6 PHILOSOPHICAL TRANSACTIONS. [aNNO 1713. 



AN — NP = 95.7244 = Aq. And in this manner, the mean motion being given, 

 by a gradual process the angle at the centre will be had, by the addition only of 

 two logs, one of which, being constant, may be preserved on the paper, to 

 spare the labour of writing it down too often. 



Now let us proceed to an orbit of the other species, such as the distance of 

 the aphelion may be to the distance of the perihelion, as 70 to J . Such nearly 

 was the orbit of that comet, which completes its period in 754- years ; as was 

 first found by that sagacious astronomer and geometrician Dr. Edmund Halley. 

 In this orbit ac or cq, will be 35.5 and cs 34.4 of such parts as sb is one. And 

 the arc Bq is to be found, when the mean motion is one lOOth part of a degree. 

 Since the middle distance exceeds the least distance about 35 times, I make 

 BQ := 0.35, when the mean motion is 0.01. In this orbit the constant log. of 

 B is found 1.7457133. Therefore this log. being added to the log. sine of the 

 arc 0.35, gives the log. of the number 0.34013, which added to the arc 0.01, 

 will make 0.35013. If this sum had been equal to 0.35, the arc ea would have 

 been rightly assumed; but the difference is 0.00013. Whence because cb is 

 to SB as 35.5 to 1, let the difference 0.00013 be multiplied by 35.5, and there 

 will arise aq = 0.004615 ; whence it will be arc Bq = 0.3546 1 5, which hardly 

 differs from the truth by 3 parts of ten thousand. 



Secondly, let the mean motion be 0.02, and suppose Ba to be 0.7]. To its 

 log. sine adding the log. of b, the sum will be the log. of the number O.O8998 ;, 

 whence bn + np = O.70998, and therefore the assumed arc bq = O.71 was 

 too much, and the difference is 0.00(X)2. Which if it be multiplied by 35.5, 

 and the product subtracted from sa, there will remain b^ = O.7092, deviating 

 from the truth hardly the ten thousandth part of a degree. 



Let the mean motion be 0.03. Suppose bq to be I.06 degrees, adding its 

 log. sine to the log. of b, the sum will be the log. of the number 1.03008. To 

 -which if bn = 0.03 be added, the sum will be I.060O8, which number is 

 greater than Ba ; wherefore if the difference 0.00008 be multiplied by 35.5, 

 and added to bq, it will be b^ = 1.06284. In like manner, when the mean 

 motion is 0.04, I suppose bq = 1.40 degrees, and find np = 1.3604; to 

 which number adding bn = 0.04, the sum is 1.4004, which exceeds 1.40 by 

 0.004. Let this difference be multiplied by 35.5, and the product 0.01420 

 will be equal to q^ ; whence b^ = 1.41420. In all these instances the errors 

 are very small, and seldom go beyond the thousandth part of a degree. 



Now let the arc Bq be to be found, when the mean motion is equal to one 

 degree. Suppose bq = 20°, and adding its log. sine to the log. of b, there 

 will be had the log. of the number 19.045; to which adding bn = I*', the sum 

 20.045 exceeds 20 by 0.045. And since in this case l — cosin. bq is to l, as 



