I have suffered from a (un)healthy[1]portion of imposter syndrome all of my life. This is the personal feeling in an academic context that one is just bluffing and doesn’t actually know anything and then any minute now somebody is going to unmask me and denounce me as an ignorant fraud. I always thought that this was a personal thing, part of my general collection of mental and emotional insecurities but in more recent years I have learned that many academics, including successful and renowned ones, suffer from this particular form of insecurity. On related problem that I have is the belief that anything I do actually know is trivial, generally known to everyone and therefore not worth mentioning[2]. I experienced an example of this recently on Twitter when I came across the following medieval illustration and its accompanying tweet.

Woman teaching geometry to monks. In the Middle Ages, it is unusual to see women represented as teachers, in particular when the students appear to be monks. Euclid’s Elementa, in the translation attributed to Adelard of Bath, 1312.

I would simply have assumed that everybody knew what this picture represents and not commented. It is not a “women teaching geometry to monks” as the tweeter thinks but a typical medieval personification of Geometria, one of the so-called Seven Learned Sisters. The Seven Learned Sisters are the personifications of the seven liberal arts, the trivium (grammar, rhetoric and dialectic)

and the quadrivium (arithmetic, geometry, music and astronomy),

which formed the curriculum in the lower or liberal arts faculty at the medieval university. The seven liberal arts, however, have a history that well predates the founding of the first universities. In what follows I shall only be dealing with the history of the quadrivium.

As a concept this four-fold division of the mathematical sciences can be traced back to the Pythagoreans. The mathematical commentator Proclus (412–485 CE) tells us, in the introduction to his commentary on the first book of Euclid’s Elements:

The Pythagoreans considered all mathematical science to be divided into four parts: one half they marked off as concerned with quantity, the other half with magnitude; and each of these they posited as twofold. A quantity can be considered in regard to its character by itself or in its relation to another quantity, magnitudes as either stationary or in motion. Arithmetic, then, studies quantities as such, music the relations between quantities, geometry magnitude at rest, spherics [astronomy] magnitude inherently moving.

The earliest know written account of this division can be found at the beginning of the late Pythagorean Archytas’ book on harmonics, where he identifies a set of four sciences: astronomy, geometry, logistic (arithmetic) and music. Archytas’ dates of birth and death are not known but he was, roughly speaking, a contemporary of Plato. He was the teacher of Eudoxus (c.390–c.337 BCE) Harmonics, by the way, is the discipline that later became known as music in the quadrivium.

Without mentioning Archytas, Plato (428/427 or 424/423 – 348/347 BCE), who was highly influenced by the Pythagoreans,takes up the theme in his Republic (c.380 BCE). In a dialogue with Glaucon, Plato explains the merits of learning the “five” mathematical sciences; he divides geometry into plane geometry (two dimensional) and solid geometry (three dimensional). He also refers to harmonics and not music.

In the CE period the first important figure is the Neo-Pythagorean, Nicomachus of Gerasa (c.60–c.120 CE), who wrote an Introduction to Arithmeticand a Manual of Harmonics, which are still extant and a lost Introduction to Geometry. The four-fold division of the mathematical sciences only acquired the name quadrivium in the works of Boethius (c.477–524 CE), from whose work the concept of the seven liberal arts was extracted as the basic curriculum for the medieval university. Boethius, who saw it as his duty to rescue the learning of the Greeks, heavily based his mathematical texts on the work of Nicomachus.

Probably the most influential work on the seven liberal arts is the strange De nuptiis Philologiae et Mercurii (“On the Marriage of Philology and Mercury“) of Martianus Capella (fl.c. 410-420). The American historian H. O. Taylor (1856–1941) claimed that On the Marriage of Philology and Mercurywas “perhaps the most widely used schoolbook in the Middle Ages,” quoted from Martianus Capella and the Seven Liberal Artsby William Harris Stahl.[3]Stahl goes on to say, “It would be hard to name a more popular textbook for Latin reads of later ages.”

Martianus introduces each of the members of the trivium and quadrivium as bridesmaids of the bride Philology.

“Geometry enters carrying a radius in her right hand and a globe in her left. The globe is a replica of the universe, wrought by Archimedes’ hand. The peplos she wears is emblazoned with figures depicting celestial orbits and spheres; the earth’s shadow reaches into the sky, giving a purplish hue to the golden globes of the sun and moon; there are gnomons of sundials and figures showing intervals weights, and measures. Her hair is beautifully groomed, but her feet are covered with grime and her shoes are worn to shreds with treading across the entire surface of the earth.”[4]

“As she enters the celestial hall, Arithmetic is even more striking in appearance than was Geometry with her dazzling peplos and celestial globe. Arithmetic too wears a robe, hers concealing an “intricate undergarment that holds clues to the operations of universal nature.” Arithmetic’s stately bearing reflects the pristine origin, antedating the birth of the Thunder God himself. Her head is an awesome sight. A scarcely perceptible whitish ray emanates from her brow; then another ray, the projection of a line, as it were, coming from the first. A third ray and a fourth spring out, and so on, up to a ninth and a tenth ray–all radiating from her brow in double and triple combinations. These proliferate in countless numbers and in a moment are miraculously retracted into the one.”[5]An allusion to the Pythagorean decade.

“Astronomy like her sister Geometry, is a peregrinator of the universe. She has traversed all the heavens and can reveal the constellations lying beneath the celestial arctic circle. […] Astronomy tells us that she is also familiar with the occult lore of Egyptian priest, knowledge hoarded in their sanctums; she kept herself in seclusion in Egypt for nearly forty thousand years, not wishing to divulge those secrets. She is also familiar with antediluvian Athens.”[6]

“Harmony herself is ineffably dazzling and Martianus is stricken in his efforts to describe her. A lofty figure, her head aglitter with gold ornaments, she walks along between Apollo and Athena. Her garment is tiff with incised and laminated gold; it tinkles softly and soothingly with every measured step She carries in her right hand what appears to be a shield, circular in form. It contains many concentric circles, and the whole is embroidered with striking figure. The circular chords encompass one another and from them pours forth a concord of all tones: Small models of theatrical instruments, wrought of gold, hang suspended from Harmony’s left hand. No know instrument produces sounds to compare with those coming from the strange rounded form.”[7]

I have included Stahl’s passages of Martianus’ descriptions of the quadrivium to make clear then when I talk of the disciplines being personified as women I don’t just mean that they get a female name but are fully formed female characters. This of course raises the question, at least for me, why the mathematical disciplines that were taught almost exclusively to men in ancient Greece, the Romano-Hellenistic culture and in the Middle Ages should be represented by women. Quite honestly I don’t know the answer to my own question. I assume that it relates to the nine ancient Greek Muses, who were also women and supposedly the daughters of Zeus and Mnemosyne (memory personified). This however just pushes the same question back another level. Why are the Muses female?

Having come this far it should be noted that although the quadrivium was officially part of the curriculum on medieval universities it was on the whole rather neglected. When taught the subjects were only taught at a very elementary level, arithmetic based on the primer of Boethius, itself an adaption of Nicomachus, geometry from Euclid but often only Book One and even that only partially, music again based on Boethius and astronomy on the very elementary Sphere of Sacrobosco. Often the mathematics courses were not taught during the normal classes but only on holidays, when there were no normal lectures. At most universities the quadrivium disciplines were not part of the final exams and often a student who had missed a course could get the qualification simple by paying the course fees. Mathematics only became a real part of the of the university curriculum in the sixteenth century through the efforts of Philip Melanchthon for the protestant universities and somewhat later Christoph Clavius for the Catholic ones. England had to wait until the seventeenth century before there were chairs for mathematics at Oxford and Cambridge.

[1]On the one hand imposter syndrome can act as a spur to learn more and increase ones knowledge of a given subject. On the other it can lead one to think that one needs to know much more before one closes a given research/learn/study project and thus never finish it.

[2]To paraphrase some old Greek geezer, the older I get and the more I learn, the more I become aware that what I know is merely a miniscule fraction of that which I could/should know and in reality I actually know fuck all.

[3]William Harris Stahl, Martianus Capella and the Seven Liberal Arts: Volume I The Quadrivium of Martianus Capella. Latin Traditions in the Mathematical Science, With a Study of the Allegory and the Verbal Disciplines by Richard Johnson with E. L. Burge, Columbia University Press, New York & London, 1971, p. 22

[4]Stahl pp. 125–126

[5]Stahl pp. 149–150

[6]Stahl p. 172

[7]Stahl p. 203