the triangles DTY, GTS there are two angles in the one equal to two angles in the other, and one side equal to one side, opposite to two of the equal angles, viz. DY to GS; for they are the halves of DE, BG: therefore the remaining sides are equal (26. 1.) each to each. Wherefore, DT is equal to TG, and YT equal to TS. Wherefore, if in a solid, &c. Q. E. D.

PROP. XL. THEOR.

If there be two triangular prisms of the same altitude, the base of one of which is a parallelogram, and the base of the other a triangle; if the parallellogram be double of the triangle, the prisms shall be equal to one another.

Let the prisms ABCDEF, GHKLMN be of the same altitude, the first whereof is contained by the two triangles ABE, CDF, and the three parallelograms AE, DE, EC; and the other by the two triangles GHK, LMN, and the three parallelograms LH, HN, NG; and let one of them have a parallelogram AF, and the other a triangle GHK for its base; if the parallelogram AF, be double of the triangle GHK, the prism ABCDEF is equal to the prism GHKLMN.

Complete the solids AX, GO; and because the parallelogram AF is double of the triangle GHK; and the parallelogram HK double

(34. 1.) of the same triangle; therefore the parallelogram AF is equal to HK. But solid parallelopipeds upon equal bases, and of the same altitude, are equal (31. 11.) to one another. Therefore the solid AX is equal to the solid GO; and the prism ABCDEF is half (28. 11.) of the solid AX; and the prism GHKLMN half (28. 11.) of the solid GO. Therefore the prism ABCDEF is equal to the prism GHKLMN. Wherefore, if there be two, &c. Q. E. D.

THE

ELEMENTS OF EUCLID.

BOOK XII.

LEMMA I.

Which is the first proposition of the tenth book, and is necessary to some of the propositions of this book.

Ir from the greater of two unequal magnitudes, there be taken more than its half, and from the remainder more than its half, and so on there shall at length remain a magnitude less than the least of the proposed magnitudes.*

Let AB and C be two unequal magnitudes, of which AB is the greater. If from AB there be taken more than its half, and from the remainder more than its half, and so on; there shall at length remain a magnitude less than C.

D

A

K

H

F

For C may be multiplied, so at length to become greater than AB. Let it be so multiplied, and let DE its multiple be greater than AB, and let DE be divided into DF, FG, GE, each equal to C: From AB take BH greater than its half, and from the remainder AH take HK greater than its half, and so on, until there be as many divisions in AB as there are in DE: and let the divisions in AB be AK, KH, HB; and the divisions in ED be DF, FG, GE. And because DE is greater than AB, and that EG taken from DE is not greater than its half, but BH taken from AB is greater than its half; therefore the remainder GD is greater than the remainder HA. Again, because GD is greater than HA, and that GF is not greater than the half of GD, but HK is greater than the half of HA; therefore the remainder FD is greater than the remainder

* See Note.

ВС

E

AK

And FD is equal to C, therefore C is greater than AK; that is, AK is less than C. Q. E. D.

And if only the halves be taken away, the same thing may in the same way be demonstrated.

PROP. I. THEOR.

SIMILAR polygons inscribed in circles are to one another as the squares of their diameters.

Let ABCDE, FGHKL be two circles, and in them the similar polygons ABCDE, FGHKL; and let BM, GN be the diameters of the circles; as the square of BM is to the square of GN, so is the polygon ABCDE to the polygon FGHKL.

Join BE, AM, GL, FN: and because the polygon ABCDE is similar to the polygon FGHKL, and similar polygons are divided into similar triangles: the triangles ABE, FGL are similar and equiangu

lar (6. 6.); and therefore the angle AEB is equal to the angle FLG: but AEB is equal (21. 3.) to AMB, because they stand upon the same circumference; and the angle FLG is, for the same reason, equal to the angle FNG: therefore also the angle AMB is equal to FNG: and the right angle BAM is equal to the right (31. 3.) angle GFN; wherefore the remaining angles in the triangle ABM, FGN are equal, and they are equiangular to one another: therefore as BM to GN, so (4. 6.) is BA to GF; and therefore the duplicate ratio of BM to GN, A F

is the same (10. def. 5. and 22. 5.) with the duplicate ratio of BA to GF: but the ratio of the square of BM to the square of GN is the duplicate (20. 6.) ratio of that which BM has to GÑ; and the ratio of the polygon ABCDE to the polygon FGHKL is the duplicate (20. 6.) of that which BA has to GF: therefore, as the square of BM to the

square of GN, so is the polygon ABCDE, to the polygon FGHKL. Wherefore, similar polygons, &c. Q. E. D.

PROP. II. THEOR.

CIRCLES are to one another as the squares of their diameters.*

Let ABCD, EFGH be two circles, and BD, FH their diameters: as the square of BD to the square of FH, so is the circle ABCD, to the circle EFGH.

For, if it be not so, the square of BD shall be to the square of FH, as the circle ABCD is to some space either less than the circle EFGH, or greater than it. First, let it be to a space S less than the circle EFGH; and in the circle EFGH describe the square EFGH: this square is greater than half of the circle EFGH: because if, through the points E, F, G, H, there be drawn tangents to the circle, the square EFGH is half (41. 1.) of the square described about the circle; and the circle is less than the square described about it; therefore the square EFGH is greater than half of the circle. Divide the circumferences EF, FG, GH, HE, each into two equal parts in the points K, L, M, N, and join EK, KF, FL, LG, GM, MH, HN, NE: therefore each of the triangles EKF, FLG, GMH, HNE is greater than half of the segment of the circle it stands in; because, if straight lines touching the circle be drawn through the points K, L, M, N, and parallelograms upon the straight lines EF, FG, GH, HE be completed; each of the triangles EKF, FLG, GMH, HNE shall be the half (41. 1.) of the parallelogram in which it is: but every segment is less than the parallelogram in which it is: wherefore each of the triangles EKF, FLG, GMH, HNE is greater than half the segment of the circle which contains it: and if these circumferences before named be divided each into two equal parts, and their extremities be joined

by straight lines, by continuing to do this, there will at length remain segments of the circle, which, together, shall be less than the

* See Note.

† For there is some square equal to the circle ABCD; let P be the side of it; and to three straight lines BD, FH, and P, there can be a fourth proportional; let this be Q: therefore the squares of these four straight lines are proportionals, that is, to the squares of BD, FH, and the circle ABCD, it is possible there may be a fourth proportional. Let this be S. And in like manner, are to be understood some things in some of the following propositions.

excess of the circle EFGH above the space S: because, by the preceding lemma, if from the greater of two unequal magnitudes there be taken more than its half, and from the remainder more than its half, and so on, there shall at length remain a magnitude less than the least of the proposed magnitudes. Let then the segments EK, KF, FL, LG, GM, MH, HN, NE, be those that remain and are together less than the excess of the circle EFGH above S: therefore the rest of the circle, viz. the polygon EKFLGMHN, is greater than the space S. Describe likewise in the circle ABCD the polygon AXBOCPDR similar to the polygon EKFLGMHN: as therefore, the square of BD is to the square of FH, so (1. 12.) is the polygon AXBOCPDR to the polygon EKFLGMHN: but the square of BD is also to the square of FH, as the circle ABCD is to the space S: therefore as the circle ABCD is to the space S, so is (11. 5.) the polygon AXBOCPDR to the polygon EKFLGMHN: but the circle ABCD is greater than the polygon contained in it: wherefore the space S is greater (14. 5.) A

than the polygon EKFLGMHN: but it is likewise less, as has been demonstrated: which is impossible. Therefore the square of BD is not the square of FH, as the circle ABCD is to any space less than the circle EFGH. In the same manner it may be demonstrated, that neither is the square of FH to the square of BD, as the circle EFGH is to any space less than the circle ABCD. Nor is the square of BD to the square of FH, as the circle ABCD is to any space greater than the circle EFGH: for, if possible, let it be so to T, a space greater than the circle EFGH: therefore, inversely, as the square of FH to the square of BD, so is the space T to the circle ABCD. But as the space T* is to the circle ABCD, so is the circle

* For, as in the foregoing note, at † it was explained how it was possible there could be a fourth proportional to the squares of BD, FH, and the circle ABCD, which was named S. So in like manner there can be a fourth proportional to this other space named T, and the circles ABCD, EFGH. And the like is to be understood in some of the following propositions.