Page:Landmarks of Scientific Socialism-Anti-Duehring-Engels-Lewis-1907.djvu/170

 building materials for new layers. But the result of the process is always positive, the restoration of a piece of ground made up of exceedingly diverse chemical elements to a condition of mechanical pulverisation, which is the cause of a most abundant and diverse vegetation.

It is the same also in mathematics. Let us take an ordinary algebraic quantity a. Let us negate it, then we have — a (minus a). Let us negate this negation, that is let us multiply — a by — a and we have + a$2$, that is the original positive quantity but in a higher form that is to the second power. It does not matter that we can attain the same a$2$ by the multiplication of a positive by itself. The negated negation is established so completely in a$2$ that under all circumstances it has two square roots a and — a. And this impossibility, the negated negation, the getting rid of the negative root in the square has much significance in quadratic equations. The negation of the negation is more evident in the higher analyses, in those "unlimited summations of small quantities," which Herr Duehring himself explains as being the highest operations of mathematics and which are usually called the differential and integral calculus. How do these forms of calculation fulfil themselves? I have for example in a given problem two variable quantities x and y, of which one cannot vary without causing the other to vary also under fixed conditions. I differentiate x and y, that is I consider x and y as being so infinitesimally small that they do not represent any real quantities, even the smallest, so that, of x and y nothing remains, except their reciprocal relations, a quantitative relation without any quantity; therefore $dx⁄dy$, the relation