Note 4 — Chapter I

Suppose, for example, that a question gives the following equation between an unknown magnitude x, and two known magnitudes, a and b,

x³ + 3ax = 2b,

as is the case in the problem of the trisection of an angle. We see at once that the dependence between x on the one side, and ab on the other, is completely determined; but, so long as the equation preserves its primitive form, we do not at all perceive in what manner the unknown quantity is derived from the data. This must be discovered, however, before we can think of determining its value. Such is the object of the algebraic part of the solution. When, by a series of transformations which have successively rendered that derivation more and more apparent, we have arrived at presenting the proposed equation under the form

x = ∛(b + √(b² + a³)) + ∛(b - √(b² + a³)),

the work of algebra is finished; and even if we could not perform the arithmetical operations indicated by that formula, we would nevertheless have obtained a knowledge very real, and often very important. The work of arithmetic will now consist in taking that formula for its starting point, and finding the number x when the values of the numbers a and b are given.