Children sometimes amuse themselves searching for the biggest number. After trying millions, billions and trillions, they realize that there is no end to the game: however big a number may be, we can always add 1 to produce a bigger number: the set of counting numbers is infinite. The concept of infinity has intrigued philosophers … Continue reading Enigmas of Infinity
Tag: History
The Beginning of Modern Mathematics
The late fifteenth century was an exciting time in Europe. Western civilization woke with a start after the slumbers of the medieval age. Johannes Gutenberg's printing press arrived in 1450 and changed everything. Universities in Bologna, Oxford, Salamanca, Paris and elsewhere began to flourish. Leonardo da Vinci was in his prime and Christopher Columbus was … Continue reading The Beginning of Modern Mathematics
The Edward Worth Library: a Treasure Trove of Maths
Infinite Riches in a Little Room. Christopher Marlowe. The Edward Worth Library may be unknown to many readers. Housed in Dr Steevens' Hospital, Dublin, now an administrative centre for the Health Service Executive, the library was collected by hospital Trustee Edward Worth, and bequeathed to the hospital after his death in 1733. The original book … Continue reading The Edward Worth Library: a Treasure Trove of Maths
Raphael Bombelli’s Psychedelic Leap
The story of how Italian Renaissance mathematicians solved cubic equations has elements of skullduggery and intrigue. The method originally found by Scipione del Ferro and independently by Tartaglia, was published by Girolamo Cardano in 1545 in his book Ars Magna. The method, often called Cardano's method, gives the solution of a depressed cubic equation t3 … Continue reading Raphael Bombelli’s Psychedelic Leap
Taylor Expansions from India
The English mathematician Brook Taylor (1685-1731) introduced the calculus of finite differences in his Methodus Incrementorum Directa et Inversa, published in 1715. This work contained the famous formula known today as Taylor's formula. In 1772, Lagrange described it as ``the main foundation of differential calculus'' (Wikipedia: Brook Taylor). Taylor also wrote a treatise on … Continue reading Taylor Expansions from India
Marvellous Merchiston’s Logarithms
Log tables, invaluable in science, industry and commerce for 350 years, have been consigned to the scrap heap. But logarithms remain at the core of science, as a wide range of physical phenomena follow logarithmic laws [TM103 or search for “thatsmaths” at irishtimes.com]. The method of logarithms was first devised by John Napier, 8th Laird … Continue reading Marvellous Merchiston’s Logarithms
Kepler’s Magnificent Mysterium Cosmographicum
Johannes Kepler's amazing book, Mysterium Cosmographicum, was published in 1596. Kepler's central idea was that the distance relationships between the six planets (only six were known at that time) could be represented by six spheres separated by the five Platonic solids. For each of these regular polyhedra, there is an inner and an outer … Continue reading Kepler’s Magnificent Mysterium Cosmographicum
The next Hamilton
William Rowan Hamilton was Ireland's greatest mathematician. His name is heard thousands of times every day throughout the world when researchers use the Hamiltonian function that encapsulates the dynamics of a vast range of physical systems. He achieved fame early in life and remains one of the all-time great scientists. [TM099, or search for “thatsmaths” at … Continue reading The next Hamilton
Computus: Dating the Resurrection
Whatever the weather, St Patrick's Day occurs on the same date every year. In contrast, Easter springs back and forth in an apparently chaotic manner. The date on which the Resurrection is celebrated is determined by a complicated convolution of astronomy, mathematics and theology, an algorithm or recipe that fixes the date in accordance with … Continue reading Computus: Dating the Resurrection
The Abel Prize – The Nobel Prize for Mathematics
There is no Nobel Prize for mathematics, but there is a close equivalent: The prestigious Abel Medal is awarded every year for outstanding work in mathematics [TM086, or search for “thatsmaths” at irishtimes.com]. This years winner, or winners, will be announced soon. When Alfred Nobel's will appeared, the absence of any provision for a prize … Continue reading The Abel Prize – The Nobel Prize for Mathematics
Franc-carreau or Fair-square
Franc-carreau is a simple game of chance, like the roll-a-penny game often seen at fairs and fêtes. A coin is tossed or rolled down a wooden chute onto a large board ruled into square segments. If the player's coin lands completely within a square, he or she wins a coin of equal value. If the … Continue reading Franc-carreau or Fair-square
Life’s a Drag Crisis
The character of fluid flow depends on a dimensionless quantity, the Reynolds number. Named for Belfast-born scientist Osborne Reynolds, it determines whether the flow is laminar (smooth) or turbulent (rough). Normally the drag force increases with speed. The Reynolds number is defined as Re = VL/ν where V is the flow speed, L the length … Continue reading Life’s a Drag Crisis
Numbering the Family Tree
The availability of large historical data sets online has spurred interest in genealogy and family history. Anyone who has assembled information knows how important it is to organize it systematically. A simple family tree showing the direct ancestors of Wanda One is shown here: This has just three generations but, as more people are added, … Continue reading Numbering the Family Tree
Melencolia: An Enigma for Half a Millennium
Albrecht Dürer, master painter and engraver of the German Renaissance, made his Melencolia I in 1514, just over five centuries ago. It is one of the most brilliant engravings of all time, and amongst the most intensively debated works of art [TM079; or search for “thatsmaths” at irishtimes.com ]. The winged figure, Melancholy, sits in a … Continue reading Melencolia: An Enigma for Half a Millennium
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. 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 … 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 … Continue reading Holbein’s Anamorphic Skull
James Joseph Sylvester
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 irishtimes.com ] Sylvester's mathematical talents became evident at … Continue reading James Joseph Sylvester
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. Raleigh needed an excellent navigator on his transatlantic voyages, and he brought … Continue reading Thomas Harriot: Mathematician, Astronomer and Navigator
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 irishtimes.com]. Robert Halpin In Wicklow town an obelisk commemorates Robert Halpin, a Master Mariner born at the nearby Bridge Tavern. Halpin, one of the more important mariners of the nineteenth … 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 ... [Hamlet] The Elements – far and away the most successful textbook ever written – is not just a great mathematics book. It is a great book. There is nothing personal in the book, nothing to give any clue as … 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 irishtimes.com). Zygmunt Janeszewski, who … 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. … 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 irishtimes.com). … 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. … Continue reading The MacTutor Archive
The Year of George Boole
This week’s That’s Maths column in The Irish Times (TM058, or search for “thatsmaths” at irishtimes.com) is about George Boole, the first Professor of Mathematics at Queen's College Cork. Mathematician and logician George Boole died just 150 years ago, on 8 December 1864, following a drenching as he was walking between his home and Queen's … 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 irishtimes.com) 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. "The ink of a scholar is holier than the blood of … 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. John Venn Venn was a logician and philosopher, born in Hull, Yorkshire in 1834. He studied at Cambridge University, … Continue reading Do you remember Venn?
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 … 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. Oliver Byrne (1810–1890), a Victorian civil engineer, was a prolific writer on science. He published more than twenty books on mathematics and several more on mechanics. The … Continue reading Euclid in Technicolor
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 … Continue reading Pythagorean triples
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 … Continue reading A Mathematical Dynasty
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 … Continue reading Sonya Kovalevskaya
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. For more than ten years from 1843 a train without a locomotive plied the 2.8 km … 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 … 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. Today (20/06/13) the results of the final examinations in mathematics will be read out at the Senate House in Cambridge University. Following tradition, the class list … 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. Group Velocity A stone dropped in a pond generates waves in a beautiful changing … 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, … 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 … 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 … 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) = 2n−1. They are called Mersenne primes. The Great Internet Mersenne Prime Search (GIMPS) is aimed at finding ever more prime numbers of this form. The search for the … Continue reading A Mersennery Quest
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