306 PHILOSOPHICAL TRANSACTIONS. [aNNO 1717. 



— nz. Make this equal to I , and we have m : w :: z : a? :: /.i — z : /. l -\- x. 

 Whence .r/. I — z + z /. 1 + ^ = O. To give an example of the application of 

 this, let 1.024 and 0-990352 be the last numbers in the table; their logarithms 

 being c and d. Then we have 1 .024 = 1 -f- ar, and O.990352 = 1 — z, 

 consequently x = 0.024, and z = O.OO9648. Whence the ratio - in the 



201 

 least numbers is — — . So that for finding the logarithms proposed we may have 



500 D + 201 c = 48510/2 — 14603 / 10 = 0, which gives /2= 0.3010307, 

 which is too great in the last figure ; but it is nearer the truth, than what is 

 got from the logarithm f supposed equal to nothing. So that by this means 

 we have saved 4 multiplications, which were necessary to find the number 

 9989595 &c. correspondent to f, and which must have been had if we would 

 make the logarithm true to the same number of places without this com- 

 pendium. 



1.280000000000 A= 7/2—2/10 /2> 0.28 



0.800000000000 B= 3/2— /lO <j 0.33 



1.024000000000 c= b + a= 10/2— 3/10 > 0.300 



0.990352031429 d= 9c + b= 93/2 — 28/10 < 0.30107 



1. 004336277664 b = 2 d + c = 196 / 2 — 59 / 10 > 0.301020 



0.998959536107 F = 2 E + D = 485 /2 — 146 / 10 < 0.3010309 



1.000162894165 G = 4 F + E = 2136/2— 643/10 > 0.30102996 



0.999936281874 H = 6 G + f = 13301 / 2 — 4004 / 10 < 0.301029997 



1.000035441215 i = 2h + G = 28738/2 — 8651 / 10 > 0.3010299951 



0.999971720830 K = I + H = 42039/2 — 12655 / 10 <J 0.3010299959 



1.000007161046 L = K + I = 70777 /2 — 21306/ 10 > 0.30102999562 



0.999993203514 M = 3 L + K = 254370/2— 76b7i / 10 <: 0.30102999567 



1.000000364511 N = M + L = 325147/2- 97879/10 > 0.3010299956635 



0.99999976^^^7 o = i 8 N -f M = 6107016 / 2 — 1838395 / 10 < 0.3010299956640 



Com. Ar. 235313 



= 364511 o + 235313 ir = 2302585825187/2 - 693147400972/ 10 > 0.301029995663987 



I have computed this table so far, that the reader may see in what manner 

 this method approximates; this whole work, as it appears, costing a little more 

 than 3 hours time. 



Some Simple Properties of the Conic Sections deduced from the Nature of the 

 Foci ; with General Theorems of Centripetal Forces ; by means of which the 

 Law of the Centripetal Forces tending to the Foci of the Sections y the Felocities 

 of Bodies revolving in themy and the Description of the Orbits, may be easily 

 determined. By Abr. Demoivre. N° 352, p. 622. 



This paper may be seen in the author's Miscellanea Analytica, p. 233, &c. ; 

 into which it was from hence copied, much enlarged and improved. 



