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base, time, amount and manner

PARADOX (from the Greek •rapa, against, and do,,a, opinion) in philosophy, a proposition seemingly absurd, because contrary to the received opinions ; but yet true in effect.

The Copernican system is a paradox to the common people; lout the learned ;ire all agreed as to its truth.

Geometricians have been accused of maintaining paradoxes ; and, it must be owned, that some use very mysterious terms in expressing themselves about as mptotes, the stuns of infi nite progressions, the areas comp'', hended between curves and their asymptotes, and the solids generated from these areas, the length of some spirals, &c. But all these para doxes and mysteries amount to no more than that the line or number may be eontinually acquiring increments, and those increments may decrease in such a manner, that the whole line or number shall never amount to a given line or number.

The necessity of admitting this is obvious from the nature of the most common geometrical figures ; thus, while the tangent of a circle increases, the area of the corresponding sector increases, but never amounts to a quadrant. Neither is it difficult to conceive, that if a figure be concave towards a base, and have an asymptote parallel to the base (as it happens hen we take a parallel to the asymptote of the logarithmic curve, or of the hyperbola, tor a base) that time ordinate in this ease always increases while the base is pro duced, but never amounts to time distance between the asymp tote and the base. In like manner, a curvilinear area may

increase while the base is produced, and approach continually to a certain finite space, but never amount to it ; and a solid may increase in the same manner, and yet never amount to a given solid.

A spiral may in like manner approach to a point con tinually, and vet in any number of revolutions never arrive at it ; and there are progressions of fractions, which may be continued at pleasure, and yet the sum of time terms shall be always less than a given number. In Maciaurin's Flax ions (book i, C11. 10. et seq.) various rules are demonstrated, and illustrated by examples, for determining time asymptotes and limits of figures and progressions, without having recourse to those mysterious expressions which have of late years crept into time writings of mathematicians. For, as that excellent author observes; elsewhere, though philosophy has, and probably always will have, mysteries to us, geometry ought to have none.