can never=0.
Straight and curved in the differential calculus are in the last resort put as equal: in the differential triangle, the hypotenuse of which forms the differential of the arc (in the tangent method), this hypotenuse can be regarded
“as a small, quite straight line which is at the same time the element of the arc and that of the tangent"-no matter whether the curve is regarded as composed of an infinite number of straight lines, or also, “whether one considers it as a strict curve; since the curvature at each point M is infinitely small, the last ratio of the element of the curve to that of the tangent is evidently a ratio of equality.”
Here, therefore, although the ratio continually approaches equality, but asymptotically in accordance with the nature of the curve, yet, since the contact is limited to a single point which has no length, it is finally assumed that equality of straight and curved has been reached. (Bossut, Calcul différentiel et intégral, Paris, An VI, 1, p. 149.) In polar curves the differential imaginary abscissae are even taken as parallel to the real abscissae and operations based on this, although both meet at the pole; indeed, from it is deduced the similarity of two triangles, one of which has an angle precisely at the point of intersection of the two lines, the parallelism of which is the whole basis of the similarity! (Fig. 17.)
When the mathematics of straight and curved lines has thus pretty well reached exhaustion a new almost infinite field is opened up by the mathematics that conceives curved as straight (the differential triangle) and straight as curved (curve of the first order with infinitely small curvature). O metaphysics!
Trigonometry. After synthetic geometry has exhausted the properties of a triangle, regarded as such, and has nothing new to say, a more extensive horizon is opened up by a very simple, thoroughly dialectical procedure. The triangle is no longer considered in and for itself but in connection with another figure, the circle. Every right-angled triangle can be regarded as belonging to a circle: if the hypotenuse =r, then the sides enclosing the right angle are sin and cos; if one of these sides =r, then the other =tan, the hypotenuse =sec. In this way the sides and angles are given quite different, definite relationships which without this relation of the triangle to the circle would be impossible to discover and use, and quite a new theory of the triangle arises, far surpassing the old and universally applicable, because every triangle can be resolved into two right-angled triangles. This development of trigonometry from synthetic geometry is a good example of dialectics, of the way in which it comprehends things in their interconnection instead of in isolation.
Identity and difference – the dialectical relation is already seen in the differential calculus, where dx is infinitely small, but yet is effective and does everything.
Molecule and differential. Wiedemann (III, p. 636) puts finite and molecular distances as directly opposed to one another.
The fact that our subjective thought and the objective world are subject to the same laws, and hence, too, that in the final analysis they cannot contradict each other in their results, but must coincide, governs absolutely our whole theoretical thought. It is the unconscious and unconditional premise for theoretical thought. Eighteenth-century materialism, owing to its essentially metaphysical character, investigated this premise only as regards content. It restricted itself to the proof that the content of all thought and knowledge must derive from sensuous experience, and revived the principle: nihil est in intellectu, quod non fuerit in sensu. It was modern idealistic, but at the same time dialectical, philosophy, and especially Hegel, which for the first time investigated it also as regards form. In spite of all the innumerable arbitrary constructions and fantasies that we encounter here, in spite of the idealist, topsy-turvy form of its result-the unity of thought and being-it is undeniable that this philosop