CO 480: Vignettes in the History of Mathematics

Steven Furino; Logan Crew

Estimated study time: 1 hr 19 min

Table of contents

This course examines the people, places, and problems that shaped the mathematics of antiquity and the early modern period through a series of vignettes. Each vignette focuses on a particular mathematician working on a particular problem within a particular historical and geographical setting. The course was developed by Steven Furino and taught in Spring 2021 by Logan Crew, Research Assistant Professor in the Department of Combinatorics & Optimization at the University of Waterloo.

Steven Furino, course author

Course author Steve Furino (left) and instructor Logan Crew (right).

Logan Crew, course instructor

Module 1: Tutte (1917–2002 CE)

Course Introduction

CO 480 begins not at the beginning of mathematical history but in the recent past, with a mathematician whose life and work are directly connected to the University of Waterloo. William Thomas Tutte was born in Newmarket, England, in 1917 and died in 2002. He spent the last four decades of his career at Waterloo, where he became one of the foundational figures in the Faculty of Mathematics. By starting with Tutte, the course establishes a direct link between the student and the broader story of mathematics: this history is not merely ancient or abstract but lives in the very institution where it is being studied.

World War II and Bletchley Park

The Second World War generated enormous demand for cryptographic talent, and Britain responded by assembling an extraordinary group of mathematicians, linguists, and engineers at Bletchley Park, a country estate in Buckinghamshire. Their task was to break the encrypted communications of Nazi Germany. The most famous success was the breaking of the Enigma machine cipher by Alan Turing and his colleagues. But there was a second, more complex cipher system that Tutte would come to dominate.

The Germans used a teleprinter cipher machine called the Lorenz SZ42 for their highest-level strategic communications. The British codebreakers nicknamed this system Fish, and the specific Lorenz traffic Tunny. Unlike Enigma, the Fish messages were transmitted by radio and intercepted without the Allies ever obtaining the machine itself. In August 1941, an operator made a catastrophic error and retransmitted a long message with only slightly different settings — giving British analyst John Tiltman enough material to recover some of the keystream. Tutte, who had joined Bletchley Park after completing a master’s degree in chemistry at Cambridge, was handed Tiltman’s work. Over the following months, using nothing but pencil and paper and extraordinary mathematical intuition, Tutte reconstructed the logical structure of the Lorenz machine and determined the number of its wheels and their interconnections — all without ever seeing the device.

This achievement, described as the greatest intellectual feat of the war, enabled the construction of Colossus, the world’s first programmable electronic computer, built to speed up the decryption of Lorenz messages. Colossus came online in 1944 and provided intelligence crucial to the success of the D-Day landings.

William T. Tutte

William Thomas Tutte was born on 14 May 1917 at Fitzroy House, Newmarket, Sussex, the son of a gardener and a housekeeper. He won a scholarship to Cambridge, where he read chemistry, but his mathematical gifts were evident from his undergraduate days: together with three fellow students — R. Leonard Brooks, Cedric Smith, and Arthur Stone — he discovered a beautiful result on squared squares (partitioning a square into smaller squares of distinct sizes) and proved the now-famous theorem that every 3-connected planar graph has a unique embedding in the sphere.

After the war, Tutte returned to Cambridge as a Fellow of Trinity College and completed a PhD in mathematics in 1948 under the supervision of Shaun Wylie. He then joined the faculty of the University of Toronto, where he rose to international prominence in combinatorics. In 1962 he moved to Waterloo, where he played a central role in establishing the Faculty of Mathematics and the Department of Combinatorics and Optimization.

Tutte’s mathematical legacy is vast. His work on graph theory — in particular his theory of matroids and his contributions to the four-colour problem — reshaped the field. The Tutte polynomial, a two-variable polynomial associated to any graph, unifies many combinatorial invariants and has applications ranging from knot theory to statistical mechanics. He was elected a Fellow of the Royal Society in 1987. His Royal Society biographical memoir, written by D. H. Younger and available as a course reading, offers a vivid portrait of a man of quiet brilliance and deep modesty.

Cryptography

The cryptography studied in this module centres on substitution ciphers, in which each letter of plaintext is replaced by another letter according to a fixed or varying key. The simplest form is the Caesar cipher, a cyclic shift of the alphabet by a fixed number of positions; Julius Caesar used a shift of three. More sophisticated is the Vigenère cipher, introduced in the sixteenth century, which uses a repeating keyword to select different shift values at different positions in the message.

To attack a Vigenère cipher, the analyst first determines the length of the key — typically using the Kasiski test or the index of coincidence — and then treats the message as a set of interleaved Caesar ciphers, each of which can be broken by frequency analysis. Since in English the letter e occurs roughly 12.7% of the time and z only 0.07%, a sufficiently long ciphertext will reveal the key letter by letter.

Assignment 1 includes actual Vigenère ciphertexts encrypted from English texts, asking students to decrypt the first line and identify the original source. The cipher text preserves the natural letter frequencies of English once the key period is known, and frequency analysis against the expected English distribution allows systematic recovery of each key character.

Looking Ahead: It is now time to turn from recent history, from someone within living memory, to the origins of mathematics thousands of years ago. But before we can do that, we need to take a look at the ancient Mediterranean civilizations where all of our stories take place.

Module 2: Ancient Mediterranean

Ancient Mediterranean

Mathematics did not emerge in a vacuum. It grew from the administrative, astronomical, and architectural needs of the great ancient civilizations that arose around the Mediterranean basin. To situate the mathematicians studied in this course, we must first understand the geography and chronology of the ancient world.

The Ancient Civilizations Timeline provided in the course maps the major historical dates for Egypt, Mesopotamia, Persia, Greece, Rome, and Islam from 3000 BCE to 1200 CE. This span of four thousand years encompasses almost the entirety of the mathematical tradition we study: from the Egyptian scribes who computed unit fractions around 1650 BCE, through the Greek geometers of Alexandria in 300 BCE, to the Islamic scholars who preserved and extended that inheritance in Baghdad around 800 CE.

The Fertile Crescent — the arc of arable land running from the Persian Gulf through Mesopotamia (modern Iraq) to the Nile Valley — was the cradle of the earliest writing and arithmetic. Agriculture in river valleys generates surplus; surplus generates administration; administration generates numbers. The earliest written numerals, from Sumerian clay tablets dated around 3500 BCE, are records of grain inventories, not philosophical speculations.

Egypt developed along the Nile, which flooded annually and deposited fertile silt across the delta. The flooding also erased property boundaries, making land surveying a recurring practical necessity and stimulating the development of geometry. The pharaonic state required large-scale construction — the Great Pyramid of Giza (c. 2560 BCE) required the coordinated labour of tens of thousands of workers and the transport of millions of stone blocks — and with it came a sophisticated administrative apparatus that demanded accurate calculation.

Greece was not a single state but a collection of city-states scattered across the Aegean. The city that matters most for the history of mathematics is Alexandria, founded by Alexander the Great in 331 BCE on the Egyptian Mediterranean coast. After Alexander’s death in 323 BCE, his general Ptolemy took control of Egypt and founded the Ptolemaic dynasty, which ruled for three centuries. The Ptolemies established the famous Library of Alexandria and the Mouseion (a kind of ancient research institute), which attracted scholars from across the Greek world. It is in this setting that Euclid, Archimedes, Apollonius, and Hypatia all worked.

Looking Ahead: With some geography and broad context under our belt, we begin at the beginning, with those societies where agriculture, writing, great cities, and administrative requirements created the demand for a new discipline: mathematics. We go to Egypt.

Module 3: Egypt

Egyptian Scribes

Egyptian mathematics is known primarily through two papyri. The Rhind Mathematical Papyrus (also called the Ahmes Papyrus), dating to around 1650 BCE but copied from an earlier Middle Kingdom original, is the most extensive source. It was purchased by the Scottish antiquarian Alexander Henry Rhind in Luxor in 1858 and later acquired by the British Museum. The scribe who copied it identified himself as Ahmes (Ahmose), and the text opens with a remarkable promise: the scroll will give an “accurate reckoning for inquiring into things, and the knowledge of all things, mysteries and secrets.”

The Rhind Papyrus contains 87 mathematical problems, covering arithmetic, algebra, and geometry. The Moscow Mathematical Papyrus, dating to roughly the same period, adds a further 25 problems, including a celebrated calculation of the volume of a truncated pyramid that implies knowledge equivalent to the formula \(V = \frac{h}{3}(a^2 + ab + b^2)\).

Egyptian scribes worked in hieratic script — a cursive form of hieroglyphics adapted for use with reed pen and papyrus. The hieratic numeration system was additive and decimal: separate symbols existed for units, tens, hundreds, thousands, and so on, and numerals were formed by repeating the appropriate symbols. Unlike the positional system we use today, where the value of a digit depends on its location, the Egyptian system required separate symbols for each power of ten. Large numbers could thus become unwieldy to write.

The Emergence of Mathematics

The most distinctive feature of Egyptian arithmetic was its treatment of fractions. Except for the special fraction \(\frac{2}{3}\), which had its own dedicated symbol, Egyptian mathematicians worked exclusively with unit fractions — fractions of the form \(\frac{1}{n}\). To express any other fraction, they decomposed it into a sum of distinct unit fractions. For example, rather than writing \(\frac{2}{5}\), an Egyptian scribe would write \(\frac{1}{3} + \frac{1}{15}\).

The Rhind Papyrus begins with a table expressing \(\frac{2}{n}\) as sums of distinct unit fractions for all odd \(n\) from 3 to 101. This 2/n table was a computational aid: since Egyptian multiplication was performed by repeated doubling, scribes frequently needed to halve odd multiples and required a precomputed table to do so. For example:

\[\frac{2}{5} = \frac{1}{3} + \frac{1}{15}\]\[\frac{2}{7} = \frac{1}{4} + \frac{1}{28}\]\[\frac{2}{11} = \frac{1}{6} + \frac{1}{66}\]

The general identity \(\frac{2}{n} = \frac{1}{\lceil n/2 \rceil} + \frac{1}{n \lceil n/2 \rceil}\) yields such decompositions, though the Egyptians used more economical forms in many cases. Students in this course derive the general identity and use it to express, for example, \(\frac{9}{23}\) as a sum of distinct unit fractions.

Egyptian multiplication itself deserves mention. To multiply 13 by 17, a scribe would double 17 repeatedly — 17, 34, 68, 136 — and then identify which powers of 2 (1, 4, 8) sum to 13: since \(13 = 8 + 4 + 1\), the product is \(136 + 68 + 17 = 221\). This is precisely the binary algorithm, and it is entirely rigorous; indeed, it is essentially the method used by modern computer hardware.

Geometry in the Rhind Papyrus is primarily mensurational. Problem 50 states that a circular field has a diameter of 9 khet and asks for its area, answering with \(64\) square khet — obtained by the rule: subtract one-ninth of the diameter, then square the result. This gives \(\left(\frac{8}{9} \cdot 9\right)^2 = 64\), which implies an effective value of \(\pi \approx \frac{256}{81} \approx 3.16\) — a remarkably good approximation for 1650 BCE.

A famous problem in the Rhind Papyrus (Problem 79) involves a geometric series: seven houses each contain seven cats; each cat has caught seven mice; each mouse would have eaten seven ears of wheat; each ear would have produced seven hekat of grain. The total of all these quantities is computed as \(7 + 7^2 + 7^3 + 7^4 + 7^5 = 19607\). This is not a practical problem but appears to be a mathematical puzzle — perhaps the ancestor of the medieval rhyme “As I was going to St. Ives.”

Looking Ahead: In Egypt, mathematics was essentially a utilitarian art. In Greece it would introduce something wholly new that would define later mathematics to this day: proof.

Module 4: Euclid (325–265 BCE): The Elements

Alexandria, 300 BCE

By 300 BCE, Alexandria had become the intellectual capital of the Mediterranean world. The Ptolemaic rulers, following the precedent set by Alexander himself, were enthusiastic patrons of learning. The Library of Alexandria, at its height holding perhaps 400,000 to 700,000 papyrus rolls, was the closest thing the ancient world produced to a universal research library. Scholars from across the Greek-speaking world came to Alexandria to study, debate, and write.

It was in this extraordinary environment that Euclid flourished. We know very little about Euclid’s life: no reliable birth or death dates survive, no portrait, no contemporary memoir. The only anecdote preserved — that when King Ptolemy asked if there were a shorter way to learn geometry, Euclid replied “there is no royal road to geometry” — is almost certainly apocryphal. What matters is his work, which shaped the course of mathematics for more than two thousand years.

Introduction to Euclid and the Elements

Euclid’s Elements (Stoicheia in Greek) is a systematic treatment of mathematics in thirteen books, covering plane geometry, proportion, number theory, three-dimensional geometry, and the theory of incommensurable (irrational) magnitudes. It is not a collection of new discoveries — much of the material derives from earlier Greek mathematicians including Eudoxus, Theaetetus, and Pythagoras — but an unprecedented synthesis organized by a single governing principle: deduction from explicit axioms.

The Elements begins with Definitions, Postulates, and Common Notions. The definitions specify the objects of geometry (a point, a line, a circle). The five Postulates state what constructions are permitted:

A straight line may be drawn from any point to any other point.
A finite straight line may be extended continuously in a straight line.
A circle may be described with any centre and any radius.
All right angles are equal to one another.
If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, then the two lines, if extended indefinitely, meet on that side.

The Common Notions assert basic logical relationships: things equal to the same thing are equal to each other; the whole is greater than the part; and so on.

The Fifth Postulate — the Parallel Postulate — stands apart from the first four in its complexity. Euclid himself avoided using it as long as possible, not invoking it until Proposition 29 of Book I. For two millennia, mathematicians suspected it ought to be provable from the other four. In the nineteenth century, Bolyai, Lobachevsky, and Gauss independently showed that removing the parallel postulate yields consistent non-Euclidean geometries.

The Mathematics of the Elements

Book I of the Elements develops the fundamental theory of triangles and parallelograms and culminates in what is perhaps the most famous theorem in mathematics:

Proposition 47 (Pythagorean Theorem): In right-angled triangles, the square on the side opposite the right angle is equal to the sum of the squares on the sides containing the right angle.

In modern notation: if a right triangle has legs \(a\) and \(b\) and hypotenuse \(c\), then \(a^2 + b^2 = c^2\). Euclid’s proof is entirely geometric, using the areas of squares and parallelograms — there is no algebra in the Elements.

Book II contains the geometric algebra that Greek mathematicians used in place of symbolic algebra. Proposition 5, for example, proves that \((a+b)(a-b) + b^2 = a^2\) — a result we would handle algebraically but that Euclid establishes by dissecting rectangles.

Books V and VI develop the theory of proportion due to Eudoxus, which handles ratios between magnitudes (including irrational ones) without requiring a numerical definition of irrational numbers. This is one of the most technically sophisticated parts of the Elements.

Book X classifies incommensurable magnitudes — what we would call irrational numbers — into thirteen distinct types. Euclid proves, for instance, that the diagonal and side of a square are incommensurable: there is no unit that measures both exactly. The proof is the classic proof by contradiction that \(\sqrt{2}\) is irrational.

Books XI–XIII treat solid geometry and conclude with the construction and classification of the five Platonic solids (tetrahedron, cube, octahedron, dodecahedron, icosahedron) and the proof that exactly five regular polyhedra exist.

Thomas L. Heath’s 1908 translation and commentary, available through the Internet Archive, remains the standard English edition. Euclid’s Elements was the second most printed book in the Western world after the Bible from the invention of printing in the fifteenth century through the early twentieth century.

Looking Ahead: It seems that the Greeks could prove almost anything using straightedge and compass. Almost. It turns out there were some things the Greeks could not prove using straightedge and compass. It took 2,000 years to prove that, though.

Module 5: Euclid: The Delian Problem

The Delian Problem

Among the three classical problems of Greek geometry, the most poetically situated is the Delian problem, also called the problem of doubling the cube. According to ancient tradition, the Athenians, suffering from a plague, consulted the oracle at Delos and were told that their troubles would end if they doubled the size of the cubic altar of Apollo. The Athenians doubled each edge of the cube — obtaining an altar eight times as large — and the plague continued. The oracle clarified: the volume, not the edge lengths, must double. This requires constructing an edge of length \(\sqrt[3]{2}\) times the original, using only straightedge and compass.

The ancient Greeks never solved this problem. Many proposed mechanical solutions — Menaechmus used intersecting conics, Nicomedes used a curve called the conchoid — but these required tools beyond the Euclidean straightedge and compass. The question of whether a pure straightedge-and-compass solution exists was not answered until 1837, when Pierre Wantzel proved definitively that it does not.

The key insight is to characterize, algebraically, exactly which lengths are constructible by straightedge and compass. Starting from a segment of length 1, a straightedge-and-compass construction can only:

Add, subtract, multiply, and divide existing lengths.
Extract square roots of existing lengths.

This means the constructible lengths are precisely those obtainable from the rationals \(\mathbb{Q}\) by a finite sequence of field operations and square-root extractions. Algebraically, they lie in field extensions of \(\mathbb{Q}\) whose degree over \(\mathbb{Q}\) is a power of two.

Now consider \(\sqrt[3]{2}\). It satisfies the polynomial \(x^3 - 2 = 0\), which is irreducible over \(\mathbb{Q}\) (by Eisenstein’s criterion). Therefore \([\mathbb{Q}(\sqrt[3]{2}) : \mathbb{Q}] = 3\). Since 3 is not a power of 2, \(\sqrt[3]{2}\) is not constructible, and the Delian problem has no straightedge-and-compass solution.

This argument relies on the concept of a field — a set equipped with addition, subtraction, multiplication, and division (by nonzero elements) satisfying the usual algebraic laws. The real numbers \(\mathbb{R}\) form a field. So do the rationals \(\mathbb{Q}\). The complex numbers \(\mathbb{C} = \{x + yi \mid x, y \in \mathbb{R}\}\), where \(i^2 = -1\), also form a field.

To verify that \(\mathbb{C}\) is a field, one checks:

Additive group: \((x + yi) + (a + bi) = (x+a) + (y+b)i\) satisfies commutativity, associativity, the zero element is \(0 + 0i\), and negatives exist.
Multiplicative group of nonzero elements: \((x+yi)(a+bi) = (xa-yb) + (xb+ya)i\); multiplicative identity is \(1 + 0i\); the inverse of \(x+yi\) is \(\frac{x}{x^2+y^2} - \frac{y}{x^2+y^2}i\).
Distributivity: inherited from the real numbers.

More generally, any set of the form \(F_k = \{x + y\sqrt{k} \mid x, y, k \in F\}\) for a field \(F\), with \(\sqrt{k} \notin F\), is a field extension of \(F\) of degree 2. A constructible number lies in a tower \(F_0 = \mathbb{Q} \subset F_1 \subset \cdots \subset F_n\) where each extension has degree 2.

The course proves that intersections of lines and circles always produce points in at most a degree-2 extension of the field generated by the existing points’ coordinates. This inductive argument, formalizing the geometry of constructions, shows that constructible lengths have degree over \(\mathbb{Q}\) equal to a power of 2 — and thus \(\sqrt[3]{2}\) is unattainable.

Looking Ahead: We next meet a great master, one of the greatest mathematicians of all time, Archimedes. Using results of Euclid and his own fantastic ingenuity, Archimedes estimated pi and we will walk through his demonstration with him.

Module 6: Archimedes (c. 287–c. 212 BCE)

Rome and Carthage

The political backdrop to Archimedes’ life is the struggle between Rome and Carthage for control of the western Mediterranean. The three Punic Wars (264–146 BCE) consumed the resources of both powers and dragged in the Greek city-states of Sicily as pawns and prizes. Syracuse, the greatest Greek city on Sicily, initially allied with Carthage, placing it directly in the path of Roman expansion. This geopolitical context explains the otherwise obscure fact that one of the greatest mathematicians of antiquity was killed by a Roman soldier during the sack of his city.

Carthage, founded by Phoenician colonists from Tyre around 814 BCE, dominated the western Mediterranean coast of North Africa and controlled key trade routes. Rome’s first confrontation with Carthage began in 264 BCE over control of the Strait of Messina. The resulting conflict drew in Syracuse under its king Hiero II, who initially opposed Rome but later made peace and became a Roman ally. It was under Hiero’s patronage that Archimedes flourished.

Archimedes

Archimedes of Syracuse was born around 287 BCE, the son of Pheidias the astronomer. He may have studied at Alexandria — he corresponds with Alexandrian scholars and dedicates works to Eratosthenes, the Library’s chief librarian — but he spent most of his life in Syracuse. Ancient sources credit him with an astonishing range of achievements: the invention of the Archimedean screw (used to raise water), the design of compound pulleys and war machines used to defend Syracuse against Roman attack, the law of the lever, the principle of buoyancy (the famous Eureka story), and mathematics so advanced that it would not be equalled for nearly two thousand years.

He died in 212 BCE when the Romans, under Marcellus, finally breached Syracuse’s defences after a two-year siege. A Roman soldier killed him despite orders from Marcellus to spare him. The exact circumstances are disputed — one tradition has him drawing mathematical figures in sand when interrupted, another has him carrying mathematical instruments to Marcellus — but the symbolic force of the image is clear: the Roman soldier represents a civilization that valued military power over intellectual achievement.

Archimedes’ Works

Archimedes’ surviving treatises include On the Sphere and Cylinder, On the Measurement of a Circle, On Spirals, On Conoids and Spheroids, On the Equilibrium of Planes, On Floating Bodies, The Quadrature of the Parabola, and the remarkable Sand-Reckoner.

Estimating π. On the Measurement of a Circle contains Archimedes’ famous bounds on \(\pi\):

\[\frac{223}{71} < \pi < \frac{22}{7}\]

He obtained these by calculating the perimeters of regular polygons with 96 sides inscribed in and circumscribed about a circle of diameter 1. The key recurrence is: if \(s_n\) denotes the side length of a regular \(n\)-gon inscribed in a unit-radius circle, then:

\[s_{2n} = \sqrt{2 - \sqrt{4 - s_n^2}}\]

Starting from a hexagon (\(s_6 = 1\), one doubles repeatedly to 12, 24, 48, and 96 sides, computing perimeters at each step. The circumscribed polygon gives an upper bound on \(\pi\) and the inscribed polygon a lower bound. The circumscribed case uses the relation:

\[S_{2n} = \frac{2 S_n s_n}{S_n + s_n}\]

where \(S_n\) is the side of the circumscribed \(n\)-gon. These recurrences require computing square roots to high precision using only rational arithmetic, and Archimedes does so with remarkable skill, using well-chosen rational approximations throughout.

The Sand-Reckoner. This treatise, addressed to King Gelon II of Syracuse, tackles the question: how many grains of sand would fill the universe? Archimedes’ motivation was partly philosophical (to refute the claim that large numbers cannot be named) and partly astronomical (to work with the dimensions of the heliocentric universe proposed by Aristarchus of Samos).

The Sand-Reckoner opens with a striking cosmological observation: Aristarchus of Samos — writing a full seventeen centuries before Copernicus — proposed that the fixed stars and the sun remain unmoved and that the earth revolves around the sun in a circular orbit, with the sphere of the fixed stars vastly larger than what contemporaries called the universe. Archimedes takes Aristarchus’s model as his working hypothesis to obtain the largest possible estimate of the universe’s size.

Archimedes constructs a system for naming very large numbers, essentially introducing a positional system in base \(10^8\). He then estimates, using astronomical measurements and geometric arguments, the diameter of the universe and the size of a grain of sand, and concludes that the number of grains needed to fill the universe does not exceed \(10^{63}\) — a finite and nameable quantity.

The Archimedes Palimpsest. In 1906 the Danish philologist Johan Ludwig Heiberg discovered a medieval prayer book in Constantinople whose parchment had been washed and reused — it was a palimpsest, and the underlying text was a tenth-century copy of Archimedes’ works. This codex, now known as the Archimedes Palimpsest, preserves the only surviving copies of The Method of Mechanical Theorems and Stomachion. The Method is particularly extraordinary: it reveals the heuristic technique Archimedes used to discover results before proving them rigorously, treating geometric figures as if they were composed of infinitely thin slices balanced on a lever — a precursor of integral calculus nearly two millennia before Newton.

Looking Ahead: We have covered Euclidean geometry and Archimedean genius, but there is one area of ancient and modern geometry we have not yet explored — one that you might have done in high school — conic sections.

Module 7: Apollonius (c. 240–c. 190 BCE)

A Brief History of Conic Sections in Greek Mathematics

Conic sections — the curves obtained by slicing a cone with a plane — were first studied by Menaechmus around 350 BCE, a pupil of Eudoxus. Menaechmus used intersections of parabolas and hyperbolas to solve the Delian problem mechanically. Euclid wrote a treatise on conics (now lost) that is known only through later references. Aristaeus, a contemporary of Euclid, also wrote on solid loci (conic sections). But the definitive ancient treatment is Apollonius’s Conics in eight books, of which seven survive.

Before Apollonius, the three conic sections were obtained by cutting a right circular cone with a plane perpendicular to a generating line — each section from a different type of cone (acute, right-angled, or obtuse). Apollonius unified them by showing that all three sections can be obtained from the same double cone by varying the angle of the cutting plane, and he gave the curves the names ellipse, parabola, and hyperbola that we still use. These names derive from the Greek words for “falling short,” “placed beside,” and “thrown beyond,” referring to the relationship between the ordinate squared and a certain rectangle in each case.

Apollonius

Apollonius of Perga was born around 240 BCE in Perga (in modern Turkey) and studied in Alexandria, probably under Euclid’s successors. He later worked in Pergamon and Alexandria. His Conics was originally in eight books; Books I–IV survive in Greek, Books V–VII in Arabic translation, and Book VIII is lost. The work was dedicated to Eudemus of Pergamon and later to one Attalus, a scholar in Pergamon.

Beyond the Conics, Apollonius wrote on tangencies (the Problem of Apollonius: given three circles, construct a circle tangent to all three), on cutting a ratio, on cutting an area, on determinate section, and on inclinations. These works survive either in fragments or through later commentaries. His astronomical work included a theory of epicycles and eccentrics — the geometric tools later developed by Ptolemy into a model of planetary motion that dominated astronomy until Copernicus.

Conic Sections

Apollonius describes each conic section by what he calls the symptoma — an intrinsic geometric property of the curve expressed as a relationship between ordinates and abscissas.

The parabola is defined by the property that, for any point on the curve, the square of the ordinate equals the product of the latus rectum \(p\) and the abscissa:

\[y^2 = px\]

In Heath’s notation, \(p\) is the length of the parameter — a specific line segment associated with the vertex of the conic. This is equivalent to the modern form \(y^2 = 4ax\) with \(p = 4a\).

The ellipse satisfies:

\[y^2 = px - \frac{p}{d} x^2\]

where \(d\) is the length of the diameter. The term \(\frac{p}{d}x^2\) is “subtracted” — the curve “falls short” (ellipsis) of the parabola.

The hyperbola satisfies:

\[y^2 = px + \frac{p}{d} x^2\]

The additional term is “thrown beyond” (hyperbolē) — the curve exceeds the parabola. The course proves this theorem of Apollonius directly, following Heath’s translation, and then verifies that the resulting curve corresponds to the modern equation \(y^2 = px + \frac{p}{d} x^2\).

Apollonius’s definition of a circle provides another perspective. Apollonius showed that the locus of points \(P\) such that the ratio \(AP/PB = r\) for two fixed points \(A\) and \(B\) and a fixed ratio \(r \neq 1\) is a circle. This is the circle of Apollonius. The proof proceeds by showing that there are exactly two points \(P_1, P_2\) on the line \(AB\) satisfying the ratio, computing their positions using the constraint \(BP_i = r \cdot AP_i\), identifying the midpoint \(O\) of \(P_1 P_2\), and then using the Law of Cosines to show that every point \(Q\) satisfying \(BQ/AQ = r\) lies at distance \(OP_1\) from \(O\).

Looking Ahead: All good things come to an end. After a long twilight, Greek scholarship came to an ignominious and brutal end. We go next to this sad finale and its echoes in today’s world.

Module 8: Hypatia (c. 355–c. 415 CE)

Hypatia

Hypatia was born in Alexandria around 355 CE, the daughter of Theon of Alexandria, a mathematician and astronomer who worked on editions of Euclid and Ptolemy. Hypatia collaborated with her father and eventually became the leading intellectual figure in Alexandria, a teacher of Neoplatonist philosophy and mathematics to students who came from across the Mediterranean world. Her students included Synesius of Cyrene, whose surviving letters to Hypatia constitute the most direct evidence for her life and influence.

Hypatia wrote commentaries on the major mathematical works of her day, including Apollonius’s Conics and Diophantus’s Arithmetica. Her commentary on Diophantus’s work on solving polynomial equations in integers — Diophantine equations — helped preserve and extend that tradition. She also constructed astronomical instruments, including an astrolabe, described in letters from Synesius. No works survive under her name alone; historians believe her contributions were incorporated into her father’s editions.

The political situation in Alexandria was increasingly unstable in the early fifth century. The Roman Empire had formally adopted Christianity in 380 CE under Theodosius I, and the old pagan philosophical traditions, however admired, had become suspect. Orestes, the Roman prefect of Alexandria, maintained a relatively tolerant administration, while Cyril, the Archbishop of Alexandria from 412, was an aggressive advocate for church authority over civic life. Hypatia, as an advisor to Orestes and a figure associated with pagan learning, became a target. In 415 CE she was seized by a mob of Christian zealots, murdered, and her body burned. Her death is widely regarded as marking the end of the great tradition of Alexandrian scholarship.

The 2009 film Agora, directed by Alejandro Amenábar and listed among the course’s additional resources, dramatizes Hypatia’s life and the destruction of the Library of Alexandria. Historians debate the extent to which the Library’s destruction was a single dramatic event rather than a gradual decline, but the cultural rupture is real: the torch of Greek mathematical learning was not kept alight in the Western Mediterranean but was preserved in the East.

Looking Ahead: The darkness that fell upon the West was, thankfully, matched by a new intellectual light from the East. A superb centre of learning in Baghdad helped secure, extend, and transmit Greek mathematical accomplishments across both time and space.

Module 9: Al-Khwarizmi (c. 780–c. 850 CE)

Al-Khwarizmi

Muhammad ibn Musa al-Khwarizmi was born around 780 CE in Khwarazm (in modern Uzbekistan) and flourished in Baghdad under the patronage of the Abbasid Caliph al-Ma’mun. He worked at the Bayt al-Hikma (House of Wisdom), a library and translation centre established in Baghdad that served a function analogous to the Library of Alexandria five centuries earlier. The House of Wisdom commissioned translations of Greek scientific and philosophical works into Arabic, engaged scholars from across the Islamic world, and produced original research in mathematics, astronomy, medicine, and geography.

Al-Khwarizmi’s name is the source of the word algorithm: the Latinized form Algoritmi (from the book title Algoritmi de numero Indorum, “Al-Khwarizmi on the Indian numerals”) was used for the procedures described therein, and from this the word algorismus and eventually algorithm evolved. His other great work gave us the word algebra: al-jabr (“completion” or “restoration”), from the treatise title al-Kitab al-mukhtasar fi hisab al-jabr wal-muqabala — translated as The Compendious Book on Calculation by Completion and Balancing, written around 820 CE.

Al-Khwarizmi’s Works

Hindu-Arabic numerals. Al-Khwarizmi’s treatise on Indian arithmetic (de numero Indorum) described the positional decimal numeral system with nine digits and zero that had been developed in India. This system — the system we use today — eventually replaced Roman numerals in Europe, though the transition took several centuries. The key features are positional notation (the value of a digit depends on its position), a symbol for zero, and simple algorithms for the four arithmetic operations. Multiplying two three-digit numbers in Roman numerals is a laborious task; in the decimal positional system, the same multiplication follows a mechanical procedure taught to every schoolchild.

Algebra. The Kitab al-mukhtasar systematically treats quadratic equations. Al-Khwarizmi classifies equations into six types according to which combinations of squares (\(x^2\), roots (\(x\), and numbers appear. His solution method for equations of the form \(x^2 + bx = c\) (where \(b, c > 0\) is geometric. He argues as follows: draw a square of side \(x\), representing \(x^2\). Attach four rectangles, each of dimensions \(x \times \frac{b}{4}\), to the four sides; together these rectangles contribute \(bx\). Fill in the four corner squares of side \(\frac{b}{4}\) to complete a larger square of side \(x + \frac{b}{2}\). The total area is:

\[\left(x + \frac{b}{2}\right)^2 = c + 4 \cdot \left(\frac{b}{4}\right)^2 = c + \frac{b^2}{4}\]

Taking square roots: \(x + \frac{b}{2} = \sqrt{c + \frac{b^2}{4}}\), so:

\[x = \sqrt{c + \frac{b^2}{4}} - \frac{b}{2}\]

This is precisely completing the square, and it is the geometric argument behind the algebraic technique of the same name. The key insight is that this construction always yields a positive solution when \(b\) and \(c\) are positive — a fact confirmed algebraically using the quadratic formula, since the discriminant \(b^2 + 4c > 0\) guarantees a real positive root.

Al-Khwarizmi also treats equations of the form \(x^2 + b = ax\) (squares and number equal roots), where the geometric demonstration is more subtle. In Frederick Rosen’s 1831 translation, the section “Demonstration of the Case: a square and twenty-one Dirhems are equal to ten Roots” illustrates the solving of \(x^2 + 21 = 10x\), which has two positive solutions (\(x = 3\) and \(x = 7\). Al-Khwarizmi’s argument identifies both solutions geometrically — the two positions at which the square can be completed in different orientations.

Looking Ahead: In leaping from Hypatia to al-Khwarizmi we leapt in time, distance, and culture: 400 years in time from Greek Alexandria on the Mediterranean to Islamic Baghdad in Persia. Our next unit begins with another large leap — another 400 years — this time to Islamic Spain at the end of the medieval period, where we will meet European Christian scholars translating Islamic works.

Module 10: Al-Andalus

Al-Andalus

Al-Andalus was the name given to the parts of the Iberian Peninsula (modern Spain and Portugal) under Islamic rule from 711 to 1492 CE. At its height in the tenth century, the Umayyad Caliphate of Córdoba controlled most of the peninsula and was one of the most sophisticated civilizations in the world. The capital, Córdoba, had a population of perhaps 500,000 — larger than any city in contemporary Christian Europe — and boasted a library rumoured to hold 400,000 volumes. The Caliph Abd al-Rahman III and his son al-Hakam II were enthusiastic patrons of learning.

The significance of al-Andalus for the history of mathematics is primarily one of transmission. When the intellectual life of the Greek and Islamic world eventually reached Western Europe, it did so largely through the translation centres of southern Spain and Sicily. The most important of these was Toledo, which had been the Visigothic capital and retained a mixed population of Muslims, Christians, and Jews even after its reconquest by Castile in 1085. The existence of Arabic-speaking scholars alongside Latin-speaking clerics made Toledo the ideal location for translation.

Gerard of Cremona (c. 1114–1187) is the most prolific of the Toledo translators. He is credited with translating approximately 87 works from Arabic into Latin, including Ptolemy’s Almagest, al-Khwarizmi’s Algebra, Euclid’s Elements (from the Arabic version), and works of Aristotle, Galen, and Avicenna. These translations gave Western European scholars access, for the first time in centuries, to the full breadth of ancient and Islamic science and mathematics.

The transmission to Europe also occurred through Fibonacci (Leonardo of Pisa, c. 1170–c. 1250), who learned the Hindu-Arabic numeral system in North Africa and introduced it to Europe in his Liber Abaci (1202). This work systematically demonstrated how the new decimal positional arithmetic vastly outperforms Roman numerals for commercial computation, and it played a decisive role in the eventual adoption of the modern numeral system in European commerce.

A second route of transmission ran through Sicily, which had been under Arab rule from 827 to 1072 CE and retained a trilingual (Arabic, Greek, Latin) scholarly community under the Norman kings. The translation activity in Sicily paralleled that in Toledo and brought further classical and Islamic works into the Latin world.

Looking Ahead: The growth in mathematics in Europe from 1150 to 1650 is nothing short of stunning. Prior to 1150, only the very rudiments of arithmetic and mathematics were known even among the learned. Access to Islamic curation and scholarship, and thus to the brilliant Greek mathematicians, expedited European learning. By 1687, mathematics was sufficiently advanced for Newton to use the calculus to describe planetary motion. But before planetary motion, we begin with something almost as hard — art.

Module 11: Dürer (1471–1528 CE)

Renaissance Europe

The Europe of the fifteenth and early sixteenth centuries was transformed by several overlapping forces. The printing press, developed by Johannes Gutenberg around 1440, enabled the rapid and cheap multiplication of books, fundamentally changing how knowledge was produced and disseminated. The rediscovery of classical texts — Greek and Latin manuscripts preserved in Byzantine and Islamic libraries and now arriving in Europe through al-Andalus, Sicily, and direct Byzantine trade — provided a new intellectual standard against which European thinkers measured themselves. The growth of Italian city-states — Florence, Venice, Milan, Rome — created wealthy mercantile and ecclesiastical patrons eager to demonstrate their sophistication through art and learning.

Perspective — the systematic representation of three-dimensional space on a two-dimensional surface — was one of the great intellectual achievements of Renaissance art. The rules of perspective had been discovered, or rediscovered, by the Florentine architect Filippo Brunelleschi around 1415 and codified by Leon Battista Alberti in his treatise Della Pittura (1435). The key insight is that parallel lines in three-dimensional space converge to a vanishing point on the picture plane, and that there is a precise geometric relationship between the positions of objects in space and their images on the canvas.

Albrecht Dürer

Albrecht Dürer (1471–1528) was born in Nuremberg, the son of a goldsmith. He trained as a painter and printmaker, travelled to Venice twice (1494–95 and 1505–07), and became the leading artist of Northern Europe. His woodcuts and engravings — including Melencolia I (1514), Knight, Death and the Devil (1513), and Saint Jerome in His Study (1514) — are among the masterpieces of Northern Renaissance art.

Dürer’s visit to Italy brought him into contact with the Italian masters of perspective, and he was determined to bring this knowledge back to Northern Europe. He reportedly learned the mathematical foundations of perspective from Jacopo de’ Barbari, an Italian artist working in Germany, though the circumstances suggest some combination of instruction and surreptitious observation. On his return to Nuremberg, Dürer produced Underweysung der Messung (Instruction in Measurement, 1525), a four-volume treatise on geometry, perspective, and proportion that made the mathematical theory of perspective available to German artists for the first time.

Dürer also designed mechanical drawing aids — devices for constructing accurate perspective drawings by physically sighting through a frame equipped with threads — that prefigure modern technical drawing instruments. The course notes that an example of such a drawing device appears in the television series The Tudors in Season 3, Episode 3.

Desargues’ Theorem

The mathematical legacy of Dürer’s preoccupation with perspective is the development of projective geometry. The fundamental insight is that properties of figures preserved under perspective projection — properties that do not depend on distance or angle — constitute a geometry in their own right.

Girard Desargues (1591–1661), a French architect and mathematician, formulated the foundational theorem of projective geometry in 1648. Desargues’ Theorem states: if two triangles are in perspective from a point — that is, the lines connecting corresponding vertices all pass through a single point — then they are in perspective from a line — that is, the intersections of corresponding sides (extended if necessary) are collinear.

The natural setting for projective geometry is the projective plane \(P(2, \mathbb{R})\), which extends the Euclidean plane by adding a “line at infinity” on which parallel lines meet. In homogeneous coordinates, a point is represented as a triple \((x : y : z)\) where \((x,y,z)\) and \(\rho(x,y,z)\) (for any \(\rho \neq 0\) represent the same point. A line is represented as \([a : b : c]\) with the same equivalence. The point \((x:y:z)\) lies on the line \([a:b:c]\) if and only if \(ax + by + cz = 0\).

The projective plane satisfies four key axioms:

Any two distinct points lie on exactly one line.
Any two distinct lines meet in exactly one point.
There exist three non-collinear points.
Every line contains at least three points.

The first axiom is verified using the cross product: given two distinct points \(P_1 = (x_1:y_1:z_1)\) and \(P_2 = (x_2:y_2:z_2)\), the line through them is \([a:b:c] = P_1 \times P_2\) (the three-dimensional cross product), which is the unique line containing both points. The second axiom follows by a dimension-counting argument: two distinct planes through the origin in \(\mathbb{R}^3\) intersect in a line through the origin, which corresponds to a unique projective point.

Looking Ahead: Dürer’s underhandedness in surreptitiously learning the art of perspective drawing certainly has a match — and more than a match — when we talk about the life of Girolamo Cardano.

Module 12: Cardano (1501–1576 CE)

Cardano, in His Own Words

Girolamo Cardano was one of the most colourful, brilliant, and disreputable figures in the history of mathematics. Born in Pavia in 1501 as the illegitimate son of a lawyer-mathematician (a friend of Leonardo da Vinci), he overcame poverty, social stigma, and his own compulsive gambling to become a leading physician, philosopher, and mathematician. He practiced astrology (casting horoscopes for emperors and popes), wrote hundreds of books on medicine, mathematics, philosophy, and natural history, and late in life composed an autobiography, De Vita Propria Liber (The Book of My Life), which is remarkable for its candour about his failings, his losses (including the execution of his eldest son for murdering his wife), and the peculiarities of his character.

Of the 131 books Cardano claimed to have written, the most mathematically significant is Ars Magna (The Great Art, 1545), which presented for the first time in print the general solutions to cubic and quartic equations. The backstory of its publication is a tale of mathematical espionage and broken promises.

A Brief History of Solutions to Equations

The ancient world knew methods for solving linear equations (\(ax = b\) and special cases of quadratic equations — al-Khwarizmi had given the general solution to quadratics. The search for a general solution to the cubic \(x^3 + px + q = 0\) occupied algebraists for centuries.

Around 1515, Scipione del Ferro, a professor at Bologna, discovered a method for solving depressed cubics (cubics with no \(x^2\) term). Following the custom of the time, he kept his method secret, communicating it only to his student Antonio Maria Fior just before his death in 1526. In 1535 Fior challenged Niccolò Fontana (known as Tartaglia, “the stammerer”) to a public mathematical contest, proposing thirty problems all of the form \(x^3 + px = q\). Tartaglia, who had independently discovered the solution to such cubics, solved all thirty in two hours and won the contest.

Cardano, learning of Tartaglia’s achievement, pressed him repeatedly for the method. In 1539 Tartaglia finally revealed the secret, under a solemn oath that Cardano would never publish it. Cardano kept this oath — until he travelled to Bologna, inspected del Ferro’s original unpublished papers, confirmed that del Ferro had discovered the method first, and concluded that he was no longer bound by his promise to Tartaglia. In 1545 he published Ars Magna, giving full credit to del Ferro and Tartaglia but presenting the method to the world. Tartaglia was furious and the ensuing controversy was bitter.

Solving Cubic Equations

Cardano’s method for solving the depressed cubic \(x^3 + px = q\) uses the algebraic identity:

\[(a - b)^3 + 3ab(a - b) = a^3 - b^3\]

If we set \(x = a - b\), this becomes \(x^3 + 3ab \cdot x = a^3 - b^3\). Comparing with \(x^3 + px = q\) gives the system:

\[3ab = p \quad \text{and} \quad a^3 - b^3 = q\]

From the first equation, \(b = \frac{p}{3a}\). Substituting into the second:

\[a^3 - \frac{p^3}{27a^3} = q\]

Multiplying through by \(a^3\):

\[a^6 - qa^3 - \frac{p^3}{27} = 0\]

This is a quadratic in \(a^3\). Solving by the quadratic formula:

\[a^3 = \frac{q}{2} + \sqrt{\frac{q^2}{4} + \frac{p^3}{27}}\]

and analogously for \(b^3\). Then \(x = a - b\) is:

\[x = \sqrt[3]{\frac{q}{2} + \sqrt{\frac{q^2}{4} + \frac{p^3}{27}}} - \sqrt[3]{-\frac{q}{2} + \sqrt{\frac{q^2}{4} + \frac{p^3}{27}}}\]

This is Cardano’s formula. For the specific example \(x^3 + 9x = 10\) (from Assignment 7), we have \(p = 9\), \(q = 10\):

\[\frac{q^2}{4} + \frac{p^3}{27} = \frac{100}{4} + \frac{729}{27} = 25 + 27 = 52\]

so \(a^3 = 5 + \sqrt{52}\) and \(b^3 = -5 + \sqrt{52}\), giving \(x = 1\) (the obvious real root, verified by inspection).

Cardano’s student Ludovico Ferrari extended this method to quartic equations \(x^4 + ax^2 + bx + c = 0\), reducing the quartic to a resolvent cubic. The general quartic thus became solvable in the 1540s — more than two millennia after the quadratic had been solved by the Babylonians and Egyptians. The question of whether the quintic (fifth-degree polynomial) could be similarly solved would remain open for nearly three more centuries, until Abel and Galois proved impossibility in the 1820s–1830s.

Looking Ahead: Cardano solved equations. But there were larger problems on the horizon — astronomically large problems. These required different techniques.

Module 13: Briggs (1561–1630 CE)

Calculation

By the late sixteenth century, the needs of astronomy, navigation, and commerce had made large-scale numerical computation indispensable — and agonizingly laborious. An astronomer computing planetary positions might need to multiply and divide fifteen-digit numbers hundreds of times. Navigation on the open ocean, which had expanded enormously since Vasco da Gama’s voyage to India (1498) and Magellan’s circumnavigation (1519–22), required accurate trigonometric tables and intricate spherical-triangle calculations.

The critical breakthrough was the invention of logarithms by John Napier (1550–1617), a Scottish nobleman and mathematician who published his discovery in Mirifici Logarithmorum Canonis Descriptio (Description of the Marvellous Rule of Logarithms, 1614). The fundamental property of logarithms — that they convert multiplication to addition — reduced the effort of large multiplications to the simpler task of table lookup and addition:

\[\log(ab) = \log a + \log b\]

A product is computed by looking up the logarithms of both factors, adding them, and looking up the antilogarithm of the result. Division becomes subtraction, powers become multiplication, and roots become division.

Briggs

Henry Briggs (1561–1630) was the first Gresham Professor of Geometry in London, appointed in 1596. Upon reading Napier’s 1614 publication, he immediately recognized its importance and wrote to Napier: “Naper, lord of Markinston, hath set my head and hands a work with his new and admirable logarithms. I hope to see him this summer, if it please God.” He did visit Napier in Edinburgh in 1615 and again in 1616 — Napier died in 1617 — and the two men agreed on a major improvement to Napier’s original construction.

Napier’s original logarithm was not the common logarithm of modern usage. His Napierian logarithm was defined by a specific geometric construction that effectively made \(\text{Nap.log}(x) = 10^7 \ln(10^7/x)\), which has base \(1/e\) in a certain sense and does not satisfy \(\text{log}(10) = 1\). Briggs proposed changing the base so that \(\log(1) = 0\) and \(\log(10) = 1\) — the definition of the common logarithm (base 10) that we use today. After Napier’s death, Briggs spent years computing a table of common logarithms to fourteen decimal places for all integers from 1 to 20,000 and from 90,000 to 100,000. This was published as Arithmetica Logarithmica (1624). The gap from 20,000 to 90,000 was filled by Adriaan Vlacq in 1628.

Logarithms

Briggs’s algorithm for computing \(\log_{10} x\) uses the fundamental property that:

\[\log_{10}(10^{1/2^k}) = \frac{1}{2^k}\]

Starting from the known values \(\log_{10}(1) = 0\) and \(\log_{10}(10) = 1\), Briggs computed:

\[10^{1/2} \approx 3.162278, \quad \log_{10}(3.162278) = 0.5\]\[10^{1/4} \approx 1.778279, \quad \log_{10}(1.778279) = 0.25\]

and so on, building up a table of logarithms of \(10^{1/2^k}\) for increasing \(k\). To find \(\log_{10}(x)\) for a general \(x\), one expresses \(x\) as a product of powers of \(10^{1/2^k}\) and uses the additive property. For example, since \(10^{13/16} = 10^{8/16} \cdot 10^{4/16} \cdot 10^{1/16}\):

\[\log_{10}(10^{13/16}) = \frac{8}{16} + \frac{4}{16} + \frac{1}{16} = \frac{13}{16}\]

The actual computation of \(10^{1/2^k}\) uses repeated square roots, and the algorithm for computing square roots uses Newton’s method (or its classical antecedent): given an initial estimate \(r_0 \approx \sqrt{x}\), refine by:

\[r_{n+1} = \frac{1}{2}\left(r_n + \frac{x}{r_n}\right)\]

This converges quadratically — each iteration roughly doubles the number of correct decimal digits — so even starting from a crude estimate, three or four iterations yield many decimal places.

The use of logarithm tables for multiplication and division persisted as the dominant computational technique until electronic calculators became widely available in the 1970s, making Briggs’s tables one of the most practically impactful works in the history of mathematics.

Looking Ahead: With common logarithms and his collaboration with Napier, Briggs was an exemplar of scientific advancement and professional relationships. Let us take a quick look at other advances, people, and relationships in that time period to set the stage for the dramatic accomplishments that were to come.

Module 14: The Making of Modern Mathematics

The Making of Modern Mathematics

The seventeenth century was an era of scientific revolution without precedent. Within a single century, Europeans dismantled the Aristotelian cosmology that had dominated intellectual life for two thousand years, created the calculus, formulated the laws of motion and universal gravitation, and developed a mathematical language adequate to describe the physical world. This module surveys the broad context of this transformation.

Galileo Galilei (1564–1642) is the pivotal figure at the century’s opening. His astronomical observations with the telescope — confirming the phases of Venus (proving it orbits the sun), observing the moons of Jupiter, studying sunspots — provided crucial evidence for heliocentrism. But his purely mathematical contributions were equally profound: in his Discorsi (1638), he laid the foundations of classical mechanics by formulating the law of falling bodies (distance proportional to time squared) and the parabolic trajectory of projectiles. He also confronted the paradoxes of the infinite, observing that the set of perfect squares can be put into one-to-one correspondence with all natural numbers — a fact that would wait two and a half centuries for Cantor’s theory of infinite sets to resolve satisfactorily.

Johannes Kepler (1571–1630) used the unprecedentedly accurate observational data of Tycho Brahe to discover his three laws of planetary motion: (1) planets travel in ellipses with the sun at one focus; (2) a line from the planet to the sun sweeps equal areas in equal times; (3) the square of the period is proportional to the cube of the semi-major axis. These laws, purely empirical in origin, were later derived by Newton from the law of universal gravitation.

Pierre de Fermat (1607–1665) worked as a lawyer in Toulouse and pursued mathematics as an amateur — one of the greatest amateurs in history. His contributions span number theory (including the statement of what became Fermat’s Last Theorem: \(x^n + y^n = z^n\) has no integer solutions for \(n \geq 3\), proved finally by Andrew Wiles in 1995), analytic geometry (independently of Descartes), the proto-calculus of tangent lines and extrema, and the foundations of probability theory (in correspondence with Pascal).

Blaise Pascal (1623–1662), a French mathematician, physicist, and theologian, contributed Pascal’s triangle (the binomial coefficients), the foundations of probability theory (in correspondence with Fermat over a gambling problem: how to divide the stakes when a game is interrupted), the invention of one of the first mechanical calculating machines, and important work in projective geometry. He is the intellectual link between Desargues’s projective geometry and its seventeenth-century development.

Isaac Newton (1643–1727) and Gottfried Wilhelm Leibniz (1646–1716) independently invented the calculus — Newton around 1665–66 (during the plague years when Cambridge closed), Leibniz around 1675. The resulting priority dispute, conducted with bitter national partisanship between English and Continental mathematicians, is one of the most famous controversies in the history of science. Newton’s notation (\(\dot{x}\) for the derivative with respect to time) and Leibniz’s notation (\(\frac{dy}{dx}\), \(\int f(x)\, dx\) both survive, though Leibniz’s is dominant. Newton published his mathematical results chiefly in Principia Mathematica (1687), which used classical geometric methods rather than the new calculus to present his physics.

Paul Erdős (1913–1996) is an outlier in this chronological tour, but the course includes his obituary (from Nature, 1996) as supplementary reading. Erdős spent six decades as a peripatetic mathematician — owning essentially no property, living from the hospitality of colleagues worldwide — and produced approximately 1,500 mathematical papers with nearly 500 co-authors. His passionate belief that mathematics is a collaborative and human activity, and his enormous output in number theory, combinatorics, and graph theory (areas directly connected to Tutte’s work), make him an appropriate figure to introduce alongside the seventeenth-century developments.

Looking Ahead: Having seen the broad social and scientific currents of the seventeenth century, it is now time to look at some specific individuals and some specific conflicts.

Module 15: Clavius (1538–1612 CE)

Marking Time

The most practical mathematical problem of the early modern period was the reform of the calendar. The Julian calendar, introduced by Julius Caesar in 46 BCE on the advice of the Alexandrian astronomer Sosigenes, assumed a solar year of exactly 365.25 days, corrected by inserting a leap year every four years. The actual tropical year (the time from one vernal equinox to the next) is approximately 365.2422 days — only 11 minutes shorter than the Julian value, but over centuries this discrepancy accumulates. By the sixteenth century, the Julian calendar had drifted by about ten days relative to the astronomical seasons, meaning that Easter — whose date is calculated relative to the vernal equinox — was falling later and later in the astronomical spring.

Pope Gregory XIII convened a commission to address this problem. The reform required solving two distinct tasks: eliminating the accumulated ten-day error, and preventing future drift.

The Gregorian calendar (1582) addressed both:

The accumulated error was corrected by removing ten days from October 1582 (the day after October 4 became October 15).
Future drift was reduced by modifying the leap-year rule: century years (1700, 1800, 1900…) would not be leap years unless divisible by 400. Thus 1900 was not a leap year but 2000 was. This gives an average year length of \(365 + \frac{97}{400} = 365.2425\) days — remarkably close to the true tropical year.

Christopher Clavius

Christopher Clavius (1538–1612) was a German Jesuit mathematician who spent most of his career in Rome and became the dominant figure in the Gregorian calendar reform. He wrote the definitive account of the calendar’s mathematical justification (Romani Calendarii a Gregorio XIII P. M. restituti explicatio, 1603) and spent years defending the reform against Protestant and mathematical criticism.

Clavius was a Euclidean traditionalist. He produced an important edition and commentary on Euclid’s Elements (1574) and was hostile to the new approach to infinity that was beginning to appear in the work of Bonaventura Cavalieri and others. Cavalieri’s method of indivisibles (1635) treated geometric figures as composed of infinitely many infinitely thin slices — a powerful heuristic that anticipated integral calculus but that Cavalieri and Clavius’s contemporaries found logically troubling.

Galileo met Clavius in Rome in 1611 and Clavius confirmed the reality of his telescopic observations. This endorsement from the leading Catholic mathematician was significant, though it did not ultimately protect Galileo from the Inquisition.

Modern Infinitesimals Versus Euclidean Tradition

The tension between infinitesimal methods and Euclidean rigour is illustrated by a deceptively simple example (from Assignment 8). Consider, for each \(n\), the staircase path \(l_n\) from \((0,0)\) to \((1,1)\) that alternates between steps of length \(1/n\) rightward and \(1/n\) upward for a total of \(n\) pairs of steps.

The area under \(l_n\) is:

\[A(l_n) = \sum_{k=1}^{n} \frac{k}{n} \cdot \frac{1}{n} = \frac{1}{n^2} \cdot \frac{n(n+1)}{2} \to \frac{1}{2} = A(l)\]

So the areas converge to the area under the diagonal. But the length of \(l_n\) is always exactly \(L(l_n) = 2\) (n steps right of length 1/n plus n steps up of length 1/n), while the length of the diagonal is \(L(l) = \sqrt{2}\). The paths converge pointwise to the diagonal, yet their lengths do not converge to the diagonal’s length: \(\lim_{n \to \infty} L(l_n) = 2 \neq \sqrt{2}\).

This paradox — that geometric limits do not commute with length — shows why Archimedes was careful to use both inscribed and circumscribed polygons to bound \(\pi\) from above and below rather than simply taking a limit of inscribed polygons. It also shows why the wholesale adoption of infinitesimal reasoning, however productive, required the rigorous foundations that were not provided until the nineteenth century with Cauchy and Weierstrass.

Looking Ahead: Clavius is an iconical example of a traditionalist. Even his enormous accomplishments in updating the calendar were intended to align the calendar with that bulwark of tradition, the Catholic Church. Ironically, it was another Catholic scholar, Descartes, who launched the modern era.

Module 16: Descartes (1596–1650 CE)

The Modern Era

By the 1630s, the new mechanical philosophy — which sought to explain all natural phenomena in terms of matter and motion, without recourse to Aristotle’s four causes or the Scholastic tradition — was transforming natural science. René Descartes was its leading philosophical architect as well as a major mathematician in his own right. His Discours de la Méthode (1637) proposed a method for attaining certainty through systematic doubt and mathematical reasoning. The Discours appeared together with three appendices: La Dioptrique (optics), Les Météores (meteorology), and La Géométrie (geometry). It is the third appendix that concerns us.

Descartes

Descartes was born in La Haye en Touraine (now called Descartes) in 1596, educated by the Jesuits at La Flèche, and spent much of his adult life in the Dutch Republic, then the most intellectually tolerant country in Europe. He moved to Stockholm in 1649 at the invitation of Queen Christina of Sweden, where he died of pneumonia in 1650.

His philosophical work — the Meditations on First Philosophy (1641), the Principles of Philosophy (1644) — establishes the famous Cogito (Cogito ergo sum, “I think therefore I am”) as the foundation of knowledge and argues for a dualistic view of mind and matter. In mathematics, his principal contribution is the systematic use of algebraic notation to represent geometric problems, and vice versa — the fusion of algebra and geometry now called analytic geometry (independently discovered by Fermat).

La Géométrie

La Géométrie is a relatively short work divided into three books. Book I introduces the basic idea of representing geometric magnitudes by algebraic letters and shows how to perform arithmetic operations geometrically. Book II studies curves and develops the method of finding tangent lines (a precursor of differentiation). Book III addresses the roots of polynomial equations.

The key innovation is the Cartesian coordinate system: given two perpendicular axes, any point in the plane is identified by a pair of numbers \((x, y)\), and any curve is the set of points satisfying a polynomial equation \(f(x, y) = 0\). This allows geometric problems to be converted into algebraic ones and solved by the powerful machinery of symbolic algebra.

Descartes also provides methods for solving polynomial equations. For quartic equations of the form \(x^4 = ax^2 + bx + c\), he proposes factoring the quartic as a product of two quadratics:

\[x^4 = ax^2 + bx + c \implies (x^2 + kx + m)(x^2 - kx + n) = 0\]

Expanding the right side and matching coefficients with the original quartic gives a system of equations in \(k\), \(m\), \(n\). Eliminating \(m\) and \(n\) yields a resolvent cubic in \(k^2\), which can be solved by Cardano’s formula. Once \(k\) is known, \(m\) and \(n\) follow, and the quartic factors into two quadratics, each solvable by the quadratic formula.

For example, to solve \(x^4 - 3x^2 + 6x - 2 = 0\) (from Assignment 8), one writes:

\[(x^2 + kx + m)(x^2 - kx + n) = x^4 + (m+n-k^2)x^2 + k(n-m)x + mn\]

Matching coefficients: \(m + n - k^2 = -3\), \(k(n-m) = 6\), \(mn = -2\). This system resolves to a cubic in \(k^2\), whose solution yields the factorization.

La Géométrie also introduced several features of modern algebraic notation that we now take for granted: using letters at the end of the alphabet (\(x, y, z\) for unknowns and letters at the beginning (\(a, b, c\) for known quantities, and using superscript exponents (\(x^2\) rather than xx).

Looking Ahead: We began the course with a direct mathematical ancestor, Bill Tutte. We will end the course with another direct mathematical ancestor, Arthur Cayley — much of what you have seen in linear algebra courses appears in Cayley’s A Memoir on the Theory of Matrices.

Module 17: Cayley (1821–1895 CE)

Universities in Victorian England

The nineteenth century saw dramatic changes in the organisation of mathematical research and education in Britain. The ancient universities of Oxford and Cambridge had, for centuries, been primarily training grounds for Anglican clergymen and gentlemen, with mathematics occupying a subordinate role subordinated to classics. The Cambridge Mathematical Tripos — the notoriously demanding undergraduate examination that ranked students as Wranglers (first class), Senior Optimes, and Junior Optimes — became, paradoxically, both the main gateway to mathematical prestige in Britain and a severe constraint on mathematical creativity, since it emphasized computational facility and geometric reasoning over abstract theory.

The leading Wranglers of the mid-nineteenth century — including Cayley, James Joseph Sylvester, George Boole, William Thomson (Lord Kelvin), and James Clerk Maxwell — demonstrated that mathematical genius could survive the Tripos, though many of them produced their most important work by breaking from its intellectual constraints. The period also saw the expansion of the London colleges and the gradual professionalization of mathematics through the founding of journals (the Cambridge Mathematical Journal, the Quarterly Journal of Pure and Applied Mathematics) and learned societies.

Arthur Cayley himself spent fourteen years as a barrister after graduating from Cambridge, during which time he produced nearly 300 mathematical papers — a testament to both his productivity and the lack of paid academic positions for mathematicians in mid-Victorian England. He was appointed Sadleirian Professor of Pure Mathematics at Cambridge in 1863, a position he held until his death.

Arthur Cayley

Arthur Cayley was born in Richmond, Surrey, in 1821 and showed exceptional mathematical talent from childhood. He entered Trinity College, Cambridge, at age seventeen and graduated as Senior Wrangler in 1842. During his years at the bar, working in Lincoln’s Inn and later as a conveyancing barrister, he collaborated extensively with James Joseph Sylvester (who was also a lawyer at the time) on what they called the theory of invariants: algebraic expressions associated with geometric forms that remain unchanged under certain coordinate transformations.

Cayley’s collected mathematical works fill thirteen large volumes and cover an extraordinary range: group theory (he gave the first abstract definition of a group), the theory of invariants and covariants, algebraic geometry, the theory of matrices, the theory of trees (in graph theory), and much else. The Cayley graph of a group, the Cayley-Hamilton theorem, and Cayley’s theorem (every group is isomorphic to a group of permutations) all bear his name.

He also made Waterloo’s history relevant: Cayley’s work in invariant theory and matrix algebra was a direct mathematical ancestor of the work on symmetric functions pursued by Logan Crew, the instructor of this course.

A Memoir on the Theory of Matrices

Cayley’s A Memoir on the Theory of Matrices (1858) is the founding document of matrix algebra as a systematic subject. Prior to this paper, determinants had been studied for over a century (by Leibniz, Cramer, and others), and linear transformations had been used extensively, but the systematic arithmetic of rectangular arrays of numbers as algebraic objects in their own right was new.

Cayley defined matrix addition and matrix multiplication in the way we still use them today. For matrices \(A\) and \(B\), multiplication is defined by the rule that the \((i,j)\) entry of \(AB\) is \(\sum_k a_{ik} b_{kj}\). He noted immediately that matrix multiplication is not commutative in general: \(AB \neq BA\) for most matrices. This was a striking departure from the algebra of numbers and foreshadowed the development of noncommutative algebra.

The Cayley-Hamilton Theorem is the central result: every square matrix satisfies its own characteristic polynomial. For a matrix \(M\), the characteristic polynomial is \(f_M(x) = \det(xI - M)\) (a polynomial in \(x\). The theorem asserts that \(f_M(M) = 0\) (the zero matrix), where we evaluate \(f_M\) at the matrix \(M\) itself using matrix powers and scalar multiples.

For a diagonal matrix \(D = \mathrm{Diag}(r_1, \ldots, r_n)\), the characteristic polynomial is:

\[f_D(x) = (x - r_1)(x - r_2) \cdots (x - r_n)\]

Evaluating at \(D\): \(f_D(D) = (D - r_1 I)(D - r_2 I) \cdots (D - r_n I)\). Each factor \(D - r_i I\) has a zero in the \(i\)-th diagonal position, so the product of all \(n\) factors gives the zero matrix — each position in the diagonal product has at least one factor that contributes a zero.

For a general diagonalizable matrix \(M = P D P^{-1}\) (where \(D\) is diagonal and \(P\) is invertible):

\[f_M(M) = f_M(PDP^{-1}) = P f_D(D) P^{-1} = P \cdot 0 \cdot P^{-1} = 0\]

using the fact that \(f_M = f_D\) (similar matrices have the same characteristic polynomial) and that the key conjugation property \(g(PDP^{-1}) = P g(D) P^{-1}\) holds for any polynomial \(g\).

Cayley wrote in his paper that he had verified the theorem for \(2 \times 2\) and \(3 \times 3\) matrices and stated it as a general result “without further proof” — relying, perhaps, on the analogy with his verified cases and the structural argument for diagonal matrices. A fully rigorous general proof had to wait for later developments in abstract algebra.

Cayley’s matrix algebra, combined with Hamilton’s quaternions and Boole’s algebra of logic, established in the mid-nineteenth century that algebra need not be the algebra of numbers — that consistent, useful algebraic systems exist with properties radically different from ordinary arithmetic. This insight underlies all of modern abstract algebra and provides the conceptual environment in which Cayley’s Waterloo successors, including Tutte himself, have worked.

CO 480 was designed to show that mathematics is not a timeless edifice dropped from Platonic heaven but a human enterprise, shaped by geography, politics, patronage, and personality. The story moves from an Egyptian scribe computing unit fractions in a Nile-valley administrative office, through the brilliant and turbulent life of the Library of Alexandria, across the deserts to the House of Wisdom in Baghdad, over the Pyrenees to the translation schools of Toledo, into the Renaissance workshops of Nuremberg, the taverns where Cardano and Tartaglia traded secrets, the Edinburgh study where Briggs and Napier agreed to change the logarithmic base, the Dutch Republic where Descartes formulated analytic geometry, the Victorian barristers’ chambers where Cayley developed matrix algebra in his spare time, and finally to the huts of Bletchley Park where Tutte broke the Fish cipher with pencil and paper. What connects them all is the conviction, evident from the very first Egyptian papyri to the most modern of Tutte’s theorems, that the world is in principle intelligible — and that mathematics is the language of that intelligibility.