CHAP. VII. On the Solution of certain Numerical Equations by means of Trigonometrical Formulæ and Tables. In this application of the Trigonometrical Analysis, the utility, whatever it be, relates to the expedition and convenience of the resolution of the equations, and not to any thing novel or curious in its principle and method; moreover, the expedition and conciseness of the resolution depends not on any essential and real abridgment, but on that sort and kind of abridgment which registered computations or tables afford: for instance, we shewed (page 68.) that a cubic equation, such as when solved by approximation, might be used for the computation of sines, or the construction of Tables. Reversely, the Tables constructed may be used for the resolution of similar cubic equations; again, we shewed (page 68.) that an equation of 5 dimensions, such as 2 sin. 5 A, or s = 5 x 5 x3 + x5 solved by approximation, might be made subservient to the construction of Tables. Reversely, Tables constructed either by the approximate solution of equations, or by other methods of approximation, may be employed in the numerical resolution of similar equations. De Moivre solved equations of the third and fifth degree by the cosines of the third part and fifth of an arc; and Vieta divided an arc into three and five equal parts, by equations of the third and fifth degree. This is sufficient, perhaps, to explain the real principle of the solution of equations by Trigonometrical Tables. The convenience or expedition of the method, as we have already said, is of the same nature as the expedition of computation by logarithms. If we do not avail ourselves of computed Tables, the whole process of the solution of a cubic equation, for instance, will be tedious; we must employ some method of approximation; now, if Trigonometrical Tables are employed, the process is short and easy, but only so, because the most laborious part of it is already done for us. In these cases, we avail ourselves of the registered computations of preceding mathematicians. Solution of a Cubic Equation. It may be proper, however, to shew more in detail the method of solving equations by the aid of Trigonometrical Tables. If we take the equations [c"] and [s"] pp. 45, 47, supply a radius r, and put c and s for cos. 3 A, and sin. 3 A, respectively, and x for cos. A and sin. A, then Hence, a being given, find in the Tables the arc 3 A corresponding to it, and, from the same Tables, take out the cosine of A; this latter value is the root of the equation [1]. If s be given, find 3 A corresponding to the sine s, and then take from the Tables the sine of A, which is the root of the equation [2]. But, cubic equations have 3 roots; now, by the Table of p. 16, or by the Cor. 6. of page 28, if c = cos. 3 A, then also c = COS. (2-34) = cos. (2 π+3 A) = cos. (4 π-3 A), &c. hence, substituting instead of the arc A, the arcs 4 π 3 2 π 2 π A, 3 + A, - A, &c. any and all, of the following equations, are true. 3 -3.r2.cos. (-4) 2 π cr=4.(cos.+A) -3.2.cos. (+4). 3 3 3 3 ......[0] ..[6] ...... cr2=4.(cos. - 4) -3..cos. (-4)......[d] 4 π A 3 &c. A π But, cos. (-4) = cos. {2-(+1)} = π 3 3 [since sin. 2 π = 0, and cos. 2 π = 1]. Hence, the equation [d] is precisely the same as the equation [b]. Again, if we take the equation that would follow the equation [d], the cosine in it would be = cos. 2 π cos. 2 π 3 (-4) : hence, this equation is precisely the same as the equation [a]; and, in like manner, succeeding equations would recur, so that, essentially, there are only 3 different equations [1], [a], [b], and hence, in the equation The same reasoning applies to the equation [2], and similar conclusions will follow. = 416.808, &c. Now the arc corresponding to this cosine, or .. A=21° 48′ nat. cos.=.9284858... to rad. 14=12.9988012, 1st root. 3 A, is 65° 24'; 120° +21° 48′= 141°.48′, nat. cos. = -.7858569; and the sum of these two last roots, that is, of the 2d and 3d roots, exactly equals the first root, which ought to be the case, since the coefficient of the 2d term in the proposed equation is = 0. But, cubic equations of any form may be solved by substituting Trigonometrical lines, sines, or tangents, for the quantities under the radical sign in Cardan's form: as Dr. Maskelyne has done in page 57 of his Introduction to Taylor's Logarithms. 1 Not only cubic, but quadratic equations may be solved by the aid of Trigonometrical Tables; and conveniently, when the coefficients are expressed by many figures. |