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XL. Notices respecting New Books. 



Table of the Reciprocals of Numbers, from 1 to 100,000, with their 

 differences, by which the reciprocals of numbers may be obtained up 

 to 10,000,000. By Lieut.-Colonel W. H. Oakes, A.I.A. Lon- 

 don : C. and E. Layton. 1865. 



TN most, if not all, tables of the reciprocals of numbers, the arrange- 

 -*- ment hitherto adopted has been to give integral numbers from 1 

 to n, and the corresponding reciprocals as decimals. In the present 

 Table both the numbers and their reciprocals are given without de- 

 cimal points, which are to be supplied according to circumstances. 

 Advantage is thus taken of the fact that the significant digits of 

 the reciprocals of, for example, 3*7256, 37*256, 372*56, &c. are 

 2684131; the reciprocals being respectively 0*2684131, 0*02684131, 

 0*002684131, &c. In other words, Colonel Oakes has arranged his 

 Table with a view to the fact that if r is the reciprocal of n, then 

 will rxlO^ be the reciprocal of rc-f-10^. In consequence he has 

 been able to render the arrangement of the Table almost identical 

 with that of an ordinary Table of Logarithms. In fact the only ex- 

 ception is that the differences reckoned in order of the numbers are 

 all negative ; as they must obviously be, since, the numbers increa- 

 sing, their reciprocals will decrease. 



Prefixed to the Table are two notices, one describing the arrange- 

 ment of the Table and exemplifying its uses, the other giving a short 

 account of the manner in which the reciprocals of the larger numbers 

 were calculated. A word or two may be said on the latter point. 



Let n denote any large number, K the sum of the arithmetical 

 complements of the logarithms of n and n-\- 1, or 



K=log— -i . 



Also let d l9 d 2 , d 3 . . . denote respectively 



log(ra+l)— log (?j— l),logrc — log(n — 2),log(?z— 1) — log(/z— 3),&.c. 



Now 



_1 1_ 1 1 n+1. 



n — 1 n n.(n — 1), n(n-\-\)' n — 1 

 Therefore 



log feb - i) = l0S K^Vr) + los ^ +1 >- >°s(»-i)=k+</, 



Similarly 



and so on. Of course d v d 2 , d 3 . . . are given by a Table of loga- 



