30 REPORT — 1873. 



for example, we pass from 39 to 41. It is quite true that the columns for 

 40 are the same as those for 4 with the addition of a ; but the awkward- 

 ness of turning to opposite ends of the book for (say) 889 and 890, and then 

 having to add a to the latter, is very great. It is a pity that a desire to 

 save a few pages should have been allowed to impair the utility (and it docs 

 so 'most seriously) of so fine a table. The matter is referred to in the 

 preface, where it is said that Crelle, " after mature reflection," decided to 

 omit these numbers. 



The original edition was published in 1820, and consisted of two thick 

 octavo volumes, the first proceeding as far as 500x1000, and the secord 

 completing the table to 1000 x 1000. The inconven'cnce refeiTcd to above 

 is felt more strongly in this than in the one-vo\imc edition, as frequently the 

 numbei-s ending in have to be sought in a different volume from the others. 

 Both editions are, we believe, very accurate. There are 3 pp. of errata 

 (pp. xvii-xix) at the beginning of the edition of 1820. De Morgan gives 

 1857 as the date of Bremiker's reprint, and says he has heard that other 

 copies bear the date 1859, and have no editor's name. 



Laundy, 1865. The first nine multiples of all numbers from 1 to 100,000, 

 given by a double arrangement : viz., if it is required to multiply 15395 by 8, 

 we enter the table on p. 4 (as 395 is intermediate to 300 and 400) at 15, 

 and in column 8 find 122 ; we enter another table on the same page at 395, 

 and in column 8 find 160; the product is therefore 123160. We take this 

 number instead of 122160 because in the column headed 8, first used, there 

 appears the note [375]*, the meaniug of which is that if the last three figures 

 of the number exceed 375 (they are 395 in the above example) the third 

 figure is to be increased bj'^ unity. The table is thus seen to be the same in 

 lyrlnciple as ERETScnNEiDER, but not quite so convenient. There are the same 

 objections to this as to the latter table. The present table occupies 10 pp. 

 4to, and Beetschneider's 99 pp. 8vo. 



Mr. Laundy remarks in his preface that Crelle's ' Erloichtcrungs-Tafel,' 

 1836, although one hundred times as largo as his, "must not bo estimated as 

 presenting advantages proportionate to its vast difference of extent." In this 

 we scarcely agree ; for it is only when the numbers are six or seven figures 

 long that one begins to feel the advantages of a table for so simple an operation 

 as multiplication by a single digit, and Crelle's table would not take much 

 longer to use than the present. 



The following is a list of references to § 4 : — 



MaJtlpUcatloa Tables.— Dovsoy, 1747, T. XXXVIII. to 9 x 999<).; Hutrox, 

 1781 [T. I.] to 100 X 1000 ; Callet, 1853 [T. VIII.] ; SchrGx, 1800, T. III. ; 

 Paekuurst, 1871, T. XXVI., XXXIII., and XXXIV.; see also Leslie, 

 1820, § 3, art. 3, and Wtjcheree, 1796, T. II. (§ 3, art. 6.) 



Art. 2. Tables of Proportional Paris. 



By a table of the proportional parts of any number x is usually under- 

 stood, a table giving -^j^x, -f^A\ . . . -fj^x true to the nearest unit. Of course 

 the assumption of 10 as a divisor is conventional, and any table giving 



X 2x (a 1),^^ 



-J — , . , . ^^ '- would equally bo called a proportional-part table. Ordi- 



(^ Gi CI 



nary proportional-part tables (viz. in which rt = 10) are given at the sides of 

 the pages in all good seven -figure tables of logarithms that extend from 

 10,000 to 100,000. The difference between consecutive logaritlims at the 

 commencement of the tables (viz. at 10,000) is 434, and at the end is there- 

 fore 43 ; so that a seven-figure table of the above extent gives the proportional 



