Ancient people used techniques of measurement to build ships, houses and temples and developed methods of calculating financial transactions. For most of the civilizations of the ancient world, these mathematical conventions were merely tools used to help people accomplish short-term tasks or solve practical engineering problems. But some clever individuals comprehended a more powerful use for these mathematical tools. The Babylonians applied the techniques of measurement on Earth to measure the heavens. The Greeks embraced a daring concept, that even the deepest mysteries of nature could be understood by human beings through the study of mathematics and geometry.
It is hard to separate the general development of mathematics from that of geometry, which the Greeks called "earth measuring." Formulas for calculating areas and volumes of geometrical shapes were known to the Babylonians, but the original concept of "proving" a geometrical theorem is usually attributed to the Greeks around the period of 600 b.c. Thales is given credit for introducing one of the first geometrical proofs, that the angle in a semicircle is a right angle. Pythagoras of Samos, who is best known for the theorem about the sides of a right triangle that bears his name, raised the study of numbers and geometry to a high art.
Mathematicians in ancient Greece were preoccupied with understanding the mysterious quantity representing the ratio of the diameter of a circle with its circumference, denoted by the Greek letter pi. Pythagoreans viewed the existence of the ratio pi as nothing less than the signature of God upon creation. To fully explore the relationships that existed among geometrical objects, relationships that the ancient Greeks realized intuitively, a systematic and logical treatment of the whole of mathematical knowledge was needed. This service was provided by Euclid, who compiled a book called The Elements around 300 b.c. Euclid's book was a brilliant synthesis of the results of centuries of mathematical work. In The Elements, Euclid systematized the entire field of plane geometry in clearly stated, rigorously logical proofs. The concise style of The Elements became the world standard for scientific discourse over the next two thousand years. Euclid's student, Apollonius of Perga, extended this work by analyzing the properties of cones, which, when cut into sections, produced examples of the most important shapes in Euclidean geometry: the circle, the ellipse, the parabola and the hyperbola.
The Greek writings were to remain the only important European innovations in mathematics and geometry for 1,500 years. During the period between the collapse of the Roman Empire and the end of the Middle Ages, meaningful mathematical progress occurred primarily in India and the Moslem empires of the Middle East. Moslem scholars translated everything they could recover from the ancient Greeks and developed a highly practical method of trigonometry. European interest in geometry reawakened in the thirteenth century after Leonardo of Pisa (known as Leonardo Fibonacci) published an important work, the Liber abaci, on Hindu-Moslem number systems. The Eastern methods of computation using abstract numerals and symbols were far superior to backward European methods, which relied on the use of counting boards and the abacus. Fibonacci also published works on Euclidean geometry which rekindled a practical interest in mathematics throughout Europe.
The final synthesis between geometry and mathematics was achieved in the works of the French philosopher and mathematician René Descartes, who in 1637 invented the discipline of analytic geometry. This discovery by Descartes was actually preceded by the work of Pierre de Fermat who, unfortunately for the history of science, never published his work. By combining the geometric concepts of lines, curves, and points with the techniques of algebra, Descartes made it possible to transfer and redefine complex events in nature into the form of much simpler, and more easily analyzable, graphic representations. After this innovation appeared, progress in mathematics and geometry advanced at a rapid pace.
Analytic geometry enabled scientists to imbue mathematical expressions with a form and shape that could be manipulated. Centuries of confusion over the meaning and nature of so-called "imaginary" numbers was clarified by representing them as points in a geometrical system, a method pioneered by Jean Robert Argand and Carl Friedrich Gauss, among others. Girard Desargues and Blaise Pascal invented synthetic projective geometry in the seventeenth century which included the architectural concept of perspective as a special case of the theory. In the late 1700s, Gaspard Monge introduced the technique of descriptive geometry to solve construction details that normally required tedious calculations. It is to Monge that architects owe their use of two projections, the "plan" and the "elevation," to represent spatial relationships. Without this innovation, engineering sciences would have developed much more slowly than they did.
During the nineteenth century, doubts about the logical validity of geometry, indeed the whole of mathematics, began to preoccupy the minds of some scientists. In the early 1800s, Jean Victor Poncelet was at the heart of a debate over the reality of space as described by geometry. The abstract concepts of "point" and " line," which were always understood as unprovable entities, began to prove problematic for certain branches of mathematics. Georg Riemann and others got around some of these obstacles by creating geometries that were not based on Euclidean axioms. The non-Euclidean geometries that were invented ultimately played an important role in the twentieth-century development of the theory of relativity by Albert Einstein and others, but in the nineteenth century, no one knew if they corresponded to anything "real" or not.
Trying to get at the heart of the matter, David Hilbert expended a huge effort to develop a formal logical basis for geometry, similar to the approach of Euclid. In this effort, Hilbert was joined by Alfred North Whitehead and Bertrand Russell. The disaster for this approach came in 1931, with Kurt Gödel 's monumental proof that a collection of axioms could never lead to a complete and self-consistent mathematical system.
Modern mathematicians no longer have the same blind faith that Euclid had in the "truth" of their logical methods. But rather than impeding further progress, this fact has only encouraged today 's scientists to work harder to understand the very foundations of nature and reality. This philosophical spirit is as much at the heart of today's mathematics and geometry as it was in the ancient world of Pythagoras.
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