The Great Rationalists - Descartes (1596 - 1650)

 (Spinoza)

( Descartes) When the Church's authority over thought was finally loosened , many people came to believe that knowledge of the world could be gained by the use of Reason alone. In philosophy this development is known as Rationalism. It was launched by Descartes , then Spinoza, and Leibntz. Descartes and Leibntz wete tbhe most gifted of all Mathematicians and for them  mathematics seemed to to provide the ideal model (Leibntz) for truly reliable knowledge. . They believed that if the methods by which mathematics such as themselves were making new discoveries and acquiring new knowledge could be applied to human attempts to understand the world , the world could be fully expained.
René Descartes ( 31 March 1596 – 11 February 1650) was a French philosopher, mathematician, and scientist. 

The house where he was born inLa Haye en Touraine

Graduation registry for Descartes at the University of Poitiers, 1616.

Descartes was born in La Haye en Touraine (now Descartes),Indre-et-Loire, France, on 31 March 1596. When he was one year old, his mother Jeanne Brochard died after trying to give birth to another child who also died. His father Joachim was a member of the Parlement of Brittany at Rennes. René lived with his grandmother and with his great-uncle. Although the Descartes family was Roman Catholic, the Poitou region was controlled by the Protestant Huguenots. In 1607, late because of his fragile health, he entered the Jesuit Collège Royal Henry-Le-Grand at La Flèche where he was introduced to mathematics and physics, including Galileo's work. After graduation in 1614, he studied two years (1615–16) at the University of Poitiers, earning aBaccalauréat and Licence in Canon and Civil Law, in accordance with his father's wishes that he should become a lawyer. From there he moved to Paris.

In his book Discourse on the Method, Descartes recalls,

I entirely abandoned the study of letters. Resolving to seek no knowledge other than that of which could be found in myself or else in the great book of the world, I spent the rest of my youth traveling, visiting courts and armies, mixing with people of diverse temperaments and ranks, gathering various experiences, testing myself in the situations which fortune offered me, and at all times reflecting upon whatever came my way so as to derive some profit from it.

Given his ambition to become a professional military officer, in 1618, Descartes joined the Dutch States Army in Breda under the command of Maurice of Nassau, and undertook a formal study of military engineering, as established bySimon Stevin. Descartes therefore received much encouragement in Breda to advance his knowledge of mathematics. In this way he became acquainted with Isaac Beeckman, principal of a Dordrecht school, for whom he wrote the Compendium of Music (written 1618, published 1650). Together they worked on free fall, catenary, conic section and fluid statics. Both believed that it was necessary to create a method that thoroughly linked mathematics and physics.

While in the service of the duke Maximilian of Bavaria since 1619, Descartes was present at the Battle of the White Mountain outsidePrague, in November 1620. He visited the labs of Tycho Brahe in Prague and Johannes Kepler in Regensburg.

Mathematical legacy

A Cartesian coordinates graph, using his invented x and y axes.

One of Descartes' most enduring legacies was his development of Cartesian or analytic geometry, which uses algebra to describe geometry. He "invented the convention of representing unknowns in equations by x, y, and z, and knowns by a, b, and c". He also "pioneered the standard notation" that uses superscripts to show the powers or exponents; for example, the 4 used in x⁴ to indicate squaring of squaring.^[69][70] He was first to assign a fundamental place for algebra in our system of knowledge, and believed that algebra was a method to automate or mechanize reasoning, particularly about abstract, unknown quantities. European mathematicians had previously viewed geometry as a more fundamental form of mathematics, serving as the foundation of algebra. Algebraic rules were given geometric proofs by mathematicians such asPacioli, Cardan, Tartaglia and Ferrari. Equations of degree higher than the third were regarded as unreal, because a three-dimensional form, such as a cube, occupied the largest dimension of reality. Descartes professed that the abstract quantity a² could represent length as well as an area. This was in opposition to the teachings of mathematicians, such as Vieta, who argued that it could represent only area. Although Descartes did not pursue the subject, he preceded Leibniz in envisioning a more general science of algebra or "universal mathematics," as a precursor to symbolic logic, that could encompass logical principles and methods symbolically, and mechanize general reasoning
 Effect on Newton Current opinion is that Descartes had the most influence of anyone on the young Newton, and this is arguably one of Descartes' most important contributions. Newton continued Descartes' work on cubic equations, which freed the subject from the fetters of the Greek and Macedonian perspectives. The most important concept was his very modern treatment of independent variables