REPORT ON CERTAIN BRANCHES OF ANALYSIS. 289 



pendent science, and not as auxiliary to the application of al- 

 gebra to geometry. It is to Euler* that we are indebted for 

 the emancipation of this most important branch of analytical 

 science from this very limited application, who first introduced 

 the functional designations sin z, cos z, tan ^, &c., to denote the 

 sine, cosine, tangent, &c., of an arc ^, whose radius is 1, which 

 had previously been designated by words at length, or by simple 

 and independent symbols, such as a, b, s, c, t, &c. The intro- 

 duction of this new algorithm speedily changed the whole form 

 and character of symbolical language, and greatly extended 

 and simplified its applications to analysis, and to every branch 

 of natural philosophy. 



The angles which enter into consideration in trigonometry 

 are generally assumed to be measured by the arcs of a cn-cle 

 of a given radius, and their sines and cosines are commonly de- 

 fined with reference to the determination of these arcs, and not 

 with refei'ence to the determination of the angles which they 

 measure. It is in consequence of this defined connexion of 

 sines and cosines with the arcs, and not immediately with the 

 angles which they measure, that the radius of the circle upon 

 which those arcs are taken must necessarily enter as an element 

 in the comparison of the sines and cosines of the same angle 

 determined by different measures : and though they Vvcre ge- 

 nerally, at least in later writers, reduced to a common standard, 

 by assuming the radius of this circle to be 1, yet formulae were 

 considered as not perfectly general unless they were expressed 

 with reference to any radius whatsoever-}-. In the application, 

 likewise, of such formulae to the business of calculation, the 

 consideration of the radius was generally introduced, producing 

 no small degree of confusion and embarrassment ; and even in 

 the construction of logarithmic tables of sines and cosines the 



• Introductio in Analysirn Injinitoruw, vol. i. cap. viii. " Quemadmodum 

 logarithmi peculiarem algorithmum vequinint, cujus in universa analyst summiis 

 extat usus, ita quantitates circulares ad certain quoque algorithmi normam 

 perduxi : ut in calculo aeqiie commode ac logarithmi et ipsae quantitates alge- 

 braicae tractari possent." — Extract from Preface. 



t We may refer to Vince's Trigonometry, a work in general use in this 

 country less than a quarter of a century ago, and to other earlier as well as 

 contemporary writers on this subject, for examples of formulse, which are uni- 

 formly embarrassed by the introduction of this extraneous element. Later writers 

 have assumed the radius of the circle to be 1, and have contented themselves 

 with giving rules for the conversion of the resulting formulae to those which 

 would arise from the use of any other radius. It is somewhat remarkable that 

 the elementary writers on this subject should have continued to encumber their 

 formulae with this element long after its use had been abandoned by Euler, 

 Lagrange, Laplace, and all the other great and classical mathematical writers 

 on the Continent. 



1833. u 



