ON MATHEMATICAL TABLES. 139 



have seen ; while in others the paper is thin and discoloured ; all arc printed 

 from the same type. 



The arrangement of T. I. (though about half the space that would be required 

 if the logarithms and differenees were written at length is thereby saved) is not 

 nearly so convenient as in Vlacq ; 1628, for there is danger of taking out a 

 wroug difference. Yega took great pains to free his tables of logarithms of num- 

 bers from error ; and he detected all the hereditary errors that had descended 

 from Vlacq which affected the first seven figures of the logarithms. But as 

 several of these errors were corrected in his errata-list and not in the text, his 

 successors, who failed to study these lists sufiiciently, were reaUy less accurate 

 than he was. The last thousand logarithms that appear for the first time in 

 this work were calculated by Lieut. Dorfmund at Vega's instigation. 



T. II. is not reprinted entirely from Vlacq's ' Trigonometria Artificialis,* 

 as the logarithms for every second of the first two degrees Avere calculated for 

 the work by Lieut. Dorfmund. Vega seems not to have bestowed on the tri- 

 gonometrical canon any thing approaching to the care he devoted to the loga- 

 rithms of numbers, as Gauss estimates the number of last-figure errors at from 

 31,983 to 47,746 (most of them only amounting to a unit, but some to as 

 much as 3 or even 4). 



Vega offered a reward of a ducat for every error found in his table ; and 

 it is to be inferred fi'om his preface that he intended to regard inaccuracies of 

 a unit as such, so that it was fortunate that no contemporary of his made an 

 examination similar to Gauss's. The paper of Gauss's in which this estimate 

 occurs is entitled " Einige Bemerkungen zu Vega's Thesaurus Logarithmo- 

 rum," and appeared in the ' Astronomische Nachrichten,' No. 756, for May 2, 

 1851 (reprinted ' Werke,' t. iii. pp. 257-264). It contains an examination 

 of the relative numbers and magnitudes of the last-figure errors that occur 

 in the sine, cosine, and tangent columns. It is easily shown that the tan- 

 gents were formed by mere subtraction from the sine and cosine columns ; 

 but Gauss was unable to explain the fact that the cosines were more accu- 

 rate than the sines, which appeared as one of the results of the examination. 

 This question is further discussed in the ' Monthly Notices of the Eoy. Ast. 

 Soe. ' for May 1873 ; and it is there shown by the reporter that this result is 

 a direct consequence of the formula by means of which Vlacq calculated the 

 table. So long as all these errors remain uncorrected, the logarithmic trigo- 

 nometrical canon cannot be considered to be in a satisfactory state, as it is 

 certainly desirable that a reliable ten-place table should exist. 



"We believe no perfect list of errors in Vega has been given : a number of 

 errors in T. I. are given by Lefort (' Annales de I'Observatoire de Paris,' 

 t. iv.) ; but this list could not, from the manner in which it was formed, in- 

 clude any errors that did not also occur in Vlacq. 



A long list of errors in the trigonometrical tables of Vega is given by 

 Gronau, ' Tafeln fiir die hyperbolischen Sectoren' &e. Dantzig, 1862, p. vi. 



Copies of Vega are still procurable (but with difficulty and delay) from 

 Germany, through a foreign bookseller, for about =£1 10s. or £1 15s. 



Vega (Manuale), 1800. T. I. Seven-figure logarithms to 1000, and 

 from 10,000 to 101,000, with proportional parts. The change in the line 

 is denoted by an asterisk prefixed to the fourth figure of all the logarithms 

 affected. A few constants are given on p. 188. 



T. II. Log sines, tangents, and arcs for the first minute to every tenth of 

 a second. Although there is a triple heading, there is but a single column of 

 tabular results, as for such small angles the sines, tangents, and arcs arc equal 

 to one another. 



