278 Mr . Woodhouse on the Integration , &c. 
vibrating in a circular arc ; and, as might have been com- 
puted, without the introduction of the Lemniscata * I have stated 
the mode by which analysis may derive aid from geometry ; the 
extent of the aid however is, I conceive, very small ; remove the 
circle, ellipse, and parabola, curves whose properties have been 
the object of so much investigation, and we only create for our- 
selves unnecessary and circuitous operations, by introducing curves 
into the discussion of questions purely analytical. For the purposes 
of classification, however, curves may not be altogether useless. 
The correspondence that has been shown, between the artifices 
of calculation and the properties of geometrical figures, may be 
thought, perhaps, curious or remarkable; and the reduction of 
several methods into one is, I presume, practically and scientifically 
useful. On similar reductions, the perfection of analysis, to a great 
degree, depends : for, a frequent result of a careful investigation 
is, the discovery that methods apparently different, because dif- 
ferently expressed, are founded on the same principle and funda- 
mental notion; but, if examination and study thus diminish the 
seeming bulk of our knowledge, they, at the same time, increase 
its precision and purity. 
• See Euler’s Memoir, Novi Comm, Tom. VI. p. 37, -&c. Likewise, relative to the 
subject of this Paper, Novi Comm. Tom. XII. Nova Acta, Tom. VII. 1 7?8. 
