An introduction to the study of the elements of the differential and integral calculus. From the German of the late Axel Harnack, With the permission of the author.

~ 1. 2. 3.

Positive and -negative num~ber~s.

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Euclid): The greatest common divisor of two numbers a and b (where
b > a) is found by forming by continued division the equations:
b = aq + -, a = r q1 + r1, r == 1q2 + etc.
The divisor of the last division, which leaves no remainder, is the
greatest common divisor of a and b.
3. The first three arithmetical operations, performed on sunis
and differences lead to the following equations:
(a + b) + (c + d) = a + b + c + d
(a + b) - (c + d) a + b - c- d
I) (a + b) + (c- d) =a + +c- d
( ) X (a + b) - (c -d) = a + b - c + d
(a - b) + ( - d) = a - b + c - d
(a b) - (c - d) =a - b- c + d
(II.) (a + b) = ac + be, (a - b) c = ac -b
(a + b) (c + d) ac- + b + ad + bd
(III.) (a + b) (c - d) -- ac + bc - ad - bd
(a - b) (c - d) = ac - be - ad - bd
The differences on the left hand sides in these equations are assumed
to be possible; some of them may be zero. A product which contains
the factor 0, is therefore, as we learn from (II) and (III), itself equal
to 0. It follows conversely from the same equations, that a product
can never be 0 unless one of its factors have the value 0.
The similarity of the results in calculations with sums and differences, suggests the advantage of regarding from the outset the difference
as a sum, for instance the difference a - b as a sum of a actual
units here to be reckoned up and b to be taken away, or as it is
better expressed, of a Positive and b Negative units. The introduction of negative unity enables us to calculate also with differences
in which the minuend is less than the subtrahend. Thus when a < b
the number a - b expresses an excess of negative units, of so many,
in fact, that (b - a) + (a - b) = 0.
In nature there are neither positive nor negative numbers in the
abstract; there exist only things which can be counted. The distinction
of positive or negative numbers - epithets which can only be understood in contrast to each other - has a meaning only for the process
of adding and thence for all other arithmetical operations. But
it is often of great advantage in applying calculation to physical
problems, to distinguish the quanitiies we calculate with, in the sense
of the positive and the negative unit.
Every subtraction is possible when we employ the negative unit,
since there is now introduced an unlimited series of negative numbers
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