Z2 REPOBT — 1873. 



therefore be tabulated to a mucb. greater extent without inconvenience. In 

 tables of quarter squares the fraction ^ which occurs when the number is 

 odd is invariably left out ; this gives rise to no difficulty, as the sum and 

 difference of two numbers must be both odd or both even. 



A product can, of course, be obtained by logarithms with about the same 

 facility as by a table of quarter squares ; but the latter is preferable when all 

 the figures of the result are required. 



LuDOLF, 1690 (see § 3, art. 4), in the preface to his 'Tetragonometria,' 

 explains the method of quarter squares completely, and shows how his table 

 is to be used for the purposes of multiplication. The earliest tabic oi quarter 

 squares De Morgan had heard of was Voisin, 1817 ; but Centnerschwer (see 

 below) refers to one by BUrger of the same date, the full title of which we 

 have quoted from Bogg. 



Crelle, in the preface to the first edition of his ' Eechentafeln ' (1820, 

 p. XV.), speaks of " Quadrat-Tafelu nach Laplace und Gergonne, mittelst 

 welchcr sicli Producte fiudcn lassen," &c. The allusion to Laplace doubtless 

 refers to the memoir in the ' Journal Polytechuique,' noticed further on in 

 this article ; but we cannot give the reference to Gergonne. 



The largest table of quarter squares that has been constructed is that 

 published by the late Mr. Laundy, which extends as far as the quarter 

 square of 100,000 ; it would be desirable, however, to have a table of double 

 this extent (viz. to 200,000), which would perform at once nnilti])lications of 

 five figures by five figures (Mr. Laundy's table is only directly available 

 when the sum of the ni;mbers to be multiplied is also of five figures). The 

 late General Shortrede constructed such a table, we believe, in India, but 

 unfortunately abandoned the idea of publishing it on his return to England, 

 where he found so much of the field already covered by Laundy's tables. 

 De Morgan, writing when it was anticipated that Shortrede's table would be 

 pubhshed, suggested that it would be convenient that the second half should 

 appear first ; and we should much like to see the publication of a quarter- 

 square table of the numbers from 100,000 to 200,000. 



Mr. Laundy, in the preface to his ' Table of Quarter Squares ' (p. vi), says 

 that Galbraith, in his ' General Tables,' 2nd edit. 1836, Avhich Avcrc intended 

 as a supplement to the second edition of his ' Mathematical and Astronomical 

 Tables,' gives a table (T. xxxiv.) of quarter squares of numbers from 1 to 

 8149. This book is neither in the British Museum nor the Cambridge Uni- 

 versity Library. The second edition of his ' Mathematical and Astronomical 

 Tables ' (1834) contains no such table. There is, however, no doubt about 

 the existence of the work, as the Babbage Catalogue contains the title 

 " Galbraith, "\Y., N"ew and concise General Tables for computing the Obhquity 

 of the Ecliptic, &c. Edinburgh, 1836." 



In 1854, Prof. Sjdvester having seen a paper in Gergonne in which the 

 method was referred to, and not being aware that tables of quarter squares 

 for facilitating multiplications had been published, suggested the calculation 

 of such tables, in two papers — " jSFote on a Formula by aid of which, and of a 

 tabic of single entry, the continued product of any set of numbers . . . may be 

 effected by additions and subtractions only without the use of Logarithms " 

 (Philosophical Magazine, S. 4. vol. vii. p. 430), and "On Muhiplication by 

 aid of a Table of Single Entry " (Assurance Magazine, vol. iv. p. 236). Both 

 these papers were probably written together ; but there is added to the former 

 a postscript, in which reference is made to Voisin and Shortrede's manuscript. 

 Prof. Sylvester gives a generalization of the formula for ah as the difference 

 of two squares, in which the product a^ a., . . . «,, is expressed as the sum of 



