Proclus

How to use this course

Proclus was a philosopher first. He headed the Platonic school in Athens for nearly fifty years and wrote long commentaries on Plato, a systematic theology and hymns to the gods. Mathematicians know him for one book: his commentary on Book I of Euclid’s Elements. Most of what we know about Greek geometry before Euclid comes through it. It names the geometers of the fifth and fourth centuries BC, credits theorems to Thales and Pythagoras, records lost proofs by Heron, Pappus and Ptolemy, and contains the most influential ancient discussion of the parallel postulate.

Proclus was not a creative mathematician. He was a careful reader who asked why each definition, postulate and proof is the way it is. This course reads him that way: as a witness to older mathematics, and as a critic whose questions led, over thirteen centuries, to non-Euclidean geometry.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. Page numbers such as “64–68” refer to Gottfried Friedlein’s Greek edition of the Commentary (1873), which Glenn Morrow’s English translation (1970) prints in its margins. References such as I.29 are to Euclid’s Elements in Heath’s numbering.

Timeline

All dates are AD. The birth date comes from the horoscope in Marinus’ Life of Proclus; Marinus’ statement that Proclus lived 75 years has led some scholars to prefer 410 or 411.

Constantinople, Xanthus and the sources

Proclus was born in Constantinople, the young capital of the eastern Roman Empire, to Patricius and Marcella, a well-off family from Lycia on the south coast of Asia Minor. His father was a lawyer in the courts of the capital. Soon after the birth the family returned to Xanthus, the chief city of Lycia, and Proclus always regarded it as his home town; the epithet Lycius records this. He went to school at Xanthus.

The sources and their reliability

• Marinus of Neapolis, his pupil and successor, wrote Proclus, or On Happiness in the year after his death. It is a eulogy organized around the virtues of a Neoplatonic sage, full of dreams, miracles and divine signs, but it also gives exact dates, names of teachers and a horoscope.
• Damascius, a later head of the school, includes anecdotes in his Philosophical History, which survives in fragments.
• Proclus’ own works, especially the prefaces of the commentaries, which name his teacher Syrianus with reverence.
• The Byzantine encyclopedia called the Suda (10th century) gives a short entry and a list of works.

The horoscope

Marinus gives the positions of the planets at Proclus’ birth. Alexander Jones (1999) showed that they fit 8 February 412, which is now the usual date. Marinus also says Proclus died in the 124th year after the accession of Julian, on 17 April, aged 75. Julian became emperor in 361, so the death year is 485, and a birth in 412 makes him 73. The discrepancy has not been fully explained.

Alexandria

Proclus was sent to Alexandria to train as a lawyer like his father. He studied rhetoric with Leonas, a sophist from Isauria, grammar with Orion, and Latin, the language of Roman law. Marinus says that on a journey to Constantinople with Leonas he received a call from the goddess Athena to take up philosophy. Back in Alexandria he studied Aristotle with Olympiodorus the Elder, a famous teacher, and mathematics with a certain Heron. This Heron is otherwise unknown and is not the engineer Heron of Alexandria, who lived some four centuries earlier.

Marinus says Proclus learned Aristotle’s logical works by heart, and that Olympiodorus wanted him to marry his daughter. Proclus never married. Alexandria in the 420s was a dangerous place for pagan intellectuals: Hypatia, the city’s leading mathematician and philosopher, had been murdered by a Christian mob in 415. Proclus soon left for Athens, where the Platonic school remained pagan.

Athens: Plutarch and Syrianus

Proclus arrived in Athens in 430 or 431, aged about eighteen. The Platonic school there was a private foundation, not a direct continuation of Plato’s Academy, though its members saw themselves as Plato’s heirs and its heads took the title “successor”. Its head was Plutarch of Athens, then very old. Plutarch read Aristotle’s On the Soul and Plato’s Phaedo with him before dying in 432. Plutarch’s daughter Asclepigenia taught Proclus the rites of theurgy, ritual practices meant to unite the soul with the gods.

Plutarch’s successor, Syrianus, became Proclus’ real teacher. Marinus says that in under two years they read all of Aristotle, then turned to Plato. Syrianus held that mathematics was a bridge between the sensible world and pure intellect, and that Aristotle was a preparation for Plato. Proclus adopted both views. Marinus says that by his 28th year Proclus had written his great commentary on the Timaeus, Plato’s dialogue on the making of the cosmos and the mathematical structure of the world soul (Lesson 10).

Syrianus died in 437. Proclus succeeded him as head of the school and inherited his house, near the theatre of Dionysus and the temple of Asclepius below the Acropolis. He was about 25.

The Commentary and the anatomy of a proof

For the general reader. Proclus’ commentary on Book I of the Elements is a long lecture course. It explains what mathematics is, what geometry studies, how Euclid’s book is built, and then goes through every definition, postulate, common notion and proposition. Along the way it gives the oldest surviving analysis of how a Greek proof is put together, a scheme still used to teach proofs.

Plan and sources

The Commentary has two prologues. The first treats mathematics in general: its objects, its place between intellect and sense, its divisions and its value. The second treats geometry and the Elements, and contains the summary of the history of geometry studied in Lesson 05. Then come the definitions, the postulates and common notions, and the 48 propositions of Book I. Proclus names his sources: Eudemus of Rhodes (via later writers), Geminus, Heron, Porphyry, Pappus, Posidonius, Carpus and Ptolemy among them. His closing words express the hope of treating the remaining books if time allows; no such commentary survives.

The six parts

Proclus explains that the passage from the particular figure back to the general conclusion is valid because nothing in the proof used the particular features of the figure chosen. This is the Greek version of universal generalization.

Problems and theorems

A problem asks for something to be constructed, and ends “which was to be done”; a theorem asserts a property, and ends “which was to be proved”. Proclus reports a debate. Speusippus and Amphinomus held that all propositions are theorems, since geometrical objects are eternal and are not really made. Menaechmus’ school held that all are problems. Proclus sides with Euclid’s practice of keeping both. He also classifies converses: a theorem and its converse may exchange the whole hypothesis and conclusion, as I.5 and I.6 do, or only part of them.

The catalogue of geometers

For the general reader. In a few pages of his second prologue, Proclus lists the Greek geometers from Thales to Euclid with a sentence or two on each. Historians call it the Eudemian summary, because much of it probably comes, through intermediaries, from the lost History of Geometry by Eudemus, a pupil of Aristotle. Together with scattered remarks elsewhere in the Commentary, it is the backbone of every history of early Greek mathematics.

The list (Proclus, 64–68)

Geometry began in Egypt with land measurement and came to Greece with Thales, who “discovered many things himself”. Then come Mamercus; Pythagoras, who made geometry a liberal education; Anaxagoras and Oenopides; Hippocrates of Chios, the first to write Elements, and Theodorus of Cyrene; Plato, who greatly advanced mathematics; the circle of the Academy, Leodamas, Archytas, Theaetetus, Neoclides and Leon, Eudoxus, Amyclas, Menaechmus and Dinostratus, Theudius, Athenaeus, Hermotimus and Philip; and finally Euclid, who lived under the first Ptolemy and put together the Elements. Historians generally take the list up to Philip to derive from Eudemus, and the paragraph on Euclid, which dates him by reference to Archimedes and Eratosthenes, to be Proclus’ own.

Attributions elsewhere in the Commentary

Proclus also reports, following Eudemus, that the Pythagoreans proved that the angles of a triangle equal two right angles (I.32), by drawing through a vertex a line parallel to the opposite side, and that they discovered the “application of areas”, the geometrical solution of quadratic problems behind the names parabola, ellipse and hyperbola, which Apollonius later used for the conic sections.

Lines, angles and horn angles

For the general reader. Euclid defines a line as “breadthless length” and an angle as the inclination of two lines. Proclus takes these definitions seriously and asks what lines exist besides straight lines and circles, and whether every angle can be compared with every other. His answers record lost Greek work on curves, and his discussion of the angle between a circle and its tangent touches a real puzzle about magnitudes that are not Archimedean.

Kinds of lines

Following Geminus, Proclus divides lines into simple and mixed, and into closed (bounding a figure) and unbounded. He mentions the conic sections, the spiric sections of Perseus (sections of a torus), the conchoid of Nicomedes, the cissoid and the cylindrical helix. He records a theorem, attributed to Geminus, singling out three lines that are “alike in all their parts” (homoeomeric): any part can be made to coincide with any other part of the same length.

In modern terms these are the curves of constant curvature and constant torsion, the orbits of one-parameter groups of rigid motions: a translation gives the line, a rotation the circle, a screw motion the helix.

The horn angle

Euclid III.16 states that the angle between a circle and its tangent is less than every acute rectilinear angle, and that the angle between the circle and its diameter is greater. Proclus calls the first the horn-like angle and uses it to argue that not every angle is a quantity of the ordinary kind.

Proof in coordinates. Take \(P=(0,0)\), \(t\) the \(x\)-axis and the circle \(x^2+(y-r)^2=r^2\). The line \(y=x\tan\theta\) meets the circle again at the point with \(x=r\sin2\theta\neq0\), which lies inside the angle between \(t\) and the line. So near \(P\) the circle lies between the tangent and any line through \(P\) at a positive angle. \(\square\)

If horn angles and rectilinear angles were magnitudes of one kind, the horn angle \(h\) would be positive yet \(nh<\theta\) for every \(n\). This breaks the axiom, used by Eudoxus and Archimedes, that some multiple of the smaller of two magnitudes exceeds the larger. Proclus does not draw this conclusion explicitly, but the problem he discusses is exactly the one that the Archimedean axiom (Euclid V, Def. 4) excludes.

Alternative proofs and Pythagorean triples

For the general reader. Many proofs from lost books survive because Proclus copied them as alternatives to Euclid’s. He also defends Euclid against critics, among them the Epicureans, who mocked a theorem as obvious even to a donkey. And on the theorem of Pythagoras he records two rules, ascribed to Pythagoras and to Plato, for making right triangles with whole-number sides.

Pappus on I.5

Euclid’s proof that the base angles of an isosceles triangle are equal extends both sides and compares two pairs of triangles. Proclus reports a shorter proof from Pappus: compare triangle \(ABC\) with the same triangle read as \(ACB\). Since \(AB=AC\), \(AC=AB\) and the angle at \(A\) is common, I.4 gives \(\angle ABC=\angle ACB\). The companion course on Euclid discusses why some find this proof troubling.

The Epicureans and I.20

I.20 states that any two sides of a triangle are together greater than the third. Proclus reports that the Epicureans ridiculed it: even an ass, put at one corner with hay at another, goes straight to the food and not along two sides. Proclus answers that perceiving a truth is not the same as knowing why it is true, and that science gives the reason. He also gives alternative proofs of I.20 from Heron and Porphyry.

Pythagorean triples

In his comment on I.47, Proclus says that “those who like to record antiquities” credit the theorem to Pythagoras and say he sacrificed an ox on its discovery; he does not endorse the story. He then gives two methods for right triangles with rational sides.

Proof. \(\big(\frac{m^2+1}2\big)^2-\big(\frac{m^2-1}2\big)^2=\frac{(m^2+1)^2-(m^2-1)^2}4=m^2\), and \((n^2+1)^2-(n^2-1)^2=4n^2\). \(\square\) In the Pythagorean rule hypotenuse and long leg differ by 1; in Plato’s they differ by 2.

The parallel postulate

For the general reader. Euclid’s fifth postulate says that two lines crossed by a third, with the inner angles on one side less than two right angles, meet on that side. Proclus argued that this is a theorem needing proof, not a postulate. He showed why intuition is not enough, criticized an earlier proof by Ptolemy, and offered his own. His proof fails, but the failure is instructive: it quietly assumes something equivalent to what it wants to prove.

Why prove it?

In his comment on Postulate 5, Proclus argues that its converse is a theorem (I.17: any two angles of a triangle are less than two right angles), so the postulate itself should be provable. Lines whose inner angles are less than two right angles certainly converge; that they must meet is only plausible. Some curves approach a line forever without meeting it, as the hyperbola approaches its asymptote. Following Geminus, he insists that geometry should not rest on plausible pictures.

Ptolemy’s proof (Proclus on I.29)

Ptolemy wrote a book trying to prove the postulate, and Proclus reports its argument for I.29. Let a transversal cross two parallel lines. If the inner angles on one side were together more than two right angles, then, since the lines are “no more parallel” on one side than the other, the inner angles on the other side would also exceed two right angles, and the four angles would exceed four right angles, which contradicts I.13. The same argument rules out “less than”. So the angles on each side make exactly two right angles.

The gap is the phrase “no more parallel on one side than on the other”. It assumes that the parallel through a point is unique, so that the line behaves the same in both directions. That is Playfair’s axiom, equivalent to the postulate.

Proclus’ proof (Proclus on I.29)

Proclus’ argument. Let \(AB\), \(CD\) be parallel and let \(FG\) cut \(AB\) at \(F\). As \(FG\) and \(FB\) are produced, their distance apart exceeds every finite magnitude, so it exceeds the distance between \(AB\) and \(CD\). Then \(FG\) must cross \(CD\).

From the lemma to the postulate. Let \(AB\), \(CD\) be cut by \(EF\) with \(\angle BEF+\angle DFE\) less than two right angles. Through \(E\) draw \(HK\) with \(\angle KEF+\angle DFE\) equal to two right angles; then \(HK\) is parallel to \(CD\) (I.28). \(AB\) cuts \(HK\) at \(E\), so by the lemma it cuts \(CD\), and on the side where the angles are less than two right angles.

Proclus also states, on I.31, that through a point only one parallel to a given line can be drawn, the form now called Playfair’s axiom. The history of later attempts, from Ibn al-Haytham and ʿUmar al-Khayyām to Saccheri, Lambert, Legendre and the founders of non-Euclidean geometry, is told in the Euclid course.

The Hypotyposis: outline of astronomy

For the general reader. The Outline of the Astronomical Hypotheses (Hypotyposis) is a survey of the astronomy of Hipparchus and Ptolemy, written for students the year after Proclus returned from exile. It explains eccentric circles and epicycles, describes instruments, and gives a method for measuring the size of the Sun with a water clock. It is also the most critical ancient reading of Ptolemy that survives: Proclus doubts that the circles are real, and he rejects the precession of the equinoxes.

Eccentric and epicycle

The Sun’s motion through the zodiac is not uniform; the seasons have unequal lengths. Greek astronomers explained this either by an eccentric, a circle whose centre is displaced from the Earth, or by an epicycle, a small circle whose centre runs on a circle about the Earth. Apollonius and Hipparchus knew the two models give the same motion; Proclus gives a proof.

Proof. In complex notation the eccentric gives \(z(t)=e+Re^{i\omega t}\). The epicycle centre is \(Re^{i\omega t}\) and the body is at \(Re^{i\omega t}+|e|e^{i\alpha}\) with \(e^{i\alpha}=e/|e|\) fixed, which is the same point. Relative to the radius \(Re^{i\omega t}\), the epicycle vector has angle \(\alpha-\omega t\), so it turns at rate \(-\omega\). \(\square\)

Proclus treated such constructions as devices for calculation. In the Hypotyposis and his Timaeus commentary he says the astronomers’ hypotheses save the appearances but may not describe how the heavens really move, and that their complexity is unworthy of the divine bodies.

The water clock

Proclus describes a method he associates with Heron’s water clock: on the day of an equinox, collect the water that flows while the Sun’s disc rises from first contact with the horizon to complete emergence, and compare it with the water that flows in a whole day. The ratio is the Sun’s angular diameter as a fraction of a full circle. Ptolemy had rejected water clocks as too imprecise for this.

Precession

Hipparchus discovered and Ptolemy measured a slow drift of the stars relative to the equinoxes, about 1° per century in Ptolemy’s value. Proclus denied it. He held that the fixed stars keep their places and that there is no evidence for their motion except Ptolemy’s and Hipparchus’ inferences. The true rate is about 1° in 72 years.

Number in the Timaeus commentary

For the general reader. In Plato’s Timaeus the creator makes the soul of the world from a mixture, divides it by the numbers 1, 2, 3, 4, 9, 8, 27, and fills the intervals with means. The result is a musical scale. Proclus’ commentary, his earliest major work, works through the arithmetic in detail, and his Elements of Theology applies the Euclidean form of definitions and propositions to metaphysics itself.

The construction (Timaeus 35b–36b)

Plato takes the doubles \(1,2,4,8\) and the triples \(1,3,9,27\). Every interval is filled with two means: the harmonic mean and the arithmetic mean, as defined by Archytas. Between \(1\) and \(2\) these are \(\frac43\) and \(\frac32\), producing ratios \(\frac43\), \(\frac98\), \(\frac43\). Each \(\frac43\) is then filled with tones \(\frac98\), leaving a remainder.

To write the whole scale in whole numbers, ancient commentators, Crantor among them according to Plutarch, began from 384. Proclus discusses these numbers and their arrangement in the shape of the letter lambda, with the doubles down one side and the triples down the other.

Mathematics as method

The Elements of Theology consists of 211 propositions, each with a proof from earlier ones, beginning with the statement that every multiplicity participates in some way in unity. The Elements of Physics does the same for parts of Aristotle’s Physics and On the Heavens. Both show what Proclus admired in Euclid: a body of knowledge deduced from first principles.

Fifty years as Successor

Proclus headed the Athenian school from 437 until his death, nearly half a century. Marinus describes a disciplined life: several lectures a day, long hours of writing, prayers to the Sun at its rising, at noon and at its setting, fasts, and the rites of the old religion, which by then were illegal in the empire. He was well off, generous to friends and to the city, and never married. Seven of his hymns survive. Among his students were Marinus, Ammonius son of Hermias, who later taught in Alexandria, and Asclepiodotus.

Athens was by then a Christian city, and Proclus’ open paganism put him at risk. At some point, perhaps in the 450s, he left for about a year and lived in Lydia in Asia Minor, where Marinus says he studied local religious traditions. He wrote the Hypotyposis the year after his return. Marinus tells that when the Christians removed the statue of Athena from the Parthenon, a woman appeared to Proclus in a dream and told him to prepare his house, because the Lady of Athens wished to live with him.

In his last years Proclus was frail, and Marinus was already acting as his successor. Marinus records a solar eclipse in January 484 as a sign of the coming death. Proclus died on 17 April 485 in Athens and was buried in the eastern part of the city near Mount Lycabettus, in the tomb of his teacher Syrianus, according to Marinus. The school survived him by less than fifty years. In 529 the emperor Justinian forbade pagans to teach, and its last head, Damascius, left Athens with several colleagues, among them the Aristotle commentator Simplicius.

Legacy: Euclid’s best reader

Mathematics

The Commentary on Euclid became a standard companion to the Elements once Greek texts reached the West. Simon Grynaeus printed it at Basel in 1533 together with the first printed Greek Euclid, and Francesco Barozzi’s Latin translation appeared at Padua in 1560. Christopher Clavius drew on it for his influential Latin Euclid. Johannes Kepler quoted it repeatedly in Harmonices mundi (1619), sharing Proclus’ belief that geometry expresses the structure of the soul and the cosmos. Thomas Taylor’s English translation appeared in 1788–89, Friedlein’s critical edition in 1873, and Morrow’s translation in 1970.

Historians of Greek mathematics, from Paul Tannery and Thomas Heath to the present, depend on the catalogue of geometers and on the attributions to Thales, the Pythagoreans and Oenopides. Proclus’ discussion of the parallel postulate framed the question for later geometers: whether the postulate could be proved, and what it was equivalent to. The answer, that it is independent, came with Lobachevsky and Bolyai.

Philosophy

The writings attributed to Dionysius the Areopagite, around 500, absorbed much of Proclus’ system into Christian theology. A ninth-century Arabic reworking of the Elements of Theology circulated in Latin as the Book of Causes under Aristotle’s name until Thomas Aquinas, using William of Moerbeke’s translation of 1268, recognized its source. Berthold of Moosburg wrote a large commentary on the Elements of Theology; Nicholas of Cusa and Marsilio Ficino read Proclus closely, and Hegel praised him as the culmination of Neoplatonism.

Honours

• The bright lunar crater Proclus, on the western edge of the Mare Crisium, is named after him.

Sources and further reading

Ancient sources and translations

1. Proclus, A Commentary on the First Book of Euclid’s Elements, transl. G. R. Morrow, Princeton University Press, 1970; reissued with a foreword by I. Mueller, 1992.
2. Proclus, In primum Euclidis Elementorum librum commentarii, ed. G. Friedlein, Teubner, Leipzig, 1873.
3. T. Taylor, transl., Proclus on Euclid I.29, transcription by D. R. Wilkins, Trinity College Dublin.
4. Proclus, Hypotyposis astronomicarum positionum, ed. K. Manitius, Teubner, Leipzig, 1909.
5. Marinus, Proclus, or On Happiness, in M. Edwards, Neoplatonic Saints, Liverpool University Press, 2000.
6. Proclus, The Elements of Theology, ed. and transl. E. R. Dodds, 2nd ed., Oxford, 1963.

Modern studies

1. C. Helmig and C. Steel, Proclus, Stanford Encyclopedia of Philosophy.
2. J. J. O’Connor and E. F. Robertson, Proclus Lycaeus, MacTutor History of Mathematics.
3. Proclus on the parallel postulate, MacTutor.
4. G. R. Morrow, Proclus, Dictionary of Scientific Biography.
5. Proclus, Biographical Encyclopedia of Astronomers.
6. Proclus, Wikipedia.
7. A. Jones, The horoscope of Proclus, Classical Philology 94 (1999).
8. T. L. Heath, The Thirteen Books of Euclid’s Elements, 2nd ed., Cambridge, 1926, vol. 1 (introduction and notes on Postulate 5 and I.29).
9. D. J. O’Meara, Pythagoras Revived: Mathematics and Philosophy in Late Antiquity, Oxford, 1989.