# Glossary

### abacus

An *abacus* is a calculating device based on either beads sliding on wires or balls or stones in grooves.

### abelian or hyperelliptic function

An *abelian or hyperelliptic function* is a generalisation of an elliptic function. It is a function of two variables with four periods.

In a similar way to an elliptic function it can also be regarded as the inverse function to certain integrals (called *abelian* or *hyperelliptic integrals*) of the form $\large{\int \frac{dz}{\sqrt{R(z)}}}$ where $R$ is a polynomial of degree *greater than* 4.

### abelian group

An *abelian group* is a group in which the group operation is commutative. i.e. it satisfies $a.b = b.a$.

### abundant number

An *abundant number* is an integer for which the sum of its proper divisors is greater than the number itself.

For example, the proper divisors of 12 are 1, 2, 3, 4, 6 which sum to 16.

See also: perfect number, deficient number.

### agrégation

The *agrégation* is a competitive examination in France which allows the holder to teach in lycées or colleges.

### Alexander the Great

Alexander the Great or Alexander III (**born:** 356 BC in Pella in Macedonia, **died:** 323 BC in Babylon) was a King of Macedonia who conquered the Persian Empire.

### algebra

An *algebra* is a set which is both a ring and a vector space satisfying, of course various special properties or axioms.

### algebraic geometry

*Algebraic geometry* studies curves, surfaces and their higher dimensional equivalents defined by systems of polynomial equations and relates their properties to the algebraic properties of the polynomial rings that they determine.

### algebraic number

An *algebraic number* is a real number which is the root of a polynomial equation with integer (or rational) coefficients.

For example, √2 is a root of $x^{2} = 2$ and so is algebraic.

### algebraic number field

An *algebraic number field* is a subfield of $\mathbb{R}$ or $\mathbb{C}$ which contains the rational numbers $\mathbb{Q}$ as well as the roots of some polynomial with rational coefficients.

For example, the field written $\mathbb{Q}$ (√2) is the set of real numbers {$a + b√2 | a, b$ are rationals}. This contains the roots of the equation $x^{2} - 2 = 0$.

### algebraic number theory

*Algebraic number theory* is number theory studied without using methods like infinite series, convergence, etc. taken from analysis.

It contrasts with analytic number theory.

### algebraic topology

*Algebraic topology* is a branch of topology in which one calculated invariants of a topological space which are algebraic structures, often groups.

For example, the fundamental group of a space is such an invariant as are homotopy groups and homology groups.

### alternating group

An *alternating group* is the group of all *even* permutations of a finite set.

A permutation is *even* if it can be written as a product of an even number of *transpositions* where a transposition swaps a pair of elements and leaves the rest fixed.

The alternating group of a set of size $n$ is denoted $A_{n}$.

The alternating group $A_{n}$ contains half the elements of the symmetric group $S_{n}$ and has order $n! /2$.

### amicable numbers

*Amicable numbers* are a pair of numbers *a, b* for which the sum of the proper divisors of *a* equals *b* and the sum of the proper divisors of *b* is equal to *a*.

For example, 220 and 284 are amicable numbers.

### Ammonius

Ammonius (**born:** about 550 AD) was a Greek philosopher who worked mainly on logic and science. He wrote many critical works on Aristotle. He was a student with Proclus then later he was appointed the head of the Alexandrian school.

### analytic number theory

*Analytic number theory* is number theory studied using methods like infinite series, convergence, etc. taken from analysis and especially from Complex analysis.

It contrasts with algebraic number theory.

### Anaximander

Anaximander (**born:** 610 BC in Miletus [now in Turkey], **died:** 546 BC) was a Greek philosopher who was a pupil of Thales. He is considered as the founder of astronomy. He was the first to develop a cosmology.

### angle trisection

One of the classic problems of Greek mathematics was to find a ruler and compass construction to *divide any angle into three equal pieces*.

The other classic problems were squaring the circle and duplicating the cube.

### annular eclipse

An *annular eclipse* is a solar eclipse in which a ring of the sun is still visible because the Moon is too close to the Earth to shield the Sun completely.

### apogee

The *apogee* is the point where a heavenly body is furthest away from the centre of its orbit.

The nearest point is called the *perigee*.

### Apollodorus

Apollodorus (**born:** about 130 BC) was a Greek scholar who is best known for his Chronicle of Greek history. He was a pupil of Aristarchus. In 146 BC he left Alexandria for Pergamum and later settled at Athens.

### apotome

An *apotome* is one of Euclid's categories of irrational numbers.

A number of the form $√(√A - √B)$ is called a *apotome*.

The corresponding number with a + sign is called a *binomial*.

### Apuleius

Apuleius (**born:** 124 AD in Madauros in Numidia, **died:** about 175 AD) was a Platonic philosopher, who was educated at Carthage and Athens, then taught rhetoric in Rome before returning to Africa. He wrote important philosophical treatises including three on Plato.

### Arcesilaus

Arcesilaus (**born:** 316 BC in Pitane in Aeolis [now in Turkey], **died:** about 241 BC) was a philosopher who became head of the Greek Academy. He was a Sceptic who believed that a wise man need know only that his actions are reasonable.

### Archimedes' principle

*Archimedes' principle* states that any body completely or partially submerged in a fluid is acted upon by an upward force which is equal to the weight of the fluid displaced by the body.

### Archimedes' screw

*Archimedes' screw* consists of a pipe in the shape of a helix with its lower end dipped in the water. As the device is rotated the water rises up the pipe.

### arithmetic progression

An *arithmetic progression* is a sequence in which each number increases by the same amount (the common difference) over the previous one.

As for example: 5, 8, 11, 14, 17, ...

### associative

An binary operation . on a set is *associative* if it satisfies the identity $(a.b).c = a.(b.c)$

Composition of functions and multiplication of matrices are important associative operations.

### astrolabe

An *astrolabe* is an early instrument for measuring the angle between the horizon and a star or planet. It was superceded by the octant and sextant

### atomic theory

*Atomic theory* is the theory that matter and space are not infinitely divisible.

### automorphism group

An *automorphism group* of a group $G$ is the group formed by the automorphisms of $G$ (bijections from $G$ to itself which preserve the multiplication).

Similarly, one can consider the automorphism groups of other structures like rings, graphs, etc. by looking at bijections which preserve their mathematical structure.

### Axiom of choice

The *Axiom of Choice* states that given any collection $A$ of mutually exclusive non-empty sets, one can create a new set by choosing an element out of each set in $A$. It is equivalent to several other important statements about infinite sets (the Well-Ordering Principle, Zorn's Lemma, ...) and has been shown to be independent of the other set-theory axioms.

### azimuth

The *azimuth* is the arc of the horizon between the meridian (the direction North) and a vertical circle passing through a heavenly body.

### Banach space

A *Banach* space is a complete normed vector space.

That is, vector space with a metric that is *complete* in the sense that a Cauchy sequence of vectors always converges to a well defined limit in the space.

### Bernoulli number

*Bernoulli numbers* were defined by Jacob Bernoulli in connection with evaluating sums of the form ∑ $i^{k}$.

The sequence $B_{0} , B_{1} , B_{2} , ...$ can be generated using the formula

$x/(e^{x} - 1) = \sum (B_{n}x^{n})/n!$

though various different notations are used for them.

The first few are: $B_{0} = 1 , B_{1} = -{{1}\over{2}} , B_{2} = {{1}\over{6}} , B_{4} = -{{1}\over{30}} , B_{6} = {{1}\over{42}} , ...$

They occur in many diverse areas of mathematics including the series expansions of $\tan(x)$, Fermat's Last theorem, ...

### Bessel's function

*Bessel functions* are the radial part of the modes of vibration of a circular drum. They satisfy Bessel's partial differential equation which is a form of Laplace's equation with cylindrical symmetry.

### beta function

The *beta function* B $(m, n)$ is defined to be the integral:

.

It can be defined in terms of the gamma function by:

.

Many integrals can be reduced to the evaluation of Beta functions.

### Bieberbach conjecture

The *Bieberbach conjecture* states that if $f$ is a complex function given by the series

$f(z) = a_{0} + a_{1}z + a_{2}z^{2} + a_{3}z^{3} + ...$

which maps the unit disc conformally in a one-one way then $|a_{n}| ≤ n |a_{1}|$ for each $n$.

It can be expressed as:

The $n$th coefficient of a univalent function can be no more than $n$.

### binary-stars

Binary Stars are star systems where two stars are orbiting each other

### binomial

A *binomial* is one of Euclid's categories of irrational numbers.

A number of the form $√(√A + √B)$ is called a *binomial*.

The corresponding number with a - sign is called a *apotome*.

### binomial coefficient

The *binomial coefficients* are the coefficients of powers of $x$ in the expansion of $(1 + x)^{n}$.

We have

where the binomial coefficient is the number of ways of choosing an (unordered) subset of size $k$ from a set of size $n$.

The binomial coefficients are the entries in the Pascal triangle.

### binomial theorem

The *Binomial Theorem* is the result which allows one to expand a *binomial*:

$(x + y)^{n} = x^{n} + a_{n-1}x^{n-1}y + a_{n-2}x^{n-2}y^{2} + ... + y^{n}$

where the coefficients *ai * are called binomial coefficients.

### biquadratic residue

A number $m$ is a *biquadratic residue* modulo $n$ if $m = r^{4}$ modulo $n$ for some $r$.

### Boolean algebra

A *Boolean algebra* is an algebra in which the binary operations are chosen to model the union and intersection operations in Set Theory.

For any set $A$, the subsets of $A$ form a Boolean algebra under the operations of union, intersection and complement.

### boundary value problem

A *boundary value problem* is an ordinary or partial differential equation given together with *boundary conditions* to ensure a unique solution.

These boundary conditions usually involve specifying the value of the function or its derivative on the edges of the region on which the equation is defined.

### brachistochrone

The *brachistochrone* is a curve such that a particle descending along the curve under gravity will travel from one point to another in minimum time.

The cycloid is such a curve.

### Brownian motion

*Brownian motion* is the random motion of particles suspended in a liquid or a gas resulting from their collision with the fast-moving atoms or molecules in the gas or liquid.

It named after the botanist Robert Brown who first observed it.

### Burnside problem

The *Burnside problem* asks whether it is possible for a finitely generated group to be infinite if all its elements have finite order. This version is usually called the *General Burnside problem* and an example of such a group was found in 1964.

The *Burnside group* $B(d, n)$ is the largest $d$ generator group in which every element satisfies $x^{n} = 1$.

The *Restricted Burnside problem* asks whether, for fixed $d$ and $n$, there is a largest **finite** $d$ generator group in which every element satisfies $x^{n} = 1$. A positive solution to the Restricted Burnside problem would show that there are only finitely many finite factor groups of $B(d, n)$.

### Calcidius

Calcidius - (or Chalcidius) (4th century) was a Christian writer who produced an important Latin translations and commentaries of Plato's works.

### calculus of variations

The *calculus of variations* is a generalisation of the calculus. It attempts to find the path, curve, surface, etc., for which a given function has a stationary value (usually a maximum or minimum).

### catenary

A *catenary* is the curve in which a heavy uniform chain hangs.

Its equation is:

$y = \cosh(x) = \frac{1} {2} (e^{x} + e^{-x}) = 1 + \frac{1}{2} x^{2} + \frac{1} {24} x^{4} + ...$

### catoptrics

*Catoptrics* is the part of optics dealing with reflection.

### Catullus

Catullus (**born:** 84 BC in Verona in Cisalpine Gaul [now Italy], **died:** 54 BC in Rome) was a Roman poet who wrote poems about love and hate which are thought to be the best lyric poetry written in ancient Rome. Some of his poems express of contempt or hatred for Cicero, Pompey and Julius Caesar.

### celestial equator

The *celestial equator* is the equator on the celestial sphere midway between the fixed points of the North and South poles.

### celestial sphere

The *celestial sphere* is the sphere on which the stars appear to move in the sky.

### Cepheid Variables

*Cepheid Variables* are very luminous stars that pulsate. Their variance in intensity is directly related to their luminosity.

### character theory

The *character theory* of a group involves studying the traces (sums of the diagonal elements) of the matrix representations of the group.

The information gained is listed in *character tables* and the properties of these can be used to give insights into the properties of the group.

### Cicero

Cicero (**born:**106 BC in Arpinum in Latium [now Arpino in Italy], **died:** 43 BC in Formiae in Latium [now Formia]) was educated in Rome and in Greece. He then became a Roman statesman, lawyer, scholar, and writer. He tried unsuccessfully to maintain republican principles in the civil wars that destroyed the Roman Republic.

### circumscribed

A circle is said to be *circumscribed* to a triangle or other polygon if the vertices of the polygon lie on the circle.

The polygon is than said to be inscribed in the circle.

The (unique) circle circumscribed to a triangle is called the *circumcircle* and its centre is the *circumcentre*.

### Cosmic microwave background radiation

*Cosmic microwave background radiation* is radiation left over from the big bang that we currently measure to be at around 2.7K

### cohomology

*Cohomology* calculates algebraic invariants of topological spaces which are formally dual to homology.

The invariants obtained are in general more powerful than those given by homology and usually have more algebraic structure.

*Generalised cohomology theories*, both for topological spaces and for purely algebraic structures, have been developed which have some of the formal properies of cohomology but which do not have the same geometric background.

### combination

A *combination* is a subset of a given set where the order of elements is ignored (as distinct from a permutation).

For example there are 6 combinations of size 2 from the set {1, 2, 3, 4}:

{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4}.

The number of combinations of size $k$ from a set of size $n$ is the binomial coefficient also written $_{n}C_{k}$.

### commensurable

Two lines or distances are *commensurable* if the ratio of their lengths is a rational number.

If the ratio is an irrational number, they are called *incommensurable*.

### commutator

The *commutator* of two elements $x, y$ of a group is the element $x^{-1}y^{-1}x y$ and is written $[x, y]$.

The *commutator subgroup* $[H, K]$ of two subgroups $H$ and $K$ is the subgroup generated by all the commutators $[x, y]$ with $x \isin H$ and $y \isin K$.

### compact

A topological space is *compact* if every covering of the space by open sets has a finite sub-cover.

In the case of metric spaces compact spaces have the property that bounded sequences have cconvergent sub-sequences.

### compass variation

*Compass variation* or magnetic declination is the amount by which compass readings differ from the true North.

### composite number

A *composite number* is an integer which is not prime numbers.

Every composite number can be written as a product of prime numbers.

### concentric spheres

*Concentric spheres* are spheres with the same centre but different radii.

Early theories of cosmology had concentric spheres for the planets to move on.

### conformal mapping

A *conformal mapping* is a map from the plane (which can be regarded as $\mathbb{R}$2 or $\mathbb{C}$) to itself which *preserves angles*. That is, the angle between any two curves is the same as the angle between their images.

Any analytic map from $\mathbb{C}$ to $\mathbb{C}$ is conformal at a point where the derivative is non-zero.

### conjunction

A *conjunction* takes place when two heavenly bodies move into the same place in the sky.

(Literally, the word means *joined together*)

### continued fraction

The *continued fraction expansion* of a number $r$ is an expression of the form:

If $r$ is a rational number this expansion terminates.

### Continuum Hypothesis

The *Continuum hypothesis* states that there is no no set whose cardinality lies between that of the Natural numbers and that of the Reals.

The *Generalised Continuum hypothesis* states that if $A$ is any set, there is no set whose cardinality lies between the cardinality of $A$ and the cardinality of the set of all subsets of $A$.

The Continuum hypothesis has been shown to be independent of the other set-theory axioms.

### control theory

*Control theory* is the study of how to adjust the parameters in the equations controlling a system to maximise its performance.

### coprime

Two integers are *coprime* if they have no factors in common.

For example, 3 and 4 are coprime, but 25 and 15 are not since both have a factor of 5.

### cosa

The word *cosa* means "a thing" and was used for the unknown in an equation.

Algebraists were called *cossists*, and algebra the *cossic art*, for many years.

The *cosa and cube* was a particular form: $x^{3} + ax = b$ of a cubic equation with $a, b > 0$.

### cosmogony

*Cosmogony* is a theory of myth of the origin of the universe, especially of the stars, nebulae, etc.

### cosmology

*Cosmology* is the science of the universe as a whole.

### cubic equation

A *cubic equation* is an equation whose highest term is of degree three:

$ax^{3} + bx^{2} + cx + d = 0$

in general.

### cuboctahedron

A *cuboctahedron* is a solid which can be obtained by cutting the corners off a cube or off an octahedron. It has 8 faces which are equilateral triangles and 6 faces which are squares.

### Curtis Shapley Debate

The *Curtis Shapley Debate* was a debate in 1920 between Herber Curtis and Harlow Shapley about whether our universe consisted of only the milky way (advocated by Shapley) or if there were other galaxies (then called 'island universes', and advocated by Curtis). At the time of the debate there was no clear winner, but astronomers looking back agree that Curtis was closer to being correct.

### Damascius

Damascius (**born:** about 480, **died:** about 550) was appointed head of the Academy in Athens about 520. He was still head when the Christian emperor Justinian closed the Academy in 529. He was a Neoplatonist philosopher and follower of Proclus.

### declination

The *declination* is the angular distance from the celestial equator.

The other coordinate is the right ascension.

### deficient number

A *deficient number* is an integer for which the sum of its proper divisors is less than than the number itself.

For example, the proper divisors of 8 are 1, 2, 4, which sum to 7.

See also: perfect number, abundant number.

### Delian League

The *Delian League* was a confederation of Greek states led by Athens, with headquarters at Delos, founded in 478 BC during the Greco-Persian wars. All Greek states were invited to join to protect themselves from Persia. After Sparta defeated Athens in 404 BC they disbanded the league.

### Delphic oracle

The most famous ancient oracle was that of Apollo at Delphi, situated on Mount Parnassus. Its fame stretched across the Hellenic world by the 7th and 6th centuries BC. Lawmakers, founders of cults, those wanting to know the outcome of wars or politicians seeking to know the outcome of their actions went there for advice. It was given in a coded form that could be interpreted in different ways.

### determinant

The *determinant* of a matrix is a combination of the entries of the matrix with the property that the determinant is 0 if and only if the matrix is not invertible.

### Dialectic

*Dialectic* is the art of conversation, of question and answer; and according to Plato, dialectical skill is the ability to pose and answer questions about the essences of things.

### difference equation

A *difference equation* is an equation involving the differences between successive values of a function of an integer variable.

A difference equation can be regarded as the discrete version of a differential equation.

For example the difference equation $f(n+1) - f(n) = g(n)$ is the discrete version of the differential equation $f'(x) = g(x)$

### differential equation

A *differential equation* is an equation involving the first or higher derivatives of the function to be solved for.

If the equation only involves first derivatives it is called an equation of *order* one, and so on.

If only $n$-th powers of the derivatives are involved, the equation is said to have *degree* $n$. Equations of degree one are called *linear*.

Equations in only one variable are called *ordinary differential equations* to distinguish them from partial differential equations.

### differential geometry

*Differential geometry* studies properties of manifolds which depend on being able to do differentiation.

Examples of the properties studied are curvature and torsion for curves and various curvatures for surfaces.

### differential topology

*Differential topology* is a branch of topology in which one investigates properties of differential geometry which are preserved by continuous transformation.

### Diogenes Laertius

Diogenes Laertius (3rd century BC) was a Greek author who wrote an important history of Greek philosophy. His major work *Lives, Teachings, and Sayings of Famous Philosophers* quotes hundreds of authorities and ranges from gossip to valuable biographical data.

### Dionysius I

Dionysius I (**born:** about 430 BC, **died:** 367 BC) was a tyrant from Syracuse who conquered Sicily and southern Italy making Syracuse the strongest Greek city west of the Greek mainland. He prevented Sicily from being conquered by Carthage but his cruel regime damaged the reputation of Greece.

### Dionysius II

Dionysius II (**born:** about 400 BC, **died:** about 330 BC) was a son of Dionysius I he succeeded his father in 367 BC and made peace with Carthage. Driven out bt Dion in 357 BC he was in exile until 346 BC when he regained control of Syracuse. He surrendered and retired to Corinth in 344 BC.

### diophantine approximation

*Diophantine approximation* is approximating real numbers by rationals (ratios of integers).

### diophantine equation

A *Diophantine equation* is one which is to be solved for integer solutions only.

### discriminant

The *discriminant* of a polynomial equation is a function of the coefficients which allows one to say when it has a pair of equal roots.

The discriminant of the quadratic equation: $ax^{2} + bx + c = 0$ is $b^{2} - 4ac$.

The discriminant of the cubic equation: $x^{3} + qx + r = 0$ is $4q^{3} + 27r^{2}$.

The *discriminant* of an algebraic number field is an important numerical invariant that (roughly) measures the size of the ring of integers of the field.

### docent

A *docent* or *dozent* or *privatdozent* was a recognised teacher at a German university who was not on the salaried staff. It was often a post held by a young mathematician working for his habilitation.

### dodecahedron

A *dodecahedron* is a regular polyhedron with 12 faces each of which is a regular pentagon.

### duplicating the cube

One of the classic problems of Greek mathematics was to find a ruler and compass construction for the cube root of 2. This was called *duplicating the cube*

This is sometimes called the *Delian problem* from the story that the oracle at Delphi demanded that this construction be performed to stop a plague.

The other classic problems were squaring the circle and trisecting an angle.

### dynamical system

A *dynamical system* gives a means of describing how one state of a system evolves into another.

Its study goes back to the work of Poincaré's work on the three body problem or even earlier to Huygen's work on the compound pendulum.

### eccentric theory

*Eccentric theory* is the theory that planets move round in circles whose centres do not coincide with the Earth

### ecliptic

The *ecliptic* is the plane in which the orbit of the Earth cuts the celestial sphere.

Hence the apparent annual motion of the Sun among the fixed stars.

### ecliptic coordinates

*Ecliptic coordinates* are the description of the position of a heavenly body in terms of its right ascension and declination.

These are given relative to the ecliptic plane.

### eigenvalue/vector/function

If $T$ is a linear transformation from a vector space to itself, then a non-zero vector $\mathbb{v}$ satisfying $T\mathbb{v} = \lambda \mathbb{v}$ is called an *eigenvector* and λ is called the *eigenvalue*.

If the space is a space of functions, the eigenvector may be called an *eigenfunction*.

### Eleatic School

The *Eleatic School* was a school of philosophy from Elea which was a Greek colony in southern Italy. The main doctrines were due to Parmenides who argued that reality was a single entity, the apparent many objects we detect with our senses are only appearances or names.

### ellipse

A *ellipse* is one of the conic sections.

It may be defined using the focus directrix property as the locus of points whose distance from a fixed point is a fixed multiple $e < 1$ from a fixed line.

or via Cartesian coordinates as the set of points in a plane satisfying the equation :

$ax^{2} + by^{2} = 1$

### elliptic function

An *elliptic function* is an analytic function from $\mathbb{C}$ to $\mathbb{C}$ which is doubly periodic. That is, for two independent values of the complex number $w$, the functions $f(z)$ and $f(w + z)$ are the same.

It can also be regarded as the inverse function to certain integrals (called *elliptic integrals*) of the form where $R$ is a polynomial of degree 3 or 4.

### entire function

In complex analysis, an *entire function* (sometimes called an *integral function*) is a complex-valued function that is holomorphic over the whole complex plane.

Such a function then has a complex derivative everywhere in **C**.

### Epictetus

Epictetus (**born:** about 55 in Hierapolis, in Phrygia [now Pamukkale in Turkey], **died:** about 135 in Nicopolisin in Epirus [Greece]) was a slave who attend lectures by the Stoic Musonius Rufus. He gained his freedom but spent his life suffering ill health. He was expelled from Rome in 90 with other Stoic philosophers by the emperor Domitian. He spent the rest of his life at Nicopolis. His teachings were popular with Christians because of their religious tone.

### Epicureans

*Epicureans* were followers of Epicurus who believed pleasure was the basis of a happy life. Although they believed that the Gods were too busy to pay attention to humans, yet men had to aspire to like the Gods since only in this way could they approach perfection.

### Epicurus

Epicurus (**born:** 341 BC in Samos in Greece, **died:** 270 BC in Athens) was a pupil of Nausiphanes, a disciple of Democritus. He spent most of the first 35 years of his life in Asia, then went to Athens in 306 and established The Garden. His philosophy was based on simple pleasure and friendship. He also proposed an atomic theory.

### epicycle

An *epicycle* is a circle whose centre is carried around the circumference of another circle.

Epicycles were introduced originally to explain the orbits of planetary bodies among the fixed stars.

### equatorial coordinates

*Equatorial coordinates* are the description of the position of a heavenly body given relative to the plane of the Earth's equator.

They differ from ecliptic coordinates because of the inclination or obliquity of the Earth's axis of rotation to the plane of the ecliptic.

### equinox

The *equinox* is the time of the year when the noon sun is overhead at the equator making day and night equal in length.

Equinoxes occur about 21st March and 23rd September.

### equivalent numbers

Two numbers *m* and *n* are called *equivalent* if sums of their proper divisors are the same.

For example, the proper divisors of 159, 559 and 703 all sum to 57.

### ergodic theory

*Ergodic theory* investigates the statistical and qualatitive behaviour of a system.

It originated in the study of statistical mechanics but now has applications in many other fields.

### Pi Chronolgy Note 10

Brahmagupta criticises Aryabhata for his earlier, and much better, value of π. He gave 3 as the approximate value for practical calculation and √10 as the exact value.

### ethics

*Ethics* is the science of morals: the branch of philosophy dealing with human character and conduct.

### Euclid of Megara

Euclid of Megara (**born:** 430 BC, **died:** about 360 BC) was a pupil of Socrates. His philosophy was that "the good is one, though it is called by many names, sometimes wisdom, sometimes God, and sometimes reason" and that "the contrary to the good has no reality".

### euclidean algorithm

The *Euclidean algorithm* is an efficient procedure to calculate the highest common factor $d$ of a pair $m, n$ and write it in the form $d = pm + qn$ with $p, q$ integers.

### Euler-Maclaurin Summation Formula

The *Euler-Maclaurin Summation Formula* relates integrals and sums of series. It can be used to evaluate integrals as sums or sums of infinite series by integrals. It was discovered independently by Euler and Maclaurin.

### Euler's conjecture

In 1769 Euler conjectured that it is impossible to exhibit three fourth powers whose sum is a fourth power, four fifth powers whose sum is a fifth power, and similarly for higher powers. Counterexamples were not found until the 20th Century

### factor group or quotient group

If $N$ is a normal subgroup of a group $G$ then one can define a *factor group* $G/N$ in which the elements of $N$ all "become the identity".

One can then try to understand the structure of $G$ in terms of the structures of $N$ and of $G/N$.

If $G$ is finite then the order of $G/N$ is $|G|/|N|$.

### factorial

The *factorial* of an integer *n* is the product 1.2.3. ... .*n*

It is denoted by $n!$

Generalised to non-integers, it is the Gamma function.

### Fermat's last theorem

*Fermat's last theorem* (so called because it was the last of the results which Fermat claimed and which had not been proved) states that if $n > 2$ then the equation $x^{n} + y^{n} = z^{n}$ has no positive integer solutions.

### Fibonacci sequence

The *Fibonacci sequence* is the sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ... generated by the rule $f_{1} = f_{2} = 1, f_{n+1} = f_{n} + f_{n-1}$.

It was introduced by Fibonacci of Pisa in *Liber abaci* in a problem involving the growth of a population of rabbits.

### Fields Medal

Fields Medals are given every four years to the most distinguished mathematicians aged 40 or under. In the absence of a Nobel prize in mathematics, they are regarded as the highest professional honour a mathematician can attain.

They were funded by the Canadian mathematician J C Fields and were first awarded in 1936.

### focus directrix

The *focus directrix* definition of a conic section is as the locus of a point which moves so that its distance from a fixed point (the *focus*) is a fixed multiple $e$ from a fixed line (the *directrix*).

If $e = 1$ then the curve is a parabola, if $e < 1$ then it is an ellipse, if $e > 1$ then it is a hyperbola.

### four colour theorem

The *four colour theorem* states that any map of regions on a plane or sphere can be coloured with at most four colours in such a way that regions sharing a boundary are coloured differently.

Although the result was conjectured as early as 1852 its proof was not completed until 1976.

### free group

A *free group* on a set $A$ of *generators* is the group consisting of all finite *words* in elements of $A$ and their inverses.

So if, for example, $A = {x, y, z}$ then words such as $x y^{2}z^{-1}x$ or $x^{2}z y^{-1}$ are elements of the free group.

One multiplies such words by *concatenating* them (writing them alongside one another) and then cancelling out any elements which are alongside thir inverses.

The *empty word* is the identity of this group.

The free group on a set of size $n$ is denoted $F_{n}$.

### Fourier analysis

*Fourier analysis* is the process by which one may write a periodic function on (say) the interval [ 0, 2π ] as the sum of multiples of the functions $\sin(nx)$ and $\cos(nx)$.

$f(x) = a_{0} + a_{1}\cos(x) + b_{1}\sin(x) + a_{2}\cos(2x) + b_{2}\sin(2x) + ...$

This is called the *Fourier series* of $f$ and the numbers $a_{i}$ and $b_{i}$ are called the *Fourier coefficients*.

### functional analysis

*Functional analysis* studies infinite dimensional vector spaces and mappings between them.

The elements of these spaces are very often themselves functions, as for example, the space of continuous functions on an interval.

### fundamental theorem of algebra

The *fundamental theorem of algebra* is the result that any polynomial with real or complex coefficients has a root in the complex plane.

### fundamental group

The *fundamental group* of a topological space $X$ is a group constructed by looking at how closed paths in $X$ can be combined to get new paths.

Under a suitable way of identifying paths ("homotopy") one can get a group structure on the set which gives an algebraic invariant of the space $X$.

### Gaius Marius

Gaius Marius (**born:** about 157 BC in Cereatae, near Arpinum [Arpino] in Latium [now in Italy], **died:** 86 BC in Rome) was a good soldier and a skilful general who was elected consul seven times on the strength of the votes from his soldiers. He was a poor politician.

### Galois theory

*Galois theory* is the study of certain groups which can be associated with polynomial equations.

Whether or not the solutions to an equation can be written down using rational functions and square roots, cube roots, etc. depends on certain group-theoretic properties of this *Galois group*.

### game theory

*Game theory* deals with the analysis of "games" (i.e., situations which involve conflicts of interests). As well as simple games, which can be analysed completely, the theory has applications in "real games" like poker, chess, etc as well as in areas like politics, economics etc.

### gamma function

The *gamma function* is a generalisation of the factorial function to the Real line and to the Complex plane.

It is defined by:

If $n$ is an integer then $\Gamma (n+1) = n!$

### Gelfond's conjecture

In 1929 Gelfond conjectured that:-

If $a_{n}$ and $b_{n}$ for $1 ≤ m ≤ n$ are algebraic numbers such that {$\log_{e} a_{n}, 1 ≤ m ≤ n$} are linearly independent overQthen

$b_{1} \log_{e} a_{1} + b_{2} \log_{e} a_{2} + ... + b_{n} \log_{e} a_{n} ≠ 0$.

A special case of this is that $a^{b}$ is transcendental if $a$ is algebraic ($a ≠ 0, 1$) and $b$ is an irrational algebraic number.

### general or point-set topology

*General topology* is a branch of topology in which one investigates how to put a structure on a set in such a way as to generalise the idea of continuity for maps from the reals $\mathbb{R}$ to itself.

A *topology* on a set $X$ is a certain set of so-called "open subsets" of the set $X$ which satisfy various axioms. The set $X$ together with this topology is called a *topological space*.

### geometric progression

A *geometric progression* is a sequence in which each number is the same multiple (the common ratio) of the previous one.

As for example: 3, 6, 12, 24, 48, ...

### Goldbach's conjecture

In 1742, in a letter to Euler, Goldbach conjectured that any even integer > 2 can be written as the sum of two primes. Despite much progress, the conjecture is still unproved.

### golden ratio

The *golden ratio* or *golden number* or *golden mean* is the real number (√5 + 1)/2 = 1.6180... .

It occurs in many areas in mathematics, including the geometry of the regular pentagon, continued fractions, the Euclidean algorithm, ...

### Gravitational Waves

Gravitational Waves

### great circle

A *great circle* is a line of shortest distance on a sphere: the intersection of the sphere with a plane through the origin.

### group

A *group* is a set together with a method of combining elements to get new ones (addition or multiplication or ...) which satisfies certain properties making it suitable for a wide variety of applications.

Groups of permutations, symmetries, matrices, ... are widely used in many areas of mathematics and physics.

### group presentation

A *group presentation* is a way of representing a group in the form $< A | R >$where $A$ is a set of generators for the group and $R$ is a set of relations which are satisfied by words in the generators.

So, for example$< a, b | a b = b a >$is a free abelian group with two generators.

### gymnasium

A *Gymnasium* is a senior secondary school in Germany and certain other countries of mainland Europe.

### habilitation

The *Habilitation* is the extra post-doctoral qualification needed to lecture at a German university. The thesis that has to be presented as part of this is the *Habilitationsschrift*.

### heliocentric theory

*Heliocentric theory* is the theory that planets move around the Sun rather than the Earth

### Heracleitus

Heracleitus (**born:** about 540 BC in Ephesus in Anatolia, **died:** about 480 BC) was a Greek philosopher who tried to explain the world based on the belief that fire forms the basic material principle of an orderly universe. He stressed that men must live together in social harmony. He is famous for his analogy of life to a river.

### Herodotus

Herodotus (**born:** 484 BC in Halicarnassus in Asia Minor [now Bodrum in Turkey], **died:** about 425 BC) travelled widely through the Persian Empire but spent time in Athens and Thurii, a colony of Athens in southern Italy. He is famed as an historian whose most famous work is his history of the wars between Greece and Persia from 499 BC to 479 BC.

### Hilbert space

A *Hilbert space* is a vector space over $\mathbb{R}$ or $\mathbb{C}$ equipped with an inner-product.

The most important real example is the space of square summable sequences together with the usual dot-product.

### hippopede

A *hippopede* is a figure of eight curve Eudoxus used in his planetary theory.

(The word means a *horse-fetter*).

### holomorphic

A *holomorphic* function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain.

### Homer

Homer (before 7th century BC) was the poet who wrote the famous Iliad and Odyssey. It is far from certain that he actually existed, but if he did he must be regarded as one of the greatest poets of all time. These works were unifying factors in the development of early Greece and Homer must have played a large part in its success.

### homocentric spheres

The *homocentric sphere system* consisted of a number of rotating spheres, each sphere rotating about an axis through the centre of the Earth.

The axis of rotation of each sphere was not fixed in space but, for most spheres, this axis was itself rotating as it was determined by points fixed on another rotating sphere.

### homology

*Homology* is a way of attaching abelian groups (or more elaborate algebraic objects) to a topological space so as to obtain algebraic invariants. In some sense it detects the presence of "holes" of various dimensions in the space.

The methods developed to handle this led to what is now called *homological algebra* and homological invariants can be calculated for many purely algebraic structures.

### homotopy

A *homotopy* is a continuous transformation from one path in a topological space to another or more generally of one function to another.

Paths which are connected by a homotopy are called *homotopic* and are said to be in the same *homotopy class*.

Properties which are left unchanged by such homotopies are called *homotopy invariants*.

Homotoy classes of paths can be composed to form the fundamental group or *first homotopy group*. Other maps can be used to form *higher homotopy groups*.

### hyperbola

A *hyperbola* is one of the conic sections.

It may be defined using the focus directrix property as the locus of points whose distance from a fixed point is a fixed multiple $e > 1$ from a fixed line.

or via Cartesian coordinates as the set of points in a plane satisfying the equation :

$ax^{2} - by^{2} = 1$

### hypergeometric function

The *hypergeometric function* introduced by Gauss, is the sum of the *hypergeometric series*:

It is the solution of the *hypergeometric equation*:

Many common functions can be written as hypergeometric functions.

### hypotenuse

The *hypotenuse* of a right-angled triangle is the side opposite the right angle.

### Iamblichus

Iamblichus (**born:** about 250 AD in Chalcis in Coele Syria [now in Lebanon], **died:** about 330) was a Syrian philosopher who played a large part in the development of Neoplatonism. His writings which have survived include *On the Pythagorean Life*; *The Exhortation to Philosophy*; *On the General Science of Mathematics*; *On the Arithmetic of Nicomachus*; and *Theological Principles of Arithmetic*. He was a major influence on Proclus.

### icosahedron

An *icosahedron* is a regular polyhedron with 20 faces each of which is an equilateral triangle.

### idempotent

An element $r$ of a ring or semigroup is *idempotent* if it satisfies $r^{2} = r$.

### infinitesimal

An *infinitesimal* is an arbitrarily small quantity which early mathematicians found it necessary to incorporate into their theories in the absence of a proper theory of limits.

*Infinitesimal calculus* is the Differential and Integral Calculus.

### inscribed

A circle is said to be *inscribed* to a triangle or other polygon if the edges are tangents to the circle.

The polygon is than said to be circumscribed to the circle.

The (unique) circle inscribed to a triangle is called the *incircle* and its centre is the *incentre*.

### integral equation

An *integral equation* is an equation which involves an integral of the function which is to be solved for.

For example, being given $g, w$ and $a$, one might wish to solve the equation for $f$.

### integral transform

An integral transform is a process involving an integral which can be applied to a function to get another function.

It is usually of the form for some "weight" function $w$ where the integral is taken over an interval $I$.

Examples of integral transforms are Laplace transforms or Fourier transforms.

### integrating factor

An *integrating factor* is something which one multiplies an ordinary differential equation by to make it possible to solve it.

### intuitionism

*Intuitionism* is a branch of logic which stresses that mathematics has priority over logic, the objects of mathematics are constructed and operated upon in the mind by the mathematician, and it is impossible to define the properties of mathematical objects simply by establishing a number of axioms.

In particular, intuitionists reject the *Law of the Excluded Middle* which allows proof by contradiction.

### Invariant theory

An *invariant* is something that is left unchanged by some class of functions. In particular, *invariant theory* studied quantities which were associated with polynomial equations and which were left invariant under transformations of the variables.

For example, the *discriminant*: $b^{2} - 4ac$ is an invariant of the quadratic form: $ax^{2} + bxy + cy^{2}$.

### involute

If $C$ is a curve and $C'$ is its evolute, then $C$ is called an *involute* of $C'$.

Any parallel curve to $C$ is also an involute of $C'$.

Hence a curve has a unique evolute but infinitely many involutes.

Alternatively an involute can be thought of as any curve orthogonal to all the tangents to a given curve.

### irrational

An *irrational number* is a real number which is not rational and so cannot be written as a quotient of integers.

For example, √2 is irrational.

### Isidorus

Isidorus (about 537 AD) was an architect from Miletus who directed the building of the Hagia Sophia in 537.

### isoperimetry

*Isoperimetry* is the comparison of the areas of figures with the same perimeter of of the volumes of solids with the same surface areas.

### Kepler's problem

Kepler speculated in that the most efficient arrangement of spheres in three-dimensions is the most obvious arrangement used to stack oranges, cannon-balls, ... . This was eventually proved to be true in 1998.

### knot theory

*Knot theory* is the study of the way a closed curve can be embedded in three dimensional space without intersecting itself.

Intuitively, one may "make a knot" by tying a knot in an ordinary piece of string and then fusing together the free ends of the string.

Associated with any knot is its *knot group* which is the fundamental group of the space obtained by removing the knot from the $\mathbb{R}$3 in which it is embedded.

### Laplace transform

The *Laplace transform* of a function *f* is defined by the integral:

### Laplace's equation

*Laplace's equation* is one of the important partial differential equations of physics.

In two dimensions it is: while in three dimensions it is .

### Lie algebra

A *Lie algebra* is an algebra in which the muliplication satisfies properties similar to the so-called *bracket operation* on matrices given by $[A, B] = AB - BA$ where the operation on the right-hand side are ordinary multiplication and subtraction of matrices.

The operation is not associative.

### Lie group

A *Lie group* is a group which is also a manifold.

Groups of real matrices give naturally occurring examples of Lie groups.

The tangent space at the identity element of a Lie group forms a Lie algebra in a natural way.

### Lie ring

A *Lie ring* is a ring in which the muliplication satisfies properties similar to the so-called *bracket operation* on matrices given by $[A, B] = AB - BA$ where the operation on the right-hand side are ordinary multiplication and subtraction of matrices.

The operation is not associative.

### linear group

A *linear group* is a group of matrices under matrix multiplication.

### Livy

Livy (**born:** 62 BC in Patavium in Venetia in Italy, **died:** 17 AD in Patavium) spent most of his life in Rome. He was one of the great Roman historians and he wrote a history of Rome in 142 books that became a classic in his lifetime. The work had a major influence on the style and philosophy of historical writing for nearly 2000 years.

### locus

A *locus* is the line or surface made up of all the positions of a point or line satisfying some condition.

For example, the locus of points equidistant from a fixed point is a circle.

### loxodrome

A *loxodrome* is a line on a sphere cutting each meridian at the same angle. It is the path followed when following a fixed compass bearing and is a straight line on the Mercator's projection.

### Lucian

Lucian (**born:** about 120 AD in Samosata in Commagene in Syria [now Samsat, in Turkey], **died:** after 180 in Athens [Greece]) was educated in western Asia Minor where he received a Greek literary education. He travelled in Greece, Italy and then Gaul. In Athens he became a famous rhetorician, pamphleteer and satirist.

### lune

A *lune* is the area cut off by one circle from the interior of a smaller one.

### magic square

A *magic square* is a set of integers (often $1, 2, ... , n^{2}$) arranged in a square in such away that each row, each column (and often the two diagonals as well) sum to the same number.

For example: is a 3 × 3 magic square.

### magnetic declination

*Magnetic declination* or compass variation is the amount by which compass readings differ from the true North.

### Absolute and Apparent Magnitude

The *absolute magnitude* is how bright the star is from 10 parsecs away.

The *apparent magnitude* is how bright the star appears on earth.

### manifold

An $n$-dimensional *manifold* is a topological space which is locally Euclidean. That is, every point lies in a region which looks like the space $\mathbb{R}$$^{n}$.

A 1-manifold is a curve, a 2-manifold is a surface, etc.

Manifolds which have a structure which allow differentiation to be performed are called *differentiable manifolds* and if differentiation can take place arbitrarily often they are called *smooth manifolds*.

### Marcellus

Marcellus (**born:** about 268 BC, **died:** 208 BC near Venusia in Apulia) was a Roman general who was consul five times. After the Romans were defeated at Cannae in 216 BC, he commanded the army and saved regions from Hannibal. From 214 to 211 he served in Sicily, where, after a two-year siege, he took Syracuse in a siege in which Archimedes died. He later fought inconclusive battles with Hannibal.

### Martingale

In probability theory, a *martingale* is a model of a fair game where knowledge of past events never helps predict the mean of the future winnings.

Originally it was a gambling system of continually doubling the stakes in the hope of an eventual win that must yield a net profit.

### matrix

A *matrix* is two dimensional array or "box" of numbers or more complicated things.

There are (easy) rules for adding and subtracting matrices of the same shape and sometimes (more complicated) rules for multiplying or even dividing them.

Sets of square matrices often form interesting algebraic objects like groups or more general structures.

Some examples:

### measure theory

*Measure theory* investigates the conditions under which integration can take place.

It considers various ways in which the "size" of a set can be estimated.

### medial

A *medial* is one of Euclid's categories of irrational numbers.

If the sides of a rectangle are commensurable in square only (e.g., 3 and √2), then the side of the square equal to the rectangle (3√2, in this case) is an irrational called a *medial*.

### Menedemus of Pyrrha

Menedemus of Pyrrha was a member of the Academy in Plato's lifetime.

### Meromorphic function

A *meromorphic function* is a function of a complex variable which has no essential singularities. That is, its worst singularities are poles

### metaphysics

*Metaphysics* is the branch of philosophy that investigates the first principles of nature and thought.

### method of exhaustion

The *method of exhaustion* is calculating an area by approximating it by the areas of a sequence of polygons.

For example, filling up the interior of a circle by inscribing polygons with more and more sides.

### Milky Way

The *Milky Way* is one of billions of galaxies that make up the universe. It is a large disk-shaped system of stars in which the sun is situated and appears as an irregular luminous band that circles the sky. The luminous band is caused by large numbers of stars each too faint to be seen in individually, but the large number gives a background glow in that part sky in the plane of the galaxy.

### model theory

*Model theory* is the study of those mathematical structures which satisfy a particular set of axioms -- particularly in the area of Logic.

### modulo

We say that numbers $m$ and $n$ are equal *modulo* $p$, if $p$ divides exactly into $m - n$.

### Neoplatonist

*Neoplatonists* were the last great ancient Greek school of philosophy. The school was founded by Plotinus in the 3rd century. Plotinus and his followers considered themselves Platonists but their philosophy contained ideas from Aristotelian and the Stoics as well as from Plato.

### nilpotent group

A *nilpotent group* is a group which has a chain of subgroups ${1} = \gamma _{n+1} \subset \gamma _{n} \subset ... \subset \gamma _{1} = G$ in which each subgroup $\gamma _{i}$ is the commutator subgroup $[ \gamma _{i-1}, G]$.

The smallest $n$ for which $\gamma _{n+1} = {1}$ is the *index of nilpotence*.

Every nilpotent group is a direct product of $p$-groups for different primes $p$.

### non-Euclidean

A *non-Euclidean geometry* is a geometry in which Euclid's parallel postulate fails, so that there is not a unique line failing to meet a given line through a point not on the line.

If there is more than one such "parallel" the geometry is called *hyperbolic*; if there is no line which fails to meet the given one, the geometry is called *elliptic*.

### non-standard analysis

*Non-standard analysis* is a theory which gives an alternative model for the Real numbers (sometimes called *hyperreals*) in which infinitesimals (numbers > 0 but < ${{1}\over{n}}$ for all $n$) can be interpreted in a different way.

### normal subgroup

A *normal subgroup* is a subgroup which is closed under the operation of conjugation.

That is, a subgroup $N$ of a group $G$ such that whenever an element $n$ is in $N$ and $g$ is in $G$, the element $gng^{-1}$ is in $N$.

### The Rhind Papyrus

Problem 50

Example of a round field of diameter 9 khet. What is its area?

Take away 1/9 of the diameter, namely 1: the remainder is 8. Multiply 8 times 8: it makes 64. Therefore it contains 64 setat of land.

Do it thus:

1 9 1/9 1 this taken away leaves 8Its area is 64 setat.

1 8 2 16 4 32 8 64

### number theory

*Number theory* is the study of the properties of the natural numbers **N**.

It includes such topics as prime numbers, including the prime number theorem, quadratic reciprocity, quadratic forms, diophantine approximation and diophantine equations, algebraic number fields, Fermat's last theorem and the methods developed to prove it.

### obliquity of the ecliptic

The *obliquity of the ecliptic* is the "tilt" of the Earth's axis of rotation relative to the ecliptic plane. It is currently about 23.4° and slowly decreasing).

### oblong number

An *oblong number* is any positive integer which is not a perfect square

### occultation

*Occultation* is the concealing of one heavenly body by another, as for example of a star by a planet.

### octahedron

An *octahedron* is a regular polyhedron with 8 faces each of which is an equilateral triangle. It looks like two square pyramids with their square faces glued together.

### Operational research

*Operational research* a method of mathematically based analysis for providing a quantitive basis for management decisions.

### opposition

*Opposition* is the situation of a heavenly body when it is directly opposite another: usually the Sun.

### osculate

A pair of curves *osculate* (from the Latin for "kiss") at a point if they both pass through the point and if their first and second derivatives at the point coincide.

For example the curve $y = x^{3}$ and the $x$-axis osculate at the origin.

### Panaetius of Rhodes

Panaetius of Rhodes (**born:** about 195 BC in Lindus in Rhodes, **died:** 109 BC) studied the philosophies of Plato and of Aristotle in Athens. For many years he lived in Rome, then he spent the last 20 years of his life in Athens as head of the school in Athens.

### parabola

A *parabola* is one of the conic sections.

It may be defined using the focus directrix property as the locus of points which are equidistant from a fixed line and a fixed point.

or via Cartesian coordinates as the set of points in a plane satisfying the equation :

$y = x^{2}$

### Parallax

*Parallax* is the angular difference from the lines of sight from two observation points to the object being measured.

### parallel postulate

The *parallel postulate* is Euclid's fifth postulate: equivalent to the idea that there is a unique parallel to any line through a point not on the line.

### Parmenides

Parmenides (**born:** about 515 BC, **died:** about 450 BC) was the Greek philosopher of Elea in southern Italy who founded the Eleatic School of philosophy.

### Parsecs

A *Parsec* is a measurement of distance frequently used in astronomy, 1 parsec is about 31 trillion km.

### partial differential equation

A *partial differential equation* is an equation involving derivatives with respect to more than one variable.

Many of the equations used to model the physics of the real world are partial differential equations.

### pascal triangle

The *Pascal triangle* (actually known long before Pascal) is a table of the binomial coefficients where the $(n, k)$th entry is .

Each entry is the sum of the pair immediately above it.

### Peloponnesian War

The *Peloponnesian War* was a war between Athens and Sparta which took place between 431 BC and 404 BC. Essentially all the Greek states were involved on one side or the other. It ended with defeat for Athens.

### perfect number

A *perfect number* is an integer for which the sum of its proper divisors is equal to the number itself.

For example, 6 and 28 are both perfect numbers.

See also: abundant number, deficient number.

### Pericles

Pericles (**born:** about 495 BC in Athens, **died:** 429 in Athens) was an important statesman who developed democracy in Athens and helped make Athens the political and cultural centre of the Greek world. He is responsible for the construction of the Acropolis on which work started in 447.

### Permanent

In linear algebra, the *permanent* of a square matrix is a function of the matrix similar to the determinant.

The definition of the permanent of $A$ differs from that of the determinant of $A$ in that the signatures of the permutations are not taken into account.

### permutation

A *permutation* of an ordered set $X$ is a rearrangement of the elements of $X$.

For example if $X$ is the ordered set (1, 2, 3) there are six permutations of $X$:

(1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), (3, 2, 1).

In general, a set with $n$ elements has $n!$ permutations.

One may permute $k$ elements out of a set of size $n$ and get $n!/k!$ permutations. This number is sometimes written $_{n}P_{k}$ to distinguish it from the number $_{n}C_{k}$ of combinations.

### permutation group

One can combine permutations by treating them as maps from a set to itself and composing the maps. Some sets of such permutations then form a *group of permutations*.

### Philolaus

Philolaus (lived about 475 BC) was a student of Pythagoras who, after Pythagors died, fled to Lucania and then to Thebes in Greece. He returned to Italy where he taught Archytas. According to Speusippus, Philolaus wrote a book containing a doctrine based on the first four numbers.

### Plateau problem

The *Plateau problem* is to show the existence of a surface of minimal area with a given boundary curve

Dipping a frame in the shape of the curve into a soap solution will produce a film which takes the form of this surface.

### Pliny

Pliny (**born:** 23 in Novum Comum in Transpadane Gaul [now in Italy], **died:** 79 in Stabiae near modern Naples) studied in Rome and then after a military career he returned to Rome and studied law. Later he became procurator in Spain, but returned to Rome. He wrote *Natural History* which is an encyclopaedic work on scientific subjects.

### Plotinus

Plotinus (**born:** 205 AD in Lyco in Egypt, **died:** 270 in Campania) began to study philosophy in Alexandria when he was 28 years old. He joined an expedition to Persia where he hoped to learn more of the philosophies of the Persians and Indians. He spent the last years of his life teaching in Rome. He was the founder of the Neoplatonic school of philosophy. His biography was written by his student Porphyry.

### Plutarch

Plutarch (**born:** about 46 AD in Chaeronea in Boeotia [Greece], **died:** after 119) studied mathematics and philosophy at Athens as a student of Ammonius. He travelled widely but kept his contacts with the Academy in Athens. He was an author writing biographies and essays on many topics such as ethics, politics, and science.

### Poincaré conjecture

In 1904 Poincaré conjectured that any closed 3-dimensional manifold which is homotopy equivalent to the 3-sphere must be the 3-sphere. Higher-dimensional analogues of this conjecture were proved first and, the original conjecture was proved by Grigori Perelman in 2002.

### polygonal number

A *polygonal number* is the number of dots that may be arranged in a regular polygon.

As, for example triangular numbers, square numbers, ...

### Pompey the Great

Pompey the Great (**born:** 106 BC in Rome, **died:** 48 BC in Pelusium in Egypt) was a statesman and Roman general. He led successful campaigns in Africa (where his soldiers called him "the Great") and then reorganised the East of the Roman Empire where he greatly strengthened control by Rome. He was involved in a power struggle with Julius Caesar which resulted in civil war. Pompey was defeated.

### potential/harmonic theory

*Potential theory* or *harmonic analysis* is the study of *harmonic* or *potential functions*.

These are solutions of the partial differential equation called Laplace's equation.

Such functions are of great use in the study of electricity, electromagnetism, gravitation and fluid flow.

### precession of the equinoxes

The *precession of the equinoxes* is a slow westward motion of the equinoctal points along the ecliptic caused by the greater attraction of the Sun and Moon on the excess of matter at the equator, so that the times at which the Sun crosses the equator come at shorter intervals than they would otherwise do.

### prime number

A *prime number* is an integer > 1 is prime if it is divisible only by itself and 1. The number 1 is *not* considered prime.

Every positive integer can be written as a product of prime numbers in a unique way (up to the order of the factors).

### prime number theorem

The *Prime Number Theorem* states that

The number of primes ≤ *n* tends to ∞ as fast as $n/\log_{e} n$.

### primitive root

A *primitive root* for a prime $p$ is one whose powers generate all the non-zero integers modulo $p$.

For example, 3 is a primitive root modulo 7 since:

3 = 31 , 2 = 32 mod 7, 6 = 33 mod 7, 4 = 34 mod 7, 5 = 35 mod 7, 1 = 36 mod 7.

### probability theory

*Probability theory* studies the possible outcomes of given events together with their relative likelihoods and distributions.

In fact there is considerable debate about exactly what *probability* means in practice. Some mathematicians regard it as simply a component of an abstract theory, while others give it an interpretation based on the frequencies of certain outcomes.

### projective geometry

*Projective geometry* is the branch of geometry dealing with the properties and invariants of geometric figures under projection from a point.

So, for example, a circle may be projected into an ellipse or a hyperbola and so these curves may all be regarded as equivalent in projective geometry.

### Propertius

Propertius (**born:** 50 BC in Assisi in Umbria [Italy], **died:** after 16 BC in Rome) is regarded as the greatest elegiac poet of ancient Rome. He published four books of poems.

### Protagoras

Protagoras (**born:** about 485 BC in Abdera in Greece, **died:** about 410 BC) lived most of his life in Athens but was exiled during the last five years of his life. He taught philosophy for 40 years and was the first of the Sophists. His teaching made him both famous and wealthy.

### pyramidal number

A *pyramidal number* is the number of dots that may be arranged in a pyramid with a regular polygon as base.

As, for example *tetrahedral numbers* (triangular base), *square pyramidal numbers*, ...

### quadratic equation

A *quadratic equation* is an equation with a second order term: $ax^{2} + bx + c = 0$ in general.

### quadratic form

A *quadratic form* is a general expression with a second order terms.

In two variables it is usually written:

$ax^{2}+ 2hxy + by^{2} + 2gx + 2fy + c$.

### quadratic number field

A *quadratic number field* is a sub field of either $\mathbb{R}$ or $\mathbb{C}$ consisting of numbers of the form $a + b√n$ with $a, b, n$ rational numbers and $n$ with no rational square root.

If $n$ is positive this is called a *real* quadratic number field, otherwise it is called an *imaginary* quadratic number field.

### quadratic reciprocity

The *Law of Quadratic Reciprocity* gives the conditions for a prime $p$ to be a quadratic residue modulo a prime $q$ in terms of whether of not $q$ is a quadratic residue modulo $p$.

### quadratic residue

An integer $m$ is a *quadratic residue* modulo $n$ if $m = r^{2}$ modulo $n$ for some $r$.

Other numbers are called *quadratic non-residues*.

### quadrature

*Quadrature* means calculating the area of a figure or the area under the graph of a function. Literally, finding a square with the same area.

### quadric

A *quadric* or *quadratic surface* is a surface given by an equation of degree 2.

Examples of quadrics are paraboloids, hyperboloids and ellipsoids.

### quantum mechanics

*Quantum mechanics* is the branch of mathematical physics treating atomic and subatomic systems and their interaction with radiation in terms of observable quantities.

It is based on the observation that all forms of energy are released in discrete units or bundles called *quanta*.

### quartic equation

A *quartic equation* is an equation with a fourth order term:

$ax^{4} + bx^{3} + cx^{2} + dx + e = 0$

in general.

### quaternion

The *quaternions* are a set of symbols of the form

$a + bi + cj + dk$

where $a, b, c, d$ are Real numbers.

They multiply using the rules

$i^{2} = j^{2} = k^{2} = -1$ and $ij = k$.

They form a *non-commutative division algebra*.

### quintic equation

A *quintic equation* is an equation with a fifth order term:

$ax^{5} + bx^{4} + cx^{3} + dx^{2} + ex + f = 0$

in general.

### radical

The word *radical* means *root* and so a radical is an $n$th root of a number.

To *solve a polynomial equation by radicals* means finding a formula for its roots in terms of the coefficients so that the formula only involves the operations of addition, subtraction, multiplication, division and taking roots.

### rational

A *rational number* is one which can be written as a quotient of integers.

For example, 1 /2 and 3 /5 are rational, but √2 is not.

The rational numbers are usually denoted by **Q** (standing for "quotient").

### rational function

A *rational function* is a quotient of two polynomials (whose coefficients need not necessarily be rational numbers).

### rationalise

To *rationalise the denominator* of a fraction is to manipulate it to remove √ signs from the denominator.

For example, 1 /√3 = √3 /3 after multiplying top and bottom by √3.

### reductio ad absurdum

The method of *reductio ad absurdum* is assuming that a proposition does *not* hold and using this assumption to deduce a contradiction.

### refraction

*Refraction* is the bending of a ray of light when it passes from one medium to another, as for example, from air to water or glass.

### regular polyhedron

A *regular polyhedron* is a solid body with plane faces in which each face is the same regular polygon and all the faces meet at the same angles.

There are five regular polyhedrons: the tetrahedron, cube, octahedron, dodecahedron and icosahedron.

They are also known as *Platonic solids*.

### representation theory

*Representation theory* seeks to understand an abstract algebraic system such as a group by obtaining it in a more concrete way as a permutation group or as a group of matrices.

### rhetoric

*Rhetoric* is the theory or practice of eloquence, whether spoken or written and the art of using language to persuade others.

### Riemann hypothesis

The *Riemann hypothesis* states that the nontrivial roots of the Riemann zeta function defined on the complex plane $\mathbb{C}$ all have real part 1 /2 .

The line $Re(z) = {{1}\over{2}}$ is called the *critical line*.

The truth (or otherwise) of the Riemann hypothesis would have important consequences for the Prime Number Theorem.

### right ascension

The *right ascension* is the coordinate of the position of a heavenly body (usually in terms of time) eastwards along the celestial equator from the First Point of Aries.

The other coordinate is the declination.

### ring

A *ring* is a set together with *two* methods (usually written as addition and multiplication) of combining elements to get new ones which satisfy certain properties or axioms.

Rings of real or complex numbers, of polynomials, of matrices, ... are widely used.

Most rings are associative, but non-associative structures such as Lie rings are also important.

### ruler and compass

*Ruler and compass constructions* use a restricted number of operations, based on an ideal compass and an unmarked straight edge.

They were the basis for the constructive forms of Greek geometry.

### Samaritan

*Samaritans* were a people who claimed be descended from Jews of ancient Samaria who were not deported by the Assyrian conquerors of Israel in 722 BC. Some believe, however, that they came from Mesopotamia and settled in Samaria after the Assyrian conquest.

### Awareness of Alcuin

by

**Siegfried Sassoon**

At peace in my tall-windowed Wiltshire room,

(Birds overheard from chill March twilight's close)

I read, translated, Alcuin's verse, in whom

A springtide of resurgent learning rose.

Homely and human, numb in feet and fingers,

Alcuin believed in angels; asked their aid;

And still the essence of that asking lingers

In the aureoled invocation which he made

For Charlemagne, his scholar. Alcuin, old,

Loved listening to the nest-near nightingale,

Forgetful of renown that must enfold

His world-known name; remembering pomps that fail.

Alcuin, from temporalities at rest,

Sought grace within him, given from afar;

Noting how sunsets worked around to west;

Watching, at spring's approach, that beckoning star;

And hearing, while one thrush sang through the rain,

Youth, which his soul in Paradise might regain.

### Sceptic

*Sceptic* or *Skeptic* - There are two forms of Skepticicism. The first is Academic Skepticism which is based on Socrates' statement that the only thing he knew was that he knew nothing. The second form follows the Pyrrhonists who claim nothing, but believe that people should stop making any claims to knowledge.

### Secular Parallax

*Secular Parallax* is the angular difference from the lines of sight from two observation points to the object being measured, using the sun's motion through space as the baseline for the observation points

### sexagesimal

In the *sexagesimal system* calculations are done in the base 60 as used by the Ancient Babylonians.

The remnants of sexagesimal notation remain in our method of telling time and measuring angles.

### Sidereal

Sidereal

### sidereal year

The *sidereal year* is the time that the sun takes to return to the same place amongst the fixed stars.

### simple group

A *simple group* is group which has no non-trivial proper normal subgroups.

Simple groups are important because the can be thought of as the "blocks" out of which other groups can be built. Much activity has been expended in the classification of all finite simple groups.

### Socrates

Socrates (**born:** about 470 BC, **died:** 399 BC) was one of the most important of the Greek philosophers who was influential in developing Western culture.

### Socrates Scholasticus

Socrates Scholasticus (**born:** about 380 in Constantinople, **died:** about 450) was a Byzantine church historian whose annotated chronicles are an important source of early Christian history.

### solar system

The *solar system* consists of the Sun and all its planets, their moons etc.

### Solar Year

The *Solar Year* is the time taken for the Sun to return to the same position in the cycle of seasons, as seen from Earth; for example, the time from spring equinox to spring equinox, or from summer solstice to summer solstice.

### solstice

The *solstice* is the time of the year when the sun is at its maximum distance from equator making day and night longest or shortest.

They occur round about 21st June and 21st December.

### soluble by radicals

A polynomial equation is *soluble by radicals* if there is a formula for its roots in terms of the coefficients, so that the formula only involves the operations of addition, subtraction, multiplication, division and taking roots (= radicals).

This is sometimes called *soluble algebraically*.

### soluble or solvable group

A *soluble group* (called a *solvable group* in the US) is a finite group which has a chain of subgroups ${1} = H_{n} \subset H_{n-1} \subset ... \subset H_{0} = G$ in which each subgroup is normal in the next one and each factor group $H_{i} / H_{i-1}$ is an abelian group.

The reason for the name is that a polynomial is soluble by radicals provided that its Galois group is soluble.

### Sosigenes

Sosigenes the Peripatetic (2nd century AD) was an Egyptian philosopher who was the tutor of the Greek philosopher Alexander of Aphrodisias.

### special_function

A *special function* is a function (often named after the person who introduced it) having a particular use in physics or some branch of mathematics.

Examples include Bessel functions, Lagrange polynomials, Beta functions, Gamma functions, Hypergeometric functions, etc.

### spectral theory

The *spectrum* of a linear map (or operator) between vector spaces is its set of eigenvalues.

*Spectral theory* attempts to understand the properties of the operator in terms of the properties of this spectrum.

The theory is most useful when handling vector spaces of functions in functional analysis.

### Spectroscopy

*Spectroscopy* is splitting the light emitted by heated atoms to form a line spectrum. Each element has its own distinctive line.

### Speusippus

Speusippus (**born:** about 400 BC, **died:** 338 BC) was a nephew of Plato who followed his uncle in his philosophical views. He became head of the Academy on Plato's death in 347 BC. He wrote *On Pythagorean Numbers* which looks at why numbers such as 10 are considered as representing perfection.

### square number

A *square number* is the number of dots that may be arranged in a square: 1, 4, 9, 16, ...

In general $n$2 .

### square the circle

*Squaring the circle* means constructing a square (ideally using only ruler and compass constructions) with the same area as a given circle.

### Staatsexamen

The *Staatsexamen* or *State examination* was an examination taken to qualify to teach in a Gymnasium

### stadium

A *stadium* (plural *stadia*) is a Greek measure of distance equal to 600 Greek feet. Opinions vary about how long it actually was, with estimates varying from 154 to 215 metres.

### statistical mechanics

*Statistical mechanics* models a system in terms of the average behaviour of the large numbers of atoms and molecules making up the system.

### stereographic projection

*Stereographic projection* is projection of the points of a sphere from the North Pole of the sphere along straight lines on to the equatorial plane or onto the tangent plane at the South Pole.

### Stoa Poikile Academy

*Stoa Poikile Academy* means *Painted Colonnade* and refers to the hall in Athens in which the Academy, founded by Zeno of Citium, held its lectures. The philosophy Stoicism takes its name from this hall.

### Stobaeus

Stobaeus (5th century AD) was a Greek anthologist.

### Stoic

A *Stoic* was a member of the school of philosophy founded by Zeno of Citium. The Stoics strongly believed in duty and considered the universe to operate with rational principles.

### Strabo

Strabo (**born:** about 64 BC, **died:** about 24 AD) was a Greek geographer and historian.

### symmetric group

A *symmetric group* is the group of all permutations of a finite set.

The symmetric group of a set of size *n* is denoted $S_{n}$ and has $n!$ (= $n$ factorial) elements.

### symmetry group

The set of all rigid motions (translations, rotations, reflections, etc) of Euclidean space which map all points of a subset $F$ into $F$ form a group which is called the *symmetry group* of the figure $F$.

### synodic month

The *synodic month* is the time between successive oppositions of the sun and moon

### Syrianus

Syrianus was a neo-Platonist philosopher from Athens who taught the Alexandrian Hermias. He was the teacher of Proclus.

### tangent

A *tangent* to a curve at a point $p$ is the best linear approximation to the curve near that point. It can be regarded as the limit of the chords from the point $p$ to other points close to $p$.

If two curves have a common tangent at a point of intersection they are said to *touch* or *be tangent*.

### tautochrone

A *tautochrone* is a curve which has a shape such that a small ball rolling on the curve will have oscillations whose period is independent of their size.

The cycloid is such a curve.

### Taylor series

The **Taylor series** allows a function to be expanded as a power series aound a point. The series appear in Brook Taylor's *Methodus incrementorum directa et inversa * (1715) but were only named "Taylor series" in 1785.

### tensor

*Tensors* are "multidimensional boxes of numbers". So, for example, a 0-tensor is a scalar, a 1-tensor is a vector, a 2-tensor is a matrix, etc.

Tensors are used in differential geometry and similar areas.

### tetrahedron

A *tetrahedron* is a regular polyhedron with 4 faces each of which is an equilateral triangle. It looks like a pyramid on a triangular base.

### Themistius

Themistius (**born:** about AD 350) taught in Constantinople and modernised the language and style of many of Aristotle's treatises.

### theodolite

A *theodolite* is a surveying instrument for measuring horizontal and vertical angles.

### Theophrastus

Theophrastus of Lesbos (**born:** 372 BC in Eresus, Lesbos, **died:** 287 BC) was taught by Aristotle in Athens and in 323 when Aristotle retired he became the head of the Lyceum. He fully accepted all of Aristotle's philosophy.

### three body problem

The *Three Body Problem* is the problem of investigating the behaviour of three mutually attracting bodies (such as the Sun, Earth and Moon) and the stability of their motion.

### topological or continuous group

A *topological group* is a set which has both the structure of a group and of a topological space in such a way that the operations defining the group structure give continuous maps in the topological structure.

Many groups of matrices with entries in $\mathbb{R}$ or $\mathbb{C}$ give topological groups.

### topology

*Topology* is the study of properties which are left unchanged by "continuous deformation".

The development of topology has had a great influence on many areas of mathematics.

Various branches of topology are: general topology (sometimes called point-set topology), algebraic topology and differential topology.

### transcendental

A *transcendental number* is a real number which is not the root of a polynomial equation with integer (or rational) coefficients.

One may prove (with difficulty!) that π and $e$ (the base of natural logarithms) are transcendental.

### transit

A *transit* is the passage of a heavenly body over the meridian or of one heavenly body in front of another.

### triangular number

A *triangular number* is the number of dots that may be arranged in an equilateral triangle: 1, 3, 6, 10, ...

In general $n(n + 1)/2$.

### tropical year

The *tropical year* is time between the repeat of the seasons. It is calculated from the length of time between the equinoxes.

### twin prime conjecture

Twin primes are pairs of primes 2 apart, as, for example: 17, 19 or 41, 43, ...

The *Twin Prime Conjecture* is the (still unproven) statement that there are infinitely many such pairs.

### twisted cubic

A *twisted cubic* is a curve in three dimensional space or projective space whose points are given by $(x(t), y(t), z(t))$ for a parameter $t$ and where $x, y, z$ are polynomials of degree at most 3.

### vector space

A *vector space* over a field $F$ is a set $V$ of vectors which can be added to get new vectors and can be multiplied by elements of $F$ (called *scalars*).

Vector spaces over $\mathbb{R}$ like $\mathbb{R}$2 or $\mathbb{R}$3 are particularly useful.

### Virgil

Virgil (**born:** 15 October 70 BC in Andes [near Mantua], in Italy, **died:** 19 BC in Brundisium) was the most important Roman poet whose major work was the Aeneid, which relates the story of the founding of Rome.

### Vitruvius

Vitruvius (1st century BC) was a Roman architect and engineer. He wrote the famous treatise *De architectura* which is a textbook written for Roman architects.

### wrangler

A *Wrangler* is the name given to someone graduating with a first class degree in Mathematics from Cambridge University. The Senior Wrangler was the person with the highest marks, followed by the Second Wrangler and so on down the list. This method of classification lasted until 1909, since when the lists have been published in alphabetical order.

### Zeno of Citium

Zeno of Citium (**born:** about 335 BC in Citium, Cyprus, **died:** 263 BC in Athens) was a Greek philosopher who founded the Stoic school of philosophy. He believed that happiness could be achieved by following rational reasoning.

### zeta function

The *Riemann zeta function* is the sum of the infinite series

$\zeta (s) = \Sigma (1/n^{s})$

thought of as either a Real or Complex series.

### Note 11

Al'Khwarizmi states in his Algebra that the practical man uses ^{22}/_{7} for π, the geometer uses √10 for π, while the astronomer uses 3.1416.

### Note 12

Fibonacci gave the approximation $^{864}/_{275} = 3.14181818...$ He used inscribed and circumscribed regular polygons of 96 sides, taking the mean, but made no reference to Archimedes.

### Note 13

Madhava was an Indian mathematician who discovered Gregory's series long before Gregory and was the first to use a series to calculate π.

### Note 14

Al'Kashi obtained his remarkably accurate value using a regular polygon with $3 × 2^{28} = 805306368$ sides. His aim was to calculate a value which was accurate enough to allow the calculation of the circumference of the universe to within a hair's breadth. He takes the universe to be a sphere 600000 times the diameter of the earth. Note that in fact 39 places of π will compute the circumference of the universe (known in 2000) to an accuracy of the radius of the hydrogen atom.

### Note 15

Valentin Otho, a pupil of Rheticus, was a German engineer. He gave the approximation $^{355}/_{113}$.

### Note 16

Vite used Archimedes method with polygons of $6 × 2^{16} = 393216$ sides to obtain 3.1415926535 <π < 3.1415926537. He is also famed as the first to find an infinite series for π.

### Note 17

Romanus (or Adriaan van Roomen) used a regular polygon with $2^{30}$ = 1073741824 sides. He gaveπ to 17 places but only 15 are correct.

### Note 18

Van Ceulen used inscribed and circumscribed polygons with $60 × 2^{33} = 515396075520$ sides. He also used what in modern notation would be essentially $1 - \cos A = 2\sin^{2} A/2$. He had discovered this result.

### Note 19

Van Ceulen used a polygon with $2^{62} = 4611686018427387904$ sides to obtain π to 35 places of accuracy. He spent most of his life on this calculation which my computer now gives essentially instantly:

3.1415926535897932384626433832795029. Snell in 1621 showed that more accurate values of π could be obtained by a clever variation in the construction so that he needed only to use a polygon with $2^{30}$ sides to obtain the accuracy which Van Ceulen obtained by using ones with $2^{62}$ sides. It was around this time that variants of Archimedes' method stopped being used and future calculations used infinite series expansions.

### Note 2

It is claimed that in a text which is now lost Archimedes gave better bounds whose average gives the value 3.141596 for π, correct to seven places.

### Note 20

Isaac Newton used the series $\arcsin(x) = x +1/(2.3) x^{3} + 1.3/(2.4.5) x^{5} + 1.3.5/(2.4.6.7) x^{7} +$ ... putting $x = ^{1}/_{2}$ to obtain a series for $\arcsin(^{1}/_{2}) = π/6$. This requires taking about 40 terms.

### Note 21

Abraham Sharp used James Gregory's series $\arctan(x) = x - x^{3}/3 + x^{5}/5 - ...$ Putting $x = 1/√3$ gives a series for $\arctan(1/√3) = π/6$. To obtain π correct to 71 places he used nearly 300 terms of the series.

### Note 24

John Machin used the formula $π/4 = 4 \arctan(^{1}/_{5}) - \arctan(^{1}/_{129})$ and James Gregory's series for $\arctan(x)$.

### Note 25

De Lagny used the same method as Sharp.

### Note 26

Takebe Hikojiro Kenko (1664-1739) was a Japanese mathematician.

### Note 27

Matsunaga Ryohitsu was a Japanese mathematician. He used the same method as Newton. This requires taking about 160 terms in the series.

### Note 28

Baron Georg von Vega used the relation $π/4 = 5 \arctan(^{1}/_{7}) + 2 \arctan (^{3}/_{79})$, and James Gregory's series for $\arctan(x)$. Euler had calculated π correct to 20 places in an hour in 1755 using this method.

### Note 29

William Rutherford published his result in the Transactions of the Royal Society in 1841. He used the relation $π/4 = 4 \arctan(^{1}/_{5}) - \arctan(^{1}/_{70}) + \arctan(^{1}/_{99})$ which Euler published in 1764.

### Note 3

Marcus Vitruvius Pollio was a Roman architect. He made his calculation of π by measuring the distance a wheel of a given diameter moved through one revolution.

### Note 30

von Strassnitzky from Vienna gave the formula $π/4 = \arctan(^{1}/_{2}) + \arctan (^{1}/_{5}) + \arctan(^{1}/_{8})$ to Zacharias Dase, a calculating prodigy. He used took about two months to calculate π to 200 places using the formula.

### Note 31

Thomas Clausen wanted to find the correct value of π to 200 places since he knew that Rutherford and Dase gave different values from the 153rd place. Clausen used Machin's relation $π/4 = 4 \arctan(^{1}/_{5}) - \arctan(^{1}/_{129})$. He discovered that Dase was correct and extended the number of correct places to 248.

### Note 32

W Lehmann from Potsdam used the formula $π/4 = \arctan(^{1}/_{2}) + \arctan(^{1}/_{3})$ which was published by Charles Hutton in 1776. The formula $π/4 = 2 \arctan(^{1}/_{3}) + \arctan(^{1}/_{7})$ was also given by Hutton in 1776 and Euler, independently, in 1779.

### Note 33

William Rutherford, having made an error in his published value of π in 1841, made another calculation. This time he used Machin's formula.

### Note 34

W Lehmann from Potsdam used the formula $π/4 = \arctan(^{1}/_{2}) + \arctan(^{1}/_{3})$ which was published by Charles Hutton in 1776. The formula $π/4 = 2 \arctan(^{1}/_{3}) + \arctan(^{1}/_{7})$ was also given by Hutton in 1776 and Euler, independently, in 1779.

### Note 35

D F Ferguson who worked at the Royal Naval College in England used the formula $π/4 = 3 \arctan(^{1}/_{4}) + \arctan(^{1}/_{20}) + \arctan(^{1}/_{1985})$ to compute π. He worked on it from May 1944 until May 1945 by which time he had calculated 530 places and found that Shanks was wrong after place 528. He continued with his efforts and published 620 correct places of π in July 1946. Ferguson then continued with his calculations, but after this he used a desk calculator. This marks the point at which hand calculations of π ended and computer assisted calculations began.

### Note 4

Chang Hong (78-139) was an astrologer from China. He was one of the first to use this approximation. He deduced it from his value for the ratio of the volume of a cube to that of the inscribed sphere was $8\over{5}$.

### Note 5

Ptolemy used a regular 360 polygon to approximate π. He actually obtained the number $3 + 8/60 + 30/60^{2}$ which written as a decimal is 3.1416666...

### Note 6

Wang Fan (229-267) was a Chinese astronomer. No method of arriving at his value of π is recorded, although it has been conjectured that he came up with the value $^{142}/_{45}$ "knowing that 3.14 was too low".

### Note 7

Liu Hui was a Chinese mathematician. He calculated π using an inscribed regular polygon with 192 sides.

### Note 8

Zu Chongzhi produced a value of π which was not bettered for nearly one thousand years. In fact although he gives $^{355}/_{113}$ as a "very close" ratio, he actually calculated 3.1415926 < π < 3.1415927 which, taking an average, would give π correct to 8 places.

### Note 9

Aryabhata made his approximation using an inscribed regular polygon with 384 sides.

### Apsides

In a heliocentric orbit, the point furthest from the Sun (S) is called *aphelion* (A), and the point nearest to the Sun is called *perihelion* (P), as shown in the figure. These two points define the line of **apsides** [singular: **apse**] which passes through the Sun, and is an axis of symmetry of the path. When the path is discovered to be an ellipse this line is identified as its major axis.

The corresponding points that arose originally in geocentric astronomy are (still) called 'apogee' and 'perigee'. Kepler invented the names 'aphelion' and 'perihelion' by analogy when he needed them for his heliocentric astronomy (from the Greek: helios = Sun, ge- referring to Earth, with prefixes ap(o)- = away from, peri- = around / near).

### Geocentric

This describes the view when the Earth is taken as centre. In astronomy, it refers to the situation when bodies move round the Earth rather than round the Sun. [Contrast heliocentric]

### Heliocentric

This describes the view when the Sun is taken as centre. In astronomy, it refers to the situation when bodies move round the Sun. [Contrast geocentric]

### Secular changes

The theoretical elliptic orbit of a planet is subject to slow changes of various kinds, due to the effect of other bodies in the universe. These secular changes are often perceptible only over centuries, and they may or may not be periodic (repeating), with a period of enormous length.

['secular' in this sense is cognate with siècle = century]

### Mensuration / stereometry

**Mensuration** (from the Latin for measurement) is the name of the branch of mathematics concerned with the measurement of the perimeter, area, volume of geometrical figures. The term can be extended to finding the position of 'centres' of particular interest in certain figures. Archimedes was the most famous exponent of this topic in antiquity.

**Stereometry** (the term is derived from Greek) deals with the particular set of results involving three-dimensional (solid) bodies. (Kepler used this term in the title of his work Stereometria Doliorum (1615), which gives a theoretical treatment of the measurement of wine barrels.)

### Note

The use of the singular "mathematic" and "physic" is archaic, and would have been so even in 1873.

### Note 1

To provide an idea of the tone that ruled throughout the letters of these two men and it is certainly the case of Mr. Bernoulli who commented very early of Mr. Euler's genius, it is sufficient here to provide the end of one of the letters selected by chance amongst those of 1739

De caetero gratissimum mihi fuit intelligere, quod ad admirationem usque Tibi placuerint quae scripsi de oscillationibus verticalibus, propter simplicitatem expressionis et insignem usum, quem praestare possunt in explicadis navium ponderibus; maluissem autem , ut ipse quoque calculum fecisses ex Tuo ingenio, quo mihi patuisset anon in ratiocinando erraverim. Nam ingenue fateor, me Tuis luminibus plus fidere quam meis. Quae uberius afferes, Vir excel de Isoperimericis, credo equidem Te omnia probe ruminasse atque ad veritatis trutinam expendisse, ita ut vix quicquam restet, quod acerrimam Tuam sagacitatem subterfugere potuerit: etc.

### Note 2

The marks of esteem to which a virtuous and enlightened man is witness abiding merit honors excessively the man who gives them and the man who receives them so that I should be bound as a duty to publish them, on this occasion that the late Mr. Turgot wrote concerning Mr. Euler by notifying him of the King's command. It is as follows:

At Fontainebleau, 15 October 1775

During the time of my appointment at the Department of the Navy, I thought that there was nothing that I could do of greater advantage for the education of young students raised in the naval and artillery schools, than to provide them with the works that you have written on these two mathematical topics. As a result of this, I have proposed to the King to authorize to have your "Treatise on the Construction and Steering of Ships" as well as your "Commentary on the Principles of Artillery" by Robins.

Had I been in closer contact with you, I would have asked for your permission prior to having disposed of these works which belong to you. However, I knew that you would be well-remunerated for the proprietary content by a sign of the King's favor. His Majesty has authorized me to indemnify you with a payment with a payment of one thousand rubles, which he requests that you accept as a token of the esteem that he has for you and which you so justly deserve.

At this time I am overjoyed by being the bearer of such news, and to seize, with a true pleasure, this opportunity to express that which I have known for a very long time to a man that honors humanity with his genius and science with his style.

I am, etc. Turgot

### Note 3

After the death of Baron de Wolff, there remained the matter of finding his replacement at the University of Halle. The king wrote to Euler concerning this matter. At first Euler suggested Daniel Bernoulli, however after the refusal of this illustrious scientist, M. de Segner who obtained this position with extremely beneficial conditions which M. Euler obtained for him and at the same time convinced the King to purchase all of the scientific apparatus of the late M. de Wolff. It was also M. Euler to whom the King asked to employ the late M. Haller to enter the University of Halle. The benefits that Haller requested displeased the King and the project failed.

### Note 4

He had sent a pair of these glasses to the King, which had been manufactured according to his principles, who applauded this useful work and addressed the following letter to Euler from Waldau, this letter is all the more remarkable since it is written in His Majesty's own hand:

I wish to thank you for the little spyglasses that followed you letter of the 14 of this month and I praise the care that you have taken to providing mankind with the theory of your study and your scientific application. As my current situation does not permit me an in depth examination of everything that has come to me from you, I reserve to do so when I have the free time. With this I pray to God that he keeps you in his holy and safety.

Waldau, this 15 September 1759

Frederick

### Note 5

Euler had communicated to Mr. Bernoulli in 1776 concerning a new method of Mr. Euler's, more general than all the others since it applied to any initial random figures , the nature of which cannot be represented by any equation. The extract of the following response will make clear the point where the controversy was then and the noblesse of the proceeding of the two great men who are of differing opinions:

The sketch of Mr. Euler's method has given me great pleasure, but it has not altered my ideas of this matter one bit, I remain convinced that my method provides for all the possible instances in abstracto. I will admit however that, in certain cases, that of Mr. Euler's is preferable to mine, but there are also points of view to the contrary since my method can be applied to such a number of finite bodies which is proposed even though in the system there is no perfect return or period to wait for. Irrespective of my pretensions, I am always prepared to drop my flag before my admiral.

### Note 6

Euler opened his home to students that the Academy sent to Berlin to study mathematics. Mssrs Kotelnikov and Rumovsky spent a number of years in this situation and enjoyed the teachings of their incomparable master.

### Note 7

As the result of frequent and familiar communication that the Prince conducted with him and the intimate friendship which was the result bonded them and he sincerely regretted the recognition of the loss which was due to everything that Mr. Euler had contributed to the cultural and spiritual enlightenment of the Margrave's daughters. He provided them with lessons and it was for them that he wrote during his stay at the Marburg Court the letters concerning various subjects of physics and philosophy which he had published after his return to Saint Petersburg.

### Note 8

Throughout his life he preserved the sweet memory of the great goodness that the King showed him and the bonding which inspired his affection towards this heartfelt and spiritual prince and this continued through a correspondence which he had the honor to maintain with him. I cannot withstand the temptation to ornament this eulogy with one of these letters that the king wrote in 1772:

Monsieur le professor Euler, In response to your letter of 4 August last , I had hoped to be able to confirm the opinion that you are enjoying different circumstances from which your friendship for me has been dictated by the expression of a virtuous and sensible heart. However ... I wish to thank you to good wishes in this regard, and I continue on to the recognition that I owe to your caring in communicating to me the observations of that the accurate astronomers that your Academy made at Bender and near the sources of the Dniestr and the Danube with the locations of some places equally important to geography. I will attempt to put them to good use as with these that are be done in this country with hard work and success, despite the troubles which blocked scientific progress. I request that you continue, as much towards the public utility as my own personal satisfaction and for you to take the opportunity to keep me apprised. I pray to God that he keeps you, dear professor Euler in his holy and worthy safety.

Written at Warsaw, 7 June 1772

King Stanislaus Augustus

### Note 9

The recognition of an enlightened nation testifies by its real merit, is at once too flattering for a great man to whom it is indebted and too encouraging to those who follow his path, so as to provide for its in its proceedings, so as to insert an extract of a letter that Mr. Euler received by the secretary of the board of longitudes.

Admiralty Office, London 13 June 1765

Sir, the Parliament of Great Britain having, by an act passed in their late sessions (a printed copy of which I herewith transmit to you) been pleased to direct, that a sum of money, not exceeding three hundred pounds in the whole, shall be paid to you, as a rewards for having furnished theorem, by the help of which the late Professor Mayer of Göttingen constructed his lunar table, by which table great progress has been made towards discovering the longitude at Sea. I am directed by the Commissioners of the longitude to acquaint you therewith and to congratulate you upon this honorary and pecuniary acknowledgement, directed to be made you by the highest Assembly of this nation, for your useful and ingenious labors towards the said discovery etc.

### Note 10

It is known that the numbers of foreign memberships of the Paris academy is set at eight. Mr. Euler was elected the ninth, without there being, by consequence a vacancy. The circumstances which have accompanied this induction are worthy to be mentioned in this eulogy. They have been included in the following letter by the Marquis d'Argenson.

At Versailles, 15 June 1755

The King has just selected you, Sir, after the wishes of His Royal Academy of Sciences to receive a foreign membership in that Academy and as it has named at the same times My Lord Macclesfield, president of the Royal Society of London, to fill a similar place, which has be vacated by the death of M. Moivre, His Majesty has decided that the first place of this type to be vacated will not be reappointed. The extreme rarity of these types of arrangement is such a marked distinction to make it necessary to bring it to your attention as to assure you of all the support that I provide for it. The Academy truly desired for you to be associated to its works, and His Majesty could only adopt an attitude of recognition that you have justly deserved. Please be convinced, Sir, that it is not possible to be more perfectly devoted as I am.

### Note 11

The notes of the paper concerning the prize were lost on this occasion, and M. Euler's son was obligated to redo all the lunar theory and to re-calculate everything for a second time.

### Note 12

On might be under the impression that due to the great number of his discoveries might have caused him to become emotional as one does when the soul perceives a novel truth, the pleasure that a mathematician has the advantage of experiencing perhaps more than others. Mr. Euler always enjoyed this type of susceptibility and he would wish that everyone experience the same. He would become seriously upset with the attitude of indifference that my modest temperament made to assume when I told him the solution to a problem or a proof of a theorem that I was able to find.

### Note 13

Another proof of the force of his memory and his imagination is worthy of mention. He gave lessons in algebra and geometry to his grandsons. He often proposes numbers for which the extractions of their roots were powers. He calculated them in his head, and when unable to sleep because of insomnia he calculated the first six powers of all the numbers under twenty and recited them to us to our great astonishment a few days later.

### Note 14

It is very befitting that I should tell the readers of this eulogy, that the King of Prussia, the King of Poland, The Prince Royal of Prussia and the Margraf of Schwendt are bereaved as is the Academy due to the death of Mr. Euler. They have offered their regrets to his eldest son with letters of condolences befitting this honorable man since they render justice in the most gracious words to with talents as to his virtues.

### Note 15

The Academy has eight mathematicians who have enjoyed the instructions of Mr. Euler: MM. J.A. Euler, Kotelnikov, Rumovsky, Krafft, Lexell, Inokhodtsov, Golovine and I, with three absences.

O my dear friends and colleagues how I have seen, by this brief stroke of the pen written by the heart, the warm tears! I have only been able to touch your hand after the pain stifled my voice, but I will never forget the memory of the mark of your sincere affliction and I hereby give public notice of the sensitivity of spirit and love that you have shown on this occasion to our dear and incomparable Master.

### Note 1

A European call-option is a contract that gives the owner the right, but not the obligation, to buy the given stock at a specified price (the strike price or the exercise price). If the market price of the stock, on the expiration date, is lower than the strike price, then a call-option expires worthless.