;>!() Mr. J. AY. L. Glaisher on Multiplication 



dam n lieu jus exercitati in Graeca lingua. Quid si occurrerit 

 quod placet? ^.etcra^OeLa dpiOfjunTiKij. Nosti enim, XP ecoy 

 aTTOKOTTa? sic dici. lnest vocabulo et emphasis et proprietas 

 et similitudinis gratia, quia me Hercule novas tabulas introdu- 

 cis, et uno ictu liberas computatores debitis multiplicandi et 

 dividendi inextricabilibus. Sed facile est exercitato, copiam 

 afferre similium compositorum, ut ex iis aptius aliquod eliga- 

 tur. Mihi jam plura non occurrunt." He then refers to the 

 Tabula Tetragonica [Venice, 1592] of Maginus, and adds in 

 a postscript " Titulus igitur talis : ^eiaaypeia sive Novae 

 Tabulae, quibus Arithmetici debitis inextricabilibus multipli- 

 candi et dividendi liberantur, ingenio, tempori viribusque ra- 

 tiocinantis consulitur." 



§ 12. It thus appears that the table was printed from a 

 manuscript that Herwart used himself, and which very likely 

 he had had made. As for the word prosthaphaeresis which 

 occurs on the title, it is shown by the correspondence that 

 Herwart used the table for multiplications in general, and that 

 it was Kepler who pointed out that by means of it spherical 

 triangles could be solved more easily than by the prosthaphae- 

 resis of Wittich, and suggested the addition of the precepts 

 for the solution of triangles, which actually occur in the pre- 

 face to the work. The title suggested by Kepler was not 

 adopted, nor was his advice about shortening it ; but it must 

 be acknowledged that Herwart succeeded in obtaining a 

 " splendid title," which also contained a Greek word. De 

 Morgan explained the use of the word " prosthaphaeresis " upon 

 the title-page of Herwart's table, thus: " Prosthaphaeresis is a 

 word compounded of prosthesis and aphaeresis, and means addi- 

 tion and subtraction. Astronomical corrections, sometimes ad- 

 ditive and sometimes subtractive, were called prosthaphaereses. 

 The constant necessity for multiplication in forming propor- 

 tional parts for the corrections, gave rise to this table, which 

 therefore had the name of its application on the title-page." 

 The correspondence, however, shows that the table derived 

 its title not from its general use in the calculation of astrono- 

 mical prosthaphaereses, but from the special prosthaphaeresis 

 of Wittich for the solution of triangles. In a paper on Her- 

 wart's Table read before the Cambridge Philosophical Society* 

 on October 25, 1875, which contained the greater part of the 

 contents of §§ 9 and 11 of the present paper, I remarked that 

 the prosthaphaeresis referred to seemed to be most likely a 

 method of solving spherical triangles, in which the product of 



* " On Herwart ab Hohenburg's Tabulce Arithmeticce TrpoadcHfiaipeo-eeds 

 universales, Munich, 1610." Proceedings of the Cambridge Philosophical 

 Society, vol. ii. pp. 386-892 (part xvi.). 



