by a Table of Single Entry. 339 



them in their calculations in 1582. Wittich in 1584 made 

 known at Cassel the calculation of one case by this prostha- 

 phaeresis ; and Justus Byrgius proved it in such a manner that 

 from his proof the extension to the solution of all triangles 

 could be deduced. Clavius generalized the method in his 

 treatise De Astrolabio (1593), lib. i. lemma liii. The lemma 

 commences as follows : — 



" Qucestiones omnes, qua per sinus, tangentes, atque secantes 

 absolvi solent, per solam prosthaphceresim, id est, per solam ad~ 

 ditionem, subtr actionem, sine laboriosa numerorum multiplica- 

 tione, divisioneque expedire. 



"Edidit ante tres, quatuorve annos Nicolaus Raymarus 

 Ursus Dithmarsus libellum quendam, in quo prseter alia pro- 

 ponit inventum sane acutum, et ingeniosuru, quo per solam 

 prosthaphaeresim pleraque triangula sphaerica solvit. Sed 

 quoniam id solum putat fieri posse, quando sinus in regula 

 proportionum assumuntur, et sinus totus primum locum ob- 

 tinet, conabimur nos earn doctrinam magis generalem efncere, 

 ita ut non solum locum habeat in sinibus, et quando sinus 

 totus primum locum in regula proportionum obtinet, verum 

 etiam in tangentibus, secantibus, sinibus versis et aliis nu- 

 meris, et sive sinus totus sit in principio regular proportionum, 

 sive in medio, sive denique nullo modo interveniat : quae res 

 nova omnino est, ac jucunditatis et voluptatis plena." 



The work of Raymarus Ursus, referred to by Clavius, is 

 his Fundamentum Astronomicum (1588). Longomontanus, 

 who also assisted Tycho Brahe, in his Astronomia Danica 

 (1622) gives an account of the method, stating that it is not 

 to be found in the writings of the Arabs or Reo-iomontanus. 

 Scheibel also mentions a manuscript " Melchior Jostelii logis- 

 tica irpoaOafyaipeais astronomica " (1609). 



With the exception of Clavius, I have not examined the 

 works referred to, but have relied on Scheibel and the 

 other writers mentioned at the beginning of this section. 

 It did not seem necessary to enter further into the matter, as 

 there can be no doubt that the method of prosthaphaaresis is 

 that to which Laplace's remark refers, and that it was used 

 for performing multiplications, even when the quantities to be 

 multiplied did not present themselves as sines and cosines. It 

 need scarcely be remarked that when the method was in use 

 (previous to the invention of logarithms) the cosine had not 

 been introduced; so that the rules for the different cases of 

 sin a sin b, sin a cos b, &c. were complicated, it being neces- 

 sary to frequently pass from the angles to their complements. 

 In Kliigel's Worterbuch it is mentioned that, if other trigono- 

 metrical functions have to be multiplied besides sines and 



Z2 



