by a Table of Single Entry. 345 



volumine, quam TrpoaOa^aipeaei Witichiana, quae saepissime 

 inducit in tabulam sinuum, crebro permutat sinus complemen- 

 tis, arcus rectis, rectas arcubus. Illaque memoriter nunquam 

 retineo, et taediosum est, toties adire scripta praecepta, prae- 

 sertim ingenioso, qui nihil sine causae cognitione cupit agere. 

 Requirit praeterea cautiones crebras, ex quibus neglectis 

 crebra offendicula. Causa? operationum etsi ingeniosissimae 

 tamen semper in abstruso sunt inter operandum. At si mul- 

 tiplicemus et dividamus simpliciter, tunc videmus quid aga- 

 mus ; et possunt varietates triangulorum talibus praeceptis 

 comprehendi, quae memoria retineri facile possunt." Then 

 follows a synopsis of the different cases of the solution of sphe- 

 rical right-angled triangles. Herwart writes on November 5, 

 and says he had found that triangles could be solved better 

 and more quickly by means of his table than by prostha- 

 phaeresis, so that Kepler was quite right. He proceeds : — 

 " Darunter ich aber Yitichium nit gesehen. Dieser abacus 

 ist universalis. Weil man pflegt den Biichern ein splendidum 

 titulum zu geben, ut reddantur venales, wollt ich gern dessen 

 Gutachten vernehmen, ob es nit thunlich ungefahr so : Nova 

 exacta, certa et omnium facillima ratio Arithmetices, per quam 

 numerus ex multiplicatione productus sine operatione multi- 

 plicationis per solam additionem, et quotiens ex divisione re- 

 sultans absque operosis ambagibus divisionis per solam sub 

 tractionem cujuscunque, etiam maximae summae, etiam ab eo, 

 qui arithmetices non admodum sit gnarus, citius quam ulla alia 

 ratione invenitur." He then asks Kepler for his advice as to 

 how the book should be entitled in Latin and German, and 

 prays him to write soon. 



Not receiving an answer, on December 2 he sends a short 

 letter to Kepler, again asking for a reply, and suggesting that 

 perhaps it was not prudent " so speciosum titulum tantillae rei 

 zu praefigiren." But immediately after, Kepler's reply [which 

 is lost] was received ; and writing on December 5, Herwart 

 explains that he does not expect an answer to his last letter, and 

 that he understands that Kepler has no suggestion to make 

 with regard to the title, but thinks it should be shortened. He 

 cannot understand the meaning of Kepler's advice, " Graeca 

 compositio imploranda, sed exercito," and asks for an expla- 

 nation. 



In the last letter of the correspondence, dated December 12, 

 1608, Kepler explains that, as the title seemed long, he had 

 advised that it should be shortened by the composition of two 

 Greek words, as in kpvcrtireka^, pi'fyoicivhvvos, &c, and con- 

 tinues: — " Mihi quidem cum non occurreret quidquam hujus- 

 modi, quod ad nostram rem faceret, consului opem imploran- 



