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argument for the possibility of attaining the end which reformers have had in view, and nothing but his predecessors' want of success to discourage the efforts of any new competitor in the same arena. The figures which form the subject of geometrical reasoning being wholly the creation of the understanding, it would seem that they can be endowed with no essential qualities except such as are derived from the plan on which the elements of figure are put together in the conception of the geometrical species. If, therefore, we were able to analyse the first step taken by the understanding in the conception of figure, and to indicate the immediate relation by which the ultimate elements of surface and of line are combined in the conception of the simplest species of figure, the propositions enouncing this primitive synthesis, together with those laying down, in like manner, the composition of the more complicated species, should constitute premises, from whence might directly be deduced every possible relation of the geometrical system. The question then arises, Have any of the proposed amendments been based on the ultimate analysis of all the species of geometrical figure?—and specially, Has the true analysis of a plane as yet been propounded?

In the complete conception of every kind of surface, each infinitesimal element of the surface must be brought successively before the mind and arranged in proper relation to the rest of the system, and whatever can be distinctly conceived may be expressed in language. It must, there. fore be inherently possible to express in words the principle of arrangement or relation between its ultimate parts, characteristic of a plane as well as of every other species of surface. It was by such considerations that the author was led to disregard the old argument, that if the thing could be done at all, it would have been done long ago; but, as soon as he began to study the analysis of figure, he found that the previous question, by what intellectual process we are originally made acquainted with figure in general, which was necessary in order to determine what was, and what was not an elementary conception, was entirely unsettled. It thus became necessary to undertake the examination of one of the most vexed questions of metaphysics, and to trace the course of action and complex exercise of our faculties, by which we originally obtain the knowledge of body, space and form.* Having carefully gone through this inquiry, and obtained certain results to his own satisfaction, the author felt it a strong corroboration of the solidity of his groundwork, when he found that the definitions to which he was led by the metaphysical investigation, including one wholly unexpected of a plane, afforded an adequate basis for the science of

* Principles of Geometrical Demonstration,” Taylor & Walton, 1844. “On the Development of the Understanding," 1848. “On the knowledge of Body and Space." “ Trans. Cambridge Phil. Soc..” Vol. ix., 1850.

geometry, enabling us to dispense as well with the axioms, as with all ex absurdo proof, which has always been regarded as an incongruity in the system.

As the only effective test of the actual attainment of the end which has so long been had in view, the system proposed is applied in the following pages to the geometry of the first three books of Euclid, marking those propositions which are simply copied out without any material alteration in the proof.

If there be no important fallacy in the reasoning of the following pages, the premises adopted in our system are not merely an improvement on those in ordinary use, but they are the ultimate expression of the mode in which the fundamental conceptions of the science are brought into intellectual existence, and must therefore be the primary source from whence all geometrical conviction is derived. No further room will then be left for essential reform, and it would be contrary to the spirit of sound philosophy if the name of Euclid were weighty enough to preserve the sway of his imperfect system in English education, when once the true foundation of the science was effectually made known.

INTRODITATION

ERRATA.

Page. Line. 46 14 for DCB read ECB. 50 1 after D F insert, the corresponding side of the other

triangle. 56 1

for 24 rend 23. 69 14

for 31 read 30. 74 10 after XL insert Euclid I. 38. 75 13, 14 for EF reail DF. 103 18 for D E reud DC.

straight line and the plane, and accordingly rectilineal figures, or figures constructed of straight lines and plane surfaces and primarily the triangle as the rectilineal figure of fewest sides), form the earliest subject of geometrical investigation. Now the form of a triangle may be varied at pleasure, by changing the proportion between the sides, without necessarily raising the question, whether there be any corresponding variation in the proportion of the angles. We may imagine a triangle

B

geometry, enabling us to dispense as well with the axioms, as with all ex absurdo proof, which has always been regarded as an incongruity in the system.

As the only effective test of the actual attainment of the end which has so long been had in view, the system proposed is applied in the follow

„sential reform, and it would be contrary to the spirit of sound philosophy if the name of Euclid were weighty enough to preserve the sway of his imperfect system in English education, when once the true foundation of the science was effectually made known.

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