PROBLEM XXVIII. To find the fecant and co-fecant of any arch, the fine and co-fine béing given. The figure as in last Prob. CD: CE:: CA: CB; or rather CD: CA: CA: CB; therefore the rectangle contained by the co-fine and fecant of any arch is equal to the fquare of the radius : Or the radius is a mean proportional between the co-fine and fecant of any archi. Hence the fecant is found by RULE I. Divide the fquare of the radius by the co-fine of any arch; and the quotient will give the fecant of that arch. RULE 2. Divide the fquare of the radius by the fine of any arch, the quotient will be the co-fecant of that arch. 86603 -= 115469 co-fecant of 60* Secants may alfo be calculated by 47. I. Euclid, if the radius and tangents are given. Thus, add the squares of the tangent aud radius together, and the fquare root of their fum will be the fecant. Ex. 2. Required the fecant and co-fecant of 24° 13' Anf. (Sec. 109649 [co-fec. 243789 Ex. 3. Required the fecant and co-fecant of 20° 35' 1co-fec. 284438 Ex. 4. Required the fecant and co-fecant of 10° o' (Sec. 1015424 Anfco-fec. 575871 Ex. 5. Required the fecant and co-fecant of 35° 40' Sec. PROBLEM. Fig. 84. 123089 co-fec. 171505 To find the areas of lunes, or the space included between the interfecting arches of two circles. RULE. Find the areas of the two fegments, which form the lune, and their difference will be the area of the lune. EXAMPLE I. The length of the chord AB is 80, the height DC 20, and DE 8, required the area of the lune, AEBCA. AD =40 Ex. 2. The chord is 20, and verfed fines 10 and 2. RequiAnf. 128.522. red the area of the lune. Ex. 3. The length of the chord is 48, and the heights of the fegments 18 and 7. What is the area? Anf. 405.8676 Note. If femicircles be defcribed on the three fides of a rightangled triangle, as diameters, then will the triangle be equal to the two lunes on the legs, taken together. MENSURATION OF SOLIDS. DEFINITIONS. are figures that have length, breadth, and thick 2. The boundaries of folids are fuperficies. 3. A folid angle is that which is made by the meeting of more than two plane angles in the fame point, and which are not in the fame plane. 4. Similar folids are fuch as have their angles fimilar, and which are contained by the fame number of fimilar planes. 5. A cube is a folid contained by fix equal fquares. Fig. 85. 6. A parallelopipedon is a solid having fix rectangular fides, every oppofite pair of which are equal and parallel each to each. Fig. 86. 7. A prifm is a folid whofe fides are parallelograms, and is either triangular, fquare, pentagonal, &c. according to the figure of its end. Fig. 87. 8. A cylinder is a round folid, whofe bafes are equal circles. Fig. 83. 9. A pyramid is a folid, whose base is a plane figure, and its fides triangles, whofe vertices meet in a point, called the vertex of the pyramid, and is either triangular, fquare, pentagonal, hexagonal, &c. according to the figure of its bafe. Fig. 89. 10. A cone is a pyramid, having a circular bafe, and is defcribed by the revolution of a right-angled triangle about one of its legs. It is either right-angled, acute-angled, or obtuseangled, according as the revolving leg is equal to, greater, or lefs than the other. Fig. 9o. 11. The |