TH

PROP. XXXIV. THEOR.

Book XI.

HE bafes and altitudes of equal folid parallelepi- See N. peds are reciprocally proportional; and if the bafes and altitudes be reciprocally proportional, the folid parallelepipeds 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 base NP, fo is the altitude of the folid CD to the altitude of the folid AB.

First, Let the infifting ftraight lines AG, EF, LB, HK; CM, NX, OD, PR be at right angles to the bases. as the base EH to the bafe NP, fo is CM to

AG. if the bafe EH be

K B

R

D

equal to the bafe NP, then

M L X

because the folid AB is

altitudes AG, CM be not

equal, neither fhall 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 base NP, so is CM to AG.

Next, Let the bases EH, NP not be equal, but EH greater than the other. fince 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

RD

folid parallelepiped CV of

which the base is NP,

and altitude CT. Because the folid AB is equal to the folid CD,

1

2. 7. 5.

Book XI, therefore the folid AB is to the folid CV, as the folid CD to the folid CV. but as the folid AB to the folid CV, fo is the base EH to the bafe NP; for the folids AB, CV are of the fame altitude; b. 32. 11. and as the folid CD to CV, fo is the bafe MP to the base PT, and fo 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.

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 the bafes. then, if the bafe EH be e

K

BR

D

MAX

e. A. 5.

therefore the altitude of CD is equal to the altitude of AB. but folid parallelepipeds upon equal bafes, and of the fame altitude are 31. 11. equal f to one another; therefore the folid AB is equal to the folid CD.

RD

M

X

But let the bafes EH, NP be unequal, and let EH be the greater of the two. therefore, fince as the base EH to the bafe NP, fo is CM the altitude of the folid CD to AG the altitude of AB, CM is greater than AG. again, take CTequal to AG, and complete, as before, the folid CV. and, because the bafe EH is to the bafe NP, as CM to AG, and that AG is equal to CT, therefore the bafe EH is to the bafe NP, as MC to

CT. but as the bafe EII is

K B

H

A

E

સિ

C

N

to NP, fob 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 PT, and the folid CD to the folid CV. and

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

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, meet-
ing thofe 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 laft 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. Be-
caufe the folid AB is equal to the folid CD, and that the folid BT
is equal to the folid BA, for they are upon the fame bafe FK, 8.29.or 39.
and of the fame altitude; and that the folid DC is equal to the

II.

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 parallelepi-
peds of which the infifting straight lines are at right angles to their
bafes, as before was proved. therefore as the base FK to the bafe
XR, fo is the altitude of the folid DZ to the altitude of the folid
BT. and the base FK is equal to the bafe EH, and the base XR
to the bafe 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 bafe NP, fo is the

Book XI. altitude of the folid CD to the altitude of the folid AB; that is, the bafes of 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 base 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 construction being made; becaufe as the bafe EH to the bafe 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 bafe FK is to the base XR, as the altitude of the folid CD to the

altitude of AB. but the altitudes 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 29. or 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. to the folid CD. Q. E. D.

30. II.

Therefore the folid AB is equal

PROP. XXXV.

THEOR.

Book XI.

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 plancs, ftraight lines be drawn to the vertices of the angles first named; these straight lines fhall 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 straight 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 thefe planes in the points L, N; and join LA, ND. the angle GAL is equal to the angle MDN.

a

Make AH equal to DM, and thro' 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 K, N, to the a. 8.11. straight lines AB, AC, DE, DF, draw perpendiculars KB, KC, NE, NF; and join HB, BC, ME, EF. because HK is perpendicular