- NUMBER SYSTEMS 207
x = \J"1 + \/—121+ \V2 - \/—121. In this case, however, Bombelli was able to work
backward, since he knew in advance that one root is 4; the problem was to make
the formula say "4." Bombelli had the idea that the two cube roots must consist
of real numbers together with his "plus of minus" or "minus of minus." Since the
imaginary parts in the sum of the two cube roots must cancel out and the real parts
must add up to 4, it seems obvious that the real parts of the cube roots must be 2.
In our terms, the cube roots must be 2 ± ß÷/-ú for some t. Then since the cube
of the cube roots must be 2 ± llv^-T (what Bombelli called 2 plus 11 times "plus
of minus"), it is clear that the cube roots must be 2 plus "plus of minus" and 2
minus "plus of minus," that is, 2 ± ÷/^Ô. As a way of solving the equation, this
reasoning is circular, but it does allow the formula for solving the cubic equation
to make sense.
In an attempt to make these numbers more familiar, the English mathematician
John Wallis (1616-1703) pointed out that while no positive or negative number
could have a negative square, nevertheless it is also true that no physical quantity
can be negative, that is, less than nothing. Yet negative numbers were accepted
and interpreted as retreats when the numbers measure advances along a line. Wallis
thought that what was allowed in lines might also apply to planes, pointing out
that if 30 acres are reclaimed from the sea, and 40 acres are flooded, the net amount
"gained" from the sea would be -10 acres. Although he did not say so, it appears
that he regarded this real loss of 10 acres as an imaginary gain of a square of land
\/-435600 = ÈÈÏ÷/^Ô feet on a side.
What he did say in his 1673 treatise on algebra was that one could represent
yj—bc as the mean proportional between —b and c. The mean proportional is
easily found for two positive line segments ä and c. Simply lay them end to end,
use the union as the diameter of a circle, and draw the half-chord perpendicular
to that diameter at the point where the two segments meet. That half-chord is
the mean proportional. When one of the numbers (—6) was regarded as negative,
Wallis regarded the negative quantity as an oppositely directed line segment. He
then modified the construction of the mean proportional between the two segments.
When two oppositely directed line segments are joined end to end, one end of the
shorter segment lies between the point where the two segments meet and the other
end of the longer segment, so that the point where the segments meet lies outside
the circle passing through the other two endpoints. Wallis interpreted the mean
proportional as the tangent to the circle from the point where the two segments
meet. Thus, whereas the mean proportional between two positive quantities is
represented as a sine, that between a positive and negative quantity is represented
as a tangent.
Wallis applied this procedure in an "imaginary" construction problem. First he
stated the following "real" problem. Given a triangle having side AP of length 20,
side PB of length 15, and altitude PC of length 12, find the length of side AB, taken
as base in Fig. 2. Wallis pointed out that two solutions were possible. Using the
foot of the altitude as the reference point C and applying the Pythagorean theorem
twice, he found that the possible lengths of AB were 16 ±9, that is, 7 and 25. This
construction is a well-known method of solving quadratic equations geometrically,
given earlier by Descartes. It always works when the roots are real, whether positive
or negative. He then proposed reversing the data, in effect considering an impossible
triangle having side AP of length 20, side PB of length 12, and altitude PC of
length 15. Although the algebraic problem has no real solution, a fact verified by