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Book XII. another, as alfo M H, HG, GF, and the multitude of them in MF is equal to the multitude of the magnitudes in xK: [by 12. 5.] it will be as KL is to F H, fo is x K to F M. But FM is greater than xK: Therefore [by 14. 5.] FG is greater than L K. But FG is equal to c: wherefore K L is equal to AD: Therefore c will be greater than A D. Which was to be demonstrated.

This lemma is the firft propofition of the tenth book of Euclid.

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Similar polygons infcribed in circles are to one another as the fquares of the diameters.

Let the circles be A B C D E F G H K L, and the fi milar polygons in them be A B C D E, F G H K L; and let BM, GN be diameters of the circles: I fay as the fquare of BM is to the fquare of GN, fo is the polygon ABCDE to the polygon F G H K L.

For join BE, A M, GL, F N. Then because the polygon ABCDE is fimilar to the polygon F G H K L, and [by def. 1. 6.] tbe angle BAE is equal to the angle G FL, and as BA is to A E, fo is GF to FL: Therefore BAE, GFL are two triangles, having one angle of the one equal to one angle of the other, viz. the angle BAE equal to the angle GFL, and the fides about the equal angles proportional: Wherefore [by 6. 6.] the triangle A BE is equiangular to the triangle FGL: and accordingly the angle A E B is equal to the angle FLG. But [by 21. 3.] the angle A E B is equal L to the angle AMB; for they ftand the fame part of the circumference; and the angle FLG is equal to the angle F N G.

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And [by 31. 3.] the right angle BAM is equal to the right angle GFN; and fo the remaining angle is equal to the remaining angle: Therefore the triangle A B M is equiangular to the triangle FGN: Therefore [by 4. 6.] as BM is to GN, fo is BA to G F. But [by 20. 6.] the ra tio of B M to GN is the duplicate of the ratio of the square of B M to the fquare of GN, and the ratio of the polygon

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ABCDE to the polygon FGHKL is the duplicate of the ratio of BA to GF: And therefore [by 11. 5.] as the fquare of BM is to the fquare of GN, fo is the polygon ABCDE to the polygon F G H KL.

Wherefore fimilar polygons infcribed in circles are to one another as the fquares of the diameters. Which was to be demonftrated.

a The circuits of fimilar polygons infcribed in circles, are allo, as the diameters of the circles.

PROP. II. THEO R.

Circles are to one another as the fquares of their diameters.

Let the circles be ABCD, EFGH, and let B D, F H be their diameters. I fay as the fquare of BD is to the fquare of FH, fo is the circle ABCD, to the circle E F G H.

For if it be not fo, it will be as the square of B D is to the fquare of F H, fo is the circle A B C D to fome fpace either greater or lefs than the circle E F G H. Firft let it be to the fpace s lefs than the circle; and in the circle E F G H infcribe a square EFGH. Then the fquare EFGH described in the circle, is greater than one half the circle E F G H ; because if tangents to the circle be drawn thro' the points E, F, G, H, the fquare EFGH will be [by 47. I. and 31. 3.] one half the fquare described about the circle, and the circle is less than the fquare described

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about it: Therefore the fquare EFGH is greater than one half the circle EFGH. Bifect the parts of the circumference EF, FG, GH, HE in the points K, L, M, N; and join EK, KF, FL, LG, GM, MH, HN, NE. Then each of the triangles EKF, FLG, HMG, HNE is greater than one half the fegment of the circle wherein it is; because if thro' the points K,L,M, N tangents be drawn to the circle, and

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Book XII. the parallelograms upon the right lines EF, FG, GH, HE be completed [by 37. 1.] each of the triangles EKF, FLG, GM H, HNE is one half of its correfpondent parallelogram. But the fegment is lefs than the parallelogram; wherefore each of the triangles E KF, FLG, GM H, HNE is greater than one half the fegment of the circle wherein it is: Therefore if the rest of the parts of the circumfereace be bifected, and right lines be joined, and this be always done, there will at last remain fome segments of the circle, which will be lefs than the excefs of the circle EFGH above the fpace s. For it is demonftrated, in the first theorem of the tenth book, that if there be two propofed unequal magnitudes, and from the greater be taken away more than its half, and from the remainder more than its half, and this be continually done, there will at laft remain fome magnitude that is lefs than the least of the proposed magnitudes: Wherefore let the segments of the circle EFGH on the right lines EK, KF, FL, LG, GM, MH, HN, NE, be left, which are greater than the excefs whereby the circle EFGH exceeds the space s. Therefore the remaining polygon EKFLGMHN will be greater than the space s. Alfo in the circle A B C D defcribe the polygon AXBOCPDR fimilar to the polygon EKFLGMHN: Then [by 1. 12.] as the fquare of B B is to the fquare of FH, fo is the polygon AX BOC PDR to the polygon EK FLG MHN. But [by fuppofition] as the fquare of BD is to the fquare of FH, fo is the circle ABCD to the space s: Therefore also as the circle A B C D is to the space s, fo is the polygon AXBOCPDR to the polygon EKF LGMHN; and inverfely, as the circle ABCD is to the polygon within it, so is the space s to the polygon EK FLGMHN. But the circle ABCD is greater than the polygon in it: Wherefore the space s is also greater than the polygon EKF LGMH N. But [by fuppofition] it is lefs too; which is impoffible: Therefore it is not as the fquare of BD is to the fquare of FH, fo is the circle ABCD to fome fpace lefs than the circle EFGH. After the fame manner we demonftrate that `it is not as the fquare of FH to the fquare of B D, fo is the circle E F G H to fome fpace lefs than the circle A B C D : I fay alfo it is not as the fquare of BD to the fquare of FH,

* See the lemma at the beginning of this book.

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fo is the circle ABCD to fome space greater than the cir cle EFGH. For if poffible, let it be fo to a greater space $; then again inverfely it will be as the fquare of FH is to the fquare of BD, fo is the fpace s to the circle ABCD: But [as is demonftrated below] as the fpace s is to the circle A B C D, fo is the circle EFGH to fome space less than the circle A B CD: Therefore as the square of F HIS to the fquare of BD, fo is the circle EFGH to fome space lefs than the circle ABCD; which has been proved to be impoffible: Therefore it is not as the fquare of BD is to the fquare of F H, fo is the circle ABCD to fome fpace greater than the circle EFGH. But it has been proved that it is not so neither to some space lefs: Wherefore as the fquate of BD is to the fquare of FH, fo will the circle ABCD be to the circle E F G H.

Therefore circles are to one another as the squares of their diameters. Which was to be demonstrated.

LEMM A.

Now I fay, if the space s be greater than the circle EFGH, it is as the space s is to the circle ABCD, fo is the circle EFGH to fonie space lefs than the circle

ABCD.

For make as the fpace s is to the circle ABCD, fo is the circle EFGH to the space T. I fay the space T is less than the circle ABCD. For because it is as the fpace s to the circle A B C D, fo is the circle EFGH to the space T; it will be [by i6. 5.1 as the fpace s is to the circle EFGH, fo is the circle ABCD to the fpace T: But [by fup. pofition] the space s is greater than the circle EFGH: Wherefore the

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accordingly as the space s is to the circle A B C D, fo is the circle EFGH to fome fpace lefs than the circle A B C D.

Altho' thefe fort of exhaustive demonftrations ad abfurdum be moft fcrupulously rigorous, and unexceptionably exact, yet their length, and nature, are too apt to give distaste to learn

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ers, and those who have not any great ftrength of mind, or much inclination to pursue these fort of fiudies far: they are generally tired and confufed before they arrive at the conclufions of thefe long demonftrations, and oftentimes receive by them but a faint knowledge and flight conviction of the truth of the theorems. It must indeed be confeffed, that all forts of demonstrations ad abfurdum do not fo much prove the truth of the positions, as the abfurdity that would follow by granting them to be falfe; and that there is a much shorter, direct, and easier way, of fufficiently confirming the truth of these sort of theorems by the method of Indivisibles of Bonaventur. Cavallerius, which, tho' liable to fome exceptions, by men of penetration and found logic, yet the idle and the lefs difcerning either fee none at all, or overlook them ; drowning all leffer faults in the eafinefs and facililty of the method, and smothering every immaterial error, to avoid trouble, tedioufnefs, and confufion.

If circles be conceived to differ from regular polygons, infcribed in them, or circumfcribed about them, of exceeding great equal numbers of very fmall fides, by magnitudes lefs than any affignable magnitudes; thofe circles, without any finite error, may be taken for fuch polygons, and their circumferences for the circumferences of fuch polygons; and fo fince all regular polygons of equal numbers of fides are fimilar figures; the circles themfelves circumfcribing or infcribed within fuch polygons, will be fimilar figures, equivalent to right lined figures; and accordingly [by 1. 12.] will be to one another as the fquares of the diameters; and the circumferences will be as the diameters.

PROP. III. THEOR.

Every pyramid having a triangular bafe may be divided into two equal and fimilar pyramids having triangular bafes, being fimilar to the whole; as alfo into two equal prifms that are greater than one half the whole pyramid.

Let there be a pyramid whofe base is the triangle ABC, and vertex the point D. I fay the pyramid ABCD may be divided into two pyramids equal and fimilar to one another, having triangular bafes, and fimilar to the whole, as alfo into two equal prifms, which two prifms are grearer than one half the pyramid.

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