- The Principle which I have always observed in my studies and which I believe has helped me the most to gain what knowledge I have, has been never to spend beyond a few hours daily in thoughts which occupy the imagination, and a few hours yearly in those which occupy the understanding, and to give all the rest of my time to the relaxation of the senses and the repose of the mind…As for me, I have never presumed my mind to be in any way better than the minds of people in general. As for reason or good sense, I am inclined to believe that it exists whole and complete in each of us, because it is the only thing that makes us men and distinguishes us from the lower animals. (Descartes)
- Philosophy is written in the great book (by which I mean the Universe) which stands always open to our view, but it cannot be understood unless one first learns how to comprehend the language and interpret the symbols in which it is written, and its symbols are triangles, circles, and other geometric figures, without which it is not humanly possible to comprehend even one word of it; without these one wanders in a dark labyrinth. (Galileo)
- If there is any thinking to be done in this Forest – and when I say thinking, I mean thinking – you and I must do it.(Milne, The House at Pooh Corner)
The Cartesian or Euclidean plane R2
We give a brief introduction to analytic geometry of the two-dimensional Cartesian or Euclidean plane. Our common school experience has given us an intuitive geometric idea of the Cartesian/Euclidean plane as an infinite flat surface without borders consisting of points, and we also have an intuitive geometric idea of geometric objects like points, straight lines, triangles and circles in the plane.
We are familiar with the idea of using a coordinate system in the Cartesian/Euclidean plane consisting of two perpendicular straight lines with each point in the plane identified by two coordinates where and are rational numbers, and is the set of all ordered pairs of rational numbers. We mostly use the name Euclidean below, remembering that it was Descartes who reinvented the Euclidean plane and gave it a new meaning beyond Euclidean geometry.
With only the rational numbers at our disposal, we quickly run into trouble because we cannot the distance between points in may not be a rational number. For example, the distance betweenthe points and , the length of the diagonal of a unit square, is equal to , which is not a rational number.
The troubles are resolved by using real numbers, that is by extending to where is the set of real numbers. We recall that rational numbers have finite or periodic decimal expansions while non-rational or irrational real numbers have infinite decimal expansion which never repeat periodically.
In this chapter, we present basic aspects of analytic geometry in the Euclidean plane using a coordinate system identified with , following the fundamental idea of Descartes to describe geometry in terms of numbers.
Below, we extend to analytic geometry in three-dimensional Euclidean space identified with and we finally generalize to analytic geometry in , where the dimension $n$ can be any natural number.
Considering with leads to linear algebra with a wealth of applications outside Euclidean geometry, which we will meet below.
The concepts and tools we develop in this chapter focussed on Euclidean geometry in will be of fundamental use in the generalizations to geometry in and and linear algebra.
The tools of the geometry of Euclid is the ruler and the compasses, while the tool of analytic geometry is a calculator for computing with numbers. Thus we may say that Euclid represents a form of analog technique, while analytic geometry is a digital technique based on numbers. Today, the use of digital techniques is exploding in communication and music and all sorts of virtual reality.
Descartes, Inventor of Analytic Geometry
The foundation of modern science was laid by Rene Descartes (1596-1650) in Discours de la method pour bien conduire sa raison et chercher la verite dans les sciences from 1637. The Methodcontained as an appendix La Geometrie with the first treatment of Analytic Geometry.
Descartes believed that only mathematics may be certain, so all must be based on mathematics, the foundation of the Cartesian view of the World.
In 1649 Queen Christina of Sweden persuaded Descartes to go to Stockholm to teach her mathematics. However the Queen wanted to draw tangents at 5 a.m. and Descartes broke the habit of his lifetime of getting up at 11 o’clock. After only a few months in the cold Northern climate, walking to the palace at 5 o’clockevery morning, he died of pneumonia.
Descartes: Dualism of Body and Soul
Descartes set the standard for studies of Body and Soul for a long time with his De homine completed in 1633, where Descartes proposed a mechanism for automatic reaction in response to external events through nerve fibrils:
In Descartes’ conception, the rational Soul, an entity distinct from the Body and making contact with the body at the pineal gland, might or might not become aware of the differential outflow of animal spirits brought about though the nerve fibrils. When such awareness did occur, the result was conscious sensation — Body affecting Soul. In turn, in voluntary action, the Soul might itself initiate a differential outflow of animal spirits. Soul, in other words, could also affect Body.
In 1649 Descartes completed Les passions de lame, with an account of causal Soul/Bodyinteraction and the conjecture of the localization of the Soul’s contact with the Body to the pineal gland.
Descartes chose the pineal gland because it appeared to him to be the only organ in the brain that was not bilaterally duplicated and because he believed, erroneously, that it was uniquely human; Descartes considered animals as purely physical automata devoid of mental states.