169-173. Equations involving the number of faces, edges, angles containing a solid angle in regular polyhe- 26-27. To find the arithmetic value of the modulus, and to construct a table of the Decimal Logarithms.. 317 28-31. Nature and use of the tables of parts proportional.. 319 32-37. On the tables of logarithmic sines, &c. ... 38-42. Formulæ for finding accurately the angle when very ...... A SYSTEM OF PLANE AND SPHERICAL TRIGONOMETRY. PART I. PLANE TRIGONOMETRY. SECTION I. ON ANGLES AND THEIR TRIGONOMETRIC FUNCTIONS. 1. LEMMA. The circumferences of circles are to one another as the radii. LET ABC and A'B'C' be two circles, of which the radii are r and r'. Place them so that their centres may coincide at O; draw two radii cutting the circles in A, P, A', P'; from P, P' draw PN, P'N' perpendiculars to OA; and from A, A' draw AT, A'T' at right angles to AO, meeting OP, OP' produced in T, T'; therefore the four triangles TAO, T'A'O, PNO, P'N'O, have each a right angle, and the angle at O common, therefore the remaining angles are equal, and the triangles are similar : B this will remain true, whatever be the magnitude of the angle AT at O; let it then be diminished without limit, and which AO PN' by similar triangles is equal to in the vanishing state, be comes AO or 1; that is, in this state AT coincides with PN. But it is evident, that AP is not greater than AT, and not less than PN; hence, in the vanishing state, it coincides with either of them. Similarly A'P' coincides either with A'T or P'N': AT AT 2. COR. 1. From this proposition it appears, that in the circle, circumference a constant number. |