ON MATHEMATICAL TABLES. 141 



The rest of the book is devoted to astronomical tables and formiilfe, except 

 two remarkable tables at the end (pp. 364-371). The first of these [T. VII.] 

 is most simply desci'ibed by stating that it gives the number of shot in a py- 

 ramidal pile on a square base, the number n of shot in the side of the base 

 being the argument ; the table extends from n=2 to n = 40. There is also 

 given the number of shot in a pyramidal pile on a rectangular base, tlie ar- 

 guments being n the number of shot in the breadth of the base, and m the 

 number of shot in the top row (so that m+n — 1 is the number in the length 

 of the base). The ranges are, for ?i, 2 to 40, andfor jji, 2 to 44, the table being 

 of double entry. 



[T. VIII.] gives the number of shot in a pyramidal pile on a triangular 

 base, the number of shot in a side of the base being the argument, which 

 extends from 2 to 40. The other portion of the table is headed " Tabula 

 pro acervis globorum oblongis, ab utraque extremitate ad pyramides quadri- 

 latei'as appositis;" and the explanation is as follows: — Suppose we have 

 two pyramidal piles of shot on square bases (n shot on each side) placed 

 facing one another, at a distance equal to the sum of the diameters of m shot 

 apart ; and suppose it is required to fill this interval up, so as to make a pyra- 

 midal pile on a rectangular base, then this table gives the number for n (latus) 

 to n=40, and for m (longitude baseos) to mi =44, the table being of double 

 entry. 



Some errata are given after the introduction. 



"We have seen the third edition (Leipzig, 1812) ; aiid though wo have not 

 compared it side by side with the second (here described), we feel no doubt 

 the contents are identical ; at all events the number of pages in each volume 

 8 the same, and the preface is dated 1797 in both. 



Vlacq (Arithmetica Logarithmica), Gouda, 1628, and London, 1631. 

 \T. I.] Ten-figure logarithms of numbers from 1 to 100,000, with differ- 

 ences. This table occupies 667 pages. 



[T. II.] Log sines, tangents, and secants for every minute of the quadrant, 

 to 10 places, with interscript differences ; semiquadrantally arranged. This 

 table occupies 90 pp. 



In the English copies, by George Miller, there is an English introduction 

 of 54 pp., and then follows a table of latitudes (8 pp.). The original edition 

 of 1G28 has 79 pp. of introduction ; and a list of errata is given, which does 

 not occur in Miller's copies (but see ' Monthly Notices of the Eoy. Ast. Soc' 

 t. xxxiii. pp. 452, 456, May, 1873). 



There were also copies with a French titlepage ; and in these there is an 

 Introduction in the same language of 84 pp. We suspect that a Dutch edition 

 was contemplated, but that the copies of the table intended for this purpose 

 afterwards formed Miller's English edition : no Dutch edition is known to 

 exist (see Phil. Mag., May 1873). The titles of the three editions are given 

 in full in § 5 ; in all, the tabular portion is from the same type. The bibli- 

 ograjjhy of this work forms an essential part of the history of logarithms ; and 

 a good many of the references occurring in the introductory remarks to § 3, 

 art. 13, have reference to it. 



The table of logarithms of numbers contains about 300 errors, exclusive 

 of those affecting the last figure by a unit ; but a good many of these have 

 reference to the portion below 10,000, which need never be used. This is 

 still the most convenient ten-figure table there is (Vega, fol. 1794, is the only 

 othei') ; but before use the known errata should be corrected. References to 

 all the places where the requisite errata-lists are to be found are given in the 

 ' Monthly Notices of the Eoy. Ast. Soc.' for May and June, 1872. We intend, 



