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MK. T. T. WILKINSON ON 



preserved to us in the Mathematical Collections of Pappus, 

 or scattered throughout the pages of our own Mathematical 

 periodicals. Many of the most attractive theories of modern 

 times were evidently within their reach, and not unfrequently 

 engaged a portion of their attention, but they suifered them- 

 selves to be led away by comparatively unimportant objects, 

 and expended those energies upon isolated questions which 

 ought to have been directed to original and systematic 

 inquiries. M. Chasles has recently shewn in his Traite de 

 Geometrie Superieure, that Anharmonic Ratio, on which all 

 Modern Geometry is found to hinge, was distinctly under- 

 stood by the later Greek Geometers, and it is also under 

 the same form to be found in several of our Mathematical 

 journals ; but previously to the appearance of his Jper(;u 

 Uistorique, in 1837, no one had been able to see its im- 

 portance as a fundamental principle in Geometry. The 

 comparative inadequacy of the methods employed by these 

 otherwise able men, when applied to the higher branches of 

 Geometry, will no doubt account for the fact that of late 

 years the taste for the strict forms of the Ancient Analj'sis 

 has much declined, even in Lancashire and in the North of 

 England, Very few of those who flourished in the palmy 

 days of the Companion are now in existence, and as the old 

 Geometers, one by one, pass from the scene, their places are 

 supplied by others who prefer the more powerful co-ordinate 

 methods to those antiquated processes which at best can 

 only accomplish, by a series of gradations, what follows at 

 once from the general expressions of Modern Geometry. 

 But it must not be imagined that we ought by any means 

 to discard the Ancient Analysis altogether. The University 

 of Cambridge is our best guide on mathematical studies, 

 and she very properly insists upon a due acquaintance with 

 both Euclidean and Cartesian principles; the former as a 

 mental discipline ; — the latter as a more powerful instrument 



