ON MATHEMATICAL TABLES. 27 



Ludolf, 1690. Squares of all numbers from ixnity to 100,000, arranged 

 in columns, so that the first three or four figures of the root are to be found at 

 the top of the column, while the final ones are given in the left-hand column of 

 the page. The table is well printed and clear, and, except Kulik, 1848, 

 which is of the same extent, is the largest table of squares that has been 

 published, and occupies about 420 pages. Some errata in it are given at 

 the end of the introduction (150 pp. in length), in which aU possible uses 

 of the table are explained. 



Lambert (Introd. ad Supplementa, 1798) speaks of the numbers in the 

 table as " satis accurati." In chapter v. (pp. 48-86) (' Do Tabularum usu 

 sou Praxi circa Multiplicationem et Divisionem ') the use of the table as one 

 of quarter squares (see § 3, art. 3) is fully explained ; as squares are given 

 in the table, the sum and difference have to be divided by 2. llules and 

 examples are also added as to how to proceed when the semisum exceeds the 

 limits of the table by any amount ; and the processes &c. arc explained with 

 such fulness as to prove that all the credit of first perceiving the utility of 

 the method and calciilating the necessary table is due to Ludolf. 



The work is said to be very scarce ; biit we have seen several copies ; there 

 is one in the Library of Trinity College, Cambridge, and another in the 

 Graves Library. 



Heilbronner (under Herwart ab Hohenbtjrg) mentions Ludolf (Hist. Math. 

 p. 827), and (referring doubtless to the method of quarter squares) says that 

 he invented a method of performing multiplications and divisions without the 

 Pythagorfcan abacus, " quae prolixe ab lUustr. Wolfio in scinen Anfangs- 

 Griinden et suis Elementis Matheseos exjjonitur.'' 



Seguin, 1786. At the end of the book is given a table of the squares and 

 cubes of numbers from unity to 10,000. The figures have heads and tails, 

 and are very clear. De Morgan states that the table was reprinted at about 

 the beginning of the century, and that it was this table which convinced him 

 of the superiority of the numerals with heads and tails, aud led him in the 

 reprint of Lalande's table, 1839, to adopt this figure — an example Avhich has 

 since been very frequently followed. 



As De Morgan does not appear to have seen it, it is possible that the ori- 

 ginal table was not reprinted, but only published separately, as the figures in 

 the table attached to Seguin answer De Morgan's description very well. 



Barlow's tables (the stereotyped edition of 1840). Squares, cubes, square 

 roots, cube roots, and reciprocals to 10,000. The square roots and cube roots 

 are to seven places, and the reciprocals to seven significant figures, viz. nine 

 places to 1000, and above this ten. The work is a reprint of the more im- 

 portant tables in Barlow, 1814 (described in § 4) ; it was suggested by De 

 Morgan, who wrote the preface (2 pp.), and edited by Mr. Farley, of the 

 Nautical-Almanac Ofiice, who also examined carefully Barlow's tables. A 

 list of ninety errors found in the latter is given on the page following the 

 preface. This reprint is, we believe, very nearly, if not quite, free from 

 error ; it is clearly printed and much nsed, We have also an edition, 1866, 

 from the plates of 1840. 



Kulik, 1848. The principal table occupies pp. 1-401, and gives the 

 squares aud cubes of all numbers //-ohi 1 to 100,000. There is a compression 

 resembling that in Ckelle's ' Eechentafeln ;' viz. the last four figures of the 

 square and cube are printed but once in each line, these figures being the 

 same for all squares and cubes in the same line across the double page. The 

 arrangement will be rendered clear b)' the description of a page — say, that 

 corresponding to 92, There are ten columns headed 92, 192, 292. . . .092, 



