468 Mr. wales on the Refolution 
along Gardiner’s tables of logarithmic fines, by which means 
it will be readily found, that the fum of the log. cofine and 
half the log. fine of 28° 53' 30'' is lefs than 19.7843181, the 
excefs of half the log. of 10 above if log. 3, by 15, and that 
the lum of the log. cofme and half the log, fine of 28 s 53' 40" 
is greater than that difference by 60. Confequently 75 
(15 + 60) : 10" :: 15 : 2". The exaCt arc, therefore, of 
which the fum of the log. cofine and half the log. fine is equal 
to 19.7843181, is 28° 53' 32"; and the log. fine of this arc, 
increafed by the log. of 3, is 0.1612153, the logarithm of 
1.44949, the value of * required, true to the laft place. 
But many equations of this form, and this example among 
the reft, admit of two pofitive values of the unknown quan- 
tity ; and by carrying the eye farther along the tables it will be 
found alfo, that the fum of the log. cofine and half the log. 
line of 41 0 48' 30^ is greater than 19.7843181 by 50, and 
that the fum of the log. cofine and half the log. line of 
41° 48' 40 n is too little by 21. Confequently, 71 (50 + 21) 
: io'' :: 50 : 7": of courfe, 41 0 48' 37^ is another arc, of 
which the fum of the log. cofme and half the log. fine is equal 
to 19.7843181, and the log. fine of this arc, increafed bv the 
log. of 3, is the logarithm of 1.999999, another value of x 9 
and which errs but by unity in the feventh place. 
The third root, as it is generally called, of this equation, 
which is neceffarily negative, and equal to the fum of the other 
two, belongs properly to the equation which is given as the firft: 
-example, of which it is the affirmative root, and may be found 
by the directions which are there given. 
£ XA M H S 
