VOL. XXVIII.] PHILOSOPHICAL TRANSACTIONS. 5 



or the arc proportional to the time, be 3 degrees. Make the arc aq 2.745 = 

 1.83 + 0-915, and to its log. sine adding the log. of b, there will be had the 

 log. of the number 0.25392 = np, and an — np = 2.74608, and therefore 

 cap = 0.00108. Whence ag = 0.001 nearly, and Aq = 2.746. Thus, by one 

 addition of two logs, the arc Aq will be found which will be true to the 

 thousandth part of a degree. 



Now if the angle ac^ is to be found, not by proceeding gradually, but per 

 saltum, when the mean motion is 45 degrees ; I make the arc Aa to be 40 de- 

 grees, and to its log. sine adding the log. of b, the sum is 0.5325125, which is 

 the log. of the number 3.4081. This number subtracted from 45, leaves an 

 — NP = 41.5919, whose excess above the arc Aa is 1.5919. Whence if it be 

 made, as l + cosin. Aca to l, so is I.5919 to another, the arc aq will be 

 found to be 1.4865 degrees. Therefore A^ = 41.4865, which differs from the 

 truth not much above the thousandth part of a degree. But without this pro- 

 portion A^ may be found, by taking a new arc aq, which is a little less than 

 AN — NP, yet nearly equal to it. For instance, make Aa = 41.50, and add- 

 ing the given log. of b to its log. sine, there will be had another np = 3.35131, 

 which subtracted from an, gives 41. 4869 for a new Aq. And this arc is derived 

 with less trouble, and comes nearer the truth than the former Aq. 



After Aq is found, corresponding to the mean motion 45 degrees, proceed- 

 ing again by steps, by one addition of two logs, will be had Aq to all the sub- 

 sequent degrees of the mean motion. For instance, when the mean motion is 

 46 degrees, J make Aa = 42.4249 > ^^^ adding its log. sine to the constant 

 log. of B, it will be an — np = 42.4249 » to which arc if a new Aa be put 

 equal, there will be had Aq, which will not differ from the true Aq by the 

 thousandth part of a degree. So when the mean motion is 47 degrees, I make 

 Aa = 43.30, equal to the former Aq added to the increment of that arc for one 

 degree of mean motion, and adding its log. sine to the log. of b, the sum will 



be the log. of the number 3.6402, which subtracted from an, leaves an 



NP = 43.3593, equal to the new Aq, which differs from the true a^ about the 

 ten thousandth part of a degree. 



If, omitting the intermediate degrees, the arc Aq is to be found when the 

 mean motion is 100 degrees ; make Aa 96°, and adding its log. sine to the log. 



of B, the sum will be equal to the log. of the number 5.273, whence an 



NP = 94.727. Therefore, secondly, make Aa = 94.72, and adding its sine 

 to log. B, there will arise the log. of 5.285, which subtracted from an leaves 

 AN — NP = 94.715 = Aq very, nearly. In like manner, if the mean motion 

 be 101^, make Aa = 95.71, whose log. sine added to the log. of b, gives the 

 log. of the number 5.2756, which number taken from 101, there will remain 



