290 THIRD REPORT — 1833. 



occurrence of negative logarithms was avoided by a fiction, 

 which supposed them to be the sines and cosines of arcs of a 

 circle whose radius was 10'". 



A very slight modification of the definition of the sine and 

 cosine would enable us to get rid of this element altogether. 

 In a right-angled triangle, the ratio of any two of its sides will 

 determine its species, and conse- 

 quently the magnitude of its angles. 

 If we suppose, therefore, a point P 

 to be taken in one (A C) of the two 

 lines A C and A B containing the 

 angle BAG (5), and P M to be 

 drawn perpendicular to the other 

 line (A B), then we may define the 



. PM 



sine of fl to be the ratio . p , and the cosine of 9 to be the 



AM 

 ratio -J— Ti. By such definitions we shall make the sine and 

 A P •' 



cosine of an angle depend upon the angle itself, and not upon 

 its measure, or upon the radius of the circle in which it is taken : 

 and upon this foundation all the formulae of trigonometry may 

 be established, and their applications made, without the neces- 

 sity of mentioning the word radius*. 



If we likewise assume the ratio of the arc which subtends an 

 angle to the radius of the circle in which it is taken, and not 

 the arc itself, for the measure of an angle, we shall obtain a 

 quantity which is independent of this radius. In assuming, 

 therefore, the angle 9 to be not only measured, but also repre- 

 sented by this ratio, we shall be enabled to compare sin & and 

 cos Q directly with 9, and thus to express one of them in terms 

 of the other. It is this hypothesis which is made in deducing 

 the exponential expressions for the sine and cosine, and the 

 series which result immediately from them-j-. 



* See A Syllabus of a Course of Lectures upon Trigonometry, and the Appli- 

 cation of Algebra to Geometry, published at Cambridge in 1833, in which all 

 the formulae of trigonometi-y are deduced in conformity with these definitions. 



t If we should attempt to deduce the exponential expressions for sin 6 and 

 cos 6 from the system of fundamental equations, 



cos" d -\- sin^ ^ = 1 (1.) 



cos 6 = cos (— 6) (2.) 



sin ^ = — sin (— 6) (3.) 



we should find, 



aoaAH , g-A('\/^ gA^A/irT_g-A^v'irT 



COS 6 = ^^^— ' and sin ^ = ;;== 



2 2 V — 1 



in which the quantity A, in the absence of any determinate measure of the 



