4 68 Mr. wales on the Refolution 
along gaediner’s tables of logarithmic lines, by which means 
it will be readily found, that the fum of the log. cofme and 
half the log. fine of 28° 53' 30'' is lefs than 19.7843181, the 
excefs of half the log. of 10 above i f log. 3, by 15, and that 
the lum of the log. cofine and half the log. line of 28’ 53' 4c/ 7 
is greater than that difference by 60. Confequently 75 
{15 + 60) : io // :: 15 : ffb The exact arc, therefore, of 
which the fum of the log. cofme 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 x required, true to the lad 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. cofme and half the log. 
fine of 41 0 48' 30" is greater than 19.7843181 by 50, and 
that the fum of the log. cofne and half the log. line of 
41 0 48' 40" is too little by 21. Confequently, 71 (50 + 21) 
: io v :: 50 t 7" : of courfe, 41° 48' 37" is another arc, of 
which the fum of the log. coline and- half the log. line is equal 
to 19.7843181, and the log. fine of this arc, increafed by the 
log. Of 3, is the logarithm of 1.999999, ^ n °th er value of ir, 
and which errs but by unity in the feventh place. 
The third root, as it is generally called, of this equation, 
which is neceflarily negative, and equal to the fum of the other 
two, belongs properly to the equation which is given as the fmft 
example, of which it is the affirmative root, and may be found 
by the directions which are there given. 
+ X A MPL£ 
