ON MATHEMATICAL TABLES. 17 



calculations were stimulating the iiivcntiou of logarillims, they Averc also 

 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." We have seen the statement that 

 while Napier solved triangles by logarithms, Herwart did so by prosthaphce- 

 rcsis, and others of the like kind, the inference being that Herwart invented 

 a method which has been superseded by logarithms ; this (if the present 

 work is the source of the statement) is incorrect, Herwart's table being 

 merely useful in facilitating the multiplications required in the formulte. 



There are in the Eritish Museum three other works of Herwart ab Hohen- 

 burg : viz., ' Thesaurus Hieroglyphicorum e museo Joanuis Georgii Herwait 

 ab Hohenburg . . .' (Obi. fol. Munich ?, 1610 ?) ; ' Novaj, verse et exacte ad cal- 

 culum . , . Chronologise e museo . . .' Small 4to, 1612; and 'Ludovicus Quartus 

 Imperator defensus . . . ab Joanne Georgio Herwarto' &c. 4to. Munich, 1618 

 (the middle one of which is given in Lalande's Bib. Ast.). "We have looked 

 at these three books in the hope that some mention might be made in them 

 of the table, or some information given about Herwart's Museum ; but they 

 appear to contain nothing of the kind. We have seen also the titles of several 

 other works of Herwart's, and references to where particulars of his life are 

 to be found ; so that, considering the attention so large a work as his table 

 must have received from contemporary mathematicians, we still have hopes 

 of being able to bring to light some information with regard to its calciilator, 

 • his objects, &c. 



It should be stated that Herwart ab Hohenburg is spoken of quite as fre- 

 quently by the name of Hohenburg as by that of HerAvart. 



The author of the anonymous table (17*J3) described below, states that 

 many errors were found in Herwabt, and that Schiibler (whose table we have 

 not seen) was much more correct. 



Riley, 1775. The first nine multiples of all numbers from 1 to 5280. 

 The multiples of the same number are placed one under the other, the factors 

 1, 2 ... 9 being three times repeated on the page, which contains ten columns 

 of results and twenty-seven lines. 



The preface is signed Geo. Riley and T. O'i?. Macmahon. Tliere is an ad- 

 vertisement of Eiley's " historical playing-cards" &c. at the end, and of several 

 works by Macmahon. On the relation of this book to another, " printed for 

 J. Plummer" (anonymous) in the same year, see De Morgan. 



Anonymous, 1793. ^Multiplication table exhibiting products from 2x13 

 to 100 X 1000, arranged so that there are 100 multiples (in two columns) of 

 four numbers on each page, which therefore contains eight columns. 



Gruson, 1798. The first part of this book contains a number of tables, 

 the description of any one of which will explain the arrangement. Take the 

 table 36 : it has ten columns, headed 0, 1, 2, . . . , 9 (as have all the other 

 tables), and 36 lines, numbered 0, 1, 2, . . . , 35 ; we find in column 6 and 

 line 21 (say) 237=6 x 36-|-21. The use of the table is as follows : — suppose 

 it required to find the number of inches in 6 yards 21 inches ; 36 in. =1 yd., 

 we find table 36, column 6, line 21, and have the result given in inches. 

 There are tables for all numbers from 1 to 100, and for primes from 100 to 

 400, the number of lines in each table being equal to the number of the 

 table. The use of the tables in performing ordinary divisions and multipli- 

 cations when there are four or more figures in the divisor or dividend, &c. is 

 fully explained by the author in the introduction. When used for division, 

 the table gives the quotient and the remainder. 



There is also given a table of all simple divisions of numbers (not divisible 

 1873. ' c 



