Christmas Day, 25th December 1758.

Amongst the many gifts given, for astronomers, the greatest was the arrival of a comet, predicted to return by Astronomer Royal Edmond Halley precisely 76 years after its last appearance and with it heralding a new era of astronomy.

For thousands of years at different points in history, from China in 240 BC to William the Conqueror in 1066 a celestial body was observed to travel across the night sky. This “fiery star” as described by Italian writer Bartolomeo Platina in 1470, was interpreted as an omen by some, the beginning of famine, drought, and pestilence by others, and potentially as the Star of Bethlehem in the biblical story.

Halley's Comet as seen in the Bayeux Tapestry

The mysterious object observed by different cultures and civilizations throughout history would appear in the night sky for days, weeks, or months but all those who witnessed it believed it to be an isolated event until brilliant English astronomer Edmond Halley began to see connections.

Halley was born in 1656 in Haggerston, England, and had a lifelong fascination with mathematics and astronomy. He studied at Oxford University, where he developed his skills in these fields, and went on to make significant contributions to the field of astronomy, navigation, geology, and more whilst also funding Isaac Newton’s Principia Mathematica.

Halley was particularly interested in the comet that bore his name, and he dedicated much of his career to studying its orbit and predicting its return. In 1705, after using Newton’s Laws of Motion, he published a paper in which he argued that the comets observed in 1531, 1607, and 1682 were actually the same comet and that it followed a predictable orbit.

At first, Halley's predictions were met with skepticism, and he didn't live to see his comet's next predicted arrival. On Christmas Day, 1758 astronomers gathered to witness the comet reappearing exactly as Halley had predicted. In his honor, the comet was named Halley’s Comet.

The reappearance of Halley's Comet in 1758 was a momentous event that had far-reaching implications for our understanding of the solar system and the universe as a whole. It was the first time that the return of a comet had been accurately predicted, and it helped to establish the idea that comets followed predictable orbits rather than passing straight through the solar system.

Halley's Comet has continued to reappear approximately every 76 years since 1758, and its next predicted appearance is in 2061. It has inspired art and literature, captivating human imagination for centuries. Its discovery and prediction of its return by Edmond Halley was a major milestone in the field of astronomy and will continue to inspire and intrigue us for generations to come.

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In 1963, American astronaut John Glenn refused to fly until NASA mathematician Katherine Johnson had personally verified the computer generated flight calculations by hand.

**"If she says they're good, then I'm ready to go" **Glenn famously remarked.

This is the story of Katherine Johnson

Born in West Virginia, 1918, Johnson showed a high aptitude for maths from a young age. Her brilliance accelerated her several years ahead of her peers and she graduated high school at 14 and college at 18 with the highest honours.

"I counted everything. I counted the steps to the road, the steps up to church, the number of dishes and silverware I washed... anything that could be counted, I did."

Following graduation she attempted to embark on a career as a maths researcher however Johnson found the opportunities for an African American woman in 1940s American limited.

In 1941 President Roosevelt banned racial discrimination in the defense industries and NASA's predecessor, NACA, began hiring black female mathematicians. Hearing of the opportunity Johnson, who by now had trained as a teacher, applied and was accepted into the Computing Department in 1953.

After a short period at NACA she was temporarily assigned to the Flight Research Division which was staffed entirely by white male engineers. Breaking the barriers of race and gender Johnson quickly became indispensable member of the team and became the first woman in her department to have her name added to a paper.

*"In the early days of NASA women were not allowed to put their names on the reports … I finished the report and my name went on it, and that was the first time a woman in our division had her name on something".*

Following the Soviet success in launching the first satellite, Sputnik, in 1957 and then first astronaut, Yuri Gagarin in 1961 the fear engendered in the American public of Soviet technological superiority sparked the US government into investing huge resources to send astronauts into space and ultimately the moon.

Johnson's ability to complete highly complex flight calculations by hand made her indispensable to the US space program and she moved to the Spacecraft Controls Branch. There she worked on orbital trajectories, sending the first American, Alan Shepherd into space, the Mercury missions and the flight calculations of the Apollo missions to land astronauts on the moon.

The perilous nature of space travel meant that small mistakes could lead to catastrophic outcomes but such was Johnson's reputation she was famously asked to manually verify computer generated calculations by astronaut John Glenn before his launch into space. Glenn instructed flight engineers to "**get the girl**" and "**if she says they're good, then I'm ready to go**".

Subjected to the prejudice of the time as a woman and secondly a black woman in Jim Crow era America, Johnson commented on her experiences

** "I don’t have a feeling of inferiority. Never had. I’m as good as anybody, but no better"**** **

Johnson died in 2020 at the age of 101.

All images Credit NASA except the second, credit Katherine Johnson

**The story of how a simple equation a schoolchild could understand vexed the world's greatest mathematicians for hundreds of years.**

In the 17th century, mathematician **Pierre de Fermat** was finding solutions to the equation

When n= 2 there are infinite solutions and can be tackled by school children learning Pythagoras.

3^{2} + 4^{2} = 5^{2} for example.

But when n = 3, 4, or any number greater than 2, were there any solutions?

Fermat wrestled with the problem and failed to find many solutions. He then asked the question: are there any solutions at all?

After struggling with the problem he found a proof that there were no solutions for x^{n} + y^{n} = z^{n} when n >2.

The evening of his discovery he read Arithmetica by the Greek mathematician Diophantus and cryptically scribbled a note in the margin stating

*"I have a truly marvellous proof which the margin of this book is too small to contain"*

Fermat never lived to write his proof and the only evidence he left was this note discovered by his son after his death.

And so began a problem tackled by the greatest mathematical minds of all time such as **Euler, Gauss, Germain, Hilbert,** and hundreds of others to solve what became known as **Fermat’s Last Theorem.**

For hundreds of years, all were defeated until the 1960s when 10-year old **Andrew Wiles** came across Fermat, and immediately he was hooked. For his teenage years and early adult life, he spent years obsessively questioning, reading, researching, anything he could do to solve Fermat.

But by his 30s he had given up until by chance a paper crossed his desk about the obscure Taniyama-Shimura Conjecture published in the 1950s. To the uninitiated this seemingly had nothing to do with Fermat however Wiles instantly realised that if he proved Taniyama-Shimura it would lead to a proof of Fermat.

Inspired, he abandoned his research and worked in complete solitude for 7 years obsessing over the problem. In 1993 after 7 long years working he realised he had proved Taniyama-Shimura and therefore had Fermat in his grasp.

He emerged and presented his findings at the Newton Institute in Cambridge in 1994 to an incredulous mathematical community.

**However...**

Upon review one small but critical mistake was found and Wiles was in danger of being another mathematician who claimed to have solved Fermat only to suffer ridicule from his peers.

Wiles retreated and for one more year worked to rectify his mistake only to find with every correction his proof unraveled even further. Close to admitting defeat he contacted his old student Richard Taylor and together they submitted a final proof and after 358 years the problem, tackled by some of the greatest minds in history, was solved and Wiles stood atop them as the conqueror of Fermat's Last Theorem.

In honour of Wiles and other giants of maths, we included Fermat's Last Theorem and many other seminal equations on our math notebook and mugs

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It's hard to win a Nobel Prize.

It's harder to win two.

It's so hard to win two in different scientific fields that only one person has done it.

And it's unimaginably hard to do all this when you're a female migrant in 1900s France.

Maria Skłodowska was born in Poland in 1867 in a country where laboratory science was outlawed in schools and women were forbidden from attending university. A keen scientist from a young age Maria was undeterred by the obstacles, joining an underground school that admitted female students; the ‘Flying University’ of Warsaw,

Recognising the limited opportunities in her beloved Poland she enrolled at the University of Paris in 1891, studying by day to earn two degrees in physics and mathematics and tutoring every evening to make ends meet.

Following graduation she was introduced to physicist Pierre Curie. Drawn together by their shared passion for science Pierre proposed but was turned down as Marie saw her future in Poland. Leaving Paris she was informed by the University of Warsaw that she, as a female, would never be allowed a professorship whereupon she returned to Paris and married Pierre.

In 1896 the physicist Henri Becquerel noticed that when uranium salts were placed on a photographic plate it would leave a dark shadow. Becquerel proposed that the rock was emitting rays, thus discovering radioactivity. Inspired, Curie decided to investigate uranium for her thesis

Tests by Curie on uranium to alter its physical and chemical make up failed to diminish the effect of the radiation leading to Curie’s first breakthrough - she theorised that something within the atom was emitting the rays, disproving the centuries old idea that atoms are indivisible.

Curie noticed that the uranium rich ore pitchblende was four times more radioactive than pure uranium and wondered if this was due to an even more radioactive element present within. Tonne after tonne of the ore arrived by horse and cart at her unventilated shed and, using basic equipment and next to no safety considerations, she analysed and tested it leading to the discovery of two new elements in 1898; Polonium, named after her native Poland and Radium.

Recognising the magnitude of the discovery of radioactivity, in 1903 the Nobel committee sought to honour Pierre and Becquerel, omitting Marie from the nomination. Pierre launched a complaint and Marie was added, becoming the first woman to win a Nobel Prize.

In 1906, Pierre would die tragically in a traffic accident, devastating Marie. The University of Paris, planning to give a professorship to Pierre instead offered it to Marie, making her the first woman to become a professor at the institute.

She won her second Nobel Prize, this time for chemistry, in 1911 for the discovery of polonium and radium. The archaic attitudes of the time were evidenced by Svante Arhenius, the Swedish chemist and head of the Nobel committee, who attempted to prevent her attendance at the ceremony.

Curie went on to found research institutes in Paris and Warsaw, and collected numerous scientific accolades and awards for her work. However unbeknownst to her, years of directly handling highly radioactive materials had adverse effects on her health. Years spent in a poorly ventilated shed, carrying radium in her pockets, and working with highly carcinogenic materials, led to her developing a blood disorder, of which she died of in 1937 at the age of 66.

Her legacy was both scientific and societal. Overcoming countless barriers she became the first of many prominent female scientists in the 20th century, including her own daughter, who won her own Nobel Prize in chemistry.

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In 1905, unknown 26 year old Albert Einstein posted 4 papers to the Annalen der Physik that changed our understanding of light, matter, space and time.

- The 1st won him a Nobel Prize
- The 2nd proved the existence of atoms
- The 3rd, Special Relativity
- And the 4th, E=mc2

By 1905 Einstein finished his PhD and after a string of failed applications to academic jobs, he settled as a patent clerk in Bern, Switzerland with his passion for physics being relegated to a pursuit undertaken during his free time.

But by the end of the year this part time physicist had published four papers becoming the greatest scientist alive.

For over 200 years physicists were convinced that light was a wave, with the different wavelengths corresponding to different colours. However, the discovery of the photoelectric effect showed that light doesn’t always behave as a wave.To solve this contradiction Einstein proposed that light “consists of a finite number of energy quanta”. In other words, light is not a wave, nor a particle, but “packets” of energy, called photons. This leap in imagination earned Einstein a Nobel Prize in 1921 for his discovery.

To physicists pre 1905 atoms were thought of as a concept and used as a placeholder for explaining the nature of matter. It wasn’t until Einstein studied Brownian Motion that the existence of atoms was accepted.

When a particle is suspended in a medium such as water it will move at random with the movement caused by the atoms within the medium knocking against it, known as Brownian motion. Einstein concluded that the mathematical explanation for this proved the existence of atoms and his work allowed physicists and chemists to precisely determine the presence and energies of atoms, thus cementing them as real entities.

For 220 years Newton’s Laws of Motion solved nearly all problems regarding the movement of objects apart from those that involved light. To account for this discrepancy physicists proposed that light moved through a medium they called “the Aether”. The existence of the aether proved elusive and Michaelson and Morley’s experiments in 1887 to detect it proved two things

1. The aether didn’t exist.

2. Light moved through a vacuum in space.

**Classical physics had no explanation for this**

Einstein proposed that light is a constant and that regardless of the speed an object moves at, light always travels at a constant speed. This led to profound consequences in that if light is a constant, time will slow when an object is travelling at speeds relative to an object at rest.

Having established light as a constant, in September Einstein posted his fourth paper to the Annalen der Physik. His 3 page paper showed that mass and energy are different forms of the same thing, leading to **the most recognisable equation in history, E= mc2.**This simple equation explains why stars shine and ushered in the atomic age by demonstrating how much energy is contained and can therefore be released within a small mass.

Along with Newton’s 1665, Einstein’s 1905 ranks amongst those in which mankind made our greatest leaps forwards in the understanding of our Universe.

**And, his greatest work, General Relativity, was yet to come…**

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There's a deadly illness sweeping the nation. You've been confined to your house. What do you do?

Well if you're like most of the world's population, probably lots of TV or jigsaws. But if it’s 1665 and you're a 23 year old Isaac Newton...

**The Great Plague of 1665** swept across Europe with London losing 100,000 people or 25% of its population leading to Newton, then 23 years old, to leave Cambridge for the safety of Woolsthorpe Manor, 60 miles north.

There, with quiet and time and unshackled from his tutors, his intellectual creativity flourished as he filled notebook after notebook with complex mathematics and ideas undertaking what has become known as his Annus Mirablis or " ** Year of Wonders**".

Firstly Newton tackled the problem of the mathematics of continuous change, now known as calculus.

Building on the ideas of Ancient Greek mathematicians and those that came after such as Descartes, Fermat, Kepler and more he wrote a paper outlining his discoveries. Circulating his ideas amongst his peers in conversation and correspondence he failed to publish it leading to his discovery being marred in controversy as Gottfried Leibniz argued he had discovered calculus first. It is now acknowledged that both men discovered it independently.

After discovering calculus at the speed that people learn it at university he turned his mind to light.

Taking a prism, previously used as a novelties and for use in chandeliers, he pondered the fact that they changed white light into a spectrum of colours. Boring a hole in the shutters of his room, now known as Newton’s Chamber, he allowed a thin shaft of light to fall onto a prism. His moment of genius came from allowing the rainbow of light to fall on a second prism causing the dispersed light to recombine and emerge as white light. This proved the light was unaltered by the glass and that white light is a combination of colours.

Having discovered calculus, and making enormous advances in the optics Newton stood in his room and one day saw an apple fall from a nearby tree.

He asked himself why the apple fell down as opposed to upwards or sideways. He theorised that there must be a force pulling the apple down and that maybe it was the same force that held the moon in orbit of the earth and the earth in orbit of the sun. Developing his ideas led to a mathematical basis of the attraction of objects to each other, now referred to as his Laws of Gravitation.

Following the end of the plague Newton returned with his notebooks to Cambridge University and presented his ideas in 1687 in one of the most important works in human history, *Philosophiæ Naturalis Principia Mathematica*, commonly referred to as Principia and then later in 1704 his book on light, known as Optics.

His Year of Wonders led to breakthroughs the scale of which weren’t seen again until the early 1900s with a young Albert Einstein. This giant of human intellect single handedly changed the path of science history, our history, forever.

Following his death in 1727, the poet Alexander Pope wrote an epitaph for Newton:

"*Nature and Nature's laws lay hid in night: God said, Let Newton be! and all was light*.”

In 1913 a letter was received by English mathematician G.H. Hardy at the University of Cambridge from India. The letter, written by 26 year old Indian genius, Srinivasa Ramanujan, contained theorems that Hardy stated *"defeated me completely; I had never seen anything in the least like them before"* comparing Ramanujan to mathematical giants Euler and Jacobi.

Born in British occupied India in 1887 Ramanujan was recognised as a child prodigy from the age of 11 and, working largely in isolation from the mathematical world, had independently discovered work previously uncovered by Bernoulli and Euler by the age of 16.

Following his initial letter a correspondence began with Hardy with Hardy arranging for Ramanujan's travel to Cambridge. On 17th March, 1914 he boarded the S.S. Nevasa to begin his work in England.The two men had contrasting personalities; Hardy requiring rigour and proofs whereas the deeply spiritual Ramanujan relied on instinct and intuition often leaving gaps in his work that Hardy struggled to fill.The importance of his work though was recognised by all and brought him membership to the London Mathematical Society and he became the first Indian to be elected a Fellow of Trinity College, Cambridge.

In 1919 Ramanujan travelled back to India and, plagued by health problems his entire life, he died of suspected tuberculosis in 1920 at the age of 32.

In his short life Ramanujan compiled 3900 results. He left behind three notebooks that were poured over following his death and as late as 2012 new areas of mathematics were being discovered in them.

Following his death Hardy wrote of the man he mentored that "*his insight into formulae was quite amazing, and altogether beyond anything I have met with in any European mathematician... it is not extravagant to suppose that he might have become the greatest mathematician of his time*"

On top of this foundation we have included math written by Gauss, Euler, Newton, Pythagoras and many more giants of mathematics.