208 8. NUMBERS AND NUMBER THEORY IN MODERN MATHEMATICS
FIGURE 2. Wallis' geometric solution of quadratic equation with
real roots (left) and complex roots (right).the geometric figure (Fig. 2), one could certainly draw the two line segments AB.
These line segments could therefore be interpreted as the numerical solutions of the
equation, representing a triangle with one side having imaginary length.
The rules given by Bombelli made imaginary and complex numbers accessible,
and they turned out to be very convenient in many formulas. Euler made free use
of them, studying power series in which the variables were allowed to be complex
numbers and deriving a famous formula
gUi/^T _ cosv _|_ y/^is\nv.Euler derived this result in a paper on ballistics written around 1727 (see Smith,
1929, pp. 95 98), just after he moved to Russia. But he had no thought of repre-
senting 85 we now do, on a line perpendicular to the real axis.
Wallis' work had given the first indication that complex numbers would have to
be interpreted as line segments in a plane, a discovery made again a century later
by the Norwegian surveyor Caspar Wessel (1745 1818). The only mathematical
paper he ever wrote was delivered to the Royal Academy in Copenhagen, Denmark
in 1797, but he had been in possession of the results for about a decade at that
time. In that paper (Smith, 1929, pp. 55-66), he explained how to multiply lines in
a plane by multiplying their lengths and adding the angles they make with a given
reference line, on which a length is chosen to represent +1:
Let +1 designate the positive rectilinear unit and +e a certain
other unit perpendicular to the positive unit and having the same
origin; the direction angle of +1 will be equal to 0°, that of -1 to
180°, that of +€ to 90°, and that of -e to -90° or 270°. By the rule
that the direction angle of the product shall equal the sum of the
angles of the factors, we have: (+1)(+1) = +1; (+1)(—1) = -1;
(-1)(-1) = +1; (+l)(+c) - +e; (+l)(-e) = -c; (-l)(+e) = -e;
(-l)(-€) = +e; (+e)(+e) = -1; (+£)(-c) = +1; (-e)(-«) = -1.
From this it is seen that e is equal to í/~À· [Smith, 1929, p. 60]Wessel noticed the connection of these rules with the addition and subtraction
formulas for sign and cosine and gave the formula (cos χ + e sin x)(cos y + e sin y) =
cos(x + y) + esin(x + y). On that basis he was able to reduce the extraction of
the nth root of a complex number to extracting the same root for a positive real
number and dividing the polar angle by n.