whence, making a = 45° in (346), (347), (348), we find (353) (354) (355) Scholium. Other forms, serviceable in turning sums into products, and vice versa, may be found; for example, if we make = 90° in (329), we get p = 1 + sin. q = 2sin(45° +49)cos(45° — q) =2sin(45° +19)sin[90° (45°-q)] = =2sin(45° +$9). 1-sing 2cos (45° + q) = 2sin2 (45° -9); So = (357) (358) (359) (360) (361) Combining (329), (330), (331), (332), and (321), we find sin2p - sin'q = cos2p - cos2q = sin(p+q)sin(p—q), also cos'psin'q= cos(p + q)cos(p −q). The student will also find sin(a+b) cotb + cota A denominate equation is one involving denominate quantities, such as length, surface, volume, weight, time, velocity, and the like, referred indeed to a unit of measure, but distinguished from abstract quantities or mere numbers. By clearing of fractions and transposing, it is evident that any abstract or numerical equation may be represented by ; abc... [n factors] + ab2c2 ... [n, factors]+ab3c3 ... [N3] + ... = 0 ; for, if there were any powers they would be embraced in the products of equal factors, such as ab = aa = a2, abc = aaa = a3 ɑ1⁄2 • b2 = ɑ2 • ɑ2 = a2, . &c.; and we may suppose the equation freed from radicals by involution. Now a, b, c,..., ɑ2, b2..., being numbers, may represent the quotients of any denominate quantities divided by their unit of measure, or we may have 2 thus, if A yard, = a line 15 feet in length and the unit of measure be one the number 5; and so on. Substituting these clearing of fractions, on the supposition that the equation has been arranged so that n>n> n3 >.... The last equation is homogeneous, being of the nth degree, or containing n factors in each term, and, as it is denominate, the proposition is demonstrated. This theorem may be serviceable to those not yet well practised in algebra, by detecting errors. For, if we begin a problem with a denominate equation, all the following equations being denominate, will be homogeneous, and if any one, as that containing the result, want this homogeneity, it is an index of error in the operation. But a more important application is the restoring of a quantity which has disappeared from a denominate equation by being assumed as the unit of measure. Thus, if M = 1, the above equation becomes ABC... [n]+ A2B2 C2 • ... [n2] + A3B3C3 • ... [n3] + ... = 0, and the homogeneity disappears; to restore it, however, it is obviously necessary and sufficient to introduce the unit of measure, M, as a factor, into each term affected by an exponent which is the deficiency of the term in degree. Suppose we are to restore the radius in (317); the first member, sinx, is of the first degree, every term of the second must be the same, which requires The first equation in (320) becomes r sin(a + c) = sina cosx + cosa sing, when the radius is restored. So (346) when radius = r, is It is recommended to the student, as an exercise, to restore the radius in all the preceding forms, and to inspect the geometrical equations which have occurred in regard to their homogeneity. PROPOSITION XI. It is required to develope the arc in terms of its tangent. Let the arc AB, as it is the function, be indicated by y and its tangent AT, being the independent variable, by r; it is required to find the function y = fr, that y is of x. Give to the vanishing increment h, to y the corresponding increment k, and draw through T perpendicular to TO and terminating in the secant drawn through the extremities of k and h. Similar triangles give us (300) Fig. 56. OA. OB r2 whence, returning to the function (246) we have = T but x and y vanish together, 0=0+ constant, .. constant = =0; which is the required relation. When radius = 1, (271) becomes y = tany — tan3y + tan3y — ✈ tan'y +, 9 .... (372) PROPOSITION XII. To compute the semicircumference π (pi), when radius is made but (353), tan45° = r = 1, .. tan(4y-45°) = tan4y - ta 45 1+tan4y tan45° 239 whence, by substitution in (372), we get and arc(4y-45°) tan (145)=2ts — t(ats)3 +, -, ..... ; (4y—45°) 3 ..π45° 4y-(4y45°) = 4[z — †(z)3 +, -,...] COMPUTATION OF T. Operation. + Terms. =2200000000000000 (t)=()()=008.'04-'00032 000064000000000 5 000000512 000000056888889 '0000000008192 000000000063016 13 00000000000131072 000000000000077 17 3 2°. (†)3 = (†) (†)2 = ‘2 • ‘04 = ‘008 002666666666667 4°. 6o. 8°. 10°. 00000002048 000000001861818 11 '000000000032768 000000000002185 15 0000000000000524288 000000000000003 19 002668497102102 dif. 197395559849880 4 4[+-+(7)3+, -,...]= + 789582239399520 004184100418410 -[2-(23)3 +, -, ... ] = +000000024416592 correcting the last digits by an extension of the work. To compute the trigonometrical lines. Combining (374) with (317) and (318) we obtain 183 The order of computation is in dicated by 1o, 2o, 3o, ... . (373) (374) |