παράδοξον, in philosophy, a proposition seemingly absurd, as being contrary to some received opinions, but yet true in fact.

The vulgar and illiterate take almost every thing, even the most important, upon the authority of others, without ever examining it themselves. Although this implicit confidence is seldom attended with any bad consequences in the common affairs of life, it has nevertheless, in other things, been much abused; and in political and religious matters has produced fatal effects. On the other hand, knowing and learned men, to avoid this weakness, have fallen into the contrary extreme: some of them believe every thing to be unreasonable or impossible, that appears to their first apprehension; not averting to the narrow limits of the human understanding, and the infinite variety of objects, with their mutual operations, combinations, and affections, that may be presented to it.

It must be owned, that credulity has done much more mischief in the world than incredulity has done, or ever will do; because the influences of the latter extend only to such as have some share of literature, or affect the reputation thereof. And since the human mind is not necessarily impelled, without evidence, either to belief or unbelief, but may suspend its assent to, or dissent from, any proposition, till after a thorough examination; it is to be wished that men of learning, especially philosophers, would not hastily, and by the first appearances, determine themselves with respect to the truth or falsehood, possibility or imposibility of things:

A person who has made but little progress in the mathematics, though in other respects learned and judicious, would be apt to pronounce it impossible that two lines, which were nowhere two inches asunder, may continually approach towards one another, and yet never meet though continued to infinity; and yet the truth of this proposition may be easily demonstrated. And many, who are good mechanics, would be as apt to pronounce the same, if they were told, that though the teeth of one wheel should take equally deep into the teeth of three others, it should affect them in such a manner, that, in turning it any way round its axis, it should turn one of them the same way, another the contrary way, and the third no way at all.

No science abounds more with paradoxes than geometry: thus, that a right line should continually approach to the hyperbola, and yet never reach it, is a true paradox; and in the same manner a spiral may continually approach to a point, and yet not reach it in any number of revolutions, however great.

The Copernican system is a paradox to the common people; but the learned are all agreed as to its truth. Geometricians have of late been accused of maintaining paradoxes; and some do indeed use very mysterious terms in expressing themselves about asymptotes, the sums of infinite proportions, the areas comprehended between curves and their asymptotes, and the folds generated from these areas, the length of some spirals, &c. But all these paradoxes and mysteries amount to no more than this; that the line or number may be continually 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 it 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 when we take a parallel to the asymptote of the logarithmic curve, or of the hyperbola, for a base), that the ordinate in this case always increases while the base is produced, but never amounts to the distance between the asymptote 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 fold may increase in the same manner, and yet never amount to a given fold. See Mr. Laurin's Fluxions. See Logarithmic Curve.