But let the solid AK be to the solid CL, as the solid EM to the solid GN: the straight line AB is to ED, as EF to GH. Take AB to CD, as EF to ST, and from ST describe (27. 11.) a solid parallelopiped SV similar and similarly situated to either of the solids EM, GN: and because AB is to CD, as EF to ST, and that from AB, CD the solid parallelopipeds AK, CL are similarly described, and in like manner the solids EM, SV from the straight lines EF, ST; therefore AK is to CL, as E F G H Q R EM to SV: but, by the hypothesis, AK is to CL, as EM to GN; therefore GN is equal" (9. 5.) to SV: but it is likewise similar and similarly situated to SV; therefore the planes which contain the solids GN, SV are similar and equal, and their homologous sides GH, ST equal to one another: and because as AB to CD, so EF to ST, and that ST is equal to GH, AB is to CD, as EF to GH. Therefore, if four straight lines, &c. Q. E. D. PROP. XXXVIII. THEOR. “If a plane be perpendicular to another plane, and a straight line be drawn from a point in one of the planes perpendicular to the other plane, this straight line shall fall on the common section of the planes."* “Let the plane CD be perpendicular to the plane AB, and let AD be their common section; if any point E be taken in the plane CD, the perpendicular drawn from E to the plane AB shall fall on AD. See Note. - For, if it does not, let it, if possible, fall elsewhere, as EF; and let it meet the plane AB in the point F; and from F draw (12. 1.) in the plane AB a perpendicular FG to DA, which is also perpendicular (4. def. 11.) to the plane CD; and join EG: then because FG is perpendicular to C the plane CD, and the straight line EG, which is in that plane, meets it; therefore FGE is a right angle (3. def. 11.): but EF is also at right an A gles to the plane AB; and therefore EFG is a right angle: wherefore two of the angles of the triangle R EFG are equal together to two right B angles ; which is absurd : therefore the perpendicular from the point E to the plane AB, does not fall elsewhere than upon the straight line AD; it therefore falls upon it. If therefore a plane,” &c. Q. E. D. PROP. XXXIX. THEOR. In a solid parallelopiped, if the sides of two of the opposite planes be divided each into two equal parts, the common section of the planes passing through the points of division, and the diameter of the solid parallelopiped cut each other into two equal parts. * Let the sides of the D F X Y rallelopiped AF, be divided each into two E equal parts in the points K, L, M, N; X, 0, P, R; and join KL, MN, XO, PŘ: and because T DK, CL are equal and parallel, KL is parallel (33. 1.) to DC : for B the same reason, MN H is parallel to BA: and BẢ is parallel to DC; P R therefore because KL, N G BA, are each of them L M S . See Note. parallel to DC, and not in the same plane with it, KL is parallel (9. 11.) to BA: and because KL, MN are each of them parallel to BA, and not in the same plane with it, KL is parallel (9. 11.) to MN; wherefore KL, MN are in one plane. In like manner, it may be proved, that XO, PR are in one plane. Let YS be the common section of the planes KN, XR; and DG the diameter of the solid parallelopiped AF: YS and DG do meet, and cut one another into two equal parts. Join DY, YE, BS, SG. Because DX is parallel to OE, the alternate angles DXY, YOE are equal (29. 1.) to one another : and because DX is D F Y O I E 1.) to the base YE, and the other angles are equal ; therefore T the angle XYD is equal to the angle OYE, and DYE is a straight (14. 1.) line: B for the same reason м H BSG is a straight line, P S R A G equal and parallel to DB, and also equal and parallel to EG, therefore DB is equal and parallel (9. 11.) to EG: and DE, BG join their extremities; therefore DE is equal and parallel (33. 1.) to BG : and DG, YS are drawn from points in the one, to points in the other; and are therefore in one plane: whence it is manifest, that DG, YS must meet one another; let them meet in T: and because DE is parallel to BG, the alternate angles EDT, BGT are equal (29. 1.); and the angle DTY is equal (15. 1.) to the angle GTS: therefore in 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 parallelogram 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 AL, 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 neces sary to some of the propositions of this book. 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 AB and C be two unequal magnitudes, of which AB is the greater. If from AB there be taken more than D its half, and from the remainder more than its half, A and so on; there shall at length remain a magnitude less than C. For C may be multiplied, so as at length to become greater than AB. Let it be so multiplied, K F 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 H the remainder AH take HK greater than its half, and so on, until there be as many divisions in AB as G 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 B C É See Note. |