Mathematics

7 DESCARTES.

The New Method

For the human mind has in it a something divine, wherein are scattered the first seeds of useful modes of knowledge. Consequently it often happens that, however neglected and however stifled by distracting studies, they spontaneously bear fruit. Arithmetic and geometry, the simplest of the sciences, are instances of it. We have evidence that the ancient geometers made use of a certain analysis which they applied to the solution of all problems, although, as we find, they invidiously withheld knowledge of this method from posterity. There is now flourishing a certain kind of arithmetic, called algebra, which endeavors to accomplish in regard to numbers what the ancients achieved in respect to geometrical figures. These two sciences are no other than spontaneous fruits originating from the innate principles of the method in question.

For truly there is nothing more futile than to occupy ourselves so much with mere numbers and imaginary figures that it seems that we could be content to rest in the knowledge of such trifles... When, however, I afterwards bethought myself how it could be that the first discoverers of philosophy refused to admit to the study of wisdom anyone not versed in mathematics, as if they viewed mathematics as being the simplest of all disciplines, and as altogether indispensable for training and preparing our human powers for the understanding of other more important sciences, I could not but suspect that they were acquainted with a mathematics very differerent from that which is commonly cultivated in our day. Not that I imagined that they had a complete knowledge of it. Their extravagant exultations, and the sacrifices they offered for the simplest discoveries, show quite clearly how rudimentary their knowledge must have been. I am convinced that certain primary seeds of truth implanted by nature in the human mind... had such vitality in that rude and unsophisticated ancient world, that the mental light by which they discerned virtue to be preferable to pleasure... likewise enabled them to recognize true ideas in philosophy and mathematics, even though they were not yet able to obtain complete mastery of them. Certain vestiges of this true mathematics I seem to find in Pappus and Diophantus, who, though not belonging to that first age, yet lived many centuries before our time.¹ These writers, I am inclined to believe, by a certain baneful craftiness, kept the secrets of this mathematics to themselves... Instead they have chosen to propound... a number of sterile truths, deductively demonstrated with great show of logical subtlety, with a view to winning an amazed admiration, thus dwelling indeed on the results obtained by way of their method, but without disclosing the method itself—a disclosure which would have completely undermined that amazement. Lastly, in the present age there have been certain very able men who have attempted to revive this mathematics. For it seems to be no other than this very science which has been given the barbarous name, algebra—provided, that is to say, that it can be extricated from the tortuous array of numbers and from the complicated geometrical shapes by which it is overwhelmed, and that it be no longer lacking in the transparency and unsurpassable clarity which, in our view, are proper to a rightly ordered mathematics.

What, on more attentive consideration, I at length came to see is that those things only were referred to mathematics in which order or measure is examined, and that with respect to measure it makes no difference whether it be in numbers, shapes, stars, sounds or any other object that such measure is sought, and that there must therefore be some general science which explains all that can be inquired into respecting order and measure, without application to any other special subject matter, and that this is what is called mathematica universalis, no specially devised designation, but one already of long standing, and of current use as covering everything on account of which the other sciences are called parts of mathematics.

Thus, for instance, if we seek the base of a right-angled triangle with the given sides 9 and 12, the arithmetician will say that it is or 15. But we shall substitute a and b for 9 and 12, and shall find the base to be In this way the two parts a and b, which in the number notation were confused, are kept distinct. Also, the realization that terms like "root," "square," "cube," "biquadratic" for proportions which follow by continuous order, are misleading.

For though a magnitude may be entitled a cube or a biquadratic, it should never be presented to the imagination otherwise than as a line or a surface... What above all requires to be noted is that the root, the square, the cube, etc., are merely magnitudes in continued proportion, which always implies the freely chosen unit of which we have spoken above [in Rule XIV].² The first proportional is related to this unit immediately or by one single relation, the second by the mediation of the first and the second, and so by three relations, etc. We therefore entitle the magnitude, which in algebra is called the root, the first proportional; that called the square we shall speak of as being the second proportional, and similarly in the case of the other.

¹ Like Viète’s, Descartes’s starting points are Pappus and Diophantus. With Viète, who speaks of improving on or rescuing this art of analysis (which he dates up to Plato), Descartes believes that this art was well developed in ancient times and kept a semisecret. The dependence of Descartes on Viète is not clear; Descartes claimed not to have seen the logistica speciosa until he himself found his own method.

² This is the place where Descartes’s new algebra is born, the algebra that can be used for coordinate geometry. When 1 is a unit length and x an arbitrary line segment, the proportion

allows us to express x² as a line segment. It is here that Descartes breaks with Viète’s condition of homogeneity. See further Selections III.3, 4.