34. REPOKT — 1873. 



the reciprocals, but also multiples of them), wo here describe Picarte's ' La 

 Division reduite a une Addition.' 



Ficarte [18G1], The principal table occnpies pp. 15-104, and gives, to ten 

 significant figures, the reciprocals of all numbers from 1000 to 10,000, and also 

 the first nine multiples of the latter (which are therefore given to 10 or 11 sig- 

 nificant figures). It is easy to see how this table reduces Division to Addition. 

 The arguments run down the left-hand column of the page ; and there are niuo 

 other columns for the multiples ; each page contains 100 lines ; so that there 

 are 10,300 figures to the page. Owing, however, to its size, and to the smallness 

 and clearness of the figures, there is no confusion, the lines being well leaded. 

 The great table is preceded by two smaller ones, the first of which (pp. 6, 7) 

 gives the figures from the ninth to the fourteenth (inclusive) of the logarithms 

 of the numbers from 101,000 to 100,409 at intervals of unity (downwards), 

 with first, second, and third differences ; and the second (pp. 10, 11) gives 

 ten-figure logarithms of numbers to 1000 ; and from 100,000 to 101,000 at in- 

 tervals of unity (with differences). There is also some explanation &c. 

 about the manner of calculating logarithms by interpolation, &c. The 

 author remarks on the increasing rarity of ten-figure tables of logarithms, 

 referring, of course, to Vlacq and Vega. The whole work was submitted by 

 its author to the French Academy, and reported on favourably by a Commit- 

 tee consisting of MM. Mathieu, Hermite, and Bienayme. The report (made 

 to the Academy Feb. 14, 1859) is printed at the beginning of the work. 

 M. Eamon Picarto describes himself as Member of the University of Chili ; 

 and the Chilian Government subscribed for 300 copies of the work. There 

 is no date ; but the "privilege" is dated ISTov. 1860, and the book was re- 

 ceived at the British Museum, April 29, 1861, so that the date we have 

 assigned is no doubt correct. On the cover of the book are advertised the 

 following tables by the same author, which we have not seen : — 



" Tables de multiplication, contenant les produits par 1, 2, 3 .... 9 et toutes 

 les quantites au-dessous de 10,000, 1 vol. in-l8 jesus." 



" Tableau Pithagoriqiie, etendu jusqu'a 100 par 100, sous une nouveUe 

 forme qui a permis de suppriraer la moitie dcs produits." 



It is scarcely necessary to remark that any trigonometrical table giving 

 sines and cosecants, cosines and secants, or tangents and cotangents, may bo 

 used (and sometimes with advantage) as a table of reciprocals. The extreme 

 facility with which reciprocals can be found by logarithms has prevented tables 

 of the fonner from being used or appreciated as much as they deserve. 



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



Tables of Reciprocals.— }S.xmmiS, 1795; Baelow, 1814, T. I. (to 10,000) ; 

 Teottee, 1841 [T. YIII.] ; Willich, 1853, T. XXI. ; Beabdmoee, 1862, T. 

 35 ; ScHLoMiLCH [1865 ?] ; Eankine. 1866, T. I. and I. A ; Wackeebaeth, 

 1867, T. IX.; Paekhtjest, 1871, T. XXV.; see also Meepact, 1832 (§ 3, 

 art. 3) ; Baelow (1840) (§ 3, art. 4). 



Art. 8. Tables of Divisors (Factor tables), and Tables of Primes. 



If a number is given, and it is required to determine whether it be prime, 

 and if not what are its factors, there is no other way of effecting this ex- 

 cept by the simple and laborious process of dividing it by every prime less 

 than its square root, or until one is found that divides it without remainder*. 

 The construction of a tabic of divisors is on the other hand very simple, as it 



* Wilson's theorem (viz. that 1 . 2 . 3. ...(«- 1) + 1 is or is not divisible by n, 

 according as n is or is not prime) theoretically affords a criterion ; b-at the labour of 

 applying it would be far greater thun the direct procedure bv trial. 



