344 Mr. J. W. L. Glaisher on Multiplication 



Library at University College, London ; the last is not quite 

 perfect. 



§11. While I was engaged in preparing the British-Asso- 

 ciation report, I endeavoured without success to find any thing 

 relating to the history of the table ; but the hope is there ex- 

 pressed that, considering the attention so large a work must 

 have received from contemporary mathematicians, some in- 

 formation might still be gained with regard to the calculator 

 of the tables, his objects, <fcc. I afterwards met with a corre- 

 spondence of six letters between Herwart and Kepler, which 

 took place at the end of 1608, and throws light upon these 

 points. The letters are printed in Dr. Frisch's Joannis Kepleri 

 Astronomi opera omnia, t. iv. pp. 527-530, 1863. Herwart, 

 who was Chancellor of the Palatinate of Bavaria, and a man of 

 mark in his time as a statesman, was a frequent correspondent 

 of Kepler's; and many of his letters upon chronology are 

 printed in Kepler's Eclogce Chronica?. There is a gap in the 

 correspondence, however, between January 12, 1608, and 

 December 5, 1608 ; and Dr. Frisch, in the notes to the Eclogce, 

 gives the six letters referred to, prefaced by the words " Kep- 

 lerus in Eclogis omisit epistolas Herwarti datas d. 13 Sept. 

 et 5 Nov. 1608, ipsiusque responsionem d. d. 18 Oct., quum 

 nihil facerent ad Chronologiam, et maxima ex parte spectarent 

 opus Herwarti arithmeticum. quod edidit Monachii 1610 .. ." 

 In the first letter, dated September 13, 1608, Herwart WTites : — 

 " Ich hab bisher in Multiplicatione et Divisione sonderbare 

 geschriebene praxin gebraucht, dadurch ich den numerum ex 

 quavis multiplicatione productum, per solam additionem, und 

 den Quotienten ex divisione resultantem per solam subtrac- 

 tionem (absque tsediosa multiplicationum et divisionum ope- 

 ratione) gefunden." He states that J. Prsetorius and others 

 who have seen it, recommend him to have it printed ; and he 

 adds that if he had not had this method (diesen modum), on 

 account of his continual occupations, and because he is not 

 a good calculator, he should long ago have had to give up all 

 mathematical matters requiring calculation. He sends a spe- 

 cimen page of the Table, the use of which he illustrates by the 

 multiplication 945,678 x 587, and he asks Kepler to give him 

 his opinion upon the subject. 



Kepler replies on October 18, 1608, and remarks that tables 

 are very useful to the sedentary man, and " ei, qui perpetuo 

 cum libris cohabitat." For 500 or 1000 pages form a large 

 volume, which cannot always be at hand. As it will facilitate 

 astronomical calculations he advises that short precepts on the 

 solution of triangles should be added. Kepler then pro- 

 ceeds : — "Nam multis partibus expeditius est uti hoc tuo 



