the latter; outside this relation, taken by themselves, they are purely imaginary. In trigonometry and analytical geometry, together with the branches of higher mathematics of which these are the basis, they express a definite direction of motion, opposite to the positive direction. But the sine and tangent of the circle can be reckoned from the upper right-hand quadrant just as well as from the lower right-hand quadrant, thus directly reversing plus and minus. Similarly, in analytical geometry, abscissae can be calculated from the periphery or from the centre of the circle, indeed in all curves they can be reckoned from the curve in the direction usually denoted as minus, (or) in any desired direction, and still give a correct rational equation of the curve. Here plus exists only as the complement of minus, and vice versa. But algebraic abstraction treats them (negative magnitudes) as real and independent, even outside the relation to a larger, positive magnitude.
Mathematics. To common sense it appears an absurdity to resolve a definite magnitude, e.g., a binomial expression, into an infinite series, that is, into something indefinite. But where would we be without infinite series and the binomial theorem?
Asymptotes. Geometry begins with the discovery that straight and curved are absolute opposites, that straight is absolutely inexpressible in curved, and curved in straight, that the two are incommensurable. Yet even the calculation of the circle is only possible by expressing its periphery in straight lines. For curves with asymptotes, however, straight becomes completely merged in curved, and curved in straight, just as much as the notion of parallelism: the lines are not parallel, they continually approach one another and yet never meet; the arm of the curve becomes more and more straight, without ever becoming entirely so, just as in analytical geometry the straight line is regarded as a curve of the first order with an infinitely small curvature. However large the x of the logarithmic curve may become, y 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 triangl