



TAlil.K. 



TA11I.F. 



1003 



Jesuit Oavpar Schott having discovered, on some grounds of theological 

 magic, that the degrees of grace of tin- Virgin Mm v were in number 

 the 256th power of 2, calculated that number. Whether or no his 

 number correctly represented the result he tinnounccd, he certainly 

 calculated it rightly, an we find by comparison with M r. Shanks. 



1853. A. Wylie, ' Compendium of Arithmetic,' Shong-hae. A 

 treatise in Chinese, having at the end a table of six-decimal logarithms 

 of nuruK'rs ll(l> 10,000. 



1850. J. K. Hind. Natural versed sines .... used in computing 

 lunar distances for the ' Nautical Almanac.' This table gives natural 

 .mil logarithmic versed sines to seven decimals : natural, 0(10") !_'."> , 

 with the ].i.ijx'itioii.i! ]<art for each second; logarithmic, O h il'):i : ', 

 with the values of the au^'- annexed. This table, though 



bearing in its title only a limited notion uf application, will be cx- 

 :gly valuable, especially to those who want nines, cosines, anil 

 tlifir squares. 



1857. George and Edward Sellout/. Specimens of tables calculated, 

 moulded, and printed by machinery,' London. The tirst )<> 

 iluction of the machine which two Swedes, father and son, constructed 

 on Mr. Babbage's principles, as suggested by Dr. Lardner's article in 

 tlu> ' Edinburgh Review' (1834), with their oxvu details; five-figure 

 logarithms 0(1)10000; with some specimens of other tables. This 

 work was reproduced at Paris, with a French preface, in 1858. In 

 1859, was published ' Mountain Barometer Tables,' calculated by the 

 same machine. 



1859. Edward Sang, ' Five-place logarithms/ (1) 10000. 



1860. Ludwig Schriin, ' Schron's Logarithms,' Tafel I., II., III. We 

 had just, as we thought, put the finishing hand to this article, when 

 the table above-named reached us. Should it turn out to possess the 

 requisite accuracy of printing, it will have decided success. It is a 

 large octavo volume of 550 pages of tables. The type, though without 

 heads and tails, is all as nearly as possible of one thickness, and thai 

 thickness not too great, so that it might be called thin Eyi/jilian ; and 

 it is very legible. The contents are all to seven decimals. Logarithms 

 of numbers (1) 108000, with subsidiary tables at the bottom of the 

 page, which by addition of two logarithms in the page give the loga- 

 rithms of sines and tangents, (0"'001) l"(0"'0l) 10"(0 y -l) 1'40" and 

 1'40" (0"'01) 16' 40" (0"'l) 3. Logarithms of sines and tangents, 

 (10") 45. The differences begin to be inserted from 3 ; and the 

 first nine multiples complete, that is, one figure more than in the 

 common table of proportional parts, are given in the same page ; first 

 for every fifth number, then for every three, &c., as the page will bear 

 it. Under 3 multiples of subsidiary numbers are given, as explained. 

 Then follows a table of proportional parts to complete hundredths, for 

 numbers from 40 to 4n9. 



1860. Galbraith and Haughton, 'Manual of Mathematical Tables. 

 The u*ual five-figure table, with Gauss's table, A ("001 ) 2 ('01 ) 3-39 (] ) 

 5. The first British five-figure table, we believe, which gives Gauss's 

 table. 



1861. A. De Morgan. Three-figure logarithms : three figures of 

 number to three of logarithm, complete,, on a sheet of 7 A by 6 inches. 

 The third figure of the number in every case by the side of the loga- 

 rithm : all numbers in red, logarithms in black. The quarter of a 

 unit in which the logarithm lies, shown by use of the four common 

 punctuating stops. Intended for the earliest instruction in logarithms, 



a substitute for the sliding rule in cei tain cases. 



>: 7. The next tables which we shall mention are those which are 

 wanted in the higher mathematics. 



Exteii.-ivu tables of elliptic funrti<,H* are in Legendre's ' TraittS des 

 Fonctions EUiptiques,' 2 vols. 4to., 1825 and 1826. The / 

 function, FJT, is tabulated in the same work; and also in the'Exer- 

 cicesdu Calcul Integral 'of the same author, Paris. 1817, in which 

 1 other definite integrals are also tabulated. An abridgment of 

 Urn table (with ready means of restoring it fully) is in the treatise on 

 the Differential Calculus (' Lib. Use. Know.'), p. 587. Tables of the 

 integer form of r.r, or 1.2.3 .... (x- 1), or rather of the logarithms of 

 are given by C. F. Degen, ' Tabularum Enneas,' Copen- 

 hagen, 1824, up to x= 1201, to 18 decimal places: this table is re- 

 printed to six decimal places at the end of the article ' Theory of 

 Probabilities '^ in the Encyclopedia Metropolitana. Tables of the 

 integral J *->'<! t were first given by Kramp, with logarithms of the 

 values, in ' Analyse des Refractions Astronomiques,' Strasburg, 

 This table is reprinted in the Encyc. Metrop., art. ' Theory 

 of Probabilities.' The form in which this integral more usually 

 occurs in the theory of probabilities (with the factor 2 : V) 

 was given (by Professor Encke, we believe) in the Berlin ' Astrono- 

 miaches Jahrbuch ' for 1834, from whence it was copied into the 

 article in the Encyc. Metrop., above noticed; and (with extensions) 

 into the ' Ei-iciy on Probabilities and Life Contingencies ' in the Cabi- 

 net Cyclopaedia, and into the article on Probability in the new edition 

 of the ' Encyclopaedia Britannica.' A few other definite integrals have 

 been tabulated : one very useful one,Jdx ; log ar, by Soldner, ' Nouvelle 

 Fonctiun Tranacciidcntc,' Munich, 1809, copied into the 'Differential 

 Calculus' (' i MOW.'), p. 062. The integrals known by the 



name of Sjn'mV ' s are m j^e work with 



that title (Edinburgh, IMiji; Sir .1. Hersc-hel's edition. I |.,n. 



There are a few of the integrals in optics scattered through the Me- 



moirs of the Institute and of the Cambridge and Philosophical Society 

 (in memoirs by Fresnd and Mr. Airy). lYi-liap* we should also 

 mention the tables for the solution of indeterminate equations of the 

 second degree. Of these there is one in l.cgcndn-'s Tlicorie des 

 Nonibres;' another has been given by Jacobi; and a third, by Degen, 

 called ' Canon IVllianus,' Copenhagen, 1817. 



There is much need of tables of mathematical results which are not 

 numerical ; such as the following, the only ones of which we are able 

 to speak. The first, the well-kin , Unite integrals 



by Mci. r Hirsch. ' Integral tafeln,' Berlin, 1810, of which an English 

 edition was published in 1823, <?vo. The second, a tali!" ,.| .Kiinite 

 integrals, with reference to their sources, by Bierens de Haan, ' Tal 

 il'Intcgrales dSfinies,' being the fourth volume of the ' Amsterdam 

 Transactions,' Amsterdam, 1858. 



As to astronomical tables, it would be impossible to give any account 

 of the enormous mass which exists or has existed ; nor would such an 

 account be of any use, except for astronomical history. They may be 

 divided into two classes : first, the tables of observations publish 

 public or private observatories ; secondly, the fundamental tables 

 deduced from observations, to ail in the deduction of future predic- 

 tions. As to the former, every well-conducted observatory in full 

 work publishes periodically (at intervals of one or two years) ite volume 

 of observations, latterly with their reductions. As to the second class, 

 they are not the daily materials of the astronomer, but of the com- 

 puter of his epbenueris, who supplies the necessary predictions for the 

 current year. In England the Nautical Almanac gives in the preface 

 full references to the tables employed in predicting places, whether of 

 sun, moon, planets, or stars. For general purposes connected v. ith 

 the elements of the solar system, see Baily, ' Astronomical Tabl< 

 Kornml.e,' London, 1827. The most complete list of the elements of 

 the solar system recently published is at the end of Dr. Mitchel's 

 ' Popular Astronomy ' (U.S.), and also of Mr. Totnlinson's English 

 edition of the same work. 



The tables in the other physical sciences are mostly collections of 

 facts, and, we believe, generally speaking, by no means so complete as 

 they might be. The value of tabular information seem to In not 

 sufficiently felt. A large portion of every book of chemistry, for 

 instance, is a detailed statement in words at length of facts which 

 might with great advantage be made the components of a table. 



8. It remains to speak of commercial tables, a subject of great 

 interest in this country, which has produced a great many. The 

 mathematical tables connected with this subject may bo divided into 

 those intended to facilitate calculations of money with regard to other 

 countries, and with regard to transactions in this country ; to which 

 we must add. as distinct heads, tables of annuities and other life con- 

 tingencies, and metrological tables, or tables of weights and measures. 

 Of all these we shall only mention a very few. 



The most complete work on foreign exchanges, and on the weights 

 and measures of England, as compared with those of other countries, is 

 ' The Universal Cambist,' &c., London, 1S21 (2nd edition). 2 rols. 

 (with supplements), by the late Dr. Patrick Kelly. We may also 

 mention Tiarks's ' Arbitration of Exchanged,' London, 1817. 



Tables of interest Of money begin with Stevinus. \\ ho in the ' 1'r-ictiip.it! 

 d'Arithmetique,' appended to his Arithmetic, Leyden. LWi, reprinted 

 by Albert Girard in Stevinus's collected works, lt>2b',gave the first 

 tables of compound interest and annuities. They precede the famous 

 tract ' La Disme,' in which decimal fractions were first proposed. And 

 as this Prarliyite should rather have been at the. beginning than at the 

 end, if rational arrangement had been studied; and as the 'D 

 again should have preceded it, on the same supposition ; we must infer 

 it to be most likely that the tracts were placed in the order in which 

 they were written. If this be the case, then it is pretty certain that 

 these tables of compound interest suggested decimal fmc/iuns, the 

 account of which speedily follows them. They are constructed as 

 follows : Ten millions being taken as the base (or root, as Stevinus 

 calls it), and a rate, say five per cent., being chosen, the present value 

 of ten millions due at the end of 1, 2, &c., up to 30 years, are put in a 

 column, to the nearest integer. By then- sides are the sums of thci r 

 values, which give the present values of the several annuities of ten 

 million, as follows : 



Table d'Intcrest dc i pour 100. 



1 9523810 9523810 



2 9070295 18594105 



3 8638376 272:::MM 



4 .S:!-J7(i25 35459506 



30 



2313774 



153921IIM 



The rates are from 1 to 16 per cent., and also for 1 in 15, 1 in 10. .Vc., 

 k> 1 in 22; or, as the French say, ilmin- juinte, dmitr wizt, &c. At 

 the end is a direction to dispense, when convenient, with some of the 

 last figures. 



There is thus a virtual use of decimal fractions preceding the formal 

 one. The same thing happens in ti Itichard Witt, pr. 



mentioned, which we believe to be the first Kn;;lish tallies of compound 

 interest, and the first English work (except a translation of the 'Disme' 

 if Stevinus) in which decimals were used: the ui-o of them being 



