TAIU.I:. 



TABLE. 



and physics in general ore frequently tabulated : with these we have 

 coiu|Kiruti\fry little to do. since they tire rather the materials for 

 the formation or verification of other tables, than of primary use as 

 tables. 



Of simple arithmetical tables we may notice the following : 



S 1. 'j'i'lJr.i >,f Hutiii/ilii-atiim. The oldest we know of is as foil. 

 T.ili\il;t- Arithmetic* nPOS8A*AlPE5En5 * I ... K Mu>,-o 



Johaiinis Ueorgii Herwart ab H. I. IWtoris. Munich, 



1610, folio. This table goes tip to loooxlOoO; each page taking one 

 multiplier complete There are then a thousand odd pages ; 

 the mi'.T is thick, the folio is almost unique in thickness. There is 

 a short preface of seven pages, containing examples of application to 

 spherical triangles. It is truly remarkable that while the difficulties 

 of trigonometrical calculation were stimulating the invention of 

 logarithms, they were also giving rise to this, the earliest work of 

 vely tabulated multiplication. Herwart [asses for the author, 

 but nothing indicates more than that the manuscript was found iu his 

 collection. The book is excessively rare ; a copy sold by auction a few 

 years ago was the only one we ever saw. 



There have been several others of great extent, but they are 



n's ' Tables of Products,' printed by the Board of Longitude. 

 1781, go tip to 100 times 1000, but have not the reputation of correct- 

 ness. An anonymous table, London, 1775, which goes up to 10,000 

 x 10. is ' Tables of Products . . . London, printed for .1. I'lmnmer.' 

 But Riley's table, published in the same year, under exactly the same. 

 form, is ' Riley's Arithmetical Tables .... London, printed for O. 

 Riley.' It is imperfect in all the copies we have seen, ending abruptly 

 at the multiplicand 6280. The numerals are of the same form and 

 size as in Plummer. but the headings and lines are different. We 

 mispect that some writer of more than usual research on the quarrels 

 of authors, or some hunter of old injunction t cases, might find some- 

 thing about the history of these two books. The type is clear, and the 

 tables are very useful. 



Dodson'S ' Calculator,' London, 1747, has the same as the last up to 

 lono x 10, not so conveniently arranged. But by far the nloSt powef 

 fill table of this kind is Crelle's ' Hechentafeln,' Berlin, 1820, in two 

 thick 8vo volumes. This contains every product up to 1000 times 

 1000, so arranged that all the multiples of one number are seen at 

 the same opening of the book. All who have used this table know how 

 to dispense with logarithms in many cases with great advantage. 

 There is no table which we so much desire to see reprinted in this 

 country, with a few alterations, which would render it more com- 

 modious. Another edition was published by Dr. 0. Bremiker, Berlin, 

 1857, folio : but we are told that other copies bear the date 

 and. we believe, no editor's name. Each page here contains the !'!>! 

 multiples of two numbers ; giving 450 pages of tables. This edition 

 is very convenient, and of very comfortable type and paper. An 

 anonymous Table, Paris (1794), goes up to 1000 x 103 ; and another, 

 Paris" (An VII.), the same; a third, Versailles (1825), the same, with 

 many meteorological tables added. Schubler's ' Rechnung's Lexicon,' 

 Nuremberg (1739), goes to 2400x100. Oyoh's Table, Paris (1S2I), 

 goes U> 509 x 500 ; that of Cadet, Paris, 1801 or 1802, An X, to 

 jo, ooo x 100. This work of Cadet was intended for a )< 

 anil percentaii? table. Each number from 1 to 10,000 has its first 

 hundred multiples in on% folio column. Citizen Cadet was a tax- 

 gatherer, and saw that, in a decimal system, a raw table of multipli- 

 cation is a llartmt, or ready-reckoner, quite complete. Accordingly, 

 the title of his book is ' Tarifs des Centimes au franc, ou Tables de 

 multiplications et comptes-faita pour la repartition des contributions, 

 et ponvant remplacer, dans le systcuie decimal, les anciens comptes- 

 faits de BarCme.' 



Bretschneider, ' Producten-Tafel," Hamburg and Gotha, 1841j goes 

 up to 100,000 x 10. There is a compression of this kind : in finding, 

 for example, the multiples of 62873, the reader must look into the page 

 headed 2800, and there, in one part of the page, opposite to (6) 28, lie 

 finds the first three figures, and in another part, opposite to 73, the last 

 three figures. The first part, belonging to 028. is repeated twice, once 

 for the cases in which the following numbers are less than 50, and once 

 for those in which they are above it ; and an asterisk iu the ): 

 of the table occurs when it is necessary to add a unit to the preceding 



* Prosthaphseresis is a word compounded of prosthecis and aphteresis, and 

 means addition or subtraction. Astronomical corrections, sometimes additive 

 and sometimes subtracting were called prosthapbeereses. The constant i 

 : ti.uUiplication, in forming proportional parts for the corrections, gave rise 

 to thtft table, which therefore had the name of its application in the title-page. 



t The following gives a strong suspicion. In our copy there is a preface of 

 fourteen pages, signed " The Editor." The lute Mr. Woolgur, who made tables 

 a special study of great depth, had a copy in which a preface of fourteen pages 

 slimed " William \Vcbb," whose name was also in the title. Our preface 



, . 



has oly " Our tables .... are carried to nearly twice the extent of any taWr- 

 of the kind hitherto published :" Mr. Woolgar't preface has "There hare likr- 

 wise been ptiblinhcd thin year by Mr. lliley, Arithmetical Tables, containing the 

 products of all numbers from 1 to 6280, and arc a set of very useful and correct 

 tables ; the errors r thr press arc very few, the form of them we have also 

 thought convenient, and it is that which we have adopted." We surmise Unit 

 u . Webb played Vlicq to Kiley's Ilrigg as soon as Riley's tables were published, 

 and that legal proceedings were compromised by an arrangement of which it 

 was a portion that handsome mention should be made of Uiley's tables iu u new 

 j.reface, with the editor'! name. 



figures. This arrangement brings the table into ninety-nine pages 

 octavo, and i very ingenious ; but there is more risk of error in using 

 it than we like. Again, multiplying five figures by one is not so ditli- 

 cult an operation that it need be avoided by using a table which 

 lequires us to look attentively at three distinct things, after turning 

 pages. Lambert's table, 1770, presently mentioned, contains the nine 

 multiples of gin x for every degree, to six Inures: and multiples of 

 primes to tho Pohlman's Table, 1813 (2nd ed.(, contains the 



first nine multiples of all numbers up to lOOo. \\ , have heard of 

 another work of I 'relic, ' Krleichterungs Tafeln,' in oblong folio, giving 

 all numbers under 10,000,000, with their nine multiples, but with an 

 arrangement not easily nor safely used. There is a double process, as 

 in Bretschneider. 



In the Royal Society's Library is a table by J. J. Centneix 

 ' Neu-erfundcne Multiplikations-uu'd - quadrat- Tafeln,' Berlin, 1825. 

 The earliest table we have seen mentioned (by l.ipenins is T! 

 Finqk, 'Tabula- Multiplicatiouis ac Divisionis,' Copenhagen (I'i04), 

 oblong form. There is also, by the same author, ' Tabulie quotidiano 

 "idi usiii accommodate,' Copenhagen (1615), liimo. 



A- h'inck is an author of some interest iu the history of tables (as 

 will presently appear), we have made some inquiry about these works,* 

 and we find that they are not in the library at Copenhagen ; but that 

 Mollerus c Cimbria Littcrata,' vol. iii., p. 254), gives them as follows. 

 It seems they were not intended for scientific purposes : 



'Tabula; Multiplications et Divisionis, neoi>im ctiam V 

 a accommodate,' Hafuise (1604), oblong folio. 



' Tre Tabeller, indrcttet til daglig forniiden Regning.' (Three 

 accommodated to necessary daily accounts). Copenhagen (1615), 1 



Under this head we ought to mention John Bernoulli (tl, 

 ' Sexcentenary Table,' London. 17 7!', and Michael Taylor's ' Sexa- 

 Table,' London, 1780, intended to save the use of logistic logarithms ; 

 the former having 10' for the iirst term, and the latter 1. Both were 

 published by the Admiralty. In the day of Is, tables of 



multiplication were common in whieh the products wci 

 sexagesimal!}' ; as iu Orontius Finams, 'Arithmetica Practica,' Paris, 

 1555, in which is a table up to GO x 60. But the multiples of 

 example, are 0' 46", 1' 32", 2' 18", &c. The bibliographer should 

 remember that when a sexagesimal table is pasted into a copy, it does 

 not follow that it formed part of the work : owners often pasted in a 

 table from another source. 



I'nder the head of Multiplication, we must notice tables of QVAHTEK- 

 SQUAHKS. The earliest we have seen (and we believe the earliest 

 known to exist) is that of A. Voisin, Paris, 1817, ' Tables de Multipli- 

 cations,' &c., 8vo, containing quarto r-squares of numbers up to - 

 Leslie reprinted this table up to 2000, in his ' Philosophy of Arith- 

 metic ' (2nd ed.), 1820, as (according to Mr. Laundy) did Galbraith, as 

 far as 3149, in the second edition (183S) of his ' Mathematical Tables,' 

 J. M. Merpaut, ' Tables Ai ithmonomiques,' Vannes, 1832, gave quart er- 

 squares up to that of 40,000. But the largest and most valuable' .-et of 

 these tables yet published" is that of Sir. S. L. Laundy, a Loin Ion 

 actuary, ' Table of Quarter-Squares,' 1856, 8vo, which goes to 100,000, 

 and is beautifully printed. Colonel Shortrede, one of the most ener- 

 getic of tabulators, has computed this table to 200,000, but has not 

 published it, though we understand he intends to do so. 

 to him to publish the second half first. 



y'.iWi.t (:f Dir'm'itiii unit /,f I'. vrs, Fr. Schoolen, in 



book v. of his Kxercitatioues,' 1657, gave all the primes under i 

 Grusoh, ' Pinacothfcque,' Berlin, 1798, gives for all numbers mnl 

 or primes under 4UO, the quotient and remainder of every number 

 under ten times the divisor, by inspection : also, primes and lowest 

 divisors up to 10,500. Lidonne, ' Tables dc tons le Diviseurs," &c., 

 Paris, 1808, gives the divisor of all numbers up to 102,000. The 

 original edition of Barlow's Tables gives the factors of all numbers up 

 to 1(1,000 and a register of prime numbers up to 100,103. Chernac, 

 ' Cribrum Arithmeticuui,' &e., Daventrkc, 1811, gives the prime num- 

 ber.- up to 1,020,0)10, and all divisors of numbers not divisible by 

 either 2, 3, or 5. Burckhardt, ' Table des ] livisenrs/ &e., Paris, 1817, 

 gives the prime numbers up to ;V h the lowest divisor of 



each number when not either 2, 3, oi- 5. These useful works are now 

 rare. Aujema's ' Tabula Ilivisormn,' Lcydcn, 17G7, goes up to 1' 

 and Pigri's Table, Pisa (17."'!-). which ('hcrnac had never heard of, the 

 same. Anjeraa enters ever.a divisor : thus, 2 is entered as having t\\o 

 divisors. 1, 2. He also separates those less than the square root 

 the others by a hyphen ; and where the number is a square, th 

 has a hyphen on both side;. Kian-e. Jena and Lcipsic ( 1 So; 

 primes and factors up to 100,000. J. Neumann, ' Tabellen,' &c., 

 Dessau (1785), has factors and primes up to 100,100. 



Guldinus is said to have given tables of prime numbers, but we have 

 neither found them nor a description of them. Thomas Branker 

 appended to his translation of Rhonius's Algebra a table of primes and 

 divisors for all numbers under 100,000. This was reprinted by Baron 

 s, and was appended to his tract ' tin tin- Doctrine of Permu- 

 tations and Combinations,' London, 17SJ5, a book very easy to be 

 obtained. 



This translation of Rhouius is of London, 1668, " much altered and 



* From Professor \\eri.iult, Koyul librarian at Copenhagen, through tho 

 kindness of the late Professor Schumacher, the universal referee of those who 

 wanted information on any point connected with astronomy, however remotely. 



