Descartes' 'La Géométrie'
Descartes' La Géométrie is a technical work written for expert mathematicians published as an appendix to Discours de la méthode (1637). We give below only some of Descartes' more general remarks in this important work. The translation we have used is that of D E Smith and M L Latham published in 1954:
Observing that the sciences of mathematics, however different their objects, all agree in considering only the various relations or proportions subsisting among them, I thought it best to consider these proportions in the most general form possible, without referring them to any objects in particular, and without restricting them, that afterwards I might be the better able to apply them to every class of objects to which they are legitimately applicable. Perceiving that in order to understand these relations I should sometimes have to consider them one by one, and sometimes only to bear them in mind, or embrace them in the aggregate, I thought that, in order the better to consider them individually, I should view them as subsisting between straight lines, than which I could find no objects more simple, or capable of being more distinctly represented to my imagination and senses; and on the other hand that in order to retain them in the memory, or embrace an aggregate of many, I should express them by certain characters, the briefest possible. In this way I believed that I could borrow all that was best both in geometrical analysis and in algebra, and correct all the defects of the one by help of the other. The accurate observance of these few precepts gave me such ease in unravelling questions embraced in these two sciences, that not only did I reach solutions of questions I had formerly deemed exceedingly difficult, but even as regards questions of the solution of which I continued ignorant, I was enabled to determine the means whereby, and the extent to which, a solution was possible. ...
If, then, we wish to solve any problem, we first suppose the solution already effected, and give names to all the lines that seem needful for its construction - to those that are unknown as well as those that are known. Then making no distinction between known and unknown lines, we must unravel the difficulty in any way that shows the relations between these lines, until we find it possible to express a single quantity in two ways. This will constitute an equation, since the terms of one of these two expressions are equal to the terms of the other.
We must find as many such equations as there are supposed to be unknown lines, but if after considering everything involved, so many cannot be found, it is evident that the question is not entirely determined. In such a case we may choose arbitrarily lines of known length for each unknown line to which there corresponds no equation.
If there are several equations, we must use each in order, either considering it alone or comparing it with the others, so as to obtain a value for each of the unknown lines: and so we must combine them until there remains a single unknown line which is equal to some known line, or whose square, cube, fourth power, fifth power, sixth power, etc., is equal to the sum or difference of two or more quantities, one of which is known, while the others consist of mean proportionals between unity and this square, or cube, or fourth power, etc., multiplied by other known lines. I may express this as follows:
z = b,That is, z, which I take for the unknown quantity, is equal to b; or, the square of z is equal to the square of b diminished by a multiplied by z; or, the cube of z is equal to a multiplied by the square of z, plus the square of b multiplied by z, diminished by the cube of c; and similarly for the others.
or z2 = b2 - az
or z3= az2+ b2z - c3, etc
or z2 = b2 - az
or z3= az2+ b2z - c3, etc
Thus, all the unknown quantities can be expressed in terms of a single quantity, whenever the problem can be constructed by means of circles and straight lines, or by conic sections, or even by some other curve of degree not greater than the third or fourth.
But I shall not stop to explain this in more detail, because I should deprive you of the pleasure of mastering it yourself, as well as of the advantage of training your mind by working over it, which is in my opinion the principal benefit to be derived from this science. Because, I find nothing here so difficult that it cannot be worked out by any one at all familiar with ordinary geometry and with algebra, who will consider carefully all that is set forth in this treatise.
I shall therefore content myself with the statement that if the student, in solving these equations, does not fail to make use of division wherever possible, he will surely reach the simplest terms to which the problem can be reduced.
And if it can be solved by ordinary geometry, that is, by the use of straight lines and circles traced on a plane surface, when the last equation shall have been entirely solved there will remain at most only the square of an unknown quantity, equal to the product of its root by some known quantity, increased or diminished by some other quantity also known. Then this root or unknown line can easily be found.