: I 



TABLE. 



TABLE. 



100t5 



example, begins and ends with the numbers 5220 and 5310, and 

 the logarithms 71767 and -72509. According, at the corner of 

 page, so as to catch the eye on opening the book, is 



SJ20 

 5310 



71767 

 72509 



in fulL This plan ought to be adopted in all tables, instead of the 

 abbreviations which are frequently employed as headings, unless the 

 plan be adopted which is recommended at the beginning of this 

 article. 



1840. Anonymous (Taylor and Walton), Four-figure logarithms on a 

 card. Stereotyped, Reprint of a table originally privately circulated 

 among practical astronomers. (See ' Companion to the Almanac ' for 

 1841.) 



: Sines and tangents to match. Stereotyped. 



1840. Farley, 'Six-figure Logarithms,' London. Stereotyped. An 

 excellent table for those who want six figures The type as in the 

 reprint of Lalande (1839). The whole was suggested by the late Mr. 

 Galloway. 



(Second edition ) Moritz Riihlmann, ' Logarithmisch-Trigono- 

 metriache . . . Tafeln,' Dresden and Leipzig. Six-decimal logarithms 

 of numbers (1 ) 1 0080 ; logarithmic sines and tangents (!') 45 ; sines 

 and tangents OHO') 45; arcs and circles. 



(1840.) Hulse's edition of Vega. Leipzig, stereotyped. This contains 

 Gauss's tables to five decimals with proportional parts, in six columns, 

 the additional three (which contain a peculiar mode of treating the 

 ]in<p<irtiiinal parts) having been also suggested by Gauss. A more 

 recent tirade of this excellent work, 1848, contains seven-decim 

 rithm.x of number* 0(1) 108000 ; logarithmic sines and tangents (0"'l) 

 I'frir82" and 0(10") 6 (l')45; angles to eleven decimals; five- 

 decimal Gauss's tables, A being ('001) 2 (-01) 3'4 ("1) 5, with the pro- 

 portional partH above alluded to. 



1841. Gregory, Woolhouse, and Hann, ' Tables for Nautical Men.' 

 Contains five-figure logarithms, neatly printed ; the only instance we 

 know in which five-figure logarithms have proportional parts. There 

 are many astronomical tables. 



."Kiddle, 'Tables,' Ac. The six-figure logarithms from Sir. 

 I'M. lie's well known work on navigation. Stereotyped. 



Of the misuse of tables, no instance is more common than that which 

 consists in taking tables of too many places of figures. Four are very 

 often enough, more than five are rarely wanted ; but when this hap 

 pens, tables of seven figures are more conveniently used than those ol 

 six, owing to the saving of calculation which is made by the presence 

 'portional parts. In purely trigonometrical calculations, the 

 advantage of six figures over five sometimes makes itself apparent. It 

 is our own practice, when five figures are suspected to be insufficient 

 c recourse to seven at once, which we are satisfied is a saving 

 l>tli ni time and thought. For navigation, however, practical opinion 

 seems to set in favour of six figures. 



1845. Warnstorff's edition of Schumacher's ' Sammlung von Hiilfs 

 tafeln ' (firxt published in 1822), Altona. This is a well-known and 

 valuable astronomical collection. What we have hre to do with is 

 the republication of Encke's four-figure logarithms, (1) 1000, and 

 (4') 10 (10") 45, and Gauss's logarithms (-01) 1-80 (-1)4. 



1846. R. Sheepshanks, ' Tables for facilitating; Astronomical Reduc- 

 l.oiidon (also issued two years before, without title, preface, or 



author's name). This Is the most complete four-ftjnn table we know 

 of, mid will do fur the purpose oftener than our orthodox septenarians 

 are aware of. Logarithms 0(1) 1000, with proportional parts, in de- 

 cads ; logarithms of sines and cosines, tfie ani/le being in time, 0(1") 24\ 

 w jtji : ! parts for 10, and 0(10') l h ; table for converting 



sidereal into mean solar time ; logarithmic sines, tangents, and secants 

 O(l') 6(100 45 ; constants for precession; tangents -and secants 

 0(10')80(1')86(1)90, with a rule for the rent ; Bessel's refrac- 

 tions ; Gauss's tables, thus arranged, log x as an agument gives log 



1 + - \ as a tabular result, and log f 1 - - J as another, log x being 

 0(001*) -909 (-01) 2 (-1)4 in the first table, and (-001) 1 (-01) 3 (-1) 4 

 in the second ; log. sin* 4 hour n gl e > > Q *J me l b (1") 9h > numbers to 

 logarithms ('001 1 1. 



1846. G. F. Vega, ' Logarithmiscb-Trigonometrisches Handbuch.' 

 Leipeic- edited by J. A. Hulsse; 0(1)10SOOO, logarithmic sines and 

 tangents 0(OM)1' and 0(1") 132, the whole canon 0(10")6 (!') 45, 

 all to seven decimals. Gauss's table to five decimals: for A, (-001) 

 2 (-01 ) 3'4 ( 1 ) 5, with proportional parts. 



Among the titles of tables which we might have said something on 

 if we had seen them, collected from different sources, are those of 



John Lauremberg, Leyden, 1628, 8vo. ; Institutio Mathematica, 



,n, 166?, 12mo. ; Strauchius, Witteberg, 1662, 12mo., and 

 Amsterdam, 1700, 8vo. ; D. R. Van Merop, Harlingen, 1671 ; Chr. 

 Griineberg, Tabula; Mathem., Berlin and Frankfort, 1690 (oblong 

 form); Chr. Griineberg, .Pandora Mathem., Berlin and Frankfort, 

 1700, 8vo. ; Chr. Wolff. Magdeburg, 1711, 8vo. ; J. G. Leibknecht, 

 Oiesa 1726, 8vo. : Raph. Levi, Hanover, 1747, 4to., and supplement 

 in 1748; J. C. Nelkenbrechers, Leipzig, 1 752, 4to.; J. Melitao da Mata, 

 Lisbon 1790, 8vo. We have a table of which the title is torn out, but 



at the back is ' Colonise, 1649." We suppose it to be the table which 

 we have seen ascribed to Lubert Middeudorff, Cologne, 1648. It is 

 of seven decimals, (1) 10000 and 0(1')46: and to the logarithms 

 of sines, cosines, and tangents are added the tangents themselves, to 

 ;hree decimals. 



1849. Lieut.-Col. Robert Shortrede's ' Logarithmic Tables,' Edin- 

 burgh, large 8vo. The whole of these tables were constructed and 

 stereotyped by the labour and at the expense of Colonel (then Captain) 

 Shortrede, of the Bombay army, attached to the trigonometrical survey 

 of India. They first appeared in 1844; but, some defects and errors 

 having been found, the edition of 1844 was cancelled, and a new 

 edition, from corrected plates, issued in 1849. The whole is to seven 

 decimals, and contains: logarithms of numbers (1)120,000, with 

 differences and their nine multiples ; numbers to logarithms '00000 

 (00001) -99999, with the same ; trigonometrical tables to every second, 

 with arguments and signs for the four quadrants, both in space and 

 time, and proportional parts. There are also some minor tables. The 

 type is a small even figure, without head or tail, good of its kind, and 

 the same throughout the common and the trigonometrical logarithms. 

 This is, so far as we know, the only set of tables to every second 

 undertaken at the expense of an individual, and it shows extraordinary 

 energy and public spirit. 



1849. P. Gray, ' Tables and Formulae for Life Contingencies," London. 

 The tables are Gauss's tables, giving log (1 -fa-) where log a; is 0('0001)2, 

 and log (1 x) where log x is "3 (-001)1. The proportional parts are 

 to hundredths. 



1849. H. E. Filipowski, ' A Table of Antilogarithms,' London. Five 

 figures of logarithms to seven of number ; proportional parts carried to 

 hundredths. Also, Gauss's tables, on a new mode of arrangement. 



1850. Hershell E. Filipowski. A table of five-figure logarithms 

 (1) 10000, all on one side of a sheet. The object is effected by help 

 of common figures at the tops of columns ; but before we could \ise 

 the table rapidly, there must be horizontal ruling, and vertical painting 

 of the columns, with regions of different tint to distinguish the loga- 

 rithms of different second figures. This beiug'done, a person of sharp 

 sight might really have the whole within his grasp without turning a 

 leaf. A common table, with indented margin, as described at the 

 beginning, would find the logarithms far more rapidly. The necessity 

 of dispensing with printed differences is alone almost fatal to the 

 attempt at giving five-figure logarithms on one sheet. 



1850. Zacharias Dase, 'Tafelder natiirlicheu logarithmen,' Vienna. 

 Hyperbolic logarithms to seven decimals (1) 105000, arranged, with 

 proportional parts, in the common way. Mr. Dase is a mental calcu- 

 lator, and, having seen his performances, we think he has more natural 

 power than any of those who have distinguished themselves in this 

 way. 



1853. William Shanks, 'Contributions to Mathematics, comprising 

 chiefly the rectification of the Circle to 607 places of tables,' London, 

 1853. [QUADRATURE OF THE CIRCLE.] Here is a table, because it 

 tabulates the results of the subordinate steps of this enormous calcula- 

 tion as far as 527 decimals ; the remainder being added as results only 

 during the printing. For instance, one step is the calculation of the 

 reciprocal of 601. 5 001 ; and the result is given. The number of pages 

 required to describe these results is 87. Mr. Shanks has also thrown 

 off, as chips or splinters, the values of the base of Napier's logarithms, 

 and of its logarithms of 2, 3, 5, 10, to 137 decimals; and the value 

 of the modulus -4342 .... to 136 decimals; with the 13th, 25th, 

 37th, .... up to the 721st powers of 2. These tremendous stretches of 

 calculation at least we so call them in our day are useful in several 

 respects : they prove more than the capacity of this or that computer 

 for labour and accuracy ; they show that there is in the community an 

 increase of skill and courage. We say in the community : we fully 

 believe that the unequalled turnip which every now and then appears 

 in the newspapers, is a sufficient presumption that the average turnip 

 is growing bigger, and the whole crop heavier. All who know the 

 history of the quadrature are aware that the several increases of num- 

 bers of decimals to which ir has been carried, have been indications of 

 a general increase in the power to calculate, and in courage to face the 

 labour. 



Here is a comparison of two different times. In the day of Cocker, 

 the pupil was directed to perform a common subtraction with a voice- 

 accompaniment of this kind : ' 7 from 4 I canuot, but add 10, 7 from 

 14 remains 7, set down 7 and carry 1 ; 8 and 1 which I carry is 9, 9 

 from 2 I cannot, &c." We have before us the announcement of the 

 following table, undated, as open to inspection at the Crystal Palaci', 

 Sydenham, in two diagrams of 7 ft. 2 in. by 6 ft. 6 in. " The figure 9 

 involved into the 9 1 2th power, and antecedent powers or involutions, 

 containing upwards of 73,000 figures. Also, the proofs of the above, 

 containing upwards of 146,000 figures. By Samuel Fancourt, of Min- 

 cing Lane, London, and completed by him in the year 1837, at the 

 age of sixteen. N.B. The whole operation performed by simple arith- 

 metic." The young operator calculated by successive squaring the 

 2nd, 4th, 8th, &c., powers up to the 512th, with proof by division. 

 But 511 multiplications by 9, in the short (or 101) way, would havu 

 been much easier. The 2nd, 32nd, 64th, 128th, 25Bth, and 512th 

 powers are given at the back of the announcement. 



The powers of 2 have been calculated for many purposes. In vol. ii. 

 of his ' Magia Universalis Naturse et Artis,' Herbipoli, 1658, 4to, the 



