Lesson 00

How to use this course

Plato proved no theorem that we can name. He was a philosopher, and he wrote dialogues, not treatises. Yet he stands at the centre of Greek mathematics. His school, the Academy, was where Eudoxus and Theaetetus did the work that became Books V, X, XII and XIII of Euclid’s Elements. His dialogues are among our earliest witnesses to the discovery of irrational lengths. His Timaeus is the first attempt to build physics from mathematics, and his challenge to explain the planets by uniform circular motions set the programme of astronomy until Kepler.

This course reads Plato as a witness, a patron and a theorist of mathematics. For each claim it says whether the evidence is Plato’s own text, the report of a near-contemporary such as Aristotle, or a much later tradition. The main ancient sources for his life are the Seventh Letter (attributed to Plato, authenticity disputed), Diogenes Laertius’ Lives, Book III (3rd century AD), and the Herculaneum papyrus of Philodemus’ History of the Academy (1st century BC).

Prerequisites for Part II

Elementary number theory: divisibility, prime factorization, proof by contradiction.
Euclidean geometry of the plane and of simple solids; the angles of regular polygons.
Ratios and simple fractions, as used in musical intervals.
For Lesson 10, the basic phenomena of the night sky; the companion course on Eudoxus gives the details of his planetary model.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, derivations and exercises with folded solutions. References such as Timaeus 53c use the standard Stephanus pages printed in the margins of every edition and translation.

Reference

Timeline

All dates are BC and approximate. The birth and death years follow the chronographer Apollodorus as reported by Diogenes Laertius. The Sicilian dates come from the Seventh Letter.

c. 428/427 BC	Born in Athens (Diogenes also reports Aegina), to Ariston and Perictione.
c. 409–404 BC	Young man of military age during the last years of the Peloponnesian War.
404–403 BC	Regime of the Thirty, which included his relatives Critias and Charmides. The Seventh Letter says their crimes turned him away from politics.
399 BC	Trial and execution of Socrates. Plato and other Socratics withdraw to Megara.
390s BC	Travels; Diogenes names Cyrene, Italy and Egypt. Early Socratic dialogues.
c. 388/387 BC	First visit to Sicily, aged about 40. Meets Dion of Syracuse. Probably meets Archytas in Tarentum.
c. 387–383 BC	Founds the Academy in the grove of Hecademus outside Athens.
c. 380s–370s BC	Meno, Phaedo, Republic.
c. 369 BC	Death of Theaetetus after a battle near Corinth; the Theaetetus is written in his memory.
367/366 BC	Second visit to Syracuse, to teach Dionysius II. Aristotle enters the Academy in Plato’s absence.
361–360 BC	Third visit to Syracuse; released through the intervention of Archytas.
c. 360–350 BC	Timaeus, Critias, Philebus, Laws.
357–354 BC	Dion seizes Syracuse and is murdered.
348/347 BC	Dies in Athens aged about 80; buried in the Academy. Speusippus, his nephew, becomes head of the school.
c. 300 BC	Euclid’s Elements ends with the construction of the five regular solids (Book XIII).
86 BC	Sulla’s siege of Athens devastates the Academy’s grove.
5th c. AD	Proclus, head of the Neoplatonic school in Athens, writes his commentaries on Euclid and the Timaeus.
1596 AD	Kepler’s Mysterium cosmographicum spaces the planetary orbits with the five Platonic solids.

Part I · Formation

Lesson 01

Athens, family and Socrates

Plato was born into one of the oldest families of Athens around 428/427 BC, at the start of the Peloponnesian War. His father Ariston claimed descent from the early kings of Athens; his mother Perictione was descended from a relative of the lawgiver Solon. His brothers Adeimantus and Glaucon are the main speakers with Socrates in the Republic, and his sister Potone was the mother of Speusippus, who would succeed him at the Academy. After Ariston’s death Perictione married her uncle Pyrilampes, a friend of Pericles, and Plato grew up in his household.

The name

Diogenes Laertius says Plato was first called Aristocles, after his grandfather, and that “Plato” was a nickname from the Greek platys, “broad”, given by a wrestling teacher for his build or by others for the breadth of his style or of his forehead. The variety of explanations suggests a later guess. Plato was a common Athenian name, and many scholars doubt the story.

Socrates and the Thirty

Aristotle says that in his youth Plato became familiar with Cratylus, a follower of Heraclitus, and learned that the sensible world is always in flux. He then joined the circle of Socrates. His mother’s brother Charmides and her cousin Critias were also in that circle. Both became leaders of the oligarchy of the Thirty, which ruled Athens by terror in 404–403 BC. The Seventh Letter describes a young man drawn to public life, invited to join them, and repelled by their crimes, especially their attempt to implicate Socrates in an arrest. The restored democracy then tried and executed Socrates in 399 BC. According to the letter, this convinced Plato that no existing city was well governed and that only philosophers could govern well.

Exercise 1.1Classify the following claims by source: (a) Plato’s distress at the Thirty and at Socrates’ execution; (b) his early contact with Cratylus; (c) the name Aristocles; (d) his family links to Critias and Charmides. Which is best attested?

Show solution

(a) comes from the Seventh Letter, written in Plato’s voice but of disputed authenticity; even if not by Plato, it was probably written by someone close to the Academy within a generation. (b) comes from Aristotle, Metaphysics A.6, a pupil of twenty years: early and well informed. (c) is reported by Diogenes Laertius some six centuries later, with several competing explanations: weak. (d) is confirmed by the dialogues themselves, notably the Charmides, which stages these relatives, and by other Athenian sources: the best attested.

Lesson 02

Megara, Italy and Sicily

After Socrates’ death, Diogenes says, quoting Hermodorus, a member of the Academy, Plato and other Socratics took refuge at Megara with Euclid of Megara, a philosopher who must not be confused with the geometer. Diogenes then lists journeys to Cyrene, to study with the mathematician Theodorus; to Italy, to meet the Pythagoreans Philolaus and Eurytus; and to Egypt. Theodorus is a speaker in the Theaetetus, which makes the Cyrene visit plausible, but the list reads like the standard itinerary of a wise man, and the Egyptian journey in particular is doubted.

The first visit to Sicily

The Seventh Letter says Plato first came to Sicily when he was about forty, which gives 388/387 BC. There he met Dion, the young brother-in-law of the tyrant Dionysius I of Syracuse, who became his devoted friend. He very likely met Archytas of Tarentum on the same journey. Archytas was a Pythagorean, the general and leading statesman of his city, and the most important mathematician of the day. He had solved the problem of doubling the cube (Lesson 05), studied the theory of means and musical intervals, and later taught Eudoxus.

Sold as a slave?

Diogenes tells that Dionysius I, offended by Plato’s frankness, handed him to the Spartan envoy Pollis, who sold him at the slave market of Aegina, and that Anniceris of Cyrene bought his freedom. In 2024 a team led by Graziano Ranocchia announced new readings of a charred Herculaneum papyrus of Philodemus’ History of the Academy. As reported, the text places the sale on Aegina but dates it either to 404 BC or to 399 BC, not to the Sicilian journey. The story is therefore old, but its date and circumstances are uncertain.

Exercise 2.1The Seventh Letter puts the first Sicilian visit at age “about forty”, and Apollodorus puts Plato’s birth in Olympiad 88 (428–425 BC by a strict reckoning, though usually given as 428/427). What range of dates for the visit follows? How does this fit the founding of the Academy “after his return”?

Show solution

Forty years after 428–425 BC is 388–385 BC; “about forty” widens this by a year or two either way. Diogenes says the Academy was founded after Plato’s return from his travels, so the founding falls in the mid-380s at the earliest, matching the modern estimate of c. 387–383 BC. The argument depends entirely on the letter and on a chronographer’s birth date, so it gives a window, not a year.

Lesson 03

The Academy

On his return Plato began teaching in the Academy, a public gymnasium with a sacred grove about 1.5 km north-west of the Dipylon gate of Athens, named after the hero Hecademus. He owned a small garden nearby. Plato is said to have charged no fees. The school had no fixed curriculum that we know of, and no formal membership. It was a community of people who met to discuss and do research, and it lasted, in one form or another, for three centuries.

Mathematicians in the Academy

Proclus, summarizing the lost history of geometry by Aristotle’s pupil Eudemus, says that Plato’s zeal made mathematics, and geometry above all, advance greatly, and that he filled his writings with mathematical examples. He names the mathematicians who “lived together in the Academy”: Leodamas of Thasos, Archytas, Theaetetus, then Neoclides and his pupil Leon, Eudoxus, Amyclas of Heraclea, Menaechmus and his brother Dinostratus, Theudius of Magnesia, Athenaeus of Cyzicus, Hermotimus of Colophon and Philip of Mende (or Medma), often identified with Philip of Opus. Not all were students of Plato. Archytas lived in Tarentum, and Eudoxus came with his own school. Two women, Axiothea of Phlius and Lasthenia of Mantinea, are recorded among Plato’s hearers. Aristotle entered in 367 BC and stayed twenty years.

The inscription

A famous story says the Academy’s door bore the words “Let no one ignorant of geometry enter”. The earliest known witnesses are late antique, chiefly Greek commentators on Aristotle of the sixth century AD, writing some nine centuries after the school was founded. No classical author mentions it. It expresses Plato’s attitude, as Lesson 07 shows, but it is not evidence of an actual sign.

Exercise 3.1Proclus’ list is often cited as proof that Plato “directed” mathematical research. Using the information in this lesson, give one reason to accept and one reason to doubt that reading.

Show solution

For: the list goes back to Eudemus, a member of Aristotle’s school writing within a generation of Plato, and it groups these men around the Academy. Plato’s dialogues show detailed knowledge of current results (Lessons 04 and 06), and his Republic sets research problems such as solid geometry. Against: several on the list, notably Archytas and Eudoxus, were established mathematicians with their own schools; Proclus was a Neoplatonist who had every reason to magnify Plato’s role; and no theorem is attributed to Plato himself by an early source. The safe conclusion is that Plato created a setting in which mathematics was valued and discussed, and that the research was done by others.

Part II · Mathematics and the cosmos

Lesson 04

Incommensurability in the dialogues

For the general reader. Some lengths cannot be measured by any common unit, however small. The diagonal of a square and its side are the classic case: their ratio is \(\sqrt2\), which is not a fraction. The Greeks discovered this in the fifth century BC. Plato’s dialogues are among the earliest texts to discuss it, and they show it moving from a single surprise to a general theory.

Goals

Work through the geometry lesson of the Meno.
Understand the report of Theodorus and Theaetetus in the Theaetetus and prove Theaetetus’ theorem.
See how the Laws treats ignorance of incommensurables as a national disgrace.

The Meno: doubling the square (82b–85b)

Socrates draws a square of side 2 feet, area 4, and asks an uneducated slave boy for the side of a square of double area. The boy first says 4 feet, which gives area 16; then 3 feet, which gives 9. Socrates then draws the diagonals of four copies of the original square arranged in a large square of side 4. The four diagonals enclose a tilted square made of four half-squares, of area \(4\times 2=8\). The answer is the square on the diagonal. Plato’s point is philosophical, that the boy “recollects” rather than learns; the mathematical point is that the required side is not a whole number of feet, and the dialogue avoids asking what number it is.

The Theaetetus: Theodorus and Theaetetus (147d–148b)

The young Theaetetus reports that his teacher Theodorus drew figures to show that the sides of squares of area 3 and 5 square feet are not commensurable in length with the one-foot line, and went on case by case as far as 17, where he stopped. Theaetetus and his friend then sought a general account. They divided numbers into “square” numbers, products of equal factors, and “oblong” numbers, products of unequal factors. Lines whose squares are square numbers they called lengths; lines whose squares are oblong numbers they called powers (dynameis), since they are commensurable with the unit only in the squares built on them.

Theorem (Theaetetus; Euclid X.9)

Let \(n\) be a positive integer. The side of a square of area \(n\) is commensurable with the unit length if and only if \(n\) is a perfect square. Equivalently, \(\sqrt n\) is rational exactly when \(n=m^2\) for an integer \(m\).

Proof. If \(\sqrt n=p/q\) in lowest terms, then \(nq^2=p^2\). Any prime \(r\) dividing \(q\) divides \(p^2\), hence \(p\), contradicting lowest terms. So \(q=1\) and \(n=p^2\). The ancient proof used Book VII’s theory of numbers in ratio, which is equivalent.

Why did Theodorus stop at 17? The dialogue does not say, and the question has produced a large literature. One family of reconstructions, developed by Wilbur Knorr, has him prove each case by arguments about odd and even numbers. These work for every non-square up to 15 and first fail at 17, the smallest non-square that leaves remainder 1 on division by 8 (Exercise 4.2). Another links the stopping point to the “spiral of Theodorus”, a chain of right triangles with hypotenuses \(\sqrt2,\sqrt3,\dots,\sqrt{17}\) that closes nearly one full turn at \(\sqrt{17}\). Both are modern conjectures.

The Laws: a national shame (819d–820c)

In his last work Plato has an Athenian stranger say that he was ashamed, on learning late in life about incommensurable lengths, areas and volumes, of the ignorance of all the Greeks, who assume that any two magnitudes can be measured by a common measure. He wants it taught to the young. Taken with the Theaetetus, the passage shows a whole generation absorbing the discovery.

Exercise 4.1In the Meno figure, let the original square have side 2. Prove that the tilted square is indeed a square and has area 8, and show directly that its side is not a whole number or a ratio of whole numbers.

Show solution

Place the large square with vertices \((0,0),(4,0),(4,4),(0,4)\). The diagonals of the four small squares that form the tilted square join \((2,0),(4,2),(2,4),(0,2)\). The four sides all have length \(\sqrt{2^2+2^2}=\sqrt8\), and adjacent sides have direction vectors \((2,2)\) and \((-2,2)\), with dot product 0, so it is a square. Its area is \(16-4\cdot\tfrac12\cdot2\cdot2=8\). If its side were \(p/q\) in lowest terms then \(p^2=8q^2\), so \(p\) is even, \(p=2p'\), and \(p'^2=2q^2\), so \(q\) is even, a contradiction.

Exercise 4.2Give a Theodorus-style proof that \(\sqrt3\) is irrational using only odd and even numbers, and explain why the same parity argument cannot work for \(\sqrt{17}\).

Show solution

Suppose \(p^2=3q^2\) with \(p,q\) not both even. If \(q\) is even then \(p\) is even, impossible; so \(q\) is odd, hence \(p\) is odd. Odd squares are \(1\bmod 8\), so \(p^2\equiv1\) and \(3q^2\equiv3 \pmod 8\), a contradiction. For 17, both sides of \(p^2=17q^2\) with \(p,q\) odd are \(1\bmod 8\), since \(17\equiv1\pmod 8\): reduction modulo 8 (or any power of 2) gives no contradiction. A different argument, such as divisibility by 17 itself, is needed. This is one reason proposed for Theodorus stopping at 17.

Exercise 4.3Theaetetus distinguished lines commensurable “in length” from lines commensurable only “in square”. Show that \(\sqrt2\) and \(\sqrt8\) are commensurable in length, while \(\sqrt2\) and \(\sqrt3\) are commensurable in square only.

Show solution

\(\sqrt8:\sqrt2=2\), a ratio of numbers, so they share the common measure \(\sqrt2\). The squares on \(\sqrt2\) and \(\sqrt3\) are 2 and 3, commensurable; but \(\sqrt3:\sqrt2=\sqrt{3/2}=\sqrt6/2\), and \(\sqrt6\) is irrational by the theorem, so the lines themselves have no common measure.

Lesson 05

The Academy’s mathematics

For the general reader. Ancient writers credit Plato with setting problems for mathematicians rather than solving them. The most famous is the doubling of the cube: build a cube with twice the volume of a given one. This lesson collects what the sources say about Plato and the geometers around him, and separates early testimony from later stories.

Goals

Know what Proclus and Diogenes credit to Plato: analysis and definitions.
Follow the Delian problem from Hippocrates’ reduction to Menaechmus’ conics.
Weigh the evidence for a “Platonic” solution.

Analysis

Diogenes and Proclus both say that Plato taught Leodamas of Thasos the method of analysis: assume the thing sought, deduce consequences until reaching something known, then reverse the steps to obtain a proof (synthesis). The method is older than Plato, so the attribution probably means that the Academy made it a conscious technique.

The Delian problem

Eratosthenes, in a letter preserved by Eutocius’ sixth-century commentary on Archimedes, tells that the people of Delos, ordered by an oracle to double an altar, consulted the geometers who were with Plato in the Academy. Plutarch adds that Plato told them the god’s real aim was to make the Greeks study geometry, and elsewhere that Plato criticized Eudoxus, Archytas and Menaechmus for using instruments and mechanical constructions, which in his view degraded geometry back to the sensible world. These are late stories, but they agree that the problem was studied intensively in Plato’s circle.

Reduction (Hippocrates of Chios, 5th century BC)

If \(x\) and \(y\) are two mean proportionals between \(a\) and \(2a\), that is \(a:x=x:y=y:2a\), then \(x^3=2a^3\): the cube on \(x\) is double the cube on \(a\).

From the proportions, \(x^2=ay\) and \(y^2=2ax\). Then \(x^4=a^2y^2=2a^3x\), so \(x^3=2a^3\). The two equations describe two parabolas, and Menaechmus, named by Proclus among the Academy’s geometers, is credited by Eutocius with solving the problem as the intersection of two conic sections. Archytas had already found the two means by a remarkable construction in three dimensions, intersecting a cylinder, a cone and a torus. Eudoxus used “curved lines” whose nature is lost (see the Eudoxus course, Lesson 07).

The “Platonic” machine

Eutocius also describes a mechanical device for finding two mean proportionals and attributes it to Plato. It uses two set-squares sliding along each other. Most historians reject the attribution, since it contradicts Plato’s reported objection to mechanical methods, and since Eratosthenes, who reviewed the earlier solutions, does not mention a solution by Plato.

Exercise 5.1Show that the parabolas \(x^2=ay\) and \(y^2=2ax\) meet, apart from the origin, at a point whose \(x\)-coordinate is \(a\sqrt[3]2\). Find the \(y\)-coordinate.

Show solution

Substitute \(y=x^2/a\) into \(y^2=2ax\): \(x^4/a^2=2ax\), so \(x^3=2a^3\) (for \(x\ne0\)) and \(x=a\sqrt[3]2\). Then \(y=x^2/a=a\sqrt[3]4\). Check: \(a:a\sqrt[3]2=a\sqrt[3]2:a\sqrt[3]4=a\sqrt[3]4:2a\), each ratio being \(1:\sqrt[3]2\).

Exercise 5.2The square was doubled in the Meno with ruler and compass. Explain in modern terms why ruler-and-compass constructions can double the square but not the cube.

Show solution

Starting from a unit length, ruler and compass construct exactly the lengths obtainable by field operations and square roots. Each such number lies in a field extension of \(\mathbb Q\) of degree a power of 2. \(\sqrt2\) has degree 2, so it is constructible. \(\sqrt[3]2\) is a root of \(x^3-2\), irreducible over \(\mathbb Q\) by Eisenstein’s criterion at 2, so it has degree 3, which is not a power of 2. Pierre Wantzel proved this in 1837. The Greek geometers sensed the obstacle and turned to conics and curves.

Lesson 06

Side and diameter numbers

For the general reader. If the diagonal of a square has no exact numerical measure, can it at least be approximated by whole numbers? The Greeks found an endless ladder of pairs of numbers, such as 5 and 7, or 12 and 17, whose ratio comes ever closer to the ratio of diagonal to side. Plato alludes to this ladder in a famously obscure passage of the Republic.

Goals

Read the “rational diameter of five” in Republic 546c.
Prove the side-and-diameter recurrence and its invariant.
Connect it to continued fractions and to Pell’s equation.

The text

In Book VIII of the Republic (546b–c) the Muses explain the decline of the ideal city with a riddle about a “geometrical number” that governs good and bad births. Part of the riddle speaks of a hundred squares “of the rational diameter of five, each less by one”, or “of the irrational diameter, less by two”. The diagonal of a square of side 5 is \(\sqrt{50}\), the “irrational diameter”. Its nearest whole number, 7, is the “rational diameter”, and \(7^2=49=50-1\). So \(7^2-1=48\) and \(50-2=48\) give the same number, and a hundred of them make 4800. The rest of the riddle is disputed, but this part shows that Plato expected readers to know that 7 approximates the diagonal of 5.

The ladder

Theon of Smyrna (2nd century AD) and Proclus give the general rule. Start with side \(s_1=1\) and diameter \(d_1=1\). Then

\[ s_{n+1}=s_n+d_n,\qquad d_{n+1}=2s_n+d_n, \]

which gives \((1,1),(2,3),(5,7),(12,17),(29,41),(70,99),\dots\)

Proposition (side and diameter numbers)

For all \(n\ge1\), \(d_n^2-2s_n^2=(-1)^n\). Consequently \(d_n/s_n\to\sqrt2\), alternately from below and above, with \(\big|d_n/s_n-\sqrt2\big|<1/(2s_n^2)\) for \(n\ge2\).

Proof. \(d_{n+1}^2-2s_{n+1}^2=(2s_n+d_n)^2-2(s_n+d_n)^2=4s_n^2+4s_nd_n+d_n^2-2s_n^2-4s_nd_n-2d_n^2=-(d_n^2-2s_n^2)\). Since \(d_1^2-2s_1^2=-1\), induction gives \((-1)^n\). Then \(\big|d_n/s_n-\sqrt2\big|=\dfrac{1}{s_n^2\,(d_n/s_n+\sqrt2)}\), and \(d_n/s_n+\sqrt2>2\) for \(n\ge2\).

Theon says the rule is proved geometrically in Book II of the Elements, and Proclus links it to II.10. In modern terms the ratios \(d_n/s_n\) are the convergents of the continued fraction \(\sqrt2=[1;2,2,2,\dots]\), and the pairs with \(n\) even solve \(d^2-2s^2=1\), the simplest case of the equation that Euler mistakenly named after John Pell. The same alternate subtraction, anthyphairesis, appears in Euclid X.2 as a test of incommensurability: if the process on two magnitudes never ends, they are incommensurable.

Exercise 6.1Compute \(s_7,d_7\) and the error \(|d_7/s_7-\sqrt2|\). How many correct decimal places does it give?

Show solution

From \((70,99)\): \(s_7=70+99=169\), \(d_7=140+99=239\). Then \(239^2-2\cdot169^2=57121-57122=-1\). The error is \(1/(169^2(239/169+\sqrt2))\approx1/(28561\times2.8284)\approx1.24\times10^{-5}\). Since \(239/169=1.414201\ldots\) and \(\sqrt2=1.414213\ldots\), it gives four correct decimal places.

Exercise 6.2Show that the anthyphairesis of the diagonal \(d\) and side \(s\) of a square never ends, and deduce that they are incommensurable.

Show solution

Since \(s<d<2s\), subtract \(s\) once from \(d\) to get \(r=d-s<s\). Next compare \(s\) with \(r\): \(s/r=1/(\sqrt2-1)=\sqrt2+1=2+(\sqrt2-1)\). So \(s\) contains \(r\) twice with remainder \(r'=s-2r\), and \(r/r'=1/(\sqrt2-1)\) again: the ratio repeats, so the process runs for ever with quotients \(1,2,2,2,\dots\). Geometrically, the remainders are the side and diagonal of ever smaller squares. If \(d\) and \(s\) had a common measure \(u\), every remainder would be a positive whole multiple of \(u\), and a strictly decreasing sequence of positive multiples of \(u\) must stop. So no common measure exists.

Lesson 07

The Divided Line and the curriculum

For the general reader. Why study mathematics at all? Plato’s answer, in the Republic, is that it turns the mind from the changing world of the senses toward things that are unchanging and can be known with certainty. He placed mathematics on a ladder of knowledge, the Divided Line, and designed a ten-year mathematical education for the rulers of his ideal city.

Goals

State the construction of the Divided Line and its proportions.
Understand Plato’s two criticisms of mathematical method.
Know the five mathematical sciences of Republic VII and his view of astronomy and harmonics.

The Line (Republic 509d–511e)

Socrates asks Glaucon to cut a line into two unequal parts, the visible and the intelligible, and to cut each part again in the same ratio. The four segments, from lowest to highest, correspond to four states of mind and their objects:

eikasia (imaging): shadows and reflections;
pistis (belief): animals, plants and made things;
dianoia (thought): mathematical objects;
noēsis (understanding): the Forms, reached by dialectic.

Proposition (the middle segments are equal)

If a line is divided so that \(\text{visible}:\text{intelligible}=AB:BC=CD:DE\), with \(AB,BC\) the parts of the visible and \(CD,DE\) the parts of the intelligible, then \(BC=CD\).

Let the common ratio be \(1:r\), and let the visible part have length \(V\), so the intelligible has \(rV\). Then \(AB=V/(1+r)\), \(BC=rV/(1+r)\), \(CD=rV/(1+r)\) and \(DE=r^2V/(1+r)\). Hence \(BC=CD\). Plato does not mention this consequence; commentators have debated whether it is intended, since it gives belief about physical things the same “length” as mathematical thought.

Two criticisms of mathematics

Plato says that mathematicians (i) use visible diagrams while reasoning about the square itself and the diagonal itself, and (ii) take their starting points, such as the odd and the even, the figures and the three kinds of angle, as hypotheses that they do not justify, then reason downward to conclusions. Dialectic instead treats hypotheses as stepping stones up to an unhypothetical first principle. The first remark describes Greek geometry accurately. The second is a programme for foundations: Euclid’s explicit definitions, postulates and common notions answer part of it, and the axiomatic method of David Hilbert answers more.

Aristotle reports (Metaphysics A.6) that Plato placed mathematical objects between Forms and sensible things: like Forms they are eternal and unchanging, but there are many of each kind, many equal squares, whereas the Form of square is one. This “intermediate” status matches the place of dianoia on the Line.

The curriculum (Republic 521c–531c, 537b–d)

The future guardians study, in order: arithmetic, plane geometry, solid geometry, astronomy (solids in motion) and harmonics. Socrates complains that solid geometry is neglected because no city honours it, and he asks for a director of research (528b–c). He wants astronomy studied as a mathematical science, using the visible heavens only as diagrams, and harmonics studied through numbers rather than by straining ears at the strings. From about twenty to thirty the students take these sciences together and learn how they are related; from thirty to thirty-five, dialectic.

Exercise 7.1Suppose the Line has total length 1 and the ratio of visible to intelligible is \(1:2\). Find all four segments. What happens if instead the ratio is the golden ratio, \(1:\varphi\)?

Show solution

With \(r=2\): visible \(V=1/3\), intelligible \(2/3\). \(AB=1/9\), \(BC=CD=2/9\), \(DE=4/9\); the segments are in geometric progression \(1:2:2:4\) with the middle pair equal. With \(r=\varphi\): \(V=1/(1+\varphi)=1/\varphi^2\), and the segments are \(V/\varphi^2,\ V/\varphi,\ V/\varphi,\ V\). Since \(1+\varphi=\varphi^2\), the visible part is to the intelligible as the intelligible is to the whole line: the Line itself is then cut in extreme and mean ratio, the “section” studied in the Academy.

Exercise 7.2Plato orders the sciences arithmetic, plane geometry, solid geometry, astronomy, harmonics. Explain the principle behind the order, and where Archytas’ list of four sciences (arithmetic, geometry, astronomy, music) differs.

Show solution

Plato orders by dimension and then by motion: number (no dimension), plane (two), solid (three), solids in motion (astronomy), and finally motion heard rather than seen (harmonics), which he calls the sister science of astronomy. Archytas, as quoted by Porphyry, already called the four sciences sisters but had no separate solid geometry. Plato inserts it explicitly, partly because it was an active research field in his circle: the doubling of the cube (Lesson 05) and the regular solids (Lesson 09).

Lesson 08

The world soul: a cosmos of ratios

For the general reader. In the Timaeus, a craftsman god makes the world by imposing order on a disorderly medium. Before building any bodies, he makes a “soul” for the world out of numbers. The numbers turn out to be the intervals of the Greek musical scale, and the soul is then bent into circles that carry the stars and planets. Music and astronomy, the sister sciences of the Republic, become a single structure.

Goals

Reconstruct the numerical series of Timaeus 35b–36b.
Compute the arithmetic and harmonic means and the resulting scale.
Identify the tone, the leimma and the circles of the Same and the Different.

The series

The craftsman blends Being, Sameness and Difference into a strip and marks off portions in the amounts

\[ 1,\ 2,\ 3,\ 4,\ 9,\ 8,\ 27, \]

that is, two geometric progressions with ratios 2 and 3, written in a lambda shape since late antiquity: \(1,2,4,8\) on one side and \(1,3,9,27\) on the other. He then fills each double and triple interval with two means.

Definitions (the means)

Between \(a<b\), the arithmetic mean \(A=(a+b)/2\) exceeds \(a\) by the same amount as it falls short of \(b\). The harmonic mean \(H=2ab/(a+b)\) exceeds \(a\) and falls short of \(b\) by the same fraction of each: \((H-a)/a=(b-H)/b\). Moreover \(a:H=A:b\), so \(H\cdot A=a\cdot b\).

Between 1 and 2 the means are \(H=4/3\) and \(A=3/2\), so the octave \(2:1\) is filled by the fourth \(4:3\), the tone \(9:8\) between \(4/3\) and \(3/2\), and the fifth. Plato then fills each interval of a fourth with tones of \(9:8\). Two tones take \(4/3\) down to \(81/64\) above its base, and the remainder is

\[ \frac{4}{3}\Big/\Big(\frac{9}{8}\Big)^2=\frac{4}{3}\cdot\frac{64}{81}=\frac{256}{243}, \]

which Plato calls the leimma, the “remnant” (36b). The result is the diatonic scale of the Pythagoreans, carried through four octaves and a major sixth, up to 27.

The circles (36b–d)

The craftsman then splits the strip lengthwise into two, lays the halves across each other in the shape of the letter chi (Χ), and bends each into a circle. The outer circle, of the Same, turns to the right and carries the fixed stars in their daily motion. The inner circle, of the Different, is tilted to it and turns the other way; it is divided six times into seven unequal circles, for the Sun, Moon and five planets. The tilt of the Χ is the obliquity of the ecliptic to the celestial equator. Three of the inner circles move at the same speed (Sun, Venus, Mercury), and four at different speeds. The mathematics of music and the geometry of the sky share one construction.

Exercise 8.1Construct Plato’s scale between 1 and 2 explicitly and verify that it consists of five tones and two leimmata. Check that the product of all the step ratios is 2.

Show solution

The notes are \(1,\ \tfrac98,\ \tfrac{81}{64},\ \tfrac43,\ \tfrac32,\ \tfrac{27}{16},\ \tfrac{243}{128},\ 2\). Steps: \(\tfrac98,\tfrac98,\tfrac{256}{243}\) (from 1 to \(\tfrac43\)), \(\tfrac98\) (from \(\tfrac43\) to \(\tfrac32\)), \(\tfrac98,\tfrac98,\tfrac{256}{243}\) (from \(\tfrac32\) to 2, which is again a fourth). Product: \((9/8)^5(256/243)^2=\dfrac{3^{10}}{2^{15}}\cdot\dfrac{2^{16}}{3^{10}}=2\).

Exercise 8.2Find the harmonic and arithmetic means between 1 and 3, and between 9 and 27. What musical intervals appear?

Show solution

Between 1 and 3: \(H=3/2\), \(A=2\). The steps are \(3/2\) (fifth), \(4/3\) (fourth), \(3/2\) (fifth). Between 9 and 27: \(H=27/2\), \(A=18\), steps \(3/2,\ 4/3,\ 3/2\) again. Means scale with their endpoints, so every triple interval is divided into fifth, fourth, fifth, and the fourths are then filled with tones and leimmata as before.

Exercise 8.3Show that six tones exceed an octave, and that the excess equals the amount by which twelve fifths exceed seven octaves. (This is the Pythagorean comma.) Why does it prevent Plato’s scale from closing up exactly?

Show solution

\((9/8)^6/2=3^{12}/2^{19}=531441/524288\approx1.01364\). Twelve fifths over seven octaves: \((3/2)^{12}/2^7=3^{12}/2^{19}\), the same number. Since \(3^a=2^b\) has no solution in positive integers (unique factorization), no stack of fifths or tones returns exactly to an octave. A scale built from the ratios 2 and 3 alone must contain unequal steps, here the tone and the leimma.

Lesson 09

The Platonic solids

For the general reader. There are exactly five solids whose faces are identical regular polygons with the same number meeting at every corner. Plato used four of them as the shapes of the smallest particles of fire, air, water and earth, and built those faces from two kinds of right triangle. It is the first mathematical atomism: matter is geometry, and chemical change is the regrouping of triangles.

Goals

Prove that there are at most five regular convex polyhedra.
Follow Plato’s construction from the two elementary triangles (Timaeus 53c–55c).
Do the arithmetic of transformations between elements (56c–57b).

Theorem (Euclid XIII.18, addendum)

A convex polyhedron whose faces are congruent regular \(p\)-gons, with \(q\) faces at every vertex, has \((p,q)\) equal to one of \((3,3),(4,3),(3,4),(5,3),(3,5)\): the tetrahedron, cube, octahedron, dodecahedron and icosahedron. Each exists.

Proof of the bound. At a vertex of a convex solid the face angles sum to less than \(360^\circ\). The angle of a regular \(p\)-gon is \(180^\circ(1-2/p)\), so \(q\cdot180^\circ(1-2/p)<360^\circ\), that is

\[ \frac1p+\frac1q>\frac12,\qquad p,q\ge3. \]

The only solutions are the five listed. Existence is shown by construction in Euclid XIII.13–17. A scholium to Book XIII says the Pythagoreans knew the cube, pyramid and dodecahedron, and that Theaetetus added the octahedron and icosahedron; the Byzantine lexicon Suda says Theaetetus first wrote on the five solids. Plato’s account in the Timaeus is our earliest description of all five together.

The elementary triangles

Timaeus says that all bodies have depth, depth is bounded by surfaces, and every rectilinear surface is made of triangles. He chooses two right triangles as the “most beautiful”: the isosceles right triangle, with sides \(1:1:\sqrt2\), and the half-equilateral, with sides \(1:\sqrt3:2\). From six half-equilaterals around a centre he builds an equilateral triangle; from four isosceles right triangles he builds a square. Then:

4 equilateral faces make the tetrahedron, assigned to fire, the sharpest and most mobile;
8 make the octahedron, for air;
20 make the icosahedron, for water;
6 squares make the cube, for earth, the most stable;
the fifth construction, the dodecahedron, “the god used for the whole”, perhaps because its twelve faces suggest the zodiac (55c).

Transformations

Because fire, air and water are made of the same half-equilateral triangles, their particles can be broken and the triangles reassembled. Plato gives the rules: a particle of water, broken up, can become one of fire and two of air; a particle of air can become two of fire (56d–e). Earth, built from the other triangle, can be broken up but cannot turn into the others. The bookkeeping is in faces: \(20=4+2\times8\) and \(8=2\times4\). Plato also says each type comes in a range of sizes, which explains varieties within each element.

Exercise 9.1Count faces \(F\), edges \(E\), vertices \(V\) for the five solids and check \(V-E+F=2\). Then count Plato’s elementary triangles in each of the four element solids.

Show solution

Tetrahedron \((4,6,4)\), cube \((8,12,6)\), octahedron \((6,12,8)\), dodecahedron \((20,30,12)\), icosahedron \((12,30,20)\), listed as \((V,E,F)\); in every case \(V-E+F=2\), the relation published by Euler in 1758. Elementary triangles at 6 per equilateral face and 4 per square: tetrahedron 24, octahedron 48, icosahedron 120 (half-equilaterals); cube 24 (isosceles).

Exercise 9.2Solve \(1/p+1/q>1/2\) with \(p,q\ge3\) completely, and explain why \(1/p+1/q=1/2\) corresponds to the three regular tilings of the plane.

Show solution

If \(p=3\): \(1/q>1/6\), so \(q\in\{3,4,5\}\). If \(p=4\): \(1/q>1/4\), so \(q=3\). If \(p=5\): \(1/q>3/10\), so \(q=3\). If \(p\ge6\): \(1/q>1/2-1/6=1/3\), impossible. So five solutions. Equality means the angles at a vertex sum to exactly \(360^\circ\): the faces lie flat, giving \((3,6),(4,4),(6,3)\), the triangular, square and hexagonal tilings.

Exercise 9.3Using elementary triangles, check Plato’s transformation of one water particle into one fire and two air particles. Is there any other way to redistribute the triangles of one water particle into whole particles of fire and air?

Show solution

Water has \(120\) half-equilaterals; fire 24 and air 48, so \(24+2\cdot48=120\). In general we need \(24a+48b=120\), i.e. \(a+2b=5\) with \(a,b\ge0\): \((5,0),(3,1),(1,2)\). So water could also become five fire particles, or three fire and one air. Plato names only the case \((1,2)\), though his rules allow the others.

Lesson 10

Saving the phenomena

For the general reader. The planets seem to wander: they speed up, slow down and sometimes move backwards. Plato insisted that this appearance must hide a perfectly regular reality. According to a late but influential report, he asked the mathematicians which uniform circular motions, combined, would “save the phenomena”, that is, reproduce what we see. That question shaped astronomy for two thousand years.

Goals

Know the evidence for Plato’s problem and its weaknesses.
Describe the astronomy of the Timaeus and the Laws.
Relate synodic and sidereal periods, and see why commensurability matters for the “perfect year”.

The report

Simplicius, in his sixth-century AD commentary on Aristotle’s On the Heavens, citing the second-century AD Peripatetic Sosigenes, who drew in turn on Eudemus, says Plato set this problem to those working on such matters: what uniform and ordered motions must be assumed to save the apparent motions of the planets? Eudoxus, Simplicius continues, was the first to answer it, with the homocentric spheres (Eudoxus course, Lesson 09). Some historians doubt that the formulation goes back to Plato himself, since it reaches us through three intermediaries over eight centuries. The title of Pierre Duhem’s 1908 essay Sōzein ta phainomena made it a slogan for the philosophy of science.

Plato’s own astronomy

The Timaeus (38c–39e) places the Earth at the centre. Around it are the circles of the Moon, then the Sun, then Venus and Mercury, which keep pace with the Sun, then Mars, Jupiter and Saturn. The planets were made to define and preserve the numbers of time, and time itself is the moving image of eternity. When all eight circuits return together to their starting point, a “perfect year” is completed (39d). In the Laws (821b–822a) the Athenian says it is impious to call the planets “wanderers”, since each in fact moves in one and the same circular path; the appearance of wandering is our error.

Proposition (synodic and sidereal periods)

For an outer planet with sidereal period \(P\) (time to go once around the zodiac) and a year \(Y\), the synodic period \(S\) (time between successive retrogradations) satisfies

\[ \frac1S=\frac1Y-\frac1P. \]

For an inner planet, \(1/S=1/P-1/Y\).

The Sun gains on an outer planet at the rate \(1/Y-1/P\) revolutions per unit time; a full lap of gain is one synodic cycle. The relation holds equally in a geocentric description, where \(P\) is the planet’s zodiacal period, since only relative angular speeds enter.

Exercise 10.1Mars goes round the zodiac in about 1.88 years. Find its synodic period in days.

Show solution

\(1/S=1-1/1.88=0.468\), so \(S\approx2.14\) years \(\approx780\) days. Mars retrogrades roughly every 26 months.

Exercise 10.2Plato’s “perfect year” requires all the periods to return together. Show that two uniform circular motions with periods \(T_1,T_2\) ever return simultaneously to their starting positions if and only if \(T_1/T_2\) is rational, and find the length of the common period when \(T_1/T_2=m/n\) in lowest terms.

Show solution

A joint return at time \(t>0\) needs \(t=aT_1=bT_2\) with positive integers \(a,b\), so \(T_1/T_2=b/a\) is rational. Conversely, if \(T_1/T_2=m/n\) in lowest terms, then \(nT_1=mT_2\), and this is the smallest common multiple: \(t=nT_1\). If the ratio is irrational, the two motions never realign exactly, though by Dirichlet’s approximation theorem they come arbitrarily close. The existence of a perfect year is thus a statement that all celestial periods are commensurable, which ties Plato’s cosmos back to the problem of Lesson 04.

Part III · Life and legacy

Lesson 11

Syracuse and the last years

In 367 BC Dionysius I of Syracuse died, and his son Dionysius II succeeded him. Dion, now the most powerful man at court, persuaded Plato that the young ruler could be educated in philosophy and that Syracuse might become a city governed by reason. Plato, by then about sixty, sailed in 367/366 BC. The Seventh Letter tells how it went wrong: courtiers set Dionysius against Dion, who was exiled within months, and Plato was kept in the palace as an honoured but unwilling guest before being allowed to leave.

In 361 BC, under pressure from Dionysius and from friends including Archytas, he returned once more. The quarrel over Dion’s property ended relations, and Plato was in real danger; the letter says Archytas sent a ship and an envoy to secure his release in 360 BC. At the Olympic games of 360 BC he met Dion, who was gathering support. In 357 BC Dion took Syracuse with a small force, but he ruled harshly and was murdered in 354 BC by Callippus of Athens, his companion in the campaign. Later writers linked Callippus to the Academy; the Seventh Letter insists that his friendship with Dion did not come from philosophy. The Seventh Letter, if genuine, is Plato’s defence of his conduct to Dion’s followers.

Late works and death

Plato’s last years were spent teaching and writing the Timaeus, the Philebus and the Laws, his longest work. Diogenes reports that Philip of Opus transcribed the Laws from wax tablets after Plato’s death and wrote the Epinomis, a sequel that makes astronomy the highest wisdom. Aristoxenus, a pupil of Aristotle, preserves a story told by Aristotle about Plato’s public lecture On the Good: the audience expected advice about wealth and happiness and instead heard about mathematics, numbers, geometry and astronomy, ending with the claim that the Good is One. Aristotle’s reports of Plato’s “unwritten doctrines”, in which Forms are derived from the One and an indefinite Dyad, are much debated.

Plato died in Athens in 348/347 BC, aged about eighty. Diogenes, citing Hermippus, says he died at a wedding feast; the Herculaneum papyrus, as read in 2024, says he died with a fever after an evening of music and was buried in his garden in the Academy, near a shrine of the Muses. His nephew Speusippus succeeded him as head of the school, then Xenocrates. Aristotle left Athens.

DiscussionThe Syracusan venture is often read as a failed attempt to make a philosopher-king. Using the Seventh Letter and the fate of Dion, discuss whether the episode refutes the political argument of the Republic, and how far we should trust a self-justifying letter whose authenticity is disputed.

Lesson 12

Plato among the mathematicians

Ancient mathematics

The Academy’s mathematicians produced the core of Euclid’s Elements: Theaetetus’ theory of irrationals (Book X) and of the regular solids (Book XIII), Eudoxus’ proportion (Book V) and exhaustion (Book XII). Proclus calls Euclid a Platonist and reads the Elements as aimed at the construction of the cosmic figures; the claim says more about Proclus than about Euclid, but it shows how the Timaeus was read. Plato’s problem of the planets led from Eudoxus’ spheres to the epicycles of Hipparchus and Ptolemy. Theon of Smyrna wrote a handbook of the mathematics needed to read Plato, our source for the side-and-diameter numbers.

The Renaissance and after

The Timaeus was the one Platonic dialogue read continuously in the Latin West, in the partial translation of Calcidius. In 1596 Johannes Kepler argued in the Mysterium cosmographicum that the six known planets have exactly five gaps between their spheres because there are exactly five regular solids, nested in the order octahedron, icosahedron, dodecahedron, tetrahedron, cube between Mercury and Saturn. The model was wrong, but the search for exact relations it inspired led him to his three laws. Galileo wrote his dialogues in a Platonic manner and spoke of nature written in the language of mathematics. In the twentieth century Werner Heisenberg recalled reading the Timaeus as a student and saw in its triangles an anticipation of the idea that elementary particles are defined by mathematical symmetries.

Mathematical Platonism

The view that mathematical objects exist independently of us, and that mathematicians discover rather than invent them, is named after Plato. Kurt Gödel and G. H. Hardy defended versions of it, and debates with formalists and intuitionists in the foundations of mathematics are still framed by it. Alfred North Whitehead summed up the European philosophical tradition as “a series of footnotes to Plato”.

Exercise 12.1In Kepler’s model a cube sits between the spheres of Saturn and Jupiter, with Saturn’s sphere through its vertices and Jupiter’s sphere touching its faces. Compute the ratio of the radii and compare it with the ratio of the mean distances of Jupiter (5.20 AU) and Saturn (9.54 AU).

Show solution

For a cube of side 2 the circumradius is \(\sqrt3\) (half the space diagonal) and the inradius is 1, so the ratio is \(1/\sqrt3\approx0.577\). The observed ratio is \(5.20/9.54\approx0.545\). The agreement is within about 6%, close enough to encourage Kepler, though he allowed his spheres a thickness to absorb the eccentricities of the orbits.

Exercise 12.2The dodecahedron has a pentagonal face. Show that the ratio of a regular pentagon’s diagonal to its side is \(\varphi=(1+\sqrt5)/2\), and deduce that the dodecahedron, like the square of Lesson 04, involves incommensurable lengths.

Show solution

Let pentagon \(ABCDE\) have side 1 and diagonal \(d\). Its vertices lie on a circle, so Ptolemy’s theorem applies to the cyclic quadrilateral \(ABCD\) and gives \(AC\cdot BD=AB\cdot CD+BC\cdot AD\), that is \(d^2=1+d\). So \(d=\varphi\), a root of \(x^2-x-1=0\). If \(\varphi=p/q\) in lowest terms, then \(p^2=pq+q^2\), so \(q\mid p^2\) and \(p\mid q^2\), forcing \(p=q=1\), which is false. So diagonal and side are incommensurable; Euclid XIII.6 says the segments of a line cut in extreme and mean ratio are irrational “apotomes”.

Final projectChoose one of the Meno, the Theaetetus passage, the Divided Line, or the Timaeus solids. Write two pages for a strong undergraduate that set out the text in paraphrase, reconstruct its mathematics in modern notation, and state what an ancient reader would have needed to know to follow it.

Reference

Sources and further reading

Ancient sources

Plato, Complete Works, ed. J. M. Cooper, Hackett, 1997. Greek and English texts at the Perseus Digital Library.
Diogenes Laertius, Lives of the Eminent Philosophers, Book III, trans. R. D. Hicks (Loeb). Wikisource text.
Proclus, A Commentary on the First Book of Euclid’s Elements, trans. G. R. Morrow, Princeton, 1970.
Theon of Smyrna, Mathematics Useful for Understanding Plato, trans. R. Lawlor and D. Lawlor, 1979.
Euclid, Elements, trans. T. L. Heath; D. E. Joyce’s online edition.

Biography and the Academy

Plato, MacTutor History of Mathematics.
Plato and Platonic Academy, Wikipedia.
R. Kraut, Plato, Stanford Encyclopedia of Philosophy.
D. Nails, The People of Plato, Hackett, 2002.
R. Pearse, More sections of Philodemus’ history of the philosophers discovered (2024), on the Herculaneum readings.

Mathematics and cosmology

D. Zeyl and B. Sattler, Plato’s Timaeus, Stanford Encyclopedia of Philosophy.
Analogy of the divided line, Wikipedia.
T. L. Heath, A History of Greek Mathematics, vol. 1, Oxford, 1921, ch. IX (“The Academy”).
D. H. Fowler, The Mathematics of Plato’s Academy: A New Reconstruction, 2nd ed., Oxford, 1999.
W. Knorr, The Evolution of the Euclidean Elements, Reidel, 1975 (on Theodorus and Theaetetus).
F. M. Cornford, Plato’s Cosmology, Routledge, 1937.
P. Duhem, To Save the Phenomena (1908), trans. E. Doland and C. Maschler, Chicago, 1969.