Emergence of Analytical Geometry

Geometry, as a deductive science, was created by the Greeks. But despite its brilliance, Greek geometry lacked operability. And this would only be achieved through algebra as the unifying principle. The Greeks, however, were not very good at algebra. Moreover, only in the seventeenth century was algebra reasonably equipped for a creative fusion with geometry.

However, the fact that there are conditions for a discovery does not exclude someone's touch of genius. And in the case of analytical geometry, the result of this fusion, the merit was not of one person. Two Frenchmen, Pierre de Fermat (1601-1665) and René Descartes (1596-1650), interestingly both law graduates, neither professional mathematician, are responsible for this great scientific advance: the first one moved basically by his great love, the mathematics and the second for philosophical reasons. And, by the way, they did not work together: analytical geometry is one of many cases in science of simultaneous and independent discoveries.

If the successful, jealous Pierre de Fermat, a competent counselor with the Toulouse Parliament, devoted many of his best leisure hours to mathematics, it was certainly not because someone in his position lacked other ways to fill the available time. In fact, Fermat simply could not escape his true calling, and despite practicing mathematics as a hobby, none of his contemporaries contributed as much to the advancement of this science as he did. In addition to analytical geometry, Fermat played a key role in creating Differential Calculus, Probability Calculus, and especially number theory, a branch of mathematics that studies the properties of integers.

Fermat's contribution to analytical geometry can be found in a short text entitled Introduction to Flat and Solid Places and dates to a maximum of 1636 which was not published until 1679, posthumously, together with his complete work. Fermat, rather modest, was averse to publishing his works. This results in part from the fact that Descartes is often best remembered as the creator of Analytic Geometry.

Descartes's interest in mathematics arose early at the “College de la Fleche”, a high-school Jesuit-run school where he will start at the age of eight. But for a very special reason that already revealed its philosophical inclinations: the certainty that mathematical demonstrations or justifications provide. At the age of twenty-one, after attending mathematical wheels in Paris (and others) already graduated in law, he voluntarily enters the career of weapons, one of the few "decent" options offered to a young man like him, coming from minor nobility of France. During the nearly nine years he served in various armies, it is not known of any military prowess performed by Descartes. For the battles that fought his thoughts and dreams were fought in the field of science and philosophy.

Descartes' Analytical Geometry appeared in 1637 in the short text called The Geometry as one of the three appendices of the Method's Discourse, considered to be the initial milestone of modern philosophy. In it, in short, Descartes defends the mathematical method as a model for the acquisition of knowledge in all fields.

Analytical geometry, as it is today, bears little resemblance to the contributions left by Fermat and Descartes. Even its most distinctive mark, a pair of orthogonal axes, not used by any of them. But each, in their own way, knew that the central idea was to associate equations with curves and surfaces. In this particular, Fermat was happier. Descartes surpassed Fermat in algebraic notation.

HYGINO H. SUNDAYS* Submitted by user Reinaldo Silva

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