraight lines which are above the planes of the angles.

Let BAC, EDF be two equal plane angles; and from the points A, D let the ftraight lines AG, DM be elevated above the planes of the angles, making equal angles with their fides. each to each, viz. the angle GAB equal to the angle MDE, and GAC to MDF; and in AG, DM let any points G, M be taken, and from them let perpendiculars GL, MN be drawn to the planes BAC, EDF meeting these planes in the points L, N;

and join LA, ND: The angle GAL is equal to the angle MDN.

Make AH equal to DM, and through H draw HK parallel to GL: But GL is perpendicular to the plane BAC; wherefore HK is perpendicular to the fame plane: From the points a 8. 15. K, N, to the straight lines AB, AC, DE, DF, draw perpendiculars KB, KC, NE, NF; and join HB, BC, MÈ, EF:

Q 2

Because