Introduction to Geometry & Geometrics

Geometry is the study of shapes. It takes its name from the Greek belief that geometry began with Egyptian surveyors of two of three millenia ago measuring the Earth, or at least the fertile expanse of it that was annually flooded by the Nile.

It rapidly became more ambitious. Classical Greek geometry, called Euclidean geometry after Euclid, who organized an extensive collection of theorems into a definitive test, was regarded by all in the early modern world as the true geometry of space. Isaac Newton used it to formulate his Principia, the book that first set out the theory of gravity. Until the mid-19th Century, Euclidean geometry was regarded as one of the highest points of rational thought, as a foundation for practical mathematics as well as advanced science, and as a logical system splendidly adapted for the training of the mind. By the 1850s geometry had evolved considerably – indeed, whole new geometrics had been discovered.

The idea of using coordinates in geometry can be traced back to Apollonius’s treatment of conic sections, written a generation after Euclid. But their use in a systematic way with a view to simplifying the treatment of geometry is really due to Fermat and Descartes. Fermat showed how to obtain an equation in two variables to describe a conic or a straight line in 1636, but his work was only published posthumously in 1679. Meanwhile in 1637 Descartes published his book Discourse on Method, with an extensive appendix entitled La Geometrie, in which he showed how to introduce coordinates to solve a wide variety of geometrical problems; this idea become so central a part of mathematics that whole section of La Geometrie read like a modern textbook.

A contemporary of Descartes, Girard Desargues, was interested in the ideas of perspective that had been developed over many centuries by artists (anxious to portray three-dimensional scenes in a realistic way on two-dimensional walls or canvases). How do you draw a picture of a building, or a staircase, which your client can understand and commission, and from which artisans can deduce the correct sizes of each stone? Desargues realized that since any two conic sections can always be obtained as section of the same cone in R3, it is possible to present the theory of conic sections in a unified way, using concepts which later mathematician distilled into the notion of the cross-ratio of four points. Desargues’ discoveries came to be known as projective geometry.

Blaise Pascal was the son of a mathematician, Etinne, who attended a group of scholars frequented by Desargues. He heard of Desargues’s work from his father, and quickly came up with one of the most famous results in the geometry of conics – Pascal’s Theorem. By the late 19th century projective geometry came to be seen as the most basic geometry, with Euclidean geometry as a significant but special case.

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