472 Mr. wales on the Refolution 
Thus, let a be 10, and b 24 ; and the difference o? the loga- 
rithms of a and \b will be 0.0791812. Now, by running the 
eye along gal diner’s Tables of logarithmic Sines and Tan- 
gents, it will be readily been, that the log. tangent of 26° 33' 50' 7 , 
when increafed by 0.0791812, is lefs than the coline of 
53° 7' 40" twice that arc ; and. that the log. tangent of 
26° 34' o' v , when increafed by the fame quantity, is too great- 
And, by actually taking out the logarithms, and making the- 
additions, the former will be found too fmall by 455, and the 
latter too great by 632. Then, 1087 (4554-632) : 10" :r 
455 : 4"; which being added to 26° 33' 50" gives 26° 33' 54 // 
for the arc of which x is the co-tangent. And if to twice the 
log. co-tangent of this arc the logarithm of a (10) he added,, 
the fum (1.6020600) is the log. of 40, the value of x fought. 
EXAMPLE VI. 
The equation refulting from a Solution of the famous problem: 
of aliiazen maybe given as another example of the ufe of this 
method. Many folutions of this celebrated problem, by huy- 
gens, slustus, and others, may be met with in the Philofo- 
phical TranfaCtions. Solutions to it may alfo be found at the 
end of Dr. robe bt simpson’s Conic Sections, in Dr. smith’s 
Optics,. Mr. robin’s Mathematical Tracis, and other places;, 
but the moft direel and obvious method is, perhaps, that which 
follows. 
Put a — DC, b — dC , r =. Cl ; x - CB andy = CE, the cofmes> 
of the arcs IA, IH, to the radius r: then will the fines of 
by yT 2 — x* and s/r z —y 2 
and, 
thole arcs, BA,- EH, be exprefled 
