Geometry

Geometry (Greek Γεωμετρια, geo = earth, metria = measure) arose as the field of knowledge dealing with spatial relationships. It was one of the two fields of pre-modern mathematics, the other being the study of numbers. In modern times, geometric concepts have been generalized to a high level of abstraction and complexity, and have been subjected to the methods of calculus and abstract algebra, so that many modern branches of the field are barely recognizable as the descendants of early geometry. (See areas of mathematics and algebraic geometry.)

The Greek period (c. 600 B.C. ? 600 A.D.)

The Greek Period must be considered in detail, since geometry, for most of its history, was what the Greeks made it. For the Ancient Greeks, geometry was the crown jewel of their sciences, reaching a completeness and perfection of methodology that no other branch of their knowledge had attained. They expanded the range of geometry to many new kinds of figures, curves, surfaces, and solids; they changed its methodology from trial-and-error to logical deduction; they recognized that geometry studies ?eternal forms?, or abstractions, of which physical objects are only approximations; and they developed the idea of an ?axiomatic theory?, which, for more than 2000 years, was regarded to be the ideal paradigm for all scientific theories.

Related Topics:
?eternal forms? - ?axiomatic theory?

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Thales and Pythagoras

Thales (635-543 B.C.) of Ionia (now southwestern Turkey), was the first to whom deduction in mathematics is attributed. There are five geometric propositions for which he wrote deductive proofs, though his proofs have not survived. Pythagoras (582-496 B.C.) of Ionia, and later, Italy, then colonized by Greeks, may have been a student of Thales, and probably traveled to Babylon and Egypt. The theorem that bears his name was not his discovery, but he was the first to give a deductive proof of it. He gathered a group of students around him to study mathematics, music, and philosophy, and together they discovered most of what high school students learn today in their geometry courses. In addition, they made the profound discovery of incommensurable lengths and irrational numbers.

Related Topics:
Thales - Ionia - Pythagoras - Incommensurable lengths - Irrational numbers

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Plato

Plato (427-347 B.C.), the philosopher most esteemed by the Greeks, had inscribed above the entrance to his famous school, ?Let none enter here who are ignorant of geometry.? Though he was not a mathematician himself, his views on mathematics had great influence. Mathematicians thus accepted his belief that geometry should use no tools but a compass and straight edge ? never measuring instruments such as a marked ruler or a protractor, because these were a workman?s tools, not worthy of a scholar. This dictum led to a deep study of the possible ruler and compass constructions, and three classic ruler-and-compass problems: how to use these tools to trisect an angle, to construct a cube twice the volume of a given cube, and to construct a square equal in area to a given circle. The proofs of the impossibility of these constructions, finally achieved in the 19th century, led to important principles regarding the deep structure of the real number system. Aristotle (384-322 B.C.), Plato?s greatest pupil, wrote a treatise on methods of reasoning used in deductive proofs (see Logic) which was not substantially improved upon until the 19th century.

Related Topics:
Plato - Ruler and compass constructions - Aristotle - Logic

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Euclid

Euclid (365?-275? B.C.), probably a student of one of Plato?s students, wrote a treatise in 13 books (chapters), titled The Elements of Geometry, in which he presented geometry in the ideal axiomatic form. The treatise is not a compendium of all that the Greeks knew at the time about geometry; Euclid himself wrote eight more advanced books on geometry. We know from other references that Euclid?s was not the first elementary geometry textbook, but it was so much superior that the others fell into disuse and were lost. He was brought to the university at Alexandria by Ptolemy I, King of Egypt.

Related Topics:
Euclid - Ptolemy I

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

The Elements began with definitions of terms, fundamental geometric principles (called axioms or postulates), and general quantitative principles (called common notions) from which all the rest of geometry could be logically deduced. Following are his five axioms, somewhat paraphrased to make the English easier to read.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

• Any two points can be joined by a straight line.
• Any finite straight line can be extended in a straight line.
• A circle can be drawn with any center and any radius.
• All right angles are equal to each other.
• If two straight lines in a plane are crossed by another straight line (called the transversal), and the interior angles between the two lines and the transversal lying on one side of the transversal add up to less than two right angles, then on that side of the transversal, the two lines extended will intersect (also called the parallel postulate).

It was soon observed, and no doubt Euclid himself knew, that his fifth axiom could be replaced by the shorter statement ?Given a line and a point not on the line, there is only one line through the given point and in the same plane with the given line that does not intersect the given line.? This is called Playfair?s Axiom, after the British teacher who proposed to make the replacement in all the school textbooks.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

The axioms, according to Plato, should be simple and self-evident principles, so clearly true that they need no proof. Euclid?s first four axioms meet this criterion, but the fifth, even if replaced by Playfair?s Axiom, is not simple, and most would say not self-evident like the first four. The fifth resembled more the theorems that Euclid proved from the axioms. Furthermore, Euclid developed a substantial part of his theory of triangles without using the Fifth Axiom. The speculation arose, probably during Euclid?s lifetime, that the Fifth Axiom can and should be proved as a theorem from the first four, and thus is unnecessary as an axiom. Thus began many centuries of attempts to prove the Fifth Axiom, and the question was not settled until the 19th century.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Archimedes

Archimedes (287-212 B.C.), of Syracuse, Sicily, when it was a Greek city-state, was the greatest of the Greek mathematicians, and often named as one of the three greatest of all time (along with Isaac Newton and Carl Friedrich Gauss). Had he not been a mathematician, he would still be remembered as a great physicist, engineer, and inventor. In his mathematics, he developed methods very similar to the coordinate systems of analytic geometry, and the limiting process of integral calculus. The only element lacking for the creation of these fields was an efficient algebraic notation in which to express his concepts.

Related Topics:
Archimedes - Syracuse - Sicily - Greek city-state - Isaac Newton - Carl Friedrich Gauss

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

After Archimedes

After Archimedes, Greek mathematics began to decline. There were a few minor stars yet to come, but the golden age of geometry was over. Proclus (410-485), author of Commentary on the First Book of Euclid, was one of the last important players in Greek geometry. He was a competent geometer, but more importantly, he was a superb commentator on the works that preceded him. Much of that work did not survive to modern times, and is known to us only through his commentary. The Roman Republic and Empire that succeeded and absorbed the Greek city-states produced excellent engineers, but no mathematicians of note.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~