VOL. LXXI.] PHILOSOPHICAL TRANSACTIONS. 145 



37' 36', and the log. secant of it is 10.0566242, which being increased by 

 0.8494850, the log. of -/50, gives O.9061092, which is the logarithm of 

 8.055810, the value of x sought, and which is true to 7 places of figures. 



Exam. 2. — To find the value of x in the equation x 3 — r 2 x = — a. 



If r be expounded by 3, and a by 10, the equation will be x 3 — Qx = — 10, 

 and may be transformed to Vx X V (9 — x~) = V 10; and therefore by the 

 tables, the square root of the sine into the cosine of an arc, of which the radius 

 is 3, is equal to the square root of 10. Consequently an arc must be found, 

 such that the sum of the log. cosine and half the log. sine is equal to half the 

 log. of 10. But because the radius of this arc must be 3, the log. sines and 

 cosines must be increased by the log. of 3; and therefore log. cos. -f- log. of 3 

 + 4- log. sine + 4- log. of 3 must be equal to half the log. of 10; or, an arc 

 must be found of which the sum of the tabular log. cosine and half the log. sine 

 is equal to the difference between half the log. of 10 and 14 the log. of 3. 

 Hence, having subtracted 14 log. of 3 from half the log. of 10, run the eye 

 along Gardiner's tables of logarithmic sines, by which means it will be readily 

 found, that the sum of the log. cosine and half the log. sine of 28° 53' 30" is 

 less than 197843181, the excess of half the log. of 10 above 14 log. 3, by 15, 

 and that the sum of the log. cosine and half the log. sine of 28° 53' 40' / is 

 greater than that difference by 60. Consequently 75(15 + 60) : 10" :: 15 : 1", 

 The exact arc therefore, of which the sum of the log. cosine and half the log. 

 sine is equal to 19.7843181, is 28° 53' 32"; and the log. sine of this arc, in- 

 creased by the log. of 3, is 0.1 6 121 53, the logarithm of 1. 44949, the value of 

 x required, true to the last place. 



But many equations of this form, and this example among the rest, admit of 

 two positive values of the unknown quantity; and by carrying the eye farther 

 along the tables it will be found also, that the sum of the log. cosine and half 

 the log. sine of 41° 48' 30" is greater than 197843181 by 50, and that the sum 

 of the log. cosine and half the log. sine of 41° 48' 40" is too little by 21. Con- 

 sequently, 71 (50 + 21) : 10" :: 50 : 7" : of course, 41° 48' 37" is another arc, 

 of which the sum of the log. cosine and half the log. sine is equal to 197843181, 

 and the log. sine of this arc, increased by the log. of 3, is the logarithm of 

 I.999999, another value of x, and which errs but by unity in the 7th place. 



The 3d root, as it is generally called, of this equation, which is necessarily 

 negative, and equal to the sum of the other two, belongs properly to the equa- 

 tion which is given as the first example, of which it is the affirmative root, and 

 may be found by the directions there given. 



Exam. 3. — To find the value of x in the equation a? -\- r*x = a. 



Let us take as examples of this equation x 3 -f 3.r = .04, x 3 -f- 3x = .08, and 

 a 3 + 3x = .12, which are 3 of the instances given by Dr. Halley, in his Synopsis 



vol. xv. U 



