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Book XI.

PRO P. XXXIV. THE O R.
HE bases and altitudes of equal folid parallelepi- See N.

peds are reciprocally proportional; and if the
bases and altitudes be reciprocally proportional, the fo-
lid parallelepipeds are equal.

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Let AB, CD be equal solid parallelepipeds; their bases are re-
ciprocally proportional to their altitudes; that is, as the base EH
is to the base NP, fo is the altitude of the folid CD to the alti-
tude 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, fo is CM to
AG. if the base EH be

K BR D
equal to the base NP, then

G
because the folid AB is
likewise equal to the folid
CD, CM shall be equal to H LP LOS
AG. because if the bases
EH, NP be equal, but the A

С N
altitudes AG, CM be not
equal, neither shall the folid AB be equal to the folid CD. but
the folids are equal, by the hypothesis. therefore the altitude CM
is not unequal to the altitude AG; that is, they are equal where-
fore 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 folid CD, CM
is therefore greater than

R D
AG. for if it be not, nei-
ther also, in this case,

M X
would the folids AB, CD K B
be equal, which, by the

G

F 1
hypothefis, are equal.
Make then CT equal to

H

L
AG, and complete the
folid parallelepiped CV of
which the base is NP,

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

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1. 7. 5

C. 25. II.

Book XI, therefore the folid AB is to the folid CV, as the folid CD to the

folid CV. but as the solid AB to the solid CV, fo is the base EH

to the base NP; for the folids AB, CV are of the same altitude ; b. 32. 11. and as the folid CD to CV, so c is the base MP to the base PT,

and so d is the straight line MC to CT ; and CT is equal to AG. d... 6.

therefore as the base IH to the base NP, fo is MC to AG. where-
fore the bases of the solid parallelepipeds AB, CD are reciprocally
proportional to their altitudes.

Let now the bases of the folid parallelepipeds AB, CD be reci-
procally proportional to their altitudes; viz. as the bafe EH to the
base NP, so the altitude of the folid CD to the altitude of the solid
AB; the folid AB is equal
to the solid CD. let the in-

K B
fisting lines be, as before, G F M

X
at right angles to the bafes.
then, if the base EH be e-
qual to the bale NP, fince H

P Р

O
EH is to NP, as the alti-
tude of the folid CD is to A

C N
the altitude of the folid AB,
e. A.5. therefore the altitude of CD is equal e to the altitude of AB. but

folid parallelepipeds upon equal bases, and of the same altitude are f. 31.11. equal f to one another; therefore the folid AB is equal to the

follid 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, fo is
CM the altitude of the folid CD to AG the altitude of AB, CM
is greater
¢ than AG. a-

R D
gain, take CT equal to AG,
and complete, as before,

X
the solid CV. and, because K E
the base EH is to the base

T
NP, as CM to AG, and
that AG is equal to CT,

L
thercfore the base EH is to
the base NP, as MC to
CT. but as the base EH is

А.

с N
to NP, fo b is the solid AB to the solid CV; for the folids AB,
CV are of the fame altitude; and as MC to CT, so is the base
MIP to the base PT, and the folid CD to the folid < CV. and

P

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

Second general Case. Let the insisting straight lines FE, BL, GA, KH; XN, DO, MC, RP not be at right angles to the bases of the folids; 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, I, U, Z; and complete the solids TV, 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 equal, their bases are 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. Because the folid AB is equal to the folid CD, and that the folid BT is equal / to the solid BA, for they are upon the same base FK, 5.29. of 30. and of the same altitude; and that the folid DC is equal to the

II.

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folid DZ, being upon the same base XR, and of the same altitude; therefore the solid BT is equal to the solid DZ. but the bases are reciprocally proportional to the altitudes of equal folid parallelepipeds of which the insisting straight lines are at right angles to their bases, as before was proved. therefore as the base FK to the base XR, fo is the altitude of the solid 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 folids DZ, DC, as also of the folids BT, BA are the same. Therefore as the base EH to the base NP, fo is the

Book XI. altitude of the solid CD to the altitude of the folid AB ; that is,

the bases of the solid 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 base EH to the base NP, as the altitude of the solid CD to the altitude of the solid AB; the folid AB is equal to the folid CD. the same construction 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 base FK; and NP to XR ; therefore the base FK is to the base XR, as the altitude of the solid CD to the

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altitude of AB. but the altitudes of the folids AB, BT are the fame, as also of CD and DZ; therefore as the base FK to the base XR, so is the altitude of the solid DZ to the altitude of the folid BT. wherefore the bases of the folids BT, DZ are reciprocally proportional to their altitudes; and their insisting straight lines are at

right angles to the bases; wherefore, as was before proved, the 3. 29. or folid BT is equal to the solid DZ. but BT is equal 3 to the solid

BA, and DZ to the folid DC, because they are upon the fame bafes, and of the fame altitude. Therefore the solid AB is equal to the folid CD. Q. E. D.

30. 11.

Book XI.

PRO P. XXXV.

THE O R.

IF from the vertices of two equal plane angles there see N.

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be drawn two straight lines elevated above the planes in which the angles are, and containing equal angles with the sides 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 arc; and from the points in which they meet the plancs, straight lines be drawn to the vertices of the angles first named; these straight lines shall 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 fides, each to each; viz. the angle GAB equal to the angle MDL, and GAC to MDF;

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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 meet-
ing these planes in the points L, N; and join LA, ND. the angle
GAL is equal to the angle MDN.

Make 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 same 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

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