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Book XI.

PRO P. XXXIV.

THEOR.

HE bases and altitudes of equal solid parallelepi. See N. T

peds, are reciprocally proportional; and if the bases and altitudes be reciprocally proportional, the solid parallelepids are equal.

P

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

First, Let the insisting straight lines AG, EF, LB, HK;
CM, NX, OD, PR be at right angles to the bases. As the base
EH to the base NP, so is
CM to AG. If the base

K B R D
EH be equal to the base

G F M NP, then because the fo

X lid AB is likewise equal to the solid CD, CM fhall

IL be equal to AG. Because, H if the bases EH, NP be equal, but the altitudes А

C N AG, CM be not equal, neither shall the folid AB be equal to the solid CD. But the solids are equal, by the hypothelis. Therefore the altitude CM is not unequal to the altitude AG; that is, they are equal. Wherefore as the base 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 : Since then the solid AB is equal to the solid CD, CM is therefore greater than AG: For,

RD if it be not, neither al. so, in this case, would the solids AB, CD be к E

M

X equal, which, by the hypothesis, are equal. G

F

T
Make then CT equal to
AG, and complete the

L solid parallelepiped CV H

P of which the base is NP, and altitude CT. А E CN Because the solid AB is equal to the folid CD, therefore the folid AB is to the

Book XI. folid CV, as a the solid CD to the solid CV. But as the fo.

lid AB to the solid CV, so is the base EH to the base NP; for a 7. S. the solids AB, CV are of the same altitude ; and as the solid b 32. II.

CD to CV, soc is the base MP to the base PT, and so d is ¢ 25. II. 1. 6. the straight line MC to CT ; and CT is equal to AG. There

fore, as the base EH to the base NP, fo is MC to ÁG. Where. fore the bases of the solid parallelepipeds AB, CD are reciprocally proportional to their altitudes.

Let now che bases of the solid parallelepipeds AB, CD be reciprocally proportional to their altitudes ; viz. as the base EH to the base NP, so the altitude of the solid CD to

K B

R D the altitude of the solid

G

M
AB; the solid AB is e-

X
qual to the solid CD. Let
the insisting lines be, as

IL
before, at right angles to H

P
the bases. Then, if the
base EH be equal to the

E

N base NP, since EH is to

NP, as the altitude of the solid CD is to the altitude of the fo¢ A. S. lid AB, therefore the altitude of CD is equal to the altitude

of AB. But folid parallelepipeds upon equal bases, and of the $35. !! fame altitude, are equalf to one another; therefore the folid AB

is equal to the solid CD.

But let the bases EH, NP be unequal, and let EH be the
greater of the two. Therefore, fince as the base EH to the base
NP, so is CM the alti-
tude of the solid CD to

R
AG the altitude of AB,
CM is greater
than K B

M

X

V
AG. Again, Take CT
equal to AG, and com- G

F

T
plete, as before, the so-
lid CV. And, because
the base EH is to the H

L

P
base NP, as CM to AG,
and that AG is equal

А
E

С N
to CT, therefore the base
EH is to the base NP, as MC to CT. But as the base EH is to NP,
so is the solid AB to the solid CV, for the folids AB, CV are of
the same altitude; and as MC to CT, so is the bafe MP to the base

Pr.

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PT, and the folid CD to the folid CV; And therefore as the Book XI. folid AB to the folid CV, so is the solid CD to the folid CV; that is, each of the folids AB, CD has the same ratio to the < 25. 12. folid CV ; and therefore the solid AB is equal to the solid CD.

Second general cafe. Let the insisting straight lines FE, BL, GA, KH; XN, DO, MC, RP not be at right angles to the bases of the solids; and from the points F, B, K, G; X, D, R, M draw perpendiculars to the planes in which are the bases EH, NP meeting those planes in the points S, Y, V, T; Q.1, 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 case, likewise, if the solids AB, CD be e. qual, their bases are reciprocally proportional to their altitudes, viz. the base EH to the base NP, as the altitude of the solid CD to the altitude of the solid 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 same base FK, and of the

II.

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{ame altitude ; and that the solid DC is equal to the folid DZ, being upon the same bale XR, and of the same altitude ; therefore the solid BT is equal to the folid DZ: But the bales are reciprocally proportional to the altitudes of equal folid parallelepipeds of which the inGfting straight lines are at right angles to their bases, as before was proved: Therefore as ihe bale FK to the base XR, so is the altitude of the folid DZ to the altitude of the solid BT: And the base FK is equal to the base EH, and the base XR to the base NP: Wherefore, as the base EH to the base NP, so is the altitude of the folid DZ to the altitude of the folid BT: But the altitudes of the solids DZ, DC, as also of the solids BT, BA are the same.' Thereføre, as the base EH to the base NP, so is the altitude of the

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

to their altitudes.

Next, Let the bases of the folids AB, CD be reciprocally proportional to their altitudes, viz. the base 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 solid CD: The same construction being made ; because, as the base EH to the base NP, so is the altitude of the folid CD to the altitude of the folid AB ; and that the base EH is equal to the base 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.

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tudes of the folids AB, BT are the same, as also of CD and DZ; therefore, as the base FK to the base XR, so is the altitude of the folid DZ to the altitude of the solid BT: Where fore the bases of the folids BT, DZ are reciprocally proportional to their altitudes; and their ingifting straight lines are at

right angles to the bases ; wherefore, as was before proved, the 3 29. or 30. solid BT is equal to the solid DZ: But BT is equal to the so.

lid BA, and DZ to the folid DC, because they are upon the same bases, and of the same altitude. Therefore the folid AB is equal to the folid CD. Q. E. D.

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PROR

Book XI.

PRO P. XXXV. THE O R. IF;

from the vertices of two equal plane angles, there Sec N. be drawn two straight lines elevated above the planes in which the angles are, and containing equal angles with the fides of those angles, each to cach; 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, straight lines be drawn to the vertices of the angles first named ; these straight lines thall contain equal angles with the straight 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 lides 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 ta. ken, and from them let perpendiculars GL, MN be drawn to the planes BAC, EDF meeting these planes in the points L, N;

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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 same plane: From the points a 8. 11, K, N, to the Itraight lines AB, AC, DE, DF, draw perpen. diculars KB, KC, NE, NF; and join HB, BC, ME, EF :

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