by a Table of Single Entry, 343 



ence to it in Leslie's c Philosophy of Arithmetic ' (2nd edit. 

 1820, p. 246). In his article on Tables in the English Cyclo- 

 paedia (1863), De Morgan, referring to this table, wrote : — 

 " The table goes up to 1000 x 1000, each page taking one 

 multiplier complete. There are then a thousand odd pages ; 

 and as the paper is thick, the folio is almost unique in thick- 

 ness. There is a short preface of seven pages, containing ex- 

 amples of application to spherical triangles. It is truly remark- 

 able that while the difficulties of numerical calculation were 

 stimulating the invention of logarithms, they were giving rise 

 to this the earliest work of extended tabulated multiplication. 

 Herwart passes for the author ; but nothing indicates more 

 than that the manuscript was found in his possession. 

 The book is excessively rare; a copy sold by auction a few 

 years ago was the only one we ever saw." Graesse (Tresor 

 des livres rares, 1859-1867) says that by the book the use of 

 logarithms was first spread in Germany, which is obviously 

 erroneous. 



§ 10. Crelle'sRechentafeln, which also extend to 1000 x 1000, 

 were first published in two octavo volumes in 1820. A second 

 (stereotype) edition in one volume (folio) was published under 

 the editorship of the late Dr. Bremiker in 1864. This work 

 is well known, and is much used. From Crelle's preface to 

 his table in 1820, it is clear that he knew nothing of Her- 

 wart's work, and was not aware that a table to 1000 x 1000 

 had ever been published ; for he writes, " Das neueste und 

 vielleicht bedeutendste Unternehmen von Tafeln, die mit den 

 gegenwartigen einerlei Idee zum Grunde haben, ist das Werk : 

 ' Tables de multiplication, a l'usage de MM. les geometres, de 

 MM. les ingenieurs veriflcateurs du Cadastre etc. Sec. edit., 

 Paris, chez Valace, 1812 ;'" and then he states that this work 

 only extends to 500 x 500, and is therefore the fourth part of 

 his own table. It is not a little remarkable that the first mul- 

 tiplication table of any extent published should have reached 

 the limit beyond which the table has never been carried. 

 That Herwart' s idea was good and practicable is proved by 

 the continual use made of Crelle's tables ; and but for the great 

 bulk of his ponderous volume, the table would in all probability 

 have come into use. The invention of logarithms four years 

 afterwards afforded another means of performing multiplica- 

 tions, and Herwart' s huge work never became generally known. 

 It is curious that from 1610 till 1820 no similar table should 

 have been published, as it seems clear that the work of 1610 

 was a failure chiefly in consequence of its inconvenient 

 size. Herwart's table is rare; but there are copies in the 

 British Museum, the Bodleian Library, and the Graves 



