THE PHILOSOPHY OF SPINOZA AND LEIBNIZ. 339 solution upon the discovery of a relation between the straight line and the circle. Somehow it must be possible to express the circle in terms of the straight line. But you cannot do it with a ruler and a pair of compasses : you cannot draw or construct any figure which will solve the problem. The nearest approach to a solution that can be made is to con- struct a polygon with so many sides that it will come very near indeed to the circle. But you can never make the sides small enough for the figure to coincide with the circle. The sides will always remain finite straight lines, while the circle is the locus of a point which is continuously changing its direction. Accordingly the Greek geometers had recourse to the method of " exhaustions ". Thus they regarded the area of a circle as being equivalent to the " limit " area of a circumscribed and an inscribed polygon, having the same number of sides, when the sides are made infinitely numer- ous. The polygons can never actually become the circle, but the ultimate difference is negligible, being as little as we like to make it, and accordingly the "limit" area to which each polygon approaches may be taken as practically equiva- lent to the area of the circle. Now this method is one of proof per impossibile or reductio ad absurdum. The area of the circle must be either equal to, greater than, or less than the limit area of the polygons. But to suppose it greater or less would be to suppose that the polygons do not yet coin- cide, i.e., that the area is not the limit area. Therefore the area of the circle must be equal to the limit area of the polygons. But all proof per impossibile is merely a negative verification. It shows that anything other than the sug- gested law or truth (the thing to be proved) would be inconsistent with the general principles or constitution of some system, such as the system of quantity or the system of space. But it does not show how these general prin- ciples apply to the particular case or how the particular case follows necessarily from them, is an organic element in the constitution of the system. Thus, in the instance we have considered, the proof depends upon an actual construction or picturing in space of two dimensions plus a general refer- ence to the nature of quantity as being such that every element in it must be either greater than, equal to, or less than any other. Space is assumed to be quantitative, and space of two dimensions is assumed to be such that straight lines and circles can be drawn in it ; but neither the relation of space to quantity nor the nature of space of two dimen- sions as expressing itself in the straight line and circle is thought out or made an explicit premiss in the argument.
