Posts Tagged 'History'

The Ubiquitous Cycloid

Puzzle: However fast a train is travelling, part of it is moving backwards. Which part?
For the answer, see the end of this post.

Timelapse image of bike with two lights on the wheel-rims. Photo from Webpage of Alexandre Wagemakers.

Timelapse image of bike with lights on the wheel-rims. [Photo from Website of Alexandre Wagemakers, with thanks]

Imagine a small light fixed to the rim of a bicycle wheel. As the bike moves, the light rises and falls in a series of arches. A long-exposure nocturnal photograph would show a cycloid, the curve traced out by a point on a circle as it rolls along a straight line. A light at the wheel-hub traces out a straight line. If the light is at the mid-point of a spoke, the curve it follows is a curtate cycloid. A point outside the rim traces out a prolate cycloid, with a backward loop. [TM076; or search for “thatsmaths” at ]

Continue reading ‘The Ubiquitous Cycloid’

Holbein’s Anamorphic Skull

Hans Holbein the Younger, court painter during the reign of Henry VIII, produced some spectacular works. Amongst the most celebrated is a double portrait of Jean de Dinteville, French Ambassador to Henry’s court, and Georges de Selve, Bishop of Lavaur. Painted by Holbein in 1533, the picture, known as The Ambassadors, hangs in the National Gallery, London.

Double Portrait of Jean de Dinteville and Georges de Selve (

Double Portrait of Jean de Dinteville and Georges de Selve (“The Ambassadors”),
Hans Holbein the Younger, 1533. Oil and tempera on oak, National Gallery, London

Continue reading ‘Holbein’s Anamorphic Skull’

James Joseph Sylvester

J. J. Sylvester as a graduate of Trinity College Dublin.

James Joseph Sylvester (1814-1897) as a graduate of Trinity College Dublin.

James Joseph Sylvester was born in London to Jewish parents in 1814, just 201 years ago today. The family name was Joseph but, for reasons unclear, Sylvester – the name of an anti-Semitic Pope from the Roman period – was adopted later. [TM075; or search for “thatsmaths” at ]

Sylvester’s mathematical talents became evident at an early age. He entered Cambridge in 1831, aged just seventeen and came second in the notorious examinations known as the Mathematical Tripos; the man who beat him achieved nothing further in mathematics!

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Thomas Harriot: Mathematician, Astronomer and Navigator

Sir Walter Raleigh, adventurer, explorer and privateer, was among most colourful characters of Tudor times. He acquired extensive estates in Waterford and Cork, including Molana Abbey near Youghal, which he gave to his friend and advisor, the brilliant mathematician and astronomer Thomas Harriot.

Thomas Harriot (1560 - 1621)

Left: Sir Walter Raleigh (1552-1618). Right: Thomas Harriot (1560?-1621)

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For Good Comms, Leaky Cables are Best

A counter-intuitive result of Oliver Heaviside showed how telegraph cables should be designed [see this week’s That’s Maths column (TM066) or search for “thatsmaths” at].

Atlantic-Telegraph-Map Continue reading ‘For Good Comms, Leaky Cables are Best’

A King of Infinite Space: Euclid I.

O God, I could be bounded in a nutshell, and count myself a king of infinite space …

Euclid. Left: panel from the Series Famous Men by Justus of Ghent. Right: Statue in the Oxford University Museum of Natural History.

Euclid. Left: panel from series Famous Men by Justus of Ghent. Right: Statue in the Oxford University Museum of Natural History.

Continue reading ‘A King of Infinite Space: Euclid I.’

Café Mathematics in Lvov

For 150 years the city of Lvov was part of the Austro-Hungarian Empire. After Polish independence following World War I, research blossomed and between 1920 and 1940 a sparkling constellation of mathematicians flourished in Lvov [see this week’s That’s Maths column in The Irish Times (TM063, or search for “thatsmaths” at

The Scottish Café, Lvov in earlier times (left), now Hotel Atlas in Lviv.(image Wikimedia Commons).

The Scottish Café, Lvov in earlier times (left), now Hotel Atlas in Lviv (right).

Continue reading ‘Café Mathematics in Lvov’

The Birth of Functional Analysis

Stefan Banach (1892–1945) was amongst the most influential mathematicians of the twentieth century and the greatest that Poland has produced. Born in Krakow, he studied in Lvov, graduating in 1914 just before the outbreak of World War I. He returned to Krakow where, by chance, he met another mathematician, Hugo Steinhaus who was already well-known. Together they founded what would, in 1920, become the Polish Mathematical Society.

A coin and a postage stamp commemorating Stefan Banach.

A coin and a postage stamp commemorating Stefan Banach.

Continue reading ‘The Birth of Functional Analysis’

MGP: Tracing our Mathematical Ancestry

There is great public interest in genealogy. Many of us live in hope of identifying some illustrious forebear, or enjoy the frisson of having a notorious murderer somewhere in our family tree. Academic genealogies can also be traced: see this week’s That’s Maths column in The Irish Times (TM062, or search for “thatsmaths” at

MGP Continue reading ‘MGP: Tracing our Mathematical Ancestry’

The MacTutor Archive

The MacTutor History of Mathematics archive is a website hosted by the University of St Andrews in Scotland. It was established, and is maintained, by Dr John O’Connor and Prof Edmund Robertson of the School of Mathematics and Statistics at St Andrews.

MacTutor contains biographies of a large number of mathematicians, both historical and contemporary. In many cases, the Wikipedia entry on a mathematician directs to MacTutor for more complete biographical information.

Screen image of the MacTutor page listing History Topics.

Screen image of the MacTutor page listing History Topics.

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The Year of George Boole

This week’s That’s Maths column in The Irish Times (TM058, or search for “thatsmaths” at is about George Boole, the first Professor of Mathematics at Queen’s College Cork.

Boole-Year-UCC-Small Continue reading ‘The Year of George Boole’

Algebra in the Golden Age

This week’s That’s Maths column in The Irish Times (TM054, or search for “thatsmaths” at is about the emergence of algebra in the Golden Age of Islam. The Chester Beatty Library in Dublin has several thousand Arabic manuscripts, many on mathematics and science.

Left: Societ stamp commemorating al-Khwārizmī's 1200th birthday. RIght: A page from al-Khwārizmī's Al-Jebr.

Left: Soviet Union postage stamp (1983) commemorating al-Khwārizmī’s 1200th birthday. RIght: A page from al-Khwārizmī’s Al-Jebr.

Continue reading ‘Algebra in the Golden Age’

Do you remember Venn?

Do you recall coming across those diagrams with overlapping circles that were popularised in the ‘sixties’, in conjunction with the “New Maths”. They were originally introduced around 1880 by John Venn, and now bear his name.

RIght: John Venn (1834–1923) with signature. Left: Stained glass window at Gonville & Caius College showing Venn diagram [images Wikimedia Commons].

Left: Stained glass window at Gonville & Caius College, Cambridge showing a Venn diagram. Right: John Venn (1834-1923) with signature [images Wikimedia Commons].

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When did Hammurabi reign?

The consequences of the Earth’s changing climate may be very grave. It is essential to understand past climate change so that we can anticipate future changes. This week, That’s Maths in The Irish Times ( TM047 ) is about the chronology of the Middle East. Surprisingly, this has important implications for our understanding of climate change.

Left:  Image of Hammurabi in the US Congress. Right: Part of an inscription of the Code of Hammurabi.

Left: Image of Hammurabi in the US Congress.
Right: Part of an inscription of the Code of Hammurabi.

Continue reading ‘When did Hammurabi reign?’

Euclid in Technicolor

 The article in this week’s That’s Maths column in the Irish Times ( TM039 ) is about Oliver Byrne’s amazing technicolor Elements of Euclid, recently re-published by Taschen.

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Pythagorean triples

The Pythagorean theorem states that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides. It can be written as an equation,

a2 + b2 = c2,

where c is the length of the hypotenuse, and a and b are the lengths of the other two sides.

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A Mathematical Dynasty

The idea that genius runs in families is supported by many examples in the arts and sciences. One striking case is the family of Johann Sebastian Bach, the most brilliant star in a constellation of talented musicians and composers.

In a similar vein, several generations of the Bernoulli family excelled in science and medicine. More that ten members of this Swiss family, over four generations, had distinguished careers in mathematics.

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Sonya Kovalevskaya

A brilliant Russian mathematician, Sonya Kovalevskaya, is the topic of the That’s Maths column this week (click Irish Times: TM029 and search for “thatsmaths”).

In the nineteenth century it was extremely difficult for a woman to achieve distinction in the academic sphere, and virtually impossible in the field of mathematics. But a few brilliant women managed to overcome all the obstacles and prejudice and reach the very top. The most notable of these was the remarkable Russian, Sonya Kovalevskya.

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The Atmospheric Railway

Atmospheric pressure acting on a surface the size of a large dinner-plate exerts a force sufficient to propel a ten ton train! The That’s Maths column ( TM027 ) in the Irish Times this week is about the atmospheric railway.
Continue reading ‘The Atmospheric Railway’

The remarkable BBP Formula

Information that is declared to be forever inaccessible is sometimes revealed within a short period. Until recently, it seemed impossible that we would ever know the value of the quintillionth decimal digit of pi. But a remarkable formula has been found that allows the computation of binary digits starting from an arbitrary  position without the need to compute earlier digits. This is known as the BBP formula.
Continue reading ‘The remarkable BBP Formula’

Wrangling and the Tripos

The Mathematical Tripos examinations, and the Wranglers who achieve honours in them, are the topic of the That’s Maths column ( TM023 ) in the Irish Times this week.
Continue reading ‘Wrangling and the Tripos’

Ducks & Drakes & Kelvin Wakes

The theme of this week’s That’s Maths column in the Irish Times ( TM021 ) is Kelvin Wakes, the beautiful wave patterns generated as a duck or swan swims through calm, deep water or in the wake of a ship or boat.
Continue reading ‘Ducks & Drakes & Kelvin Wakes’

More Equal than Others

In his scientific best-seller, A Brief History of Time, Stephen Hawking remarked that every equation he included would halve sales of the book, so he put only one in it, Einstein’s equation relating mass and energy, E=mc2.

There is no doubt that mathematical equations strike terror in the hearts of many readers. This is regrettable, as equations are really just concise expressions of precise statements. They are actually quite user-friendly and more to be loved than feared. Continue reading ‘More Equal than Others’

Archimedes uncovered

The That’s Maths column in this week’s Irish Times ( TM012 ) describes the analysis of the ancient codex known as the Archimedes Palimpsest.

Archimedes of Syracuse

Archimedes (Ἀρχιμήδης, 287-212 BC) was a brilliant physicist, engineer and astronomer, and the greatest mathematician of antiquity. He is famed for founding hydrostatics, for formulating the law of the lever, for designing the helical pump that bears his name, for devising engines of war, and for much more. Continue reading ‘Archimedes uncovered’

Pons Asinorum

The fifth proposition in Book I of Euclid’s Elements states that the two base angles of an isosceles triangle are equal (in the figure below, angles B and C).

For centuries, this result has been known as Pons Asinorum, or the Bridge of Asses, apparently a metaphor for a problem that separates bright sparks from dunces. Euclid proved the proposition by extending the sides AB and AC and drawing lines to form additional triangles. His proof is quite complicated. Continue reading ‘Pons Asinorum’

A Mersennery Quest

The theme of That’s Maths (TM008) this week is prime numbers. Almost all the largest primes found in recent years are of a particular form M(n) = 2n1. They are called Mersenne primes. The Great Internet Mersenne Prime Search (GIMPS) is aimed at finding ever more prime numbers of this form. Continue reading ‘A Mersennery Quest’

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