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another, as also MH, HG, GF, and the multitude of them in me is equal to the multitude of the magnitudes in xk: [by 12. 5.) it will be as KL is to FH; fo is x K to FM. But FM is greater than xk: Therefore [by 14. 5.) FG is greater than I K.
But FG is equal to c: wherefore ki is equal to AD: Therefore c will be greater than AD. Which was to be demonstrated.
This lemma is the first proposition of the tenth book of Euclid.
PROP. I. THEOR. Similar polygons inscribed in circles are to one another
as the squares of the diametersa. Let the circles be A B C D E, F G H K L, and the fre milar polygons in them be A B C D E, F G let BM, GN be diameters of the circles: I say as the square of BM is to the square of GN, fo is the polygon ABCDE to the polygon FGHKL.
For join B E, AM, GL, FN. Then because the polygon A B C D E is similar to the polygon F G H K L, and (by def. 1. 6.] tbe angle B A E is equal to the angle GFL, and as B A is to A E, so is GF to Fl: Therefore B A E, GFL are two triangles, having one angle of the one equal to onaangle of the other, viz. the angle B A E equal to the angle GFL, and the sides 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 F
But [by 21. 3.]
the angle A EB is equal BA
L to the angle A MB; for G
In they stand upon the same M
part of the circumference; H K
and the angle FLG is
equal to the angle FN G. And [by 31. 3.] the right angle BAM is equal to the right angle GFN; and so the remaining angle is equal ta the remaining angle: Therefore the triangle A B M is equiangular to the triangle FGN: Therefore [by 4. 6.) as BM is.to GN, so i$ B A tO G F. But [by 20. 6.) the ratio of BM to GN is the duplicate of the ratio of the square of BM to the square of GN, and the ratio of the polygon
A B CD E to the polygon FGHKL is the duplicate of the ratio of B A to GF: And therefore [by 11. 5.] as the square of BM is to the square of GN, so is the polygon ABCDE to the polygon FGHKL.
Wherefore similar polygons inscribed in circles are to one another as the squares of the diameters. Which was to be demonstrated.
The circuits of similar polygons inscribed in circles, are allo, as the diameters of the circles.
PRO P. II. THEOR.
diametersb, Let the circles be ABCD, EFGH, and let BD, FH be their diameters. I say as the square of bd is to the square of FH, so is the circle ABCD, to the circle EFGH.
D For if it be not fo, it will be as the square of BD is to the square of
P FH, so is the circle ABCD to some space either greater or less than the circle E F GH. First let it be to the space s less than the circle ; and in
K the circle E F G H inscribe a square EFGH. Then the square E F G H
F described in the circle, is greater than one half the circle EFGH; be
M cause if tangents to the circle be drawn thro' the points E, F, G, H, the square EFGH will be [by 47. I. and 31, 3.) one half the square de
S scribed about the circle, and the circle is less than the square described about it: Therefore the square EFGH is greater than one half the circle EFGH, Bifect the parts of the circumference ET, FG, GH, He in the points K, L, M, N; and join EK, KF, FL, LG, GM, MH, HN, NE. Then each of the trie angles EKF, PLG, HMG, HNE is greater than one half the segment of the circle wherein it is ; because if thro' the points K,L,M, N tangents be drawn to the circle, and Bb
the parallelograms upon the right lines E F, FG, GH, HE be completed [by 37. 1.) each of the triangles EKF, FLG, GMH, HN E is one half of its correspondent parallelogram. But the segment is less than the parallelogram; wherefore each of the triangles E K F, FLG, GMH, HNE is greater than one half the segment of the circle wherein it is: Therefore if the rest of the parts of the circumference be bisected, and right lines be joined, and this be always done, there will at last remain fome segments of the circle, which will be less than the excess of the circle EFG H above the space s. For it is * demonstrated, in the first theorem of the tenth book, that if there be two propused 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 last remain some magnitude that is less than the least of the proposed magnitudes : Wherefore let the segments of the circle ergh on the right lines EK, KF, FL, LG, GM, MH, HN, NE., be left, which are greater than the excess whereby the circle EFGH exceeds the space s. Therefore the remaining polygon EKFLG MHN will be greater than the space s.
Also in the circle ABCD describe the polygon AX B O C PDR fimilar to the polygon EKFLGMHN: Then [by 1. 12.) as the square of BD is to the square of FH, so is the polygon AXBOC PDR to the polygon E KFLG MHN. But [by fuppofition) as the square of BD is to the square of FH, so is the circle ABCD to the space s: Therefore also as the circle ABCD is to the space s, so is the polygon AX BOC PDR to the polygon EKFLCMHN; and inversely, as the circle ABCD is to the polygon within it, so is the space s to the polygon EKFLGMHN.
But the circle ABCD is greater than the polygon in it: Wherefore the space s is also greater than the polygon E KF LGMHN. But [by fuppofition] it is less too; which is impossible: Thereforo it is not as the square of BD is to the square of FH, fo is the circle ABCD to some space less than the circle EFGH. After the same manner we demonftrate that it is not as the square of Fh to the square of BD, fo is the circle E F G H to fome space less than the circle ABCD: I say also it is not as the square of BD to the square of FH, * See the lemma at the beginning of this book.
To is the circle ABCD to fome space greater than the circle EFGH. For if possible, let it be lo to a greater space $; then again inveisely it will be as the square of this to the square of BD, fo is the space s to the circle ABCD: But [as is demonstrated below] as the space s is to the circle ABCD, so is the circle EFGH to some space less than the circle ABCD: Therefore as the square of śH is to the square of BD, so is the circle EFG H to some space less than the circle ABCD; which has been proved to be impoffible: Therefore it is not as the square of BD is to the square of F#, so is the circle A B C D toʻsome space greater than the circle EFGH. But it has been proved that it is not so neither to some space less: Wheretore as the square of BD is to the square of FH, so will the circle ABCD be to the circle EFGH,
Therefore citcles are to one another as the squares of their diameters. Which was to be demonstrated.
L E M M A.
For make as the space s is to the circle ABCD, so 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 space s to the circle ABCD, so is the circle EFG H có the space t; it will be [by i6. s.)
B space r; But [by sup. position) the space s is greater than the circle EFGH: Wherefore the
S circle ABCD is greater
T than the space T; and accordingly as the space s is to the circle ABCD, fo is the circle EFGH to some space less than the circle ABCD.
Altho' these fort of exhaustive demonftrations ad absurdum be most fcrupulously rigorous, and unexceptionably exact, yet their length, and nature, are too apt to give diftalte to ItarnBb 2
ers, and those who have not any great strength of mind, or much inclination to pursue these sort of fiudies far: they are generally tired and confused before they arrive at the conclusions of these long demonstrations, and oftentimes receive by them but a faint knowledge and flight conviction of the truth of the theorems. It must indeed be confessed, that all forts of demonstrations ad absurdum do not so much prove the truth of the positions, as the absurdity that would follow by granting them to be false; and that there is a much sorter, dire&, and easier way, of sufficiently confirming the truth of these sort of theorems by the method of Indivisibles of Bonaventur. Cavallerius, which, tho' liable to some exceptions, by men of penetration and found logic, yet the idle and the less discerning either see none at all, or overlook them ; drowning all lefser faults in the easiness and facililty of the method, and smothering every immaterial error, to avoid trouble, tediousness, and confufion.
if circles be conceived to differ from regular polygons, inscribed in them, or circumscribed about them, of exceeding great equal numbers of very small sides, by magnitudes less than any assignable magnitudes; those circles, without any finite error, may be taken for such polygons, and their circumferences for the circumferences of such polygons; and so since all regular polygons of equal numbers of lides are similar fi. gures; the circles themselves circumscribing or inscribed within such polygons, will be similar figures, equivalent to right lined figures; and accordingly [by 1.12.] will be to one another as the squares of the diameters; and the circumferences will be as the diameters.
PROP. III. THEOR. Every pyramid having a triangular base may be di
vided into two equal and similar pyramids having triangular bases, being similar to the whole; as also into two equal prisms that are greater than one half the whole pyramid.
Let there be a pyramid whole base is the triangle ABC, and vertex the point D. I say the pyramid A BCD may be divided into two pyramids equal and similar to one another, having triangular bases, and fimilar to the whole, as also into two equal prifms, which two prisms are grearer than one half the pyramid.