12G REPORT— 1873. 



from 40 to 410. The table occupies 75 pages ; and on each double page are 

 given the proportional parts to hundredths of 1, 2, 3, 4, and 5 (viz. the first 

 100 multiples divided by 100 and contracted to ouo decimal place). The 

 last page of the book is devoted to a table for the calculation of logarithms, 

 and contains common and hyperbolic logarithms of n, 1-On, I'OOn, &c., n 



being any single digit (or in other words, of 1 + — ^^ from .v = 1 to .r = 9 



and n = 1 to n = 10), to 16 places. The figures arc beautifully clear, and 

 the paper very good. The tables are of their land very complete indeed. 



We have seen errata in this work advei-tised in different numbers of 

 Grunert's ' Archiv der Mathematik und Physik.' See Schron, 1865, below. 



Schrbn (London edition), 1865. De Morgan remarked that in England, 

 though tliere existed minute- and second-tables of trigonometrical functions, 

 there was no good ten-second table ; and on learning from the publishers 

 that an English edition of kSchkon was contemplated, he offered to write a 

 short preface, as, accuracy being taken for granted, these appeared to him to 

 be the most powerful and best ten-second tables ho had seen : his oft'er, how- 

 ever, was accompanied by the condition that a careful examination should be 

 made by Mr. Farley, sufficient to judge of the accuracy of the work, and that 

 the result should bo satisfactory. Mr. Farley accordingly examined 24 pages 

 selected at hazard, wholly by differences and partly by comparison with 

 Callet ; and the pages were found to be totally free from error ; so that the 

 general accuracy of the tables was assured. They arc printed from the 

 same plates as in the German edition described above ; and the tabular matter 

 in the two seems identical in all respects. 



Schulze, 1778. [T. I.] Seven-figure logarithms to 1000, and from 

 10,000 to 101,000, with differences and proportional parts. The proportional 

 parts at the beginning of the table, which are very numerous, arc printed on 

 a folding sheet. 



. A page at the end of this table contains the first nine multiples of the 

 modulus and its reciprocal, to 48 places ; also c to 27 places, and its square, 

 cube .... to its 25th power, also its 30th and 60th powers, the number of 

 decimals decreasing as the integral portion increases. Log tt (hyperbolic and 

 Briggian) is also given. 



[T. II.] Wolfram's hyperbolic logarithms of numbers to 48 places. The 

 numbers run from unity to 2200 at intervals of unity, and thence to 10,009, 

 only not for all numbers ; " von 2200 bis 10,000 ist sie hingegen nur f iir die 

 Prim- imd etwas stark componirtc Zahlen berechnet, weil das Uebrige durch 

 leichtes Addiren kaun gefundcn werdcn " (Preface). De Morgan says " for 

 all numbers not divisible by a single digit;"' but this is incorrect, as 2219, 

 2225, &c. are divisible by single digits, while 9S09 (least factor 17), 9847 

 (least factor 47) do not occur. In fact, at first a great many composite 

 numbers are tabulated, and near the end very few, if any. All the primes, 

 however, seem to be given ; and by the aid of Wolfram's tables we may 

 regard all hyperbolic logarithms of numbers below 10,000 as known. Space 

 is left for six logarithms, which Wolfram had been prevented from computing 

 by a serious Ulness. These were supplied in the ' Eerliner Jahrbuch,' 1 783, 

 p. 191. Mr. Gray points out an error in Wolfram's table; viz. in log 14()9, 

 .... 1660 ... . should be .... 1 696 .... (' Tables for the formation &c.,' 1 865, 

 p. 38). 



On Wolfram, see § 8, art. 16. 



[T. III. J Log sines and tangents for every second from 0° to 2°, to seven 

 places ; the sines are on the left-hand pages, the tangents on the right-hand ; 

 no differences. 



