981 TABLE. 



augmented by D. P. : " this D. P. is Dr. Pell. The table of primes 

 Ac., to 100,000 is computed under Pell's advice and direction ; and i 

 great part, it is believed, by himself. But there is preserved in severa 

 places the title of a work of Pell which we have never seen, and which 

 we take from Lipenius : ' Tabula decem millium difficilium Numerorum 

 eorum nempe omnium qui ab ad centum Milliones [mille ?] habenl 

 dimcultatea,' English, London, folio, (1666). This looks like a table oi 

 prime numbers, and the number of primes under 100,000 is about 

 10,000. But we must leave it to those who can see the work, if it stil 

 exist. Branker's table was reprinted in the second volume of Harris's 

 ' Lexicon Technicum,' London, 1710. 



i (Octavo Logarithms, vol. ii.), 1797, gave primes and divisors 

 up to 102,i 00, and further primes up to 400,000, in which Chernac 

 found but 39 errors. 



Chernac mentions primes and divisors to 1 0,000, in the Arithmetic 

 of T. M. Poelius, Leipsic (1728), 8vo, reprinted in Richter's ' Lexicon 

 Mathematicum ; ' primes to 101,000 in Kruger's Algebra (17Ui>; 

 primes to 100,000 from Kruger, in a separate work of Lambert, not the 

 logarithmic and other tables of 1770 ; primes to 400,000, by A. F. 

 Marcus, Amsterdam (1772), 8vo. 



Murhard mentions the first part of a table (by A. Felkel) ot the 

 factors of all numbers not divisible by 2, 3, or 5, from 1 to a hundred 

 milli'int, Vienna (1776). Chernac alludes to this table as mentioned by 

 i*e, but had never seen it. 



I, Cu/jcs, fiquqrc Knots, Cube JRooto, and Powers 

 ro/. Perhaps the oldest printed table of squares is that in p. 30 

 !oli'* ' Sumina,' Ac. [ViETA, in Bloc. Div.], printed in 1494, and 

 again in 15v3, which, however, goes only to luo 1 . Cosmo Bartoli, 

 Modo di misurare le DLatantie,' Ac., Venice, 1504, has squares up 

 to 66 1 s . Maginus's 'Tabula Tetragonica,' Venice, 1;'>92, is not a sepa- 

 rate work, but a chapter in his work on triangles, presently mentioned : 

 it gives squares up to 10,iOu 2 , but not cubes. It was, however, pub- 

 lished separately, at the same time with the work on triangles, as well 

 as in it ; the only difference being that the separate publication has its 

 headings and explanations Italian instead of Latin. The number of 

 so-called books, which are only chapters of other books, is large enough 

 to make a big catalogue. 



Guldinux, ' De Centro Gravitatis,' Vienna, 1635, gives the squares 

 and ciilies up to those of lu,000. The founder of the modern Ency- 

 clopaedia, Alated, 1649, gives squares and cubes up to 1000. Pell, 

 n (1600), (Murh.-ii - to that of 1 0,000. William Hunt, 



Magazine,' London, 1687, gives a table of squares up to that 

 of lo. no. A i this work are Newton's. Ludolf, ' Tetra- 



Tabularia, Frankfort and Leipsic, 1690, 4to, gives the 

 squares up to that of a ' ; the largest table of squares 



in existence, and very little known. J. P. Buchner, ' Tabula Radicum,' 

 Ac., .Nuremberg, 17"1, gives squares and cubes up to those of 12,000. 

 . ctni'le in 1) gives squares up to that of ^5,400, and cubes up 

 to that of In. HUM. l.i[ii-niuM mentions 'Tabula; numerorum qtiadra- 

 toruni decies millium,' Londini (1672), which is Pell's table, though it 

 has not bis name. It has also an English title, contains the first ten 

 thou-aud squares, and also the number of pairs, triads, and quaternions 

 of figures with which a square can end. Henischius, 

 ' Arit! ' lecta, Augsburg, 1609, begins with squares and cubes 



of all numbers up to 3oO. Heilbronner (p. 627) mentions a IH/HI/H 

 Call-in a a which gives squares up to 1000-. Detached tables of powers 

 are given in various works. John Hill's Arithmetic-, of which the 

 seventh edition bears London, 1745, has all the powers of 2, up to the 

 144th, for the purpose of solving question^ about thc-^.-I. nurds and 

 horseshoe nails. We have also the title G. C. Sartoriii.i, ' Cubische 

 Tabellen, 1 Eisenach (1- 



I>. *lon's ' Calculator' (1) gave square and cube roots up to those 

 of I-M: Hutton afterwards gave tne same up to those of 1000. 

 Barlow, ' Table*,' London, 1814, gave squares, cubes, square roots, 

 cube roots, and reciprocals, up to those of 10,000 ; the roots to seven 

 decimal places ; the reciprocals to seven significants. These were 

 reprinted (London, Taylor and Walton, 1840), from the original, after 

 re-examination by Mr. Farley. Tables of squares and cubes, up to 

 those of 1 0,000, were reprinted from Se"guin's ' Manuel d'Architec- 

 with a descriptive preface, at Paris, about the beginning of the 

 v. This table has beautifully clear figures, of 24 to the inch, 

 i and even body, with heads and tails. It was this table which 

 first suggested to the author of this article the superiority of the 

 numerals with heads and tails, and gave rise to his suggestion to the 

 Useful Knowledge Society, 139, to reprint Lalande's Table in such a 

 figure. The example in now extensively followed. Meinert's Loga- 

 rithm^, Halle (17!'i>) contain i cubes M[> to those of 1000. 

 Boebert, ' Tafeln,' Ac., "Leipsit- (IM-'. goes to the square of i!:'.,--"". 

 >, and the square ami cube root of 1000. Beyens, 

 : (1827), goes to the square of 10,000, and the cube of 100U. 

 Tafeln,' Ac., Rohn om Rheim (1827), has squares up to that 

 MiTpaut, in the work above mentioned, gave reciprocals up 

 ignificants. 



Jom-onrt, l>,- la Nature do Nombres Trigonaux,' Hague, 1762, 



gave triangular numbers up to that of 20,000, cubes up to that of 600, 

 and showed how to use the former in the construction of squares 

 and square roots. As to higher powers than the third, Hutton and 

 Barlow, in works above cited, give every power of every number up 



TABLE. 



082 



to the tenth power of 100. Barlow gives also the fourth and fifth 

 powers of numbers from those of 100 to those of 1000. Bowditch 

 (work of 1834, cited under ' Logarithms,' 1757) gives square roots 

 to five decimals "000 (-001)4-200 (-01)10-19 with tables of proportional 

 parts. 



Maseres, at the end of the tracts on Combinations, London, 1795, 

 has reprinted Huttou's square roots 0(1)1000 to ten decimals, and 

 reciprocals to seven. We believe that Hutton first gave them in his 

 'Miscellanea Mathematical 4 vols. 12mo, 1775. In Jonas Moore's 

 Arithmetic, 1650, there are the squares and cubes of all numbers up 

 to 1000, the fourth powers up to 300, and the fifth and sixth up to 

 200. These were reprinted in the edition of 1S60. 



Rogg mentions 'Art gantz neu-entdeckte,' &c., Dessau (1755), 8vo, 

 containing the cubes of all numbers up to 100,000, or at least pro- 

 fessing in the title-page to give the cube root of every number under 

 a thousand millions of millions : perhaps the cubes went to that of 

 10,000, with a rule for the fifth figure. And here we may mention 

 that we have been several times deceived by a title-page stating, not 

 the extent of the table, as it ought to do, but the extent to which 

 operations of interpolation or accession will be effective. 



4. Pure Decimal Operations. Besides Barlow's reciprocals ( 3), 

 the only remarkable tables of which we know under this head are 

 Goodwyn, ' Table of the Circles arising from the Division of a Unit,' 

 London, 1 823, and ' Tabular Series of Decimal Quotients,' London, 

 1823 (both anonymous). The first gives all the circulating decimals 

 which can arise from any fraction whose denominator is under 1024 ; 

 the second arranges all fractions which in their lowest terms have a 

 numerator not exceeding 99, and a denominator not exceeding 10UO, 

 in order of magnitude, and gives their equivalent decimals to eight 

 places. Mr. Woolgar is our authority for saying that there was a 

 previous work by Goodwyn, ' First Centenary of concise and useful 

 Tables of Decimal Quotients' (1518), 4to. Mr. Goodwyn (of Black- 

 heath) was an indefatigable calculator, and the preceding tables are 

 the only ones of the kind which are published. His manuscripts, an 

 enormous mass of similar calculations, came into the possession of Dr. 

 Olinthus Gregory, and were purchased by the Royal Society at the 

 sale of his books in i842. 



R. Picarte, ' La Division re'duite Ji une Addition,' Paris, 1861, 4to, 

 gives, for all numbers up to 10,ijOO, ten significant figures of recipro- 

 cal, with the nine multiples of each. 



An anonymous work, ' Tal'elu zur Verwandlung aller Bru'che,' &c., 

 Oldenburg, 1842, gives every fraction less than unity whose denomi- 

 nator does not exceed three figures, nor its numerator two, to seven 

 places of decimals. It is arranged by numerators : that is, all fractions 

 of one numerator are in one double page. This is a useful table. We 

 may also mention (but not as having seen it) W. F. Wucherer, 

 ' Beytriige zum allgerneinen Gebrauch der Decimalbriiche,' &c., Karls- 

 ruhe (1795), 8vo. 



The oldest table we have found printed in English is in ' This boke 

 showeth the maner of measurynge of all maner of lande, as well of 

 woodlande, as of lande in the felde, and comptyuge the true noiubre of 

 acres of the same. Nen'lye invented and compyled by Syr Rycharde 

 Benese Chanon of Marton Abbay besyde London. Printed in South- 

 warke in Saynt Thomas his hospital! by me James Nicolson.' There 

 is no date, but Nicolson's dated works run from 1536 to 1538. There 

 is another edition (which omits the tables), printed by Thomas Colwell, 

 who printed from 1558 to 1575. They are double-entry tables of the 

 rudest character, for finding the number of acres in a given length and 

 rei'dth, and for casting up payment at per perch, per acre, etc. 



S r>. 1'in-i Ti-ii/iiiiniiii-/,'ini/ Tn/ili <. This section and the next form 

 almost the whole of the article; and for a sufficient reason. The 

 bistory of the trigonometrical canon, and of the logarithmic table, are 

 constituent parts of the history of the progress of mathematics : the 

 history of other tables has nothing to do with that progress, except 

 only in the case of Stevinus's tabled of compound interest, which, as 

 will appear, suggested decimal fractions to their author. The biblio- 

 graphical history of the early part of the trigonometrical canon is so 

 incorrectly given, as well as ambiguously, even by the best authorities, 

 :hat it will be worth while to collect the several heads, distinguishing 

 oetween what we know from the books themselves and what we are 

 obliged to take from other sources, by putting the name of an authority 

 'of which we have usually two or three) to the latter. Much confusion 

 las arisen from the double meaning of the word publication in regard 

 to works of the century following the invention of printing, when it 

 was applied equally to the issue of a printed book and of a manuscript. 

 We are here only concerned with the former ; and it is sometimes diffi- 

 cult to distinguish between the two. 



The- earliest trigonometrical table existing is the table of chords in 

 the first book of Ptolemy's Syntaxis or Almagest. It is by half-degrees 

 ,ip to 90, and thence by degrees. The chords are given sexagesimally, 

 M a radius of 60 : thus, the chord of 90 J is 84 51' 10". The thirtieth 

 jarts of the differences are annexed : thus, the earliest table has its 

 tif creates, and differences given by the convenient sub-multiple which 

 would probably be thought very modern. 



That Albategnius [ALBATKGNIUS, in Bioo. Div.] had substituted sines 

 .'or Ptolemy's chords, that he had also used versed sines and tangents, 

 that Purbach and Regiomontanus had constructed and issued (in 

 manuscript at least) tables of sines to two separate railii, 6,000,000 



