Bisect the arcs EF, FG, GH, HE at the pts. K, L, M, N, and join EK, KF, FL, LG, GM, MH, HN, NE. Then each of the As EKF, FLG, GMH, HNE, is greater than half of the segment of the circle in which it stands. For ▲ EKF = half of the ☐, formed by drawing a st. line to touch the at K, and parallel st. lines through E and F ; and the thus formed is greater than the segment FEK; .. ▲ EKF is greater than half of the segment FEK, and similarly for the other As. .. sum of all these triangles is greater than half of the sum of the segments of the O, in which they stand. Next, bisect EK, KF, etc., and form As as before. Then the sum of these As is greater than half of the sum of the segments of the O, in which they stand. If this process be continued, and the As be supposed to be taken away, there will at length remain segments of Os, which are together less than the excess of the EFGH above the space S, by the Lemma. Let segments EK, KF, FL, LG, GM, MH, HN, NE be those which remain, and which are together less than the excess of the O of the above S. Then the rest of the O, i.e. the polygon EKFLGMHN, is greater than S. In ABCD inscribe the polygon AXBOCPDR similar to the polygon EKFLGMHN. The polygon AXBOCPDR is to polygon EKFLGMHN as ..the polygon EKFLGMHN is less than the space S; V.14. but it is also greater, which is impossible; .. sq. on BD is not to sq. on FH aso ABCD is to any space less than EFGH. In the same way it may be shown that sq. on FH is not to sq. on BD as o EFGH is to any space less than O ABCD. Nor is sq. on BD to sq. on FH as ABCD is to any space greater than EFGH. For, if possible, let it be as C ABCD is to a space T, greater than EFGH. Then, inversely, sq. on FH is to sq. on BD as space T is to ABCD. But as space T is to © ABCD so is © EFGH to some space, which must be less than greater than o EFGH. .. sq. on FH is to sq. on BD as ABCD, because space Tis V. 14. less than ABCD; which has been shewn to be impossible.. EFGH is to some space ABCD is to any .. sq. on BD is not to sq. on FH as space greater than EFGH. And it has been shown that sq. on BD is not to sq. on FII as ABCD is to any space less than EFGH. .. sq. on BD is to sq. on FH as ABCD is to © EFGH. Q. E. D. Papers on Euclid (Books VI., XI., and XII.) set in the 1849. 1850. 1851. 1852. VI. 4. Apply this proposition to prove that the rectangle, contained by the segments of any chord, passing through a given point within a circle, is constant. XI. 11. Prove that equal right lines, drawn from a given point to a given plane, are equally inclined to the plane. VI. 10. AB is a diameter, and P any point in the circumference of a circle; AP and BP are joined and produced, if necessary; if from any point C of AB a perpendicular be drawn to AB, meeting AP and BP in points D and E respectively, and the circumference of the circle in a point F, shew that CD is a third proportional to CE and CF. VI. 3. If A, B, C be three points in a straight line, and D a point, at which AB and BC subtend equal angles, show that the locus of the point D is a circle. XI. 8. From a point E draw EC, ED perpendicular to two planes CAB, DAB, which intersect in AB, and from D draw DF perpendicular to the plane CAB, meeting it in F: shew that the line, joining the points C and F, produced if necessary, is perpendicular to AB. VI. 2. If two triangles be on equal bases, and between the same parallels, any line, parallel to their bases, will cut off equal areas from the two triangles. 1852. XI. 11. ABCD is a regular tetrahedron, and, from the vertex A, a perpendicular is drawn to the base BCD, meeting it in 0: shew that three times the square on AO is equal to twice the square on AB. 1853. 1854. 1855. VI. 6. If the vertical angle C, of a triangle ABC, be bisected by a line, which meets the base in D, and is produced to a point E, such that the rectangle, contained by CD and CE, is equal to the rectangle, contained by AC and CB: shew that if the base and vertical angle be given, the position of E is invariable. XI. 21. If BCD be the common base of two pyramids, whose vertices A and A' lie in a plane passing through BC, and if the two lines AB, AC, be respectively perpendicular to the faces. BA'D, CA'D, prove that one of the angles at A, together with the angles at A', make up four right angles. VI. 16. EA, EA' are diameters of two circles, touching each other externally at E; a chord AB of the former circle, when produced, touches the latter at C, while a chord A'B' of the latter touches the former at C: prove that the rectangle, contained by AB and A'B', is four times as great as that contained by BC and B'C. XI. 20. Within the area of a given triangle is described a triangle, the sides of which are parallel to those of the given one: prove that the sum of the angles, subtended by the sides of the interior triangle, at any point, not in the plane of the triangles, is less than the sum of the angles, subtended at the same point by the sides of the exterior triangle. vi. 2. A tangent to a circle, at the point A, intersects two parallel tangents in B, C, the points of contact of which with the circle are D, E, respectively: shew that if BE, CD, intersect in F, AF is parallel to the tangents BD, CE. 1855. XI. 16. From the extremities of the two parallel straight lines AB, CD, parallel lines Aa, Bb, Cc, Dd, are drawn, meeting a plane in a, b, c, d : prove that AB is to CD as ab is to cd, taking the 1856. 1857. 1858. case, in which A, B, C, D are on the same side of the plane. VI. Def. 1. Enunciate the propositions, which prove that in the case of triangles the conditions of similarity are not independent. XI. 11. Shew that the perpendicular, dropped from the vertex of a regular tetrahedron upon the opposite base, is treble of that dropped from its own foot upon any of the other bases. VI. 19. Any two straight lines, BB', CC', drawn parallel to the base DD', of a triangle ADD', cut AD in B, C, and AD' in B', C ; BC, B'C, are joined. prove that the area ABC or AB'C varies as the rectangle, contained by BB',CC. XI. 16. A triangular pyramid stands on an equilateral base, and the angles at the vertex are right angles: shew that the sum of the perpendiculars on the faces, from any point of the base, is constant. vi. 15. Find a point in the side of a triangle, from which two lines, drawn one to the opposite angle, and the other parallel to the base, shall cut off, towards the vertex and towards the base, equal triangles. XI. 11. Two planes intersect: shew that the loci of the points, from which perpendiculars on the planes are equal to a given straight line, are straight lines; and that four planes may be |