analytical and geometrical Methods of Investigation. 123 
voured to shew, that the geometrical calculus was competent to 
the explanation of natural phenomena; and with astonishing 
perseverance applied it to many investigations in physical astro- 
nomy. The labours of such a man are not hastily to be judged: 
yet every one must determine for himself ; and to me it seems, 
his reasonings, from their intricacy, call up so great a contention 
of the mind, that they prove, in no small degree, the unfitness 
of the geometrical method in all abstruse and intricate inves- 
tigations. 
XXIX. It may, however, be asked, are not there some sub- 
jects of inquiry to which the geometrical method is better adapted 
than the analytical ? and is not the theory of angular functions 
one of these subjects ? * I apprehend not : for, if the conditions 
Intentionally. Dr. Stewart, (Preface to Sun’s Distance,) and after him his ingenious 
biographer, for the purpose of holding up the superior simplicity of the geometrical 
calculus, has said, that in order to understand his solution, a knowledge of the ele- 
ments and conic sections only is requisite. But, in fact, the solution is effected by 
proposition heaped on proposition ; and with equal truth and justness it might be 
said, that in order to understand the analytical solution, a knowledge only of common 
algebra is requisite ; since the methods by which the solution is effected, are really and 
prqperly branches of algebra. 
• D’Alembert says, “there are cases in which analysis, instead of expediting, 
embarasses demonstration. These cases happen in the computation of angles : for angles 
can analytically be expressed only by their sines ; and the expression of the sines of 
angles is often very complicated,” &c. He adds, that it must depend on mathema- 
ticians, whether the method of the ancients or the modern analysis is to be employed, 
since it would be difficult to give on this head exact and general rules.” In the very 
case adduced, I think demonstration expedited by the analytical calculus ; and, although 
a""" 1 1 -f | j s not so speedily put down as cos. x ; yet all processes of 
, evolution, differentiation, integration, &c. are much more easily performed with the 
former expression than with the latter. Other instances of subjects of inquiry, to which 
the geometrical method is said to be peculiarly well adapted, have been adduced ; but 
I still find no convincing reason, why a mathematician must submit to the necessity of 
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