DM;

and thro' H draw HK parallel to GL. But GL is perpendicular to the plane, paffing thro' BAC: Wherefore [by 8. 11.] HK will be perpendicular to the plane paffing thro' BAC. From the points K, N draw the perpendicu lars K B, KC, NF, NE to the right lines AB, AC, DF, DE, and join H C, C B, MF, FE. Then becaufe by 47. 1.] the fquare of HA is equal to the fquares of HK, KA, and the fquares of KC, CA are equal to the fquare of KA; the fquare of HA will be equal to the fquares HK, CK, CA: But the fquare of His equal to the fquares of HK, KC: Therefore the fquare of HA will be equal to the fquares of HC, CA; and fo the angle HCA is a right angle. By the fame reafon the angle D F M. is a right angle: Therefore the angle ACH is equal to DFM: But the angle HAC is equal to the angle MDE: Therefore M D F, HAC are two triangles having two angles of the one equal to two angles of the other, each to each, and one fide of the one equal to one fide of the other, viz. the fides oppofite to the equal angles, viz. AH [by conftr.] equal to DM: Therefore [by 26 the remaining fides of the one will be equal to the remaining fides of the other, each to each: Wherefore A cis equal to DF. After the fame manner we demonstrate that A B is equal to DE Join HB, ME. Then because the fquare of AH is equal to the fquares of A, KH, and the fquares of A B, BK are equal the f fquare of AK: the fquares of AB, BK, KH, will be equal to the fquare of a H. But the fquare of BH is equal to the fquares of BK, H, for HK B is a right angle, fince

to

Kis perpendicular to the plane of the triangle. BAG Therefore the fquare of A His equal to the fquares of AB BH and for the angle ABH, is a right angle. By the fame reason the angle DEM is alfo a right angles and the angle RA is equabta the angle DM, for-fo it is made: But A H is equal to DM; therefore A Bis alfo equal to DE: Therefore because Ac is equal to DF, and A Beto DE, and fo the two fides c A, A B are equal to the twe fides FD, DE, and the angle CAE equal to the angle FDE. Therefore [by. 4. 1.] the bafe BC is equal to the bafe EF, and one triangle equal to the other, and the reft of the angles equal to the reft of the angles: Wherefore the angle ACB is equal to the angle DFE. But [by conftr.] the right angle ACK is equal to the right angle DFN; and fo the remaining angle BCK is equal to the

remaining

remaining angle EFN. By the fame reafon the angle CBK is equal to the angle FEN: Wherefore c BK, FEN are two triangles having two angles of the one equalto two angles of the other, each to each, and one fide of the one equal to one fide of the other, which is adjacent to the equal angles, viz. BC to EF: Therefore [by 26. 1.] the reft of the fides of the one are equal to the reft of the fides of the other: Wherefore c K is equal to F N. But Acis alfo equal to DF: Wherefore the two fides A ̊C, CK are equal to the two fides D F, FN; and they contain right angles. Therefore [by 4. 1.] the bafe A K is equal to the bafe DN; and fince A H is equal to DM, the fquare of A H will be equal to the fquare of DM But [by 47% N] the squares of A K, KH are equal to the fquare of AH; for AKH is a right angle; and the fquares of D N, NM, are equal to the fquare of DM; because the angle DN M is a right angle: Therefore the fquares of AK, KH are equal to the fquares of DN, NM. But the fquare of A K is equal to the fquare of DN: Therefore the remaining fquare of KH is equal to the remaining fquare of NM; and fo the right line HK is equal to M N. And fince the two fides HA, AK are equal to the two fides MD, D N, each to each, and the bafe HK has been proved to be equal to the bafe NM; [by 8. 1.] the angle HAK will be equal to the angle MD N. Which was to be demonftrated.

Corollary. From hence it is manifeft, if there be two equal right lined plane angles, and equal right lines be erected from them, containing equal angles with the fides of the first proposed angles, each to each; the perpendicuJars drawn from them to the planes wherein are the first propofed angles, are equal to one another.

PROP. XXXVI. THEOR.

If three right lines be proportional, I fay the folid parallelepipedon made of all three of them, is equal to the folid parallelepipedon made of the middle line, being equilateral and equiangular to the first mentioned one w.

Let the three right lines A, B, C be proportional, as A is to B, fo is в to c: I fay the folid parallelepipedon made A a 4

with

Book XI. with A, B, C is equal to the folid parallelepipedon made from B, equilateral and equiangular to that firft mentioned.

Let E be any folid angle contained under the three plane angles DE G, GEF, FED; make DE, GE, EF each equal to B, and complete the folid parallelepipedon E K. Make LM equal to A, and at the right line L M and point L in it, make [by 26. 11.] the folid angle contained under NL X, XL M, M L N equal to the folid angle

Then

at E. Make L x equal to B, and L N equal to c. because A is to B, as B is to C, and A is equal to L N; and B to either LX, EF, EG, or E D, and c is equal to L M; It will be as LM is to EF, fo is DE to L N, and the fides about the equal angles ML N, DEF are reciprocally proportional: Therefore [by 14. 6.] the paralleloAnd begram MN is equal to the parallelogram D F. caufe, the two right lined plane angles DEF, NLM are equal and the equal right lines LX, EG are elevated from them, containing equal angles with the fides of the angle, DEF, NLM, each to each; the perpendiculars drawn from the points G,x to the planes paffing thro' N LM, DEF, [by cor. 35. 11.] are equal to one another: Wherefore the folids LH, EK have the fame altitude. But [by 31. 11.] folid parallelepipedons that ftand upon the fame bafe, and have the fame altitude, are equal to one another: Wherefore the folid HL is equal to the folid E K.

But the folid H L is that made from A, B, C, and the folid E K that made from B. Therefore the folid made from A, B, C, is equal to the folid made from в, being equilateral and equiangular to HL.

If therefore three right lines be proportional, the folid parallelepipedon made of all three of them, is equal to the folid parallelepipedon made of the mean line, being equilateral and equiangular to the firft mentioned folid. Which was to be demonftrated.

w It might be better to fay, If three right lines be proportional, the folid parallelepipedon, whofe three fides are those three right lines, will be equal to the folid parallelepipedon, whofe fide is the mean line, being equilateral and equiangular to the folid firft mentioned.

Here it may be observed, that it would have been tnore fimple and natural for Euclid to have called one of the faces of a parallelepipedon, the fide of the folid, than to call one of the right lines which is the fide of one of the faces, the fide of the folid.-If right lines, being the bounds of a fuperficies, be called their fides, I fhould think it would be proper to call the fuperficies, being the bounds of folids, their fides ways done in common fpeech amongst us; for we for inftance, the corner of a houfe its fide. But upright faces is always called the fide of a house. lepipedon, according to Euclid, has twelve fides.

PROP. XXXVII.

This is al never call, one of its A paralle

THEOR.

If four right lines be proportional, the folid paralle lepipedons made from them, being fimilar and alike defcribed, will be proportional; and if the folid parallelepipedons, fimilar and alike defcribed, be proportional: thofe right lines they are defcribed from will be proportional.

Let four right lines A B, C D, E F, G H be proportional; and let A B be to CD, as E F is to GH, and let the fimilar and alike fituate parallelepipedons K A, L C, ME, NG be described from A B, C D, E F, GH: I fay as K A is to LC, fo is ME to N G.

For because the folid parallelepipedon K A is fimilar to LC; KA [by 33. 11.] to L C will be in the triplicate ratio of AB to CD: By the fame reason the folid M E to the

folid NG will be in the triplicate ratio of E F to GH. But

[by

Book XI. [by fuppofition] as A B is to C D, fo is E F to G H: Therefore as A K is to L C, fo is ME to N G.

Now let the folid A be to the folid Lc, as the folid ME is to the folid NG: Ffay as the right line AB is to the right line CD, fo is the right line EF to the right

line GH.

For because again A K is to L C in the triplicate ratio of AB to C D, and ME is to NG in the triplicate ratio of F to G H, and AK is to LC, as M E is to NG; it will be as A B is to CD, fo is EF to GH.

Therefore if four right lines be proportional, the folid parallelepipedons made from them, being fimilar and alike defcribed, will be proportional; and if the folid parallelepipedons being fimilar and alike defcribed, be proportional; thefe right lines upon which they are defcribed, will be proportional. Which was to be demonftrated.

x This propofition is univerfally true, as is the fixteenth of the fixth book, when the four fimilar folids are all of the fame kind. But when two are of one kind, and two of another, it will not be true, unless the two folids of the one kind be defcribed from the first and fecond, or the firft and third, and the two of the other kind from the third and fourth, or fecond and fourth of the four proportional right lines; the firit of thefe lines being to the fecond, as the the third is to the fourth. See the note upon prop. 16. of the fixth book.

PROP. XXXVII.

THEOR.

If one plane be perpendicular to another; and from any point in one plane, a perpendicular be drawn to the other plane; this will fall in the common fection of the planes

For let the plane CD be perpendicular to the plane A`B, and let AD be their common fection, and let any point E be taken in the plane CD; I fay a perpendicular drawn from E to the plane A B will fall in a D.

For if not, let it, if poffible, fall without, as EF meeting the plane AB in the point F: Draw F G [by 10. 1.] from the point in the plane A B, perpendicular to A D, which [by def. 4. 11.] will be perpendicular to the plane CD; and join EG. Then becaufe FG is at right angles so the plane'c p; and the right life EG, which touches it,