by a Table of Single Entry. 347 



two sines, or a sine and cosine, was avoided by the use of a 

 formula such as sin a sin 6 = -^{cos (a — &)— cos (a + b)}, adding 

 that Laplace referred to such a method. This inference is 

 shown to have been correct by the contents of § 7. TTittich 

 does not appear to have published the method himself, though 

 from the writings mentioned in § 7, and from Kepler's letters, 

 it is clear that it was generally attributed to him : he ought, 

 I suppose, to be considered the discoverer of the formulas 

 sin a sin b = ^{cos (a— h) — cos (« + &)}, &c, which are really 

 the prosthaphaeretical formulae. Kepler's remarks upon the 

 difficulty of using the prosthaphaeresis for spherical angles, on 

 account of the confusion between sides and angles and their 

 complements, is interesting ; and it is for this reason that I 

 have quoted so much of the letter of October 18. The word 

 prosthaphseresis often means the difference between the true 

 and mean places of a body in longitude or latitude ; but it 

 seems to have been vaguely used, very much as " correc- 

 tion " is now, to denote small quantities to be added or sub- 

 tracted to quantities obtained by theory, or by a first approxi- 

 mation, &c. : so that without a context its signification is not 

 precise ; but I have not examined this point. In Kliigel the 

 word is derived from irpoaOev and dcfraipeo-Ls; but, at all events, 

 as far as the mathematical and astronomical use of the word 

 is concerned, De Morgan's derivation from irpoaOea^ and 

 d(paip€(TLs seems to be certainly the true one. 



As it happened, Herwart's employment of the word irpo- 

 crOcKpaipeaecDs upon his title-page was not fortunate ; for only 

 four years after the publication of his table logarithms were 

 invented, all the processes of calculation were changed, and 

 Witfcich's prosthaphaeresis passed out of notice. 



Kepler, as is well known, greatly admired INapier's inven- 

 tion, and in 1624 published himself a table of Xapierian loga- 

 rithms. 



It will have been noticed that Herwart describes his table 

 as enabling multiplications to be performed " per solam addi- 

 tionem," and division " per solam subtractionem." These 

 words would immediately suggest to a writer of the last or 

 present century the method of logarithms ; and it is for this 

 reason, no doubt, that not unfrequently the methods of pros- 

 thaphaeresis and quarter squares have been confounded with 

 applications of logarithms. Yoisin. as mentioned in § 2, actu- 

 ally called his quarter squares logarithms ; and this has added 

 to the confusion. 



Trinity College, Cambridge. 

 'July 12th, 1878. 



