 the power of zero, and therewith the sole number the logarithm of which is the same in all systems, viz.,=0. Thus one is the boundary that divides all possible systems of logarithms into two parts: if the base is greater than one, then the logarithms of all numbers more than one are positive, and of all numbers less than one negative; if it is smaller than one, the reverse is the case.
Hence, if every number contains unity in itself in as much as it is compounded entirely of ones added together, unity likewise contains all other numbers in itself. This is not only a possibility, in as much as we can construct any number solely of ones, but also a reality, in as much as one is a definite power of every other number. But the very same mathematicians who, without turning a hair, interpolate into their calculations, wherever it suits them, x0=1, or a fraction whose numerator and denominator are equal and which therefore likewise represents one, who therefore apply mathematically the plurality contained in unity, turn up their noses and grimace if they are told in general terms that unity and plurality are inseparable, mutually penetrating concepts and that plurality is not less contained in unity than unity is in plurality. How much this is the case we see as soon as we forsake the field of pure numbers. Already in the measurement of lines, surfaces, and the volumes of bodies it becomes apparent that we can take any desired magnitude of the appropriate order as unity, and the same thing holds for measurement of time, weight, motion, etc. For the measurement of cells even millimetres and milligrams are too large, for the measurement of stellar distances or the velocity of light even the kilometre is uncomfortably small, just as the kilogram for planetary or, even more so, solar masses, Here is seen very clearly what diversity and multiplicity is contained in the concept of unity, at first sight so simple.
Zero, because it is the negation of any definite quantity, is not therefore devoid of content. On the contrary, zero has a very definite content. As the border-line between all positive and negative magnitudes, as the sole really neutral number, which can be neither positive nor negative, it is not only a very definite number, but also in itself more important than all other numbers bounded by it. In fact, zero is richer in content than any other number. Put on the right of any other number, it gives to the latter, in our system of numbers, the tenfold value. Instead of zero one could use here any other sign, but only on the condition that this sign taken by itself signifies zero, =0. Hence it is part of the nature of zero itself that it finds this application and that it alone can be applied in this way. Zero annihilates every other number with which it is multiplied; united with any other number as divisor or dividend, in the former case it makes this infinitely large, in the latter infinitely small; it is the only number that stands in a relation of infinity to every other number. 0/0 can express every number between – ∞ and +∞, and in each case represents a real magnitude.
The real content of an equation first clearly emerges when all its members have been brought to one side, and the equation is thus reduced to zero value, as already happens for quadratic equations, and is almost the general rule in higher algebra. The function F (x, y)=O can then also be put equal to z, and this z, although it is =0, differentiated like an ordinary dependent variable and its partial derivative determined.
The nothing of every quantity, however, is itself quantitatively determined, and only on that account is it possible to calculate with zero. The very same mathematicians who are quite unembarrassed in reckoning with zero in the above manner, i.e., in operating with it as a definite quantitative concept, bringing it into quantitative relation to other quantitative concepts, clutch their heads in desperation when they read this in Hegel generalised as: the nothing of a something is a determinate nothing.
But now for (analytical) geomet