THE GENESIS OF LOGARITHMS 149 



canon faecundus'. — computed for every degree and to radius 

 100,000. 



Barely mentioning the canon of sines given by Copernicus, 

 that of tangents by Reinhold (1553) and, about the same date, 

 of secants by Francis Maurolye, Abbot of Messina in Sicily, 

 we come to the striking work of Francis Vieta (1 540-1603), 

 without doubt the foremost algebraist of his day. 



In a folio volume published at Paris in 1579 he gives 

 tables of sines, tangents and secants for every minute of the 

 quadrant to radius 100,000. And it is to be noted that he 

 regards the "trigonometrical ratios" not as being obtained 

 from a series of lines drawn in or about a circle but as being 

 obtained from a series of plane right-angled triangles in which 

 (1) the hypotenuse has the constant value 100,000, the other 

 two sides being variable and giving the values of the sine 

 and cosine; (2) the base has the constant value 100,000, when 

 the other two variable sides give the tangent and secant ; 

 (3) when the perpendicular is kept constant the variable sides 

 give the cotangent and cosecant. 



A second table given by Vieta is something of a curiosity, 

 as in it he gives a canon of accurate sines, cosines, etc., ex- 

 pressed in integers and rational vulgar fractions. In general, 

 the numbers which express the trigonometrical ratios are 

 irrational but for certain particular angles the values are 

 rational; it is these values which are tabulated. The corre- 

 sponding angles are not given but in their place appears a 

 series of numbers called by Vieta numeri primi baseos. Let 

 one of these numbers be called p and let r be the constant 

 radius which, as before, is taken as 100,000, then, if r be taken 

 as the hypotenuse of the right-angled triangle, the sine or 

 perpendicular will be given by 



pr 



the base or cosine will be 



4 +I 



7-< 



4 +I 



with similar expressions for the tangent and secant, cotangent 



