6i REPORT — 1873. 



flegree. This was published by Tlacq at liis own expense at Gouda in 

 1633, under the title •' Trigonometria Britannica ' (see below) : the intro- 

 duction was written by Gellibraud, by whose name the book is sometimes 

 cited. In the same year Vlacq published his ' Trigonometria Artiiicialis,' 

 containing a ten-aecond canon to ten decimals. Guntee's original table 

 contains a good many errors in the last figures ; and a very slight comparison 

 shows whether any particular table was copied from Gunter or Vlacq ; 

 IIenrion, 1626, and de Decker, 162G (§ 4), are from the former, Faulhaber 

 (§ 4), 1631, from the latter. Briggs appreciated clearly the advantages of 

 a centesimal division of the quadrant, and, by taking a hundredth of a degree 

 instead of a minute, made a step towards a reformation in this respect ; 

 and Hutton has truly remarked that, but for the appearance of Vlacq's 

 work, the decimal division of the degree might have become recognized, 

 as is the case with the corresponding division of the second*. 



The next great advance on the ' Artificialis' was more than a century and 

 a half afterwards, when Michael Taylor (1792) published his seven-decimal 

 canon to every second (§ 4). On account of its great size, and for other reasons, 

 it never came into very general use, Bagay's 1829 (§ 4) being preferred ; 

 the latter is now, however, very difficult to procure. The only other canon 

 to eveiy second we have seen or heard of is Shorirede's, 1844 and 1849 

 (§ 4), which is the most complete as regards proportional parts &c. that we 

 know of. The canon is in modern editions issued separately. 



Lalande (' Encyclopedic Methodiquc. if atheraatiques,' Ast. Tables) states 

 that in April 1784 he received from M. Robert, cure of St. Genevieve at 

 Toul, a volume of sines for every second of the quadrant, and soon after 

 the tangents ; but he had heard that Taylor, in England, was engaged in 

 publishing log sines and cosines to every second, and that the Board of 

 Longitude had contributed £300 to the expense. These volumes were pur- 

 chased by Babbage at the sale of Delambre's library, and they appear in the 

 Babbage Catalogue (only the title of the table of sines is given ; but it is to 

 be presumed that the library contains both, as two volumes arc spoken of). 

 Babbage lent them in 1828 to the Board of Longitude ; and some errata in 

 Taylor, 1792, were found by means of them. [They are now (1873) in the 

 possession of Lord Lindsay, who has purchased the whole of Mr. Babbage 's 

 mathematical library.] 



No ten-decimal canon to every second has been calculated. The French 

 manuscript tables are described in § 3, art. 13. Of logarithmic trigonometrical 

 canons that have appeared the number is very great. We may especially 

 mention Callet, 1853; Bremiker's Vega, 1857; Hutton, 1858; Schron, 

 1860; Dupuis, 1868; and Bruhns, 1870. 



_ * The centesimal division of the degree is of paramount imporiance, wliereas the cente- 

 simal division of the right angle is of next to none at all ; and had the French mathemati- 

 cians at the end of the last century been content with the former, it is not unlikely that their 

 tables woidd have superseded the sexagesimal ones still in use, instead of liaving been almost 

 totally ignored by computers. Thehundredlh part of a right angle is almost as arbitrary a 

 unit as the ninetieth ; and no advantage (but on the contrary great inconvenience) would re- 

 sult from the change ; but to divide the nonagcsimaldegree into centesimal minutes, and these 

 into centesimal seconds, &c., in other words to measure angles by degrees and decimals of 

 a degree, wo\ikl ensure all the advantages of a decimal system (a saving of work in interpo- 

 lations, multiphcations, &e.). This Briggs and his followers. Roe, Oughtred, John Newton, 

 &c., perceived and acted upon two hundred and fifty years ago ; and they seem to liave 

 shown a truer appreciation of the matter than did the French mathematicians. It may 

 be taken for granted that the magnitude of the degree will never be altered; but there is 

 no reason why sexagesimal minutes and seconds should not be replaced by decimals of a 

 degree ; and this is a change which might, and we hope will hereafter be made. 



