ANCIENT ANALYSIS. POEISMS. 65 



merits of the modern method, it is certain that those geo- 

 metricians would have regarded the course taken by some of 

 its votaries in more recent times as exceptionable, whether 

 with a view to clearness or to good taste : a course to the full 

 as objectionable as woiild be the banishing of algebraical and 

 substituting of geometrical symbols in the investigations of 

 the higher geometry. La Place's great work, the ' Mecanique 

 Celeste,' and La Grange's ' Mecanique Analytique,' have 

 treated of the whole science of dynamics and of physical 

 astronomy, comprehending all the doctrine of trajectories, 

 dealing with geometrical ideas throughout, and ideas so 

 purely geometrical that the algebraic symbols, as far as 

 their works are concerned, have no possible meaning apart 

 from lines, angles, surfaces ; and yet in their whole compass 

 they have not one single diagram of any kind. Surely, 



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we may ask if -%-*/ dx* -f- dy*, T A7y\ can possibly 



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bear any other meaning than the tangent and the radius of 

 curvature of a curve line : that is, a straight line touching a 

 curve, and a circle whose curvature is that of another curve 

 where they meet ; any meaning, at least, which can make it 

 material that they should ever be seen on the page of the 

 analyst. These expressions are utterly without sense, except 

 in reference to geometrical considerations ; for although x 

 and y are so general that they express any numbers, any lines, 

 nay, any ideas, any rewards or punishments, any thoughts of 

 the mind, it is manifest that the square of the differential of a 

 thought, or the differential of the differential of a reward or 

 punishment, has no meaning ; and so of everything else but 

 of the very tangent or the osculating circle's radius : conse- 

 quently the generality of the symbols is wholly useless ; the 

 particular case of two lines being the only thing to which 



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