Surprising Discoveries in the Family Tree of Mathematics

The German word Doktorvater for a doctoral advisor indicates the close relationship between a PhD student and his or her supervisor. The relationship is often pivotal in determining the future career of the student, and the advisor also gains much from the interaction. Just as a genealogical tree can reveal fascinating information, a mathematical family … Continue reading Surprising Discoveries in the Family Tree of Mathematics →

The Doppler Effect: Simple but Remarkably Useful

We have all noticed how the horn of a speeding car changes as it approaches: each wave-peak is emitted from a closer point, so the wave is “squeezed” and the pitch increases. As the car recedes, the reverse effect stretches the wave, making it sound lower. The changing pitch of the note is called the … Continue reading The Doppler Effect: Simple but Remarkably Useful →

The Many Schools of Mathematical Thought

Mathematics is used widely, playing a central role in science and engineering and, increasingly, in the social and biological sciences. But users seldom consider the fundamental nature of mathematics. Many cannot improve on the vacuous definition: mathematics is what is done by mathematicians. We could try harder, with something like “mathematics is the language of … Continue reading The Many Schools of Mathematical Thought →

Yin and Yang — and East and West

The duality encapsulated in the concept of yin-yang is at the origin of many aspects of classical Chinese science and philosophy. Many dualities in the natural world --- light and dark, fire and water, order and chaos --- are regarded as physical manifestations of this duality. Yin is the receptive and yang the active principle. … Continue reading Yin and Yang — and East and West →

A Memorable Memo: Responding to Over-assiduous Administrators

Anyone who has worked in a large organization, with an over-loaded Administration Division, will sympathise with the actions of two scientists at the Los Alamos National Laboratory (LANL) in issuing a spoof Memorandum. They had become frustrated with the large number of mimeographed notes circulated by Administration and Services, or A&S, ``to keep laboratory members … Continue reading A Memorable Memo: Responding to Over-assiduous Administrators →

Hamilton’s Semaphore Code and Signalling System

Sir William Rowan Hamilton (1805-1865) was Ireland's most ingenious mathematician. When he was just fifteen years old, Hamilton and a schoolfriend invented a semaphore-like signalling system. On 21 July 1820, Hamilton wrote in his journal how he and Tommy Fitzpatrick set up a mark on a tower in Trim and were able to view it … Continue reading Hamilton’s Semaphore Code and Signalling System →

Sixth Irish History of Mathematics (IHoM) Conference

I attended the sixth conference of the Irish History of Mathematics (IHoM) group at Maynooth University yesterday (Wednesday 30th August 2023). What follows is a personal summary of the presentations. This summary has no official status. If speakers or attendees spot any errors, please let me know and I will correct them. [1] After a … Continue reading Sixth Irish History of Mathematics (IHoM) Conference →

Maths in the Time of the Pharaohs

Why would the Ancient Egyptians have any interest in or need for mathematics? There are many reasons. They had a well-organised and developed civilisation extending over millennia. Science and maths must have played important or even essential roles in this culture. They needed measurement for land surveying and for designing irrigation canals, arithmetic for accounting … Continue reading Maths in the Time of the Pharaohs →

Margules’ Tendency Equation and Richardson’s Forecast

During World War One, long before the invention of computers, the English Quaker mathematician Lewis Fry Richardson devised a method of solving the equations and made a test forecast by hand. The forecast was a complete failure: Richardson calculated that the pressure at a particular point would rise by 145 hPa in 6 hours. This … Continue reading Margules’ Tendency Equation and Richardson’s Forecast →

Herman Melville and Ishmael’s Cycloid

Many authors use mathematical metaphors with great effect. A recent book, “Once Upon A Prime” by Sarah Hart, describes the wondrous connections between mathematics and literature. As a particular example, she discusses the relevance of the cycloid curve in the work of Herman Melville. The book Moby-Dick, first published in 1851, opens with the words … Continue reading Herman Melville and Ishmael’s Cycloid →

Christopher Wren and the Cycloid

The remarkable polymath Christopher Wren died in March 1723, just 300 years ago. Sarah Hart, Professor of Geometry at Gresham College, recently presented a lecture, The Mathematical Life of Sir Christopher Wren; a video of her presentation in available online (see sources below). The illustration above is from the Gresham College website. Christopher Wren In … Continue reading Christopher Wren and the Cycloid →

Bach and Euler chat in Frederick’s Court

Frederick the Great of Prussia, a devoted patron of the arts, had a particular interest in music, and admired the music of Johann Sebastian Bach. In 1747, Bach visited Potsdam, where his son Carl Philipp Emanuel was the Kapellmeister in Frederick’s court. When Frederick learned of this, he summoned ‘Old Bach’ to the palace and … Continue reading Bach and Euler chat in Frederick’s Court →

The Rich Legacy of Indian Mathematics

For more than three thousand years, mathematics has played an important role in Indian culture. Sometimes it was studied for practical reasons and sometimes for pure intellectual delight. The earliest traces of mathematics are found in the Indus Valley, around 3000 BC. There is clear evidence of a structured system of weights and measures and … Continue reading The Rich Legacy of Indian Mathematics →

Finding the Area of a Field

It is a tricky matter to find the area of a field that has irregular or meandering boundaries. The standard method is to divide the field into triangular parts. If the boundaries are linear, this is simple. If they twist and turn, then a large number of triangles may be required. When we have the … Continue reading Finding the Area of a Field →

Infinitesimals: vanishingly small but not quite zero

A few weeks ago, I wrote about Hyperreals and Nonstandard Analysis , promising to revisit the topic. Here comes round two. We know that 2.999… is equal to three. But many people have a sneaking suspicion that there is “something” between the number with all those 9’s after the 2 and the number 3, that is not … Continue reading Infinitesimals: vanishingly small but not quite zero →

Mathematical Equations are our Friends

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. This cynical view is a disservice to science; we should realize that, far from being inimical, … Continue reading Mathematical Equations are our Friends →

Letters to a German Princess: Euler’s Blockbuster Lives On

The great Swiss mathematician Leonhard Euler produced profound and abundant mathematical works. Publication of his Opera Omnia began in 1911 and, with close to 100 volumes in print, it is nearing completion. Although he published several successful mathematical textbooks, the book that attracted the widest readership was not a mathematical work, but a collection of … Continue reading Letters to a German Princess: Euler’s Blockbuster Lives On →

Euler’s Journey to Saint Petersburg

It all began with an invitation to Leonhard Euler to accept a chair of mathematics at the new Imperial Academy of Science in the city founded by Peter the Great. Euler’s journey from Basel to Saint Petersburg was a highly influential factor for the development of the mathematical sciences. The journey is described in detail … Continue reading Euler’s Journey to Saint Petersburg →

Bernoulli’s Golden Theorem and the Law of Large Numbers

Jakob Bernoulli, head of a dynasty of brilliant scholars, was one of the world’s leading mathematicians. Bernoulli's great work, Ars Conjectandi, published in 1713, included a profound result that he established “after having meditated on it for twenty years”. He called it his “golden theorem”. It is known today as the law of large numbers, … Continue reading Bernoulli’s Golden Theorem and the Law of Large Numbers →

Émilie Du Châtelet and the Conservation of Energy

A remarkable French natural philosopher and mathematician who lived in the early eighteenth century, Émilie Du Châtalet, is generally remembered for her translation of Isaac Newton's Principia Mathematica, but her work was much more than a simple translation: she added an extensive commentary in which she included new developments in mechanics, the most important being … Continue reading Émilie Du Châtelet and the Conservation of Energy →

Mathematical Scandals and Scoundrels

Edna St Vincent Millay’s sonnet “Euclid alone has looked on beauty bare” evokes the ethereal, otherworldly quality of mathematics. Scandalous behaviour is not usually associated with mathematicians, but they are human: pride, overblown ego and thirst for fame have led to skulduggery, plagiarism and even murder. Some of the more egregious scandals are reviewed here … Continue reading Mathematical Scandals and Scoundrels →

Changing Views on the Age of the Earth

In 1650, the Earth was 4654 years old. In 1864 it was 100 million years old. In 1897, the upper limit was revised to 40 million years. Currently, we believe the age to be about 4.5 billion years. What will be the best guess in the year 2050? [TM217 or search for “thatsmaths” at irishtimes.com]. Ussher's Chronology … Continue reading Changing Views on the Age of the Earth →

Al Biruni and the Size of the Earth

Abu Rayhan al-Biruni (AD 973--1048) The 11th century Persian mathematician Abu Rayhan al-Biruni used simple trigonometric results to estimate the radius and circumference of the Earth. His estimate has been quoted as 6,340 km, which is within 1% of the mean radius of 6,371 km. While al-Biruni's method was brilliant and, for its era, spectacular, … Continue reading Al Biruni and the Size of the Earth →

Entropy and the Relentless Drift from Order to Chaos

In a famous lecture in 1959, scientist and author C P Snow spoke of a gulf of comprehension between science and the humanities, which had become split into “two cultures”. Many people in each group had a lack of appreciation of the concerns of the other group, causing grave misunderstandings and making the world's problems … Continue reading Entropy and the Relentless Drift from Order to Chaos →

Samuel Haughton and the Humane Drop

Samuel Haughton was born in Co. Carlow in 1821. He entered Trinity College Dublin aged just sixteen and graduated in 1843. He was elected a fellow in 1844 and was appointed professor of geology in 1851. He took up the study of medicine and graduated as a Doctor of Medicine in 1862, aged 40 [TM182 … Continue reading Samuel Haughton and the Humane Drop →

The “extraordinary talent and superior genius” of Sophie Germain

When a guitar string is plucked, we don't see waves travelling along the string. This is because the ends are fixed. Instead, we see a standing-wave pattern. Standing waves are also found on drum-heads and on the sound-boxes of violins. The shape of a violin strongly affects the quality and purity of the sound, as … Continue reading The “extraordinary talent and superior genius” of Sophie Germain →

The Vastness of Mathematics: No One Knows it All

No one person can have mastery of the entirety of mathematics. The subject has become so vast that the best that can be achieved is a general understanding and appreciation of the main branches together with expertise in one or two areas [TM174 or search for “thatsmaths” at irishtimes.com]. In the sciences, old theories fade away … Continue reading The Vastness of Mathematics: No One Knows it All →

Zeroing in on Zeros

Given a function $latex {f(x)}&fg=000000$ of a real variable, we often have to find the values of $latex {x}&fg=000000$ for which the function is zero. A simple iterative method was devised by Isaac Newton and refined by Joseph Raphson. It is known either as Newton's method or as the Newton-Raphson method. It usually produces highly … Continue reading Zeroing in on Zeros →

George Salmon, Mathematician & Theologian

As you pass through the main entrance of Trinity College, the iconic campanile stands before you, flanked, in pleasing symmetry, by two life-size statues. On the right, on a granite plinth is the historian and essayist William Lecky. On the left, George Salmon (1819–1904) sits on a limestone platform. Salmon was a distinguished mathematician and … Continue reading George Salmon, Mathematician & Theologian →

Stokes’s 200th Birthday Anniversary

Next Tuesday, the 30th of August, is the 200th anniversary of the birth of George Gabriel Stokes. This extended blog post is to mark that occasion. See also an article in The Irish Times. Whether we are designing aircraft, modelling blood flow, studying propulsion, lubrication or the dynamics of swimming, constructing wind turbines or forecasting … Continue reading Stokes’s 200th Birthday Anniversary →

What did the Romans ever do for Maths?

The ancient Romans developed many new techniques for engineering and architecture. The citizens of Rome enjoyed fountains, public baths, central heating, underground sewage systems and public toilets. All right, but apart from sanitation, medicine, education, irrigation, roads and aqueducts, what did the Romans ever do for maths? [TM166 or search for “thatsmaths” at irishtimes.com]. It might … Continue reading What did the Romans ever do for Maths? →

Cumbersome Calculations in Ancient Rome

“Typus Arithmeticae” is a woodcut from the book Margarita Philosophica by Gregor Reisch of Freiburg, published in 1503. In the centre of the figure stands Arithmetica, the muse of mathematics. She is watching a competition between the Roman mathematician Boethius and the great Pythagoras. Boethius is crunching out a calculation using Hindu-Arabic numerals, while Pythagoras … Continue reading Cumbersome Calculations in Ancient Rome →

Bernard Bolzano, a Voice Crying in the Wilderness

Bernard Bolzano, born in Prague in 1781, was a Bohemian mathematician with Italian origins. Bolzano made several profound advances in mathematics that were not well publicized. As a result, his mathematical work was overlooked, often for many decades after his death. For example, his construction of a function that is continuous on an interval but … Continue reading Bernard Bolzano, a Voice Crying in the Wilderness →

Discoveries by Amateurs and Distractions by Cranks

Do amateurs ever solve outstanding mathematical problems? Professional mathematicians are aware that almost every new idea they have about a mathematical problem has already occurred to others. Any really new idea must have some feature that explains why no one has thought of it before [TM155 or search for “thatsmaths” at irishtimes.com]. It is both difficult and … Continue reading Discoveries by Amateurs and Distractions by Cranks →

The “Napoleon of Crime” and The Laws of Thought

A fascinating parallel between a brilliant mathematician and an arch-villain of crime fiction is drawn in a forthcoming book – New Light on George Boole – by Des MacHale and Yvonne Cohen. Professor James Moriarty, master criminal and nemesis of Sherlock Holmes, was described by the detective as “the Napoleon of crime”. The book presents … Continue reading The “Napoleon of Crime” and The Laws of Thought →

Grandi’s Series: A Second Look

In an earlier post, we discussed Grandi's series, originally studied by the Italian monk Dom Guido Grandi around 1703. It is the series $latex \displaystyle G = 1 - 1 + 1 - 1 + 1 - 1 + \dots &fg=000000$ This is a divergent series: the sequence of partial sums is $latex {\{ 1, … Continue reading Grandi’s Series: A Second Look →

Grandi’s Series: Divergent but Summable

Is the Light On or Off? Suppose a light is switched on for a half-minute, off for a quarter minute, on for one eighth of a minute and so on until precisely one minute has elapsed. Is the light on or off at the end of this (infinite) process? Representing the two states ``on'' and … Continue reading Grandi’s Series: Divergent but Summable →

Optical Refinements at the Parthenon

The Parthenon is a masterpiece of symmetry and proportion. This temple to the Goddess Athena was built with pure white marble quarried at Pentelikon, about 20km from Athens. It was erected without mortar or cement, the stones being carved to great accuracy and locked together by iron clamps. The building and sculptures were completed in … Continue reading Optical Refinements at the Parthenon →

Mathematics at the Science Museum

The new Winton Gallery at London's Science Museum in South Kensington holds a permanent display on the history of mathematics over the past 400 years. The exhibition shows how mathematics has underpinned astronomy, navigation and surveying in the past, and how it continues to pervade the modern world [see TM139, or search for “thatsmaths” at … Continue reading Mathematics at the Science Museum →

Fourier’s Wonderful Idea – II

Solving PDEs by a Roundabout Route Joseph Fourier, born just 250 years ago, introduced a wonderful idea that revolutionized science and mathematics: any function or signal can be broken down into simple periodic sine-waves. Radio waves, micro-waves, infra-red radiation, visible light, ultraviolet light, X-rays and gamma rays are all forms of electromagnetic radiation, differing only … Continue reading Fourier’s Wonderful Idea – II →

Fourier’s Wonderful Idea – I

Breaking Complex Objects into Simple Pieces ``In a memorable session of the French Academy on the 21st of December 1807, the mathematician and engineer Joseph Fourier announced a thesis which inaugurated a new chapter in the history of mathematics. The claim of Fourier appeared to the older members of the Academy, including the great analyst … Continue reading Fourier’s Wonderful Idea – I →

Cubic Skulduggery & Intrigue

Babylonian mathematicians knew how to solve simple polynomial equations, in which the unknown quantity that we like to call x enters in the form of powers, that is, x multiplied repeatedly by itself. When only x appears, we have a linear equation. If x-squared enters, we have a quadratic. The third power of x yields … Continue reading Cubic Skulduggery & Intrigue →

Subtract 0 and divide by 1

We all know that division by zero is a prohibited operation, and that ratios that reduce to ``zero divided by zero'' are indeterminate. We probably also recall proving in elementary calculus class that $latex \displaystyle \lim_{x\rightarrow 0} \frac{\sin x}{x} = 1 &fg=000000$ This is an essential step in deriving an expression for the derivative of … Continue reading Subtract 0 and divide by 1 →

Galileo’s Book of Nature

In 1971, astronaut David Scott, standing on the Moon, dropped a hammer and a feather and found that both reached the surface at the same time. This popular experiment during the Apollo 15 mission was a dramatic demonstration of a prediction made by Galileo three centuries earlier. Galileo was born in Pisa on 15 February … Continue reading Galileo’s Book of Nature →

Hardy’s Apology

Godfrey Harold Hardy's memoir, A Mathematician's Apology, was published when he was 63 years old. It is a slight volume at just 90 pages, but is replete with interesting observations and not a few controversial opinions. After 78 years, it is still in print and is available in virtually every mathematics library. Though many of … Continue reading Hardy’s Apology →

Quadrivium: The Noble Fourfold Way

According to Plato, a core of mathematical knowledge – later known as the Quadrivium – was essential for an understanding of the Universe. The curriculum was outlined in Plato's Republic. The name Quadrivium means four ways, but this term was not used until the time of Boethius in the 6th century AD [see TM119 or search … Continue reading Quadrivium: The Noble Fourfold Way →

Robert Murphy, a “Brilliant Meteor”

“A brilliant meteor that flared intensely but all too briefly”; this was how Des MacHale described the Cork-born mathematician Robert Murphy in his biography of George Boole, first professor of mathematics in Cork. Murphy was a strong influence on Boole, who quoted liberally from his publications [see TM118 or search for “thatsmaths” at irishtimes.com]. Robert Murphy was … Continue reading Robert Murphy, a “Brilliant Meteor” →

Patterns in Poetry, Music and Morse Code

Suppose we have to ascent a flight of stairs and can take only one or two steps at a time. How many different patterns of ascent are there? We start with the simplest cases. With one step there is only one way; with two, there are two: take two single steps or one double step. … Continue reading Patterns in Poetry, Music and Morse Code →