How Geometry Entered Greek Education: A Literature Review

N. Froese summarizes the rise of demonstrative mathematics and especially geometry as a cultural asset in his paper Pythagoras & Co - Die Erfindung der beweisenden Mathematik with these words:

"Die Entwicklung der beweisenden Mathematik und insbesondere der beweisenden Geometrie hat die griechische Kultur bereits im 5. Jahrhundert v.Chr. deutlich geprägt. Im 4. Jahrhundert v.Chr. steigt die Geometrie zur gepriesenen Schule des Verstandes auf. Die Auseinandersetzung mit den Methoden, Problemen und Resultaten der beweisenden Geometrie wird ein zunehmend selbstverständlicher Teil der gehobenen Bildung. Eine Einführung in die Mathematik (insbesondere in die Geometrie) gehörte im Athen des 4. Jh. v.Chr. zum Lehrstoff für 'Gymnasiasten'." (Seite 44)

[Translation: "The development of demonstrative mathematics and especially demonstrative geometry shaped Greek culture as early as the 5th century BCE. In the 4th century BCE, geometry rose to become the praised school of understanding. Engagement with the methods, problems and results of demonstrative geometry became an increasingly natural part of higher education. An introduction to mathematics (especially geometry) belonged to the curriculum for 'gymnasium students' in 4th century BCE Athens."]

What do we know about the details of this rise of geometry into the educational curriculum?

Educational Structures Before Geometry's Prominence

Before geometry achieved its cultural prominence, Greek education followed a relatively simple pattern. For those families who could afford it, boys received instruction in three main areas: letters (reading and writing), music (including poetry and literature), and gymnastics (physical training). This basic education, sometimes called the arkhaia paideia or "ancient education," had a long tradition in Greek cities, particularly Athens.

The teaching of letters fell to the grammatistes, the music teacher was the kitharistes, and physical education was supervised by the paidotribes. These teachers worked independently, and families would hire them as needed and as finances allowed. For the wealthy, education might also include instruction from sophists, the traveling teachers who commanded substantial fees for their services.

Mathematics in this early period was primarily practical arithmetic, taught to merchants, artisans, and others who needed calculation skills for their trades. This form of mathematical knowledge was distinctly separated from what later came to be called the "science of numbers" - the theoretical and demonstrative mathematics that developed among the Pythagoreans and other philosophical communities.

The Gymnasium: From Athletic Ground to Intellectual Center

The gymnasium began as a training facility for athletic competition. The name itself derives from gymnos (naked), reflecting the Greek practice of exercising without clothing. By the 6th century BCE, gymnasia had become established institutions in Greek cities, typically located in suburban areas where space was available and access to water was good.

The transformation of the gymnasium from purely athletic facility to intellectual center occurred gradually during the 5th and 4th centuries BCE. Several factors contributed to this development. First, gymnasia provided open spaces where people naturally gathered and conversed. The physical layout - typically including a courtyard surrounded by colonnades - created environments conducive to discussion. Second, the gymnasia attracted young men of the social class most likely to pursue higher education. These were the sons of wealthy families who had leisure for both athletic training and intellectual pursuits. Third, the gymnasia were public institutions, unlike the private homes where traditional elementary education occurred. This public character made them natural venues for teachers seeking students.

The philosopher Socrates, as portrayed in Plato's dialogues, frequented Athenian gymnasia and engaged young men in philosophical discussion there. This was not unusual behavior but reflected the gymnasia's role as gathering places for intellectual exchange. By the late 5th century BCE, some gymnasia had developed associations with particular philosophical schools or teachers. The Academy, founded by Plato around 387 BCE, was located at the site of a gymnasium outside Athens. Similarly, Aristotle's Lyceum took its name from a gymnasium dedicated to Apollo Lyceus.

The Role of Sophists in Spreading Mathematical Education

The sophists - professional teachers who emerged in the 5th century BCE - played a significant role in making mathematical education more widely available. These traveling intellectuals offered instruction in rhetoric, philosophy, and various specialized subjects, including mathematics and geometry. Unlike the Pythagorean communities, which maintained secrecy about their mathematical discoveries, sophists taught openly to anyone who could pay their fees.

Hippias of Elis exemplifies the sophist mathematician. Active in the late 5th century BCE, Hippias claimed expertise in astronomy, geometry, arithmetic, and harmonics, along with more traditional sophistic subjects. He reportedly invented the curve known as the quadratrix, which could be used to solve the problem of squaring the circle. Hippias represents a type of teacher who made advanced mathematical knowledge accessible beyond exclusive philosophical circles.

The sophists' contribution to mathematical education was not primarily in advancing mathematical knowledge itself but in creating a market for mathematical instruction. They demonstrated that mathematical expertise could be taught, learned, and valued by a broader segment of society than had previously engaged with it. Unlike the esoteric Pythagorean community or the exclusive philosophical schools, sophists openly advertised their services and taught anyone who could pay. They offered more flexible arrangements than traditional institutions - students might attend a series of lectures without committing to years of study. They also connected mathematical knowledge to practical concerns of civic life, showing how geometric thinking could apply to problems of measurement, construction, and rational argumentation.

Not everyone approved of the sophists' commercialization of education. Socrates and Plato criticized them for teaching skills without concern for truth or virtue. Nevertheless, the sophists' role in spreading mathematical education beyond small circles of initiates was significant.

Geometry's Rising Cultural Prestige

During the 5th century BCE, geometry achieved a cultural prominence unprecedented for any form of mathematical knowledge. Several factors contributed to this development. The discovery of incommensurable magnitudes - the recognition that some geometric ratios cannot be expressed as ratios of whole numbers - had profound intellectual impact. This discovery, traditionally attributed to the Pythagorean Hippasus of Metapontum, demonstrated that geometry could reveal truths inaccessible to ordinary arithmetic. The diagonal and side of a square, for instance, have no common measure.

This discovery actually enhanced geometry's prestige rather than diminishing it. Geometry proved capable of handling relations that arithmetic could not express. The Greek mathematical community recognized that geometric methods provided access to a broader realm of mathematical relationships than number theory alone.

The three classical problems of Greek geometry - squaring the circle, doubling the cube, and trisecting an angle - also contributed to geometry's prominence. These problems attracted attention from leading intellectuals throughout the 5th and 4th centuries BCE. The problems were easy to state but resistant to solution, and attempts to solve them drove significant mathematical innovation.

Geometry also acquired prestige through its association with philosophy. The geometric proof exemplified the kind of certain, demonstrable knowledge that philosophers sought. When Socrates wanted to illustrate the theory of recollection in Plato's Meno, he chose a geometric problem, leading an uneducated slave boy through a demonstration about doubling the area of a square. This famous scene illustrates how geometry had become the paradigmatic example of knowledge that the mind could discover through pure thought.

Geometry in the Gymnasium Curriculum

By the 4th century BCE, geometry had become part of the standard curriculum for young men in Athenian gymnasia. The evidence, though fragmentary, allows us to sketch the general outlines of what this education involved.

The geometric instruction offered in gymnasia differed significantly from the advanced study pursued in philosophical schools. Gymnasium students learned basic geometric concepts: the properties of triangles, the construction of perpendiculars and parallels, the properties of circles, and the relationships between geometric figures. They learned to work with ruler and compass, the tools that Oenopides of Chios (5th century BCE) had established as the proper instruments for geometric construction.

The goal was not to train research mathematicians but to provide mental discipline and to equip young citizens with useful knowledge. Geometric training taught careful reasoning, attention to precise definitions, and the ability to follow logical arguments. These skills transferred to other domains, particularly rhetoric and philosophy. A young man who could follow a geometric proof had developed habits of thought valuable in analyzing political arguments or philosophical positions.

The teaching approach emphasized learning through problem-solving rather than passive listening. Students worked through constructions and proofs themselves, guided by teachers. This active engagement with geometric problems provided both intellectual training and a kind of mental gymnastics complementing physical exercise.

Not all families could afford this level of education, and not all young men pursued it even when families could pay. Geometry remained part of the education of the leisured class. Nevertheless, the incorporation of geometric instruction into the gymnasium curriculum marked a significant development. Mathematical knowledge had moved from the exclusive preserve of small philosophical communities to become part of the expected education of Athens' elite young men.

Plato and the Systematization of Mathematical Education

Plato's Academy represented a more systematic approach to mathematical education than the gymnasia provided. While the Academy was not exclusively or even primarily a school of mathematics, mathematical study formed an essential component of its curriculum. Plato believed that mathematical training provided necessary preparation for philosophical study.

In the Republic, Plato outlined an ideal educational program that included extensive mathematical instruction. Students would study arithmetic, plane geometry, solid geometry, astronomy (understood as the geometry of moving bodies), and harmonics (the mathematical theory of music). This curriculum reflected Plato's conviction that mathematical study trained the mind to apprehend abstract truths and prepared students for philosophical understanding.

The Academy attracted talented mathematicians. Theaetetus, who made significant contributions to the theory of irrational magnitudes and to solid geometry, was associated with Plato's school. Eudoxus of Cnidus, whose theory of proportion provided the foundation for much of Euclid's Elements, spent time at the Academy. These mathematicians pursued research while also teaching, creating an environment where advanced mathematical work and mathematical education reinforced each other.

The famous inscription reportedly displayed at the Academy's entrance - "Let no one ignorant of geometry enter" - may be legendary rather than historical, but it captures the school's emphasis on mathematical preparation. Whether or not the inscription actually existed, the Academy clearly required geometric knowledge of its students and cultivated mathematical study among its members.

Evidence and Its Limitations

Our knowledge of geometry's entry into Greek education relies on fragmentary and sometimes problematic evidence. No complete educational textbook from the 5th or 4th century BCE survives. Euclid's Elements (composed around 300 BCE) survives complete. We know of earlier textbooks, including works by Hippocrates of Chios and Leon, only through references in later sources. For the 5th and 4th centuries BCE, we must piece together evidence from scattered references in philosophical dialogues, later historical accounts, and occasional mentions in other literary works.

Plato's dialogues provide valuable evidence but must be used cautiously. While they show geometry being taught and discussed in 4th century Athens, the dialogues are literary works, not documentary records. Plato sometimes uses mathematical examples to make philosophical points, and we cannot always be certain whether a particular teaching method or curriculum he describes reflects actual practice or represents his own educational ideals.

Later sources, including commentaries by Proclus and others, preserve information about earlier mathematicians and their teaching, but these accounts were written centuries after the events they describe. They sometimes reflect the interests and assumptions of their own times rather than accurate historical understanding.

Archaeological evidence helps but remains limited. We have identified ruins of ancient gymnasia, which confirm their large size and complex architecture. However, physical structures cannot tell us what was taught there or how teaching was conducted.

Given these limitations, some questions remain open to scholarly debate. The exact content of gymnasium-level mathematical instruction remains somewhat unclear. The extent to which geometry was taught to young men of middle-class rather than aristocratic background is uncertain. The relationship between practical mathematical training for specific occupations and the more theoretical geometric education of the gymnasium requires further clarification.

Further Reading:

• Beck, Frederick A. G. Greek Education, 450-350 B.C. London: Methuen, 1964.
• Heath, Thomas L. A History of Greek Mathematics. 2 vols. Oxford: Clarendon Press, 1921.
• Jaeger, Werner. Paideia: The Ideals of Greek Culture. Translated by Gilbert Highet. 3 vols. New York: Oxford University Press, 1939-1944.
• Knorr, Wilbur R. The Ancient Tradition of Geometric Problems. Boston: Birkhäuser, 1986.
• Marrou, Henri-Irénée. A History of Education in Antiquity. Translated by George Lamb. New York: Sheed and Ward, 1956.
• Netz, Reviel. The Shaping of Deduction in Greek Mathematics: A Study in Cognitive History. Cambridge: Cambridge University Press, 1999.
• Netz, Reviel, and William Noel. The Archimedes Codex. Philadelphia: Da Capo Press, 2007.
• Too, Yun Lee, ed. Education in Greek and Roman Antiquity. Leiden: Brill, 2001.