342 Mr, J. W. L. Glaisher on Multiplication 



figures, and in the solution of spherical triangles, is ex- 

 plained. 



The mathematical bibliographers and historians afford very 

 little information with regard to the work. Heilbronner (His- 

 toria Matheseos, 1742, p. 801) gives the title not quite cor- 

 rectly, and adds, " Docet in his tabulis sine abaco multiplies- 

 tionem atque divisionem perficere." Kastner (Geschichte der 

 MatJtematik, 1796-1800, t. iii. p. 8) quotes the title from Heil- 

 bronner and his remark, and adds that Heilbronner could not 

 have known Herwart's method, or he would have described it. 

 He remembers to have read somewhere that the book con- 

 tained a number of tables of products,, arranged by factors, 

 like a great multiplication table. Scheibel (Einleitung zur 

 rnathematlschen Bilcherkenntniss : eilftes Stuck, 1779, pp. 417— 

 420) gives the full title of the work, a description of it, and 

 an example worked out to show how it is to be applied to 

 multiply numbers of more than three figures. He commences 

 his account: — " Der Herausgeber dieses colossalischenEinmal 

 Eins ist also kein anderer, als der so berukmte Staatsmann 

 und Gesehichtschreiber, dessen sehr wichtiges und seltenes 

 Werk : Ludovicus IV Imperator defensus zu Mtinchen 

 1618. 1619. in gr. 4. in der Hamburg. Histor. Bibliothek 

 Cent. VIII. Art. 97, beschrieben ward;" and concludes, " So 

 viel von diesem ungeheuren Folianten, den man bloss zur Cu- 

 riositat und seiner Seltenheit wegen, in einer mathematischen 

 Buchersammlung aufbewahret." Montucla (Histoire des Ma- 

 thematiques, t. ii. pp. 13, 14) gives a short account of the table 

 and the mode of using it when the numbers to be multiplied con- 

 tain more than three figures. This is introduced by the words: — 

 " Nous terminons cet article, en faisant connoitre un livre 

 assez obscur, dont l'objet etoit d'abreger les calculs arithme- 

 tiques, et qui n'auroit pas ete inutile, sans l'invention des lo- 

 garithmes ;" and Montucla adds further on, u Tel est l'esprit 

 de cette invention, qui, sans la decouverte des logarithmes, 

 auroit pu etre de quelque utilite aux calculateurs, si toutefois 

 la peine de chercher ces produits, au nombre de 6, 7, 8 ou 9, 

 disperses dans un enorme in-folio, n'eut pas paru plus fatigante 

 et non moins laborieuse que le calcul meme, pour un homme 

 exerce." Murhard (Bibl. Math. 1797-1804, t. ii. p. 199) 

 gives the title correctly, and marks it with an asterisk to show 

 that he has seen the book itself. Rogg (Bibl. Math. 1830, 

 p. 142) merely has, " Hohenburg, Gregor (sic) Herwardt ab, 

 tabular arithmetica3 irpoaOa^aipeaew^ universales, 1610." 

 Neither Weidler (Bill. Ast. 1755), Deschales (Cur. seuMund. 

 Math. 1690), Lalande (Bibl. Ast. 1803), nor Delambre (Hist 

 de V Ast. Mod. 1821) mention the work; but there is a refer- 



