TOL. XIX.] PHILOSOPHICAL TRANSACTIONS. 23 



thereto, nor can any thing more easy be desired. And to encourage him, I 

 here give the logarithms of the first prime numbers under 20, to 6o places, 

 computed by the accurate pen of Mr. Abraham Sharp, from whose industry 

 and capacity the world may in time expect great performances, as they were 

 communicated to me by our common friend, Mr. Euclid Speidall. 



Number. Logarithm. 



2 0,301029995663981 19521373889-172-i4930267681898814.62108541310427 



3 0,477121254719f>624372950279032551 15309200128864190695864829866 

 7 0,8450980400142568307 122l6258592636l93483572396323965406503835 



11 1,041392683158225040750199971243024241706702190466453094596539 



13 l,1139-13352306837769206541S95G2624625456l 189005053673288598083 



17 1,230448921378273028540169894328337030007567378425046397380368 



19 1,278753600952828961536333475756929317951129337394497598906819 



The next prime number is 23, which I will take for an example of the fore- 

 going doctrine ; and by the first rules, the logarithm of the ratio of 22 to 23 

 will be found to be either 



1 1,1 1,1. 



— ^ ^—^ -4- — -4- &c or 



22 968 ' 319-i4 937024 ^ 25768I6O 



1 J. _L. j_ _J_ _i. __L_ _!. L_ &c 



23 ' 1058 ~ 36501 ^ 1119364 ^ 32181715 



As likewise that of the ratio of 23 to 24 by a like process, 



1 1,1 1,1a 



23 1058 ^ 36501 1119364 ^ 32181715 



1 , _L. , \ I 1 [_ 1 &c 



24 ' 1152 "•" 41472 "•" 1327104 """ 39813120 



And this is the result of the doctrine of Mercator, as improved by the 

 learned Dr. Wallis. But by the second theorem, viz. ^ -j- — -f- — &c. 

 the same logarithms are obtained by fewer steps. To wit, 



II I fij 



45 "*" 273375 ~^ 922640625 "*" 26l5686l71875 



2 2 2 2 



_ 4. ^ 4. ~ _L Z gjc 



47 ^ 311469 1146725035 ^ 3546361843241 



which was invented and demonstrated in the hyperbolic spaces analogous to the 

 logarithms, by the excellent Mr. James Gregory, in his Exercitationes Geome- 

 tricae, and since farther prosecuted by the aforesaid Mr. Speidall, in a late 

 treatise in English by him published on this subject. But the demonstration, as 

 I conceive, was never till now perfected without the consideration of the hyper- 

 bola, which in a matter purely arithmetical, as this is, cannot so properly be 

 applied. But what follows I think I may more justly claim as my own, viz. 



