432 14. EQUATIONS AND ALGORITHMSThis is a cubic equation called the resolvent cubic. Once it is solved, the original
quartic breaks into two quadratic equations upon taking square roots and adding
an ambiguous sign.
A few aspects of the solution of cubic and quartic equations should be noted.
First, the problem is not a practical one. Second, the Cardano recipe for solving
an equation sometimes gives the solution in a rather strange form. For example,
Cardano says that the solution of x^3 + 6x = 20 is V^VW+IO- \/\/Ú08- 10. The
expression is correct, but can you tell at a glance that it represents the number 2?
Third, the procedure does not always work. For example, the equation x^3 +6 =
7x has to be solved by guessing a number that can be added to both sides so as to
produce a common factor that can be canceled out. The number in this case is 21,
but there is no algorithm for finding such a number. For equations of this type the
algebraic procedures for finding ÷ involve square roots of negative numbers. The
search for an algebraic procedure using only real numbers to solve this case of the
cubic continued for 300 years, until finally it was shown that no such procedure can
exist.
6.6. Consolidation. There were two natural ways to build on what had been
achieved in algebra by the end of the sixteenth century. One was to find a notation
that could unify equations so that it would not be necessary to consider so many
different cases and so many different possible numbers of roots. The other was to
solve equations of degree five and higher. We shall discuss the first of these here
and devote Chapter 15 to the quest for the second and its consequences.
All original algebra treatises written up to and including the treatise of Bombelli
are very tiresome for the modern student, who is familiar with symbolic notation.
For that reason we have sometimes allowed ourselves the convenience of modern
notation when doing so will not distort the thought process involved. In the years
between 1575 and 1650 several innovations in notation were introduced that make
treatises written since that time appear essentially modern. The symbols + and —
were originally used in bookkeeping in warehouses to indicate excess and deficien-
cies; they first appeared in a German treatise on commercial arithmetic in 1489
but were not widely used in the rest of Europe for another century. The sign for
equality was introduced by a Welsh medical doctor, physician to the short-lived
Edward VI, named Robert Recorde (1510-1558). His symbol was a very long pair
of parallel lines, because, as he said, "noe 2. thynges, can be moare equalle." The
use of abbreviations for the various powers of the unknown in an equation was
eventually achieved, but there were two other needs to be met before algebra could
become a mathematical subject on a par with geometry: a unified way of writing
equations and a concept of number in which every equation would have a solution.
The use of exponential notation and grouping according to powers was discussed
by Simon Stevin (see Section 7 of Chapter 6). Stevin used the abbreviation Ì for
the first unknown in a problem, sec for the second, and ter for the third. Thus (see
Zeuthen, 1903, p. 95), what we would write as the equation6x^3 2 3x^2
— 4- 2xz^2 = —?
y yz^2
was expressed as follows: If we divide6 Ì © D sec © by 2 Ì © ter © ,