ry. Here zero is a definite point from which measurements are taken along a line, in one direction positively, in the other negatively. Here, therefore, the zero point has not only just as much significance as any point denoted by a positive or negative magnitude, but a much greater significance than all of them: it is the point on which they are all dependent, to which they are all related, and by which they are all determined. In many cases it can even be taken quite arbitrarily. But once adopted, it remains the central point of the whole operation, often determining even the direction of the line along which the other points-the end points of the abscissae – are to be inserted. If, for example, in order to arrive at the equation of the circle, we choose any point of the periphery as the zero point, then the line of the abscissae must go through the centre of the circle. All this finds just as much application in mechanics, where likewise in the calculation of the motions the point taken as zero in each case forms the main point and pivot for the entire operation. The zero point of the thermometer is the very definite lower limit of the temperature section that is divided into any desired number of degrees, thereby serving as a measure both for temperature stages within the section as also for higher or lower temperatures. Hence in this case also it is a very essential point. And even the absolute zero of the thermometer in no way represents pure abstract negation, but a very definite state of matter: the limit at which the last trace of independent molecular motion vanishes and matter acts only as mass. Wherever we come upon zero, it represents something very definite, and its practical application in geometry, mechanics, etc., proves that - as limit - it is more important than all the real magnitudes bounded by it.
Zero powers. Of importance in the logarithmic series:
√—1. The negative magnitudes of algebra are real only in so far as they are connected with positive magnitudes and only within the relation to 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 