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HENSLEIGH WEDGWOOD, M.A.,
LATE FELLOWRIST COLLEGE CAMBRIDGE.
WALTON AND MABERLY,
UPPER GOWER STREET, AND IVY LANE, PATERNOSTER ROW.
183. C. G.
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