In this first lecture (with two parts, 1a & 1b) we first give a very rough outline of world history from a mathematical point of view, position the work of the ancient Greeks as following from Egyptian and Babylonian influences, and introduce the most important theorem in all of mathematics: Pythagoras’ theorem.
Two interesting related issues are the irrationality of the ‘square root of two’ (the Greeks saw this as a length, but not as a number), and Pythagorean triples, which go back to the Babylonians. These are closely related to the important rational parametrization of a circle, essentially discovered by Euclid and Diophantus.
The Greeks thought of mathematics differently than we do today, and all students can benefit from a closer appreciation of the difficulties which they saw, but which we today largely ignore.
The ancient Greeks loved geometry and made great advances in this subject. Euclid’s Elements was for 2000 years the main text in mathematics, giving a careful systematic treatment of both planar and three dimensional geometry, culminating in the five Platonic solids. Apollonius made a thorough study of conics. Constructions played a key role, using straightedge and compass.
The ancient Greeks studied squares, triangular numbers, primes and perfect numbers. Euclid stated the Fundamental theorem of Arithmetic: that a natural number could be factored into primes in essentially a unique way. We also discuss the Euclidean algorithm for finding a greatest common divisor, and the related theory of continued fractions. Finally we discuss Pell’s equation, arising in the famous Cattle-problem of Archimedes.
We discuss primarily the work of Eudoxus and Archimedes, the founders of calculus. Archimedes in particular discovered formulas that are only found in advanced calculus courses, concerning the relations between the volumes and surface areas of a sphere and a circumscribing cylinder. We also discuss his work on the area of a parabolic arc, Heron’s formula (improved using ideas of Rational Trigonometry), hydrostatics, and the Principle of the Lever. He was a true genius.
After the later Alexandrian mathematicians Ptolemy and Diophantus, Greek mathematics went into decline and the focus shifted eastward. This lecture discusses some aspects of Chinese, Indian and Arab mathematics, in particular the interest in number theory (Pell’s equation, the Chinese remainder theorem, and algebra. Most crucial was the introduction of the Hindu-Arabic number system that we use today.
We also discuss the influence of probably the most important problem of the mathematical sciences from a historical point of view: understanding the motion of the night sky, in particular the planets. This motivated work in trigonometry, particularly spherical trigonometry, of both Indian and Arab mathematicians.
Prominent mathematicians whose work we discuss include Sun Zi, Aryabhata, Brahmagupta, Bhaskara I and II, al-Khwarizmi, al-Biruni and Omar Khayyam.
We now move to the Golden age of European mathematics: the period 1500-1900, in this course on the History of Mathematics. We discuss hurdles that the Europeans faced before this time and how they emerged, with the help of Arab algebra and translations of Greek works, to harness the Hindu-Arabic number system and a host of novel symbols including Vieta’s new use of letters to represent unknowns to tackle new problems.
Quadratic equations had been solved by almost all earlier mathematical civilizations; cubic equations was a natural step, taken by Tartaglia and Cardano and others. Tartaglia also discovered a formula for the volume of a tetrahedron, and Vieta a trigonometric way of solving cubics.
The development of Cartesian geometry by Descartes and Fermat was one of the main accomplishments of the 17th century, giving a computational approach to Euclidean geometry. Involved are conics, cubics, Bezout’s theorem, and the beginnings of a projective view to curves. This merging of numbers and geometry is discussed in terms of the ancient Greeks, and some problems with our understanding of the continuum are observed; namely with irrational numbers and decimal expansions. We also discuss pi and its continued fraction approximations.
Projective geometry began with the work of Pappus, but was developed primarily by Desargues, with an important contribution by Pascal. Projective geometry is the geometry of the straightedge, and it is the simplest and most fundamental geometry. We describe the important insights of the 19th century geometers that connected the subject to 3 dimensional space.
Calculus has its origins in the work of the ancient Greeks, particularly of Eudoxus and Archimedes, who were interested in volume problems, and to a lesser extent in tangents. In the 17th century the subject was widely expanded and developed in an algebraic way using also the coordinate geometry of Descartes. This is one of the most important developments in the history of mathematics.
Calculus has two branches: the differential and integral calculus. The former arose from the study by Fermat of maxima and minima of functions via horizontal tangents.
The integral calculus computes areas and volumes beyond the techniques of Archimedes. It was developed independently by Newton and Leibnitz, but others contributed too. Newton’s focus was on power series, for which differentiation and integration can be done term by term using a formula of Cavalieri, and which gave remarkable new formulas for pi and the circular functions. He had a dynamic view of the subject, motivated in large part by physics.
Leibnitz was more interested in closed forms, and introduced the notation which we use today. Both used infinitesimals, in the form of differentials.
We discuss various uses of infinite series in the 17th and 18th centuries. In particular we look at the geometric series, power series of log, the Gregory-Newton interpolation formula, Taylor’s formula, the Bernoulli’s, Eulers summation of the reciprocals of the squares as pi squared over 6, the harmonic series, product expansion of sinx, the zeta function and Euler’s product expansion for it, the exponential function, complex values and finally the circular functions too!
The main historical problem in the history of science is: to explain what is going on with the night sky, in particular what the planets are doing. The resolution of this was the greatest achievement of the 17th century. The key figures were Copernicus, Galileo, Brahe, Kepler and most famously Isaac Newton. This interesting story, culminating with Kepler’s Laws and their explanation by Newton’s laws of motion and Law of gravitation, ought to be studied in depth by all undergraduate students of mathematics!
The development of non-Euclidean geometry is often presented as a high point of 19th century mathematics. The real story is more complicated, tinged with sadness, confusion and orthodoxy, that is reflected even the geometry studied today. The important insights of Gauss, Lobachevsky and Bolyai, along with later work of Beltrami, were the end result of a long and circuitous study of Euclid’s parallel postulate. But an honest assessment must reveal that in fact non-Euclidean geometry had been well studied from two thousand years ago, since the geometry of the sphere had been a main concern for all astronomers.
This lecture gives a somewhat radical and new interpretation of the history, suggesting that there is in fact a much better way of thinking about this subject, as perceived already by Beltrami and Klein, but largely abandoned in the 20th century. This involves a three dimensional linear algebra with an unusual inner product, looked at in a projective fashion. This predates and anticipates the great work of Einstein on relativity and its space-time interpretation by Minkowski.
Next : a fuller account of this improved approach: Universal Hyperbolic Geometry
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