## Posts Tagged 'History'

### Changing Views on the Age of the Earth

[Image credit: NASA]

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].

### Al Biruni and the Size of the Earth

Al Biruni at Persian Scholars Pavilion in Vienna.

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, the accuracy claimed must be regarded with suspicion.

Al-Biruni assumed that the Earth is a perfect sphere of (unknown) radius ${a}$. He realised that because of the Earth’s curvature the horizon, as viewed from a mountain-top, would appear to be below the horizontal direction. This direction is easily obtained as being orthogonal to the vertical, which is indicated by a plumb line.

### 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 more difficult to solve. Snow compared ignorance of the Second Law of Thermodynamics to ignorance of Shakespeare [TM209 or search for “thatsmaths” at irishtimes.com].

### Samuel Haughton and the Humane Drop

Samuel Haughton (1821-1897).

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 or search for “thatsmaths” at irishtimes.com].

In addition to his expertise in geology and medicine, Haughton was a highly talented applied mathematician. His mathematical investigations included the study of the motion of solid and fluid bodies, solar radiation, climatology, animal mechanics and ocean tides. One of his more bizarre applications of mathematics was to demonstrate a humane method of execution by hanging, by lengthening the drop to ensure instant death.

### 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 it determines the mixture of standing wave harmonics that it can sustain [TM179 or search for “thatsmaths” at irishtimes.com].

French postage stamp, issued in 2016, to commemorate the
250th anniversary of the birth of Sophie Germain (1776-1831).

### 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].

The Princeton Companions to Maths and Applied Maths

### Zeroing in on Zeros

Given a function ${f(x)}$ of a real variable, we often have to find the values of ${x}$ 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 accurate approximations to the roots of the equation ${f(x) = 0}$.

A rational function with five real zeros and a pole at x = 1.

### George Salmon, Mathematician & Theologian

George Salmon (1819-1904) [Image: MacTutor]

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 (18191904) sits on a limestone platform.

Salmon was a distinguished mathematician and theologian and Provost of Trinity College. For decades, the two scholars have gazed down upon multitudes of students crossing Front Square. The life-size statue of Salmon, carved from Galway marble by the celebrated Irish sculptor John Hughes, was erected in 1911. Next Wednesday will be the 200th anniversary of Salmon’s birth [TM171 or search for “thatsmaths” at irishtimes.com].

### The Brief and Tragic Life of Évariste Galois

On the morning of 30 May 1832 a young man stood twenty-five paces from his friend. Both men fired, but only one pistol was loaded. Évariste Galois, a twenty year old mathematical genius, fell to the ground. The cause of Galois’s death is veiled in mystery and speculation. Whether both men loved the same woman or had irreconcilable political differences is unclear. But Galois was abandoned, mortally wounded, on the duelling ground at Gentilly, just south of Paris. By noon the next day he was dead [TM169 or search for “Galois” at irishtimes.com].

French postage stamp issued in 1984.

### 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.

### 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].

Roman aqueduct at Segovia, Spain.

### 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 uses a counting board or abacus (tabula) and – presumably – a less convenient number system. Arithmetica is looking with favour towards Boethius. He smiles smugly while Pythagoras is looking decidedly glum.

The figure aims to show the superiority of the Hindu-Arabic number system over the older Greek and Roman number systems. Of course, it is completely anachronistic: Pythagoras flourished around 500 BC and Boethius around AD 500, while the Hindu-Arabic numbers did not arrive in Europe until after AD 1200.

### Bernard Bolzano, a Voice Crying in the Wilderness

Bernard Bolzano (1781-1848)

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 nowhere differentiable, did not become known. Thus, the credit still goes to Karl Weierstrass, who found such a function about 30 years later. Boyer and Merzbach described Bolzano as “a voice crying in the wilderness,” since so many of his results had to be rediscovered by other workers.

### The Rise and Rise of Women in Mathematics

Sonya Kovalevskya (1850-1891)

The influential collection of biographical essays by Eric Temple Bell, Men of Mathematics, was published in 1937. It covered the lives of about forty mathematicians, from ancient times to the beginning of the twentieth century. The book inspired many boys to become mathematicians. However, it seems unlikely that it inspired many girls: the only woman to get more than a passing mention was Sofia Kovalevskaya, a brilliant Russian mathematician and the first woman to obtain a doctorate in mathematics [TM163 or search for “thatsmaths” at irishtimes.com].

### 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].

Pierre de Fermat and Srinivasa Ramanujan, two brilliant “amateur” mathematicians.

### 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 convincing evidence that Moriarty was inspired by Professor George Boole [TM151, or search for “thatsmaths” at irishtimes.com].

### 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

$\displaystyle G = 1 - 1 + 1 - 1 + 1 - 1 + \dots$

This is a divergent series: the sequence of partial sums is ${\{ 1, 0, 1, 0, 1, 0, \dots \}}$, which obviously does not converge, but alternates between ${0}$ and ${1}$.

### 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 “off” by ${1}$ and ${0}$, the sequence of states over the first minute is ${\{ 1, 0, 1, 0, 1, 0, \dots \}}$. But how do we ascertain the final state from this sequence? This question is sometimes known as Thomson’s Lamp Puzzle.

### 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 just 15 years, between 447 and 432 BC. [TM141 or search for “thatsmaths” at irishtimes.com].

### 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 irishtimes.com].

Central Display at the Science Museum

### Fourier’s Wonderful Idea – II

Solving PDEs by a Roundabout Route

Joseph Fourier (1768-1830)

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 in frequency  [TM136 or search for “thatsmaths” at irishtimes.com].

### 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 Lagrange, entirely incredible.”

Introduction

Joseph Fourier (1768-1830)

The above words open the Discourse on Fourier Series, written by Cornelius Lanczos. What greatly surprised and shocked Lagrange and the other academicians was the claim of Fourier that an arbitrary function, defined by an arbitrarily capricious graph, can always be resolved into a sum of pure sine and cosine functions. There was good reason to question Fourier’s theorem. Since sine functions are continuous and infinitely differentiable, it was assumed that any superposition of such functions would have the same properties. How could this assumption be reconciled with Fourier’s claim?

### Cubic Skulduggery & Intrigue

Solution of a cubic equation, usually called Cardano’s formula.

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 a cubic equation, the fourth power a quartic and so on [TM135 or search for “thatsmaths” at irishtimes.com].

### 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

$\displaystyle \lim_{x\rightarrow 0} \frac{\sin x}{x} = 1$

This is an essential step in deriving an expression for the derivative of ${\sin x}$.

### 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 1564, just 454 years ago today [TM133 or search for “thatsmaths” at irishtimes.com].

Image: NASA

### 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 Hardy’s opinions are difficult to support and some of his predictions have turned out to be utterly wrong, the book is still well worth reading.

### 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 for “thatsmaths” at irishtimes.com].

Image from here.

It is said that an inscription over the entrance to Plato’s Academy read “Let None But Geometers Enter Here”. This indicated that the Quadrivium was a prerequisite for the study of philosophy in ancient Greece.

### 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].

### 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. With three steps, there are three possibilities. We can now proceed in an inductive manner.

### Enigmas of Infinity

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 since antiquity, and it leads to many surprises and paradoxical results [TM110 or search for “thatsmaths” at irishtimes.com].

### 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 discovering a new world.

Illustrations by Leonardo da Vinci in Pacioli’s De Divina Proportione.

### 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 shelves and cases remain as they were in the 1730s. The collection is catalogued online. [TM105 or search for “thatsmaths” at irishtimes.com].

### 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 + p t + q = 0. The general cubic equation can be reduced to this form by a simple linear transformation of the dependent variable. The solution is given by

Cardano assumed that the discriminant Δ = ( q / 2 )2 + ( p / 3 )3, the quantity appearing under the square-root sign, was positive.

Raphael Bombelli made the psychedelic leap that Cardano could not make. He realised that Cardano’s formula would still give a solution when the discriminant was negative, provided that the square roots of negative quantities were manipulated in the correct manner. He was thus the first to properly handle complex numbers and apply them with effect.

### Taylor Expansions from India

FIg. 1: Brook Taylor (1685-1731). Image from NPG.

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 linear perspective (see Fig. 1).

It is noteworthy that the series for ${\sin x}$, ${\cos x}$ and ${\arctan x}$ were known to mathematicians in India about 400 years before Taylor’s time.
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].

Android app RealCalc with natural and common log buttons indicated.

### 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 sphere. The inner sphere is tangent to the centre of each face and the outer sphere contains all the vertices of the polyhedron.

Figure generated using Mathematica Demonstration [2].

### 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 irishtimes.com].

### 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 the motions of the Sun and Moon [TM087, or search for “thatsmaths” at irishtimes.com].

Iona Abbey, the last Celtic monastery to hold out against  Easter reform [Image Wikimedia Commons]

### 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 in mathematics gave rise to rumours of discord between Nobel and Gösta Mittag-Leffler, the leading Swedish mathematician of the day. They are without foundation, the truth being that Nobel had little interest in the subject, and probably didn’t appreciate the practical benefits of advanced 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 coin crosses a dividing line, it is lost.

The playing board for Franc-Carreau is shown above, together with a winning coin (red) contained within a square and a loosing one (blue) crossing a line. As the precise translation of franc-carreau appears uncertain, the name “fair square” would seem appropriate.

The question is: What size should the coin be to ensure a 50% chance of winning?

### 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 scale and ν the viscosity coefficient. The transition from laminar to turbulent flow occurs at a critical value of Re which depends on details of the system, such as surface roughness.

### 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:

### 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 mood of lassitude and brooding dejection, weighed down by intellectual cares. Her head rests on her left hand while her right hand holds a mathematical compass, one of many symbols and motifs in the work that reflect Dürer’s interest in mathematics.

### 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 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 irishtimes.com ]

### 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 (“The Ambassadors”),
Hans Holbein the Younger, 1533. Oil and tempera on oak, National Gallery, London

### James Joseph Sylvester

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 irishtimes.com ]

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!

### 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.

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

### 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].

### 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]

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

### 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).

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