Pasts and Presences
Geometrical Constructions in Antiquity
and the Middle Ages

3 November 1997

In the Sixth Century B.C. travel was brisk in the Eastern Mediterranean. People got around as tourists and for business. Two Ionian Greems who travelled to Egypt are worth noting.

Thales (640-550 B.C.) was a merchant noted for his shrewdness. Accord ing to Aristotle he achieved a monopoly on olive presses in his native region and was able thereby to control the production of olive oil. He went to Egypt for business and in his leisure studied astronomy and geometry. When he returned years later to his home in Miletus he devoted himself to science.

Pythagoras (ca. 569-500 B.C.) went as a young person to study in Egypt and remained there several years. He too returned to his native land, the island of Samos in his case, but at age 40 emigrated to Sicily with his mother and a student. In Croton, a Dorian colony in the south of Italy, he established schools that were vastly successful. Women attended his lectures and he married on of his most gifted students.

Thales and Pythagoras established the geometry that was so esteemed by Plato (ca. 425-347 B.C.) and that was compiled by Euclid (about whose life almost nothing is known) in the form that has been studied in schools ever since. When Alexandria fell to the followers of Mohammed in about 641 A.D., most of the many thousands of books that encompassed a millenium of Greek science were destroyed, but copies of many works, including Euclid's geometry, survived at Constantinople. In the Ninth Century A.D. arabic translations of Euclid and other books were made at Constantinople under commission from various caliphs. Some of these made their way into Moslem Spain and were there discovered by Europeans, including Gerbert (953-1003), who later became Pope Sylvester II.

In spite of the stimulus that Gerbert provied, geometry received scant attention from the professors in the cathedral school and universities that were established in the middle ages. Their interests lay in rhetoric, philosophy, and theology, very little in mathematics or natural science.

Constantinople fell to the Turks in 1453, and the last semblance of a Greek school of mathematics then disappeared. Numerous Greeks took refuge in Italy. In the West the memory of Greek science had all but vanished and only a few names were remembered; the books brought by the refugees gave a great impetus to the study of science and the emergence of the renaissance.

The word geometry means literally "earth measure," and yet in the Greek geometric tradition rulers with marks on them were not allowed. The instruments used were an unmarked straightedge and a collapsing compass, i.e., a stretched string with which circles were drawn. The geometric elements were points, lines, and circles; geometric constructions proceeded by intersecting lines and/or circles in order to produce new points. The beginning of every construction was a line on which two points were placed, whose separation served to define a unit distance.

The spirit of this undertaking is familiar to us from school geometry. For example the most elementary contruction is an equilateral triangle; a square is more demanding but still quite elementary.

At the other end of the scale is the construction of a regular pentagon, possibly known or discovered by Pythagoras. It derives from the "golden rectangle," which has the remarkable property that, when the square is removed, the rectangle that remains has the same proportions as the original.

The pentagon is important too because it is the face of a regular dodecahedron, one of the five so-called "Platonic Solids," known to the Pythagoreans long before Plato was born.

Most fundamental in geometry is the theorem that bears Pythagoras's name: that a square contructed on the hypotenuse of a right triangle has an area which is the sum of the areas of triangles constructed on the other two sides. Special cases of this theorem were known in still more ancient times, in Egypt and Babylonia; the fact that line segments of lengths 3, 4, 5 will form a right triangle is an important principle in surveying and the building trades. But Pythagoras's theorem is much more farreaching and characterizes ordinary geometry in a universe of possible alternatives, e.g., spherical geometry where (on the surface of a sphere), Pythagoras's theorem fails. (There exists on the sphere a triangle which is at one at the same time a right triangle and an equilateraI triangle.)

Thales developed much of the geometry the modern world has inherited, but Pythagoras had ambitions which went far beyond school mathematics. He and his collaborators wanted to give all knowledge a geometric foundation. The greatest success came with the theory of proportions: any integral ratio could be computed by geometric construction. Moveover, any two such ratios could be added, subtracted, multiplied or divided geometrically. Even square roots were obtainable.

What is more, musical harmony fit neatly into the geometric framework, for simple ratios gave the amounts by which a string on an instument would be shortened or lengthened in order to produce an interval, e.g., the ratios 1:2 and 2:1 corresponded to an octave, 3:4 and 4:3 to fifth, and 3:5 and 5:3 to a fourth.

The Pythagoreans believed two that the planets were arranged harmonically and that the key to understanding the universe lay in properties of numbers. They held a strong belief that pure thought was sufficient for understanding its properties; modern success in science has been the result of similar inclinations, but moderated and strengthened by observation. Johannes Kepler, for example, was a theoretician very much in the Pythagorean tradition, but whose later work was based on the astronomical observations of the Tycho Brahe. And, although Pythagoras and Kepler were mistaken regarding special orbits for planets, the same dominates the motion of particles in atoms and nuclei.

Along the same lines, a century after Pythagoras, Leucippus and Democritus put forward the a theory of atomism which also attempted to describe the world in integral units. One sees in the general tendencies not only an enormously precocious preview not only of modern physical science, but also of computer science, where again success is achieved by combinations of integers, but on a very large scale.

Another point is that science has progressed largely from the construction of what Copernicus called "theories more pleasing to the mind," whose worth subsequently was judged by patient experimentation.

We know much about Thales achievements in mathematics and philosophy, but regarding the Pythagorean school we know also a good bit about the disappointments and failures.

The greatest disappointment was that even closely related geometric quantities turned out to be incommensurable, that is, not related to each other by proportions of whose quantities. In particular, the diagonal of a square is incommensurable with the side of the square. In modern terms this means that N/7 is not the quotient of integers. It is thought that the Pythagoreans were so shocked and disappointed by this discovery that they kept it secret and later expelled one of their members who divulged it to the larger world. But the very discovery and especially its method of proof were in themselves great achievements which are surprising in their modernity.

Two failures are also worth noting: (1) their inability to construct a regular heptagon (seven sided), which was proved by Gauss at the end of the Eighteenth Century to be impossible under the classical rules for construction. Archimedes found a nonstandard construction in the Third Century b.c. (2) their inability to construct a square which had the same area as a given circle. Archimedes gave approximate results (a series of approximations of arbitrarily high precision, but still not exact) and Lindeman in the second half of the Nineteenth Century proved that an exact solution was in fact impossible.

In relation to this second failure, a curious partial sucess which gave false hope was made by Hippocrates of Chios (different from his contemporary, the physcian, Hippocrates of Cos) who discovered that a lune-shaped portion of a circle has an area equal to a square on its radius. Herodotus stated that geometry arose because in Egypt the spring floods each year eradicated boundary markers along the Nile's banks, so that property ownership need to be reestablished annually by meticulous reconstructions of parcel boundaries.

This may be correct respecting the primitive origins of geometry, but it was also certainly brought to a higher level in Egypt by the contruction of monuments which required three-dimensional concepts and an extraordinary degreee of precision and planning. It is quite plausible that the brotherhood of masons in Egypt were as conversant with geometry as were Pharoah's land agents. It is also quite likely that this knowlege was imported to Greece in the Sixth century and helped make possible the high degree of precision and sophistication in Fifth Century architecture.

Roman construction also is highly geometrical, at the very least because, like all good architecture through the ages, it is based on an initial unit corresponding to two points being given on a line as the starting place in geometry. In the Pantheon and many other Roman buildings a fundamental element is a spherical vault.

Toward the end the the Roman period the Hagia Sophia in Constantinople shows extremely complex geometrical forms, all however based on elements of the sphere, the cylinder, the circle and the square. Moreover, in the case of the Hagia Sophia we in fact know that the master builders were highly accomplished mathematically and had the accumulated mathematical knowledge of Greece at their disposal.

Regarding the next great movement in architecture, the medieval master masons certainly were aware of precedents in Greece, Egypt, and Byzantium, as well as the the Roman style architecture which was abundant around them. The freemasonry movement, which is thought to have begun in the early middle ages, claims oriental origins in masonry and geometry. It is quite likely, moreover, that there was travel to Byzantium and even to Egypt. It is absolutely clear, from the churches and other buildings constructed in the late middle ages, that a high degree of familiarity with geometry was achieved and put into practice. The geometric elements and the modularity is evident. What is not so evident, from the scant records that remain from the building of the great churches, is the manner in which the geometric principles were put into practice.