42 EEPORT — 1873. 



lished by Eegiomontauus (1504) ; and tho first complete canon giving all the 

 six ratios of the sides of a right-angled triangle is due to llheticns (1551), 

 -who also introduced the semiquadrantal arrangement. Eheticus's canon was 

 to every ten minutes to 7 places ; and Vieta first extended it to evenj minute 

 (1579). The first complete canon published in England was by Blundevile 

 (1594), although a table of sines had appeared four years earlier. 



It may be added that Eegiomontauus (1504) called his table of tangents (or 

 rather cotangents) Tabula foscimda, on account of its great use ; and till the in- 

 troduction of the word tangent by Finck (1583), a table of tangents was called 

 a Tabula fcecunda or Canon foecundus ; Finck also introduced the term secant, 

 the table of secants having previously been called Tabula benefica by Mauro- 

 lycus (1558), and Talula fommdissima by Yieta. 



The above historical sketch has been compiled from Hutton and De Morgan ; 

 so that most of tho statements contained in it arc not derived from our own 

 inspection of the works mentioned. It is only inteaded to give an idea of the 

 history of the natural canon ; and from the experience we have had of the value 

 of second-hand information in mathematical bibliography, we should not re- 

 commend great reliance to be placed ou any one of the facts. A good deal of 

 information about llheticus, Vieta, &c. is given by De Morgan, whom we have 

 scarcely ever found inaccurate, even in trifling details, Avhen describing works 

 he has examined himself. We have seen several of the works noted, but not 

 BulRcient to make any corrections of importance to the current histories. 



The next author of importance to Eheticus was Pitiscus (1G13), whose im- 

 portant canon, which stiU remains unsuperseded, is described below. The in- 

 vention of logarithms in the following year changed all the methods of calcula- 

 tion ; and it is worthy of note that Napier's original table of 1614 (see § 3, art. 

 17) was a logarithmic canon of sines and not a table of the logarithms of 

 numbers. Almost at once the logarithmic superseded the natural canon ; 

 and since Pixiscus's time no really extensive table of pure trigonometrical 

 functions has appeared. Natural canons are now most common in Nautical 

 collections, where the tabular results are generally given to 5 or places only. 



Traverse tables (multiples of sines and cosines) have not been included 

 (see § 2, art. 12). Massalotjp (described below), however, is really a table 

 of this kind, although constructed for a different purpose. 



Finck [1583]. Canon of sines, tangents, and secants in separate tables, 

 quadrantally arranged, for every minute of the quadrant, to 7 decimal ])laces. 

 The sines occupy pp. 138-173, the tangents pp. 176-221, and the secants 

 pp. 224-269. De Morgan says that Finck calculated his own secants. There 

 is no date on the titlepage ; but the i^reface and the colophon are both dated 

 1583. The name tangent is introduced by Finck on p. 73, and that of 

 secant on p. 76. These names were speedily adopted : thus Clavius, at the 

 end of his edition of ' Theodosius ' (Eome, 1586), reprints Finck's tables, and 

 uses his terms both in the headings of the tables and in the trigonometry. 

 He does not mention either Finck or Eheticus by name, but speaks of them 

 as recentiores (p. 188). Pitiscus, in his trigonometry appended to Abraham 

 Shultet's ' Sphsericorum ' (Heidelberg, 1595), uses the names tangent and 

 secant, and refers to Finck or Eheticus for the requisite canons ; and in his 

 larger trigonometry (Augsburg, 1600) he reprints Finck's tables to five deci- 

 mals, placing the sines, tangents, and secants together in one table. Blun- 

 devile, in his ' Exercises ' (London, 1594), reprinted the tables from Clavius. 

 All these woiks are before us ; but a more detailed account would be of only 

 historical or bibliographical interest. 



