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A E is to EK, fo is + the Parallelogram A G to the
Parallelogram GK; but as G E is to E L, so is GK
to KL; and as H E is + to EM, so is PE to KM:
Therefore, as the Parallelogram A G is to the Paral-

lelogram GK, fo is G K to KL, and PE to K M. 132 of tbis. But as A G is to G K, fo is † the Salid A B to the So

Jid E X; and as GK is to KL, so is the Solid E X to the Solid PL; and as PE is to KM, fo is the Solid PL to the Solid KO: Therefore, as the Solid AB is to the Solid E X, fo is * E X to PL, and P L to

KO: But if four Magnitudes be continually propor+ Def. 11.5. tiomal, the first to the fourth hath + a triplicate Pro

portion of that which it has to the second. Therefore, also, the Solid A B to the Solid KO, hath a triplicate Proportion of that which A B has to EX: But as A B is to E X, so is the Parallelogram A G to the Parallelogram. G K; and fo is the Right Line A E to the Right Line EK: Wherefore the Solid A B to the Solid K O, bath a Proportion triplicate of that which A E has to EK. But the Solid K O is equal to the Solid CD, and the Right Line EK equal to the Right Line CF: Therefore, tbe Solid AB, to the Solid CD, has a Proportion triplicate of that which the homologous Side A E has to the bomologous Side CF; which was to be demonstrated.

Coroll

. From hence it is manifeft, if four Right Lines be continually proportional, as the first is to the fourth, fo is a folid Parallelop pedon described upon the first, to a similar folid Parallelopipedon, alike fxuate, described upon the second; because the first to the fourth, has a Proportion triplicate of that which it has to the second.

PROI

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PROPOSITION XXXIV.

THEOREM.
The Bases and Altitudes of equal solid Parallelopi.

pedons are reciprocally proportional; and those
solid Parallelopipedons, whose Bases and Alti-
tudes are reciprocally proportional, are equal.

ET AB, CD, be equal solid Parallelopipedóns.

I say, their Bases and Altitudes are reciprocally proporcional , that is, as the Base E H is to the Base NÉ, fo is the Altitude of the Solid CD to the Alditude of the Solid A B..

First, let the infiftent Lines AG, EF, LB, HK, CM, NX, OD, PR, be at Right Angles to their Bales: I say, as the Bale E H is to the Bale N P, so is CM to AG. For, if the Base EH be equal to the Bale N P, and the Solid A B is equal to the Solid CD; the Altitude. CM Ihall also be equal to the Altitude AG: For if, when the Bases EH, NP, are equal, the Altitudes AG, CM, are not so; then the Solid AB will not be equal to the folid CD, but it is put equal to it: Therefore the Altitude C M is not unequal to the Altitude A G, and fo they are neceffarily equal to one another ; and, consequently, as the Base E H is to the Base N Peso Mall CM be to AG. But now let the Base E H be unequal to the Base NP, and let EH be the greater ; then, since the Solid A B is equal to the Solid CD, C M is greater than A G; for otherwise, it would follow, that the Solids A B, CD, are not equal, which are put such : Therefore, make CT equal to AG, and compleat the solid Parallelopipedon V. C upon the Base NP, having the Altitude CT: Then, because the Solid A B is equal to the Solid CD, and V C is some other Solid; and since equal Magnia fudes have * the same Proportion to the same Magni-*7.50 tudes ;-it shall be, as the Solid A B is to the Solid CV, fo is the Solid C D to the Solid CV: But as the Soid AB is to the Solid CV, so is + the Base E H to the t 32 of ilise Bale NP; for AB, CV, are Solids having equal Alticudes : And as the Solid C D is to the Solid CV, lo

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+25 of tbis. is I the Base MP to the Base PT, and so is MC to CT:

Therefore, as tbe Base EH is to the Base NP, fo is MC to C T. But CT is equal to AG; wherefore, as the Bale EH is to the Base NP, ro is MC to AG: Therefore the Bases and Altitudes of the equal solid Parallelopipedons A B, C D, are reciprocally proportional.

Now, let the Bales and Altitudes of the solid Pasallelopipedons AB, CD, be reciprocally proportional; that is, let the Bale E H be to the Bale N P, as the Antitude of the Solid CD is to the Aliitude of the Solid AB. I say, the Solid A B is equal to the Solid CD.

For, let again the insistent Lines be at Right Angles to the Bases; then, if the Base EH be equal to the Bale NP, and EH is to NP as the Altitude of the Solid C D is to the Altitude of the Solid A B; the Altitude of the Solid CD Mall be equal to the Altitude of the Solid A B. But Solid Parallelopipedons,

that stand upon equal Bases, and have the same Alti130bis. tude

are equal to each other; therefore the Solid A B is equal to the Solid CD.

But now let the Base E H not be equal to the Base NP, and let E H be the greater ; then the Altitude of the Solid C D is greater than the Altitude of the Solid AB; that is, C M is greater than AG: Again, put CT equal to A G, and compleat the Solid CV, as before; and then, because the Base E His to the Base NP, as M C is to A G, and A G is equal to CT; ic fhall be as the Base E H is to the Base N P, so is MC to CT: But as the Bale E H is to the Base N P, so is the Solid AB to the Solid CV; for the Solids AB, CV, have equal Altitudes ; and as MC is to CT, fo is the Bafe M P to the Base P'T, and so the Solid C D to the Solid CV: Therefore as the Solid A B is to the Solid CV, fo is the Solid C D to the Solid CV: But since each of the Solids AB, CD, has the same Proportion to CV; the Solid A B shall be equal to the Solid CD; whence, the two folid Parallelopipedons AB, CD, whose Bases and Altitades are reciprocally proportional, are equal; which was to be demonstrated.

Now, let the infiitent Lines F E, BL, GA, KH, XN, DO, MC,RP, not be at Right Angles to the Bases; and from the Points F, G, B, K, X, M, D, R, let there be drawn Perpendiculars to the Planes

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of the Bases EH, NP, mecting the same in the
Points S, T, Y, V, Q, Z, 2, 0, and compleat the
Solids F V. X.2. Then, I say, if the Solids AB,
CD, be equal, their Bases and Altitudes are recipro-
cally proportional; viz, as the Base E H is to the Base
NP, so is the Altitude of the Solid CD to the Alti-
tude of the Solid A B.

For, because the Solid AB is equal to the Solid CD,
and the Solid AB is * cqual to the Solid BT; for they * 30 of ibis.
stand upon the same Base FK, and have the fame All
titude; and the Solid DC is * equal to the Solid DZ,
since they stand upon the fame Base XR, and have the
same Altitude ; therefore the Solid BT shall be equal to
the Solid D Z. But the Bases and Alutudes of those
equal Solids, whose Altitudes are at Right Angles to
their Bases, are + reciprocally proporcional; therefore as +
the Bale F K is to the Base XR, fo is the Altitude of wbar bas
the Solid D Z to the Altitude of the Solid B T. But been before

provit. the Base FK is equal to the Base E 4, and the Base XR to the Base NP; wherefore, as the Base E H is to the Base N P, fo is the Altitude of the Solid D Z to the Altitude of the Solid BT. But the Solids D Z, DC, have the same Altitude, and so have the Solids BT, BA; therefore the Base E H is to the Balc NP, as the Altitude of the Solid D C is to the Aluude of the Solid A B; and so, the Bases and Aliitudes of qual folid Paralielopipedons are reciprocally propertional.

Again, let the Bases and Altitudes of the folid Parallelopipedons AB, CD, be reciprocally proportional; viz. as the Base E H is to the Basc NP, io let the Aititude of the Solid C D be to the Altitude of the Solid AB: I say, the Solid A B is equal to ihe Solid CD.

For, the same Construction remaining, because the Bale E H is to the Base NP, as the Altitude of the Solid CD is to the Altitude of the Solid AB; and since the Base EH is equal to the Base FK and ŅP to XR; it shall be as the Bale FK is to the Base XR, fo is the Altitude of the Solid CD to the Aliitude of the Solid AB. But the Altitudes of the solids AB, BT, are the fame; as also of the Solids CD, DZ; therefore the Base F K is to the Base X R, as the Altitude of the Solid DZ is to the Altitude of the Solid B T; wherefore the Bases and Altitudes of the solid Parallelopipe

dons

1

2.3

wbat bas

dons BT, DZ, are reciprocally proportional ; but those solid Pa allelopipedons, whose Altitudes are at

Right Angles to their Bases, and the Bases and Alti+ From

tudes are reciprocally proportional, are equal to † each been before other. But the Solid B T is equal to the Solid B A, proved. for they ftand upon the same Base F K, and have the

same Altitude ; and the Sojid D Z is also equal to the Solid DC, since they stand upon the fame Base XR, and have the same Altitude : Therefore the Solid AB is equal to the Solid CD; whence solid Parallelopipedons, whoje Bases and Altitudes are reciprocally proportional, are equal, which was to be demonstrated.

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PROPOSITION XXXV.

THEOR E M.
If there be two plane Angles equal, and from the

Vertices of those Angles two Right Lines be ele.
vated above the Planes, in which the Angles
are, containing equal Angles wilb the Lines first
given, each to its correspondent me ; and if in
those elevated Lines ony Points be taken, from
which Lines be drawn perpendicular to the
Planes in zehich ibe Angles first given art, and
Right Lines be drawn 10 the Angles first given
from the Points made by ibe Perpendiculars in
she Planes ; those Right Lines will contain
equal Angles to the elevated Lines.
ET BAC, EDF, be two equal Right-lined plane

Angles, and from A and D, the Vertices of those Angles, let two Right Lines, A G and D M be elevated above the Planes of the said Angles, making equal Angles with the Lines first given, each to its correspondent one; viz. the Angle MD E equal to the Angle G AB, and the Angle MDF to the Angle GAC: and take any Points G and M in the Right Lines AG, DM; from which let GL and M N be drawn perpendicular to the Planes paffing thro' BAC, EDF, meeting the same in the Points L and N; and join LA and ND. I say, the Angle GAL is equal to the Angle MDN

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