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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as he began to study the analysis of figure, he found that the previous question, by what intellectual process we are originally made acquainted with figure in general, which was necessary in order to determine what was, and what was not an elementary conception, was entirely unsettled.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

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"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.

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lineal 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

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PREFACE.

THE attempts at a reformation of the Premises of Geometry have been so numerous, and have met with so little success, that another essay in the same direction will doubtless be classed by many with the endeavours to square the circle or find perpetual motion. A little consideration, however, will shew that the circumstances are widely different. The notion of irrational quantities, or quantities whose proportions cannot be exactly expressed by means of numbers, is one which causes difficulty only to the uninstructed. However extended the numbers of a fraction may be by which we attempt to express a proportion, it is readily seen, after a little familiarity with arithmetical conceptions, that the numerator may be a little too great or too small, while the addition or subtraction of an unit may make too great a difference in the opposite direction. There is then no reason to expect that any particular proportion, as that between the circumference and the diameter of a circle, should be capable of exact numerical expression; or, in other words, that the squaring of the circle should be a possible problem. In geometry, on the other hand, there is positive à priori

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