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EASIER THAN YOU THINK...

Biographies (001-020)

The mathematicians and scientists here have inspired a chocolate drink, invented the “=” sign, investigated nuclear physics, music and the mind, calculated the size of the Earth, and named absolutely gigantic numbers!  I hope you enjoy learning about them and are inspired to read and research further.

Milo Carring a BullRobert RecordeJohn von NeumannErnest RutherfordJoseph John ThompsonRichard FeynmanManfred ClynesBlaise PascalEdward KasnerEratosthenesSofia Vasilyevna Kovalevskaya

Chocolate and Mathematics ~ The Story of Milo c. 500 BC

When I was a child, Milo was a common night-time drink during winter. Just secretly, we used to love sprinking it on ice cream, too!  I am aware that this fortified chocolate drink (fortified with extra vitamins and minerals) is not sold everywhere in the world, so let me explain.

It is an Australian drink, invented by Nestlé in Sydney in 1934, and is very similar to Ovaltine. Generations of children in Australia are very familiar with this product. For me, however, it has come to represent a great deal more! As a teenager, I discovered that there was a real "Milo" and that the drink was named after him.

Map of Magna Graecia showing Crotona (where Pythagoras and Milo lived)

Milo lived in a Greek colony called Croton in southern Italy just over 500 years BC (i.e. 2500 years ago!).  There were many Greek colonies along southern Italy at that time.

You need to understand that he was a remarkably strong man ... possibly the strongest historical Greek figure. He competed in the open wrestling event in the original Olympic Games and WON AT SIX SUCCESSIVE OLYMPIC GAMES! Can you imagine someone today winning the wrestling as, say, an eighteen year old, and still winning against all competitors at 38 years of age? This is an extraordinary story and Nestlé named their drink after him because they wanted to market it as a "strength" drink for children.

What has this got to do with mathematics? You will be amazed! One of the most famous mathematicians of all time was Milo's teacher and friend. They were, by all accounts, "good buddies." I cannot see this product or hear the name "Milo" without thinking of these two remarkable men, the mathematics, philosophy, music and science of their time, and the history of the world during the sixth century BC. I hope, after watching this video, that you will be similarly enriched!

Who was his friend and teacher? None other than Pythagoras! I will have a lot more to say about Pythagoras at a later time. Our culture is greatly indebted to Pythagoras and his community (including Milo).

There are stories about Milo and a bull.  One was that he started squatting the calf just after it had been born, and kept doing so until it was a 4 year old bull.  Another was that he carried a bull through the archway into the arena at the Olymic Games one year in order to intimidate the opposition!  I have read a variety of stories and encourage you to read about Milo on Wikipedia.  You might even like to Google his name and do some further reading and discovering.

Here is a close-up of the label on the 'original' Milo tin so that you can see Milo of Croton carrying the bull.

A Scan of an 'Original' Milo Tin Showing Milo of Croton Carrying a Bull

Robert Recorde and His Invention of the "Equals Sign" in 1557

I did things all "back the front" for this video.

Having received a facsimile of Robert Recorde's original book, The Whetstone of Witte in the mail, I rushed to share it with you before fully researching the material. I added the refinements in translation to the video later by adding text! I hope you don't mind the rawness of it all.

Robert Recorde

This is a most amazing work, and Robert Recorde was quite an amazing person. He was born in Wales and lived during the exciting, but dangerous, times of Henry VIII, Edward VI, Lady Jane Gray and Mary I (Bloody Mary). His primary training at Oxford University was as a medical doctor. Robert Recorde was a supporter of Reformation and this carried over to his mathematics. He wrote a number of texts on arithmetic, algebra, geometry, navigation and medicine in English (not Latin) and he wrote with the intention that everyone could learn mathematics and gave lots of descriptions and common examples from everyday life to illustrate his work.

In doing so, he created/used English terms for many concepts and ideas (such as the tangent to a circle being called a "touching line"). He popularised the use of the 'Hindu-Arabic' numeral system in England, as well as the "+" and "-" signs that had been developed in Europe.

In some ways, he was to English mathematics what Leonardo of Pisa (Fibonacci) was to European mathematics (Fibonacci introduced Arabic numerals to Europe and showed merchants and others how to use them).

Robert Recorde is also famous for inventing the "=" sign which made the solving of algebraic equations so much easier. His first book, the Grounde of Artes (1543), was primarily an arithmetic book, but is also remembered as the first English book of algebra. By 1548 he had also written and published his medical work, The Urinal of Physick. He published a book of geometry and astronomy in 1551 titled, The Pathway to Knowledge. This was the same year that he was appointed under Edward VI to govern the silver mines in Ireland, and shortly before Edward produced the first English silver coin to bear a date in Hindu-Arabic numerals rather than Roman numerals. In 1556 he published The Castle of Knowledge, a book in which he discusses the Ptolemaic universe (a treatise on circles and spheres) and mentions the work of Copernicus. And, the following year (1557), he published his last book, the one that I discuss in this video, called The Whetstone of Witte. It is in this work that he introduced his "=" sign.

Sadly, he had made a significant political enemy in those hard times and, in the same year that he published The Whetstone of Witte, he lost his court battle (he was charged with libel) and was sent to prison where he died around the middle of the following year (1558). It is said of him that it was he who established the English school of mathematics ... and it was he who first introduced algebra to England.

What a legacy he has left! He was obviously a fine and experienced teacher of mathematics. That much shines through his work in the care he takes to ensure that the reader UNDERSTANDS WHY certain things are as they are, and why problems are solved in certain ways.

Let me add two more observations: The first is that, when he wrote of "a pair of parallels, or Gemowe lines of one length" in presenting his "=" sign, the word Gemowe means 'Gemini' or 'twin.' The second comment concerns his final book, "The Whetstone of Witte." The title is, in fact, a clever pun. In those days, no one spoke of 'algebra.' What we call algebra today was known as The Cossick Art. Mathematicians who studied algebra were called 'cossists.' This is because, in Latin, 'cosa' meant 'thing,' and algebra is a language that describes and manipulates 'things.' But the (similar) Latin word 'cos' means 'a whetstone' which is a stone used for sharpening knives, axes, and other such blades and tools. So, playing on the Latin cos/cosa, Robert Recorde presented his text as a book on which his readers could sharpen their wits!

If you wish to examine a fairly detailed copy of the pages from The Whetstone of Witte that contains Robert Recorde's first use of the equals sign, you may right-click on the image below and save the file.

A Scan of a page from Robert Recorde's 1557 book, The Whetstone of Witte, in which he introduces the equals sign

John von Neumann (1903-1957)

A very great name in mathematics is John von Neumann.

He was an extraordinary child prodigy in the areas of language, memorization, and mathematics. Devine and Cohen, in Absolute Zero Gravity (page 84), relate that "he combined a photographic memory with breathtaking mathematical power. He could quote page after page of a novel read twenty years before, or invent on the spur of a moment a proof of someone else's brand-new result."

A Portrait of John von Neumann

As a 6-year-old, he could divide two 8-digit numbers in his head. By the age of 8, he was familiar with differential and integral calculus. At the age of 15, he began to study advanced calculus under the renowned analyst Gábor Szegő. On their first meeting, Szegő was so astounded with the boy's mathematical talent that he was brought to tears. By the age of 19, von Neumann had published two major mathematical papers, the second of which gave the modern definition of ordinal numbers, which superseded Georg Cantor's definition.

Von Neumann was part of a Budapest generation noted for intellectual achievement: he was born in Budapest around the same time as Theodore von Kármán (b. 1881), George de Hevesy (b. 1885), Leó Szilárd (b. 1898), Eugene Wigner (b. 1902), Edward Teller (b. 1908), and Paul Erdős (b. 1913).

Now, I want you to take a deep breath before you read the next three paragraphs. Most, but not all of his major contributions are listed here (he made numerous other contributions as well). Please read this list and wonder why you have never heard of John von Neumann:

  • He was responsible for major contributions to the development of set theory and founded the mathematical discipline of game theory. He also founded the field of continuous geometry, introduced the study of rings of operators through the von Neumann algebras, was the first to rigorously establish a mathematical framework for quantum mechanics and, with Garrett Birkhoff, was the first to prove that quantum mechanics requires a propositional calculus substantially different from all classical logics (they rigorously isolated a new algebraic structure for quantum logics), he raised the intellectual and mathematical level of economics in several stunning publications (i.e. mathematical economics), he made fundamental contributions to mathematical statistics, was leading authority of the mathematics of shaped charges and was a key contributor to the Manhatten Project to develop the atomic bomb, and he made major contributions to measure theory and ergodic theory, and lattice theory.
  • Not only that, but he made major breakthroughs in fluid dynamics and numerical hydrodynamics and (with Robert D Richtmyer) is especially remembered for an improved understanding of shock waves. It is possible that we would not understand much of astrophysics, and might not have highly developed jet and rocket engines without the work of von Neumann.
  • He was also a founding figure in computing and invented the theory of duality in linear programming. It is said that George Dantzig was once describing his work to von Neumann. Impatiently, von Neumann asked him to get to the point, whereupon he summarised his work in a few minutes. Dantzig then listened dumbfounded while von Neumann provided an hour lecture on convex sets, fixed-point theory, and duality, conjecturing the equivalence between matrix games and linear programming.

I warned you to take a deep breath!

Von Neumann's ability to instantaneously perform complex operations in his head stunned other mathematicians. Paul Halmos stated that "von Neumann's speed was awe-inspiring," and Israel Halperin said, "Keeping up with him was ... impossible. The feeling was you were on a tricycle chasing a racing car." Edward Teller wrote that von Neumann effortlessly outdid anybody he ever met, and said "I never could keep up with him". Teller also said "von Neumann would carry on a conversation with my 3 year old son, and the two of them would talk as equals, and I sometimes wondered if he used the same principle when he talked to the rest of us." Jacob Bronowski wrote "He was the cleverest man I ever knew, without exception. He was a genius." George Pólya, whose lectures at ETH Zürich von Neumann attended as a student, said "Johnny was the only student I was ever afraid of. If in the course of a lecture I stated an unsolved problem, the chances were he'd come to me at the end of the lecture with the complete solution scribbled on a slip of paper."

I mentioned earlier that Von Neumann had a very strong eidetic memory, commonly called "photographic" memory. Herman Goldstine writes: "One of his remarkable abilities was his power of absolute recall. As far as I could tell, von Neumann was able on once reading a book or article to quote it back verbatim; moreover, he could do it years later without hesitation. He could also translate it at no diminution in speed from its original language into English. On one occasion I tested his ability by asking him to tell me how A Tale of Two Cities started. Whereupon, without any pause, he immediately began to recite the first chapter and continued until asked to stop after about ten or fifteen minutes."

Please follow the link and read his Wikipedia biography.

Ernest Rutherford (1871-1937)

You should set some time aside to learn a little about the famous Kiwi (New Zealand) physicist Ernest Rutherford (his English parents had emigrated to New Zealand).  This man was responsible for revolutionising our understanding of the atom.  He worked out the mathematics of radioactivity and half-lives, studied the electrical charges in an atom, and was responsible for naming the proton!  Because of these and many other things, he is commonly known as the father of nuclear physics.

He was a significant figure for me as a school student because he came from my part of the world (I am Australian and he was from New Zealand) and was such a famous and highly regarded scientist!

A Portrait of Ernest Rutherford

According to his Wikipedia biography, Encyclopædia Britannica considers him to be the greatest experimentalist since Michael Faraday!

Let me recommend that you read his biography on Wikipedia. In particular, I want you to examine the photographs and note the equipment that he used to make such revolutionary breakthroughs in physics and chemistry ... and earn him the Nobel Prize in Chemistry in 1908.  Amazingly, most scholars agree that he did his best work AFTER he won the Nobel Prize!

When asked to explain the preeminence of British science at the time (early in the 20th century), he rather modestly replied, "We haven't much money, so we've got to use our brains."

That is something to think about!

Joseph John (J J) Thompson (1856-1940)

After graduating from Canterbury College in New Zealand, Ernest Rutherford was awarded a Research Fellowship to travel to England and engage in postgraduate study at the Cavendish Laboratory, Cambridge University. At that time he studied under the very gifted teacher Sir Joseph John (J J) Thompson.

In this post I wish to tell you a little about J J Thompson. Lest you be in any doubt about the magnitude of this man's influence on modern physics, not only did he earn the 1906 Nobel Prize in Physics for the discovery of the electron and for his work on the conduction of electricity in gases ... but SEVEN of his students, as well as his son, George Paget Thomson, also became Nobel Prize winners! Try to imagine the excitement of working in such a rich learning environment with him.

A Portrait of Joseph John Thompson

When J J Thompson was a boy in Manchester, among his neighbours was another boy who left school early and became a very successful manufacturer. Many years later, a banquet was given in honour of this manufacturer. He used the occasion to denounce schooling and to remark upon the futility of pursuing an education. "When I was a boy in school," he said, "the teachers were always praising a student named Johnny Thompson. The rest of us were constantly told to try to be more like little Johnny Thompson. Well, who ever hears of little Johnny Thompson NOW?"

Oh, the irony!

Please set a few minutes aside now to read the Wikipedia biography of this amazing man.

Richard Feynman (1918-1988)

Some of you may have studied physics at school. Some of you may even be studying it now. Many of you, I suspect, were alienated from studying the world in this way by poor school experiences.

Let me introduce all of you to Richard Feynman, joint winner of the Nobel Prize for Physics in 1965. I do not use the word lightly, but he was a GENIUS and, more than that, was actually able to communicate his passion for learning and the concepts of physics to anyone who would listen. Some described him as "The Great Explainer." Let me recommend that you Google his name and learn something about him.

A Portrait of Richard Feynman

During the early 1960s he gave a series of lectures to students at CalTech which were published as the Feynman Lectures on Physics. These publications (three volumes) were highly sought after, even when I was studying physics in the early 1970s.

These books are STILL considered by many to be the best and most popular books on physics EVER. They have been printed in a dozen languages. More than 1.5 million copies have sold in English, and probably even more copies in foreign-language editions! A recent review in Nature (3013) described the book as having "simplicity, beauty, unity … presented with enthusiasm and insight."

Why do I share all this? If you are studying physics, or are willing to learn a little and have your attitudes changed, the good news is that, you no longer have to spend about $200AUS on buying the set. This is because, in 2013, Caltech made the book freely available on the Internet!

Let me recommend that you click on this link and, by way of introduction, read sections 1-1 (Introduction) and 1-2 Matter is Made of Atoms. The combined sections will not take you long to read, but will give you some insight into how this fascinating man communicated his love of physics to students.

I am saving for a copy of The Feynman Lectures on Physics: The Definitive and Extended Edition (2nd edition, 2005). If you read reviews about that edition on Amazon.com you might discover why it might be highly prized.

Over to you, friends. I hope you enjoy your excursion into the world of physics (how our world works "mechanically").

Manfred Clynes (1925- )

Let me introduce you to a wonderful gentleman, Manfred Clynes. I met him in Sydney during the late 1970s and have a signed copy of his book before me as I type. But I get ahead of myself. Let me tell you a little about him first.

He grew up in Vienna, Austria, during the late 1920s and early 1930s. In 1938, his family emigrated to Australia to escape the Nazis (they were Jewish). Within a couple of years, now aged 15 and having just learned calculus, Manfred invented an inertial guidance system for aircraft that used piezoelectric crystals. Authorities dismissed his invention, but an almost identical system was used in aircraft during the latter half of WWII.

Portrait of a Younger Manfred Clynes      An Almost Tolstoy-Like Portrait of an Older Manfred Clynes

He subsequently trained in engineering and music, becoming a very accomplished concert pianist. He also pursued studies in the psychology of music and it was at about this time that he became a friend of Albert Einstein (they were both at Princeton University).

Shortly after this, he performed research in the fields of neurophysiology and neuroscience, inventing the CAT computer for electrical brain research (not to be confused with the CAT scan ... a different thing altogether). In collaboration with Nathan Kline, he explored how technology and the human body could work together. It was Manfred Clynes and Nathan Kline who invented/coined the term CYBORG in 1960! You have him to thank for that. So, if you ever watched Six Million Dollar Man, or have seen the Terminator movies, or something similar, you will be familiar with the concept of cyborg (although these movies do not quite depict what Manfed had in mind).

I hope you can see that you really need to read the Wikipedia article about him!

Now to the mathematics ... because I haven't yet told you what I think is his most amazing discovery!

When I met him in the late 1970s, he had just begun what was to be a very fulfilling ten years of research and music making at the NSW Conservatorium of Music in Sydney, Australia. He had been carrying out some fascinating research that I want to share with you here (in summary form). I apologise to Dr Clynes if I have incorrectly described some of the process. This is my simplification of it.

He, and others, had noticed that, when people conducted a musical work by Beethoven, for example, there was a similarity in their hand movements. Also, the movements seemed to change pattern when pieces were played by a different conductor. There was a different pattern for Mozart, another one for Mendelssohn, and so on. He asked the questions, "Is there something of the composer in the music itself?" and "Can this 'something' be identified?"

Being an engineer, he constructed a small button that recorded vertical and horizontal pressures and asked people to press the button in time with music so that he could record their reactions, just like tapping one's finger on a table. He found that the vertical and horizontal pressure curves were consistent for a given composer whether or not the subject had music training. The equipment meant that he was now able to MEASURE reponses.

Because of his background in psychology, he hypothesised that it was something of the EMOTIONAL makeup of the composers that was conveyed through their music. So, he asked people to think of a time in their life when they experienced a particular raw emotion ... love, joy, anger, hatred, awe, etc. ... all the time while pressing on the sensor button. He found that the pressure curves/patterns for each emotion were quite distinct and were universally shared across gender, nationality and cultures. This suggested that we ALL experience such emotions in the same way.

Excited by this, he then wondered if he had stumbled upon some way of identifying or measuring emotion in humans. He decided to turn the process around and apply it in reverse! He asked volunteers to sit and have their finger attached to the sensor button. This time, he only asked them to report what they were feeling when he made the button vibrate/oscillate according to the pattern for specific emotions. He found that vibrating part of a person's body in this way CAUSED THEN TO FEEL THAT EMOTION. He would play the pressure patterns for anger/rage, for example, and subjects would report feeling unaccountably enraged (without being aware of what signals were being given to them)!

Don't you find this to be exciting research?

He called these patterns "Sentics." His book is called Sentics: The Touch of Emotions ~ A revolution in understanding how we experience and communicate emotion. The enthusiastic forword was written by Yehudi Menuhin (I hope you know of him).

Manfred combined his love of engineering and mathematics with his passion for music, psychology and the human body to discover these wonderful things. He has since applied his understanding of these sentic patterns to synthesised music, so that computer-generated music 'feels' warm and emotional, as though a human played it, instead of clinical. Some examples are on YouTube.

I hope you enjoyed learning about how mathematics has been used, in this way, to investigate the wonderful world of music and emotions.

Don't forget to read about this fascinating man, who turned 90 this year (2015), and is still 'going strong!'

Blaise Pascal (1623-1662)

I was recently reading about Blaise Pascal, a famous French mathematician. Blaise, by the way, is pronounced 'blaze.' You may have encountered "Pascal's Triangle" at some point:

Pascal's Triangle

This deceptively simple triangle is used to solve a bewildering array of problems in mathematics. Can you work out what the next two lines would look like? The pattern is really quite a simple one (once you 'see' it).

We can be greatly inspired by what we know of Blaise as a young man, living in Paris ...

Portrait of Blaise Pascal

His father was a friend of Marin Mersenne who has been described as "the center of the world of science and mathematics during the first half of the 1600s." As an aside, I recommend that you read about Mersenne on Wikipedia. I remember learning of this great man while studying number theory at school.

Blaise had started teaching himself geometry at the age of twelve. At fourteen years of age, he began attending meetings of the mathematical academy of Paris with his father. Mersenne, Pierre de Fermat, Pierre Gassendi, and other famous mathematicians also participated. Even René Descartes was linked to this learned group that, apparently, later became the Académie des Sciences in Paris.

At sixteen years of age, he produced a paper at one of these meetings (Mersenne was one of the leaders) which contained a number of theorems in projective geometry. One of the items was his proof of what is still known as "Pascal's Mystic Hexagon." You may like to Google the term to discover what this precocious young mathematician was producing as a teenager (after learning geometry for only four years). His proof was so startling that Descartes believed that Blaise could not have produced it ... and attributed it to his father ... until Mersenne convinced him that Blaise was the genuine author!

In 1654, when Blaise was about 31 years old, Chevalier de Méré challenged him with a gambling puzzle: If two equally skilled players are interrupted while playing a game of chance for a certain amount of money, how should one divide the stakes based upon the score at that time? Blaise discussed the problem in his regular correspondence with Fermat and the two of them realised that the stakes should be shared, not based upon the history to that point, but based upon the chances that each player would win had the game not been interrupted! These two mathematicians jointly laid the foundations for the modern discipline of probability.

Sadly, Blaise lived with poor health and died just after his 39th birthday.

Please take the time to read about him on Wikipedia. He is famous for so many other things during his short life. For instance:

  • He invented one of the world's first mechanical calculators (fore-runner of our computers ... they are called Pascalines). He actually started work on these machines when he was just eighteen or nineteen years old!
  • He was also a philosophical and deeply religious thinker. Some of you may have heard of his book, Pensées (Thoughts).
  • We even measure pressure using his name. Next time you check the pressure in your car's tyres, you may find that the gauge measures the pressure in either pounds per square inch, or in kiloPascals (kPa).

He was an amazing and intriguing mathematician.

I think you will find that it is worth learning a little about him on this biography site as well.

Edward Kasner (1878-1955), Milton Sirotta (1929-c.1980), and the Googol

Mathematicians and scientists often find themselves dealing with very large numbers.  They arise from asking questions like, "How many raindrops would fall from a storm cloud during a heavy rainstorm?" or "How many living cells are there in all the people on Earth?" Such questions leads me to think, sometimes, of the number 'googol.'

There is such a number, and it is massive! It is one followed by 100 zeros and that, my friends, is greater than the total number of subatomic particles and photons of electromagnetic radiation in the entire observable universe! And, believe it or not, the name was given to this number by a nine-year-old boy (Milton Sirotta) in 1938!

Milton Sirotta's uncle was an accomplished mathematician, Edward Kasner. Dr Kasner coauthored a book with James Newman in 1940 called Mathematics and the Imagination. It was destined to become a 'classic' and the copy that I have dates from 1948, by which time it was in its eleventh printing!  The image below is of Edward Kasner in 1907.  Sadly, I cannot find any image of Milton Sirotta at all.

Portrait of Edward Kasner

It was in Mathematics and the Imagination that Edward Kasner shares the story of the naming of the googol ... and it is that story which I share with you here. I hope you like it.  By the way, let me recommend that you obtain a copy of this wonderful book (from Amazon or the Book Depository) and read it. There are many great classics in mathematics, and this would be one of them. 'Old' does not mean 'useless' or 'irrelevant.' There is much to learn and enjoy here!

This account of the 'invention' if the googol is taken from page 23 of Mathematics and the Imagination by Edward Kasner and James Newman (Simon and Schuster, New York, 1948).

You may now read it for yourself (or share it with others).


Text from Kasner's Book about the Invention of the Googol by Milton Sirotta in 1938

If you are a teacher, you may wish to keep a copy to share with students occasionally ... that young students (like Milton Sirotta) can "do mathematics" and become famous, and that fun can be had with very large numbers!

Eratosthenes of Alexandria (276 BC - 194 BC)

As a school student, I was fascinated to learn of the Library of Alexandria in Egypt. I remember reading about it when I was in Primary School (when I was about 11 years old).

Alexander the Great took about eleven years to create his huge Macedonian Empire. Near the beginning of this period (in 332 BC) his army conquered Egypt and he founded a city near the mouth of the Nile. It was called Alexandria ... named after him ... and it is still one of Egypt's major cities today.

After his death in Babylon in 323 BC, the Macedonian Empire divided into four power blocks. One of his generals, Ptolemy I Soter, ruled Egypt and his "Ptolemaic Dynasty" included Cleopatra (who was therefore Greek/Macedonian by family origin rather than Egyptian).

This first Ptolemy established a great library in Alexandria and it became a major centre of learning. The Wikipedia article explains that the library was part of the Museum of Alexandria, which functioned as a sort of research institute. The Museum contained not just the Library, but included rooms for the study of astronomy, anatomy, and even a zoo of exotic animals. The classical thinkers who studied, wrote, and experimented at the Museum include the great names of mathematics, astronomy, physics, geometry, engineering, geography, physiology, and medicine. These included notable thinkers such as Euclid, Archimedes, Eratosthenes, Herophilus, Erasistratus, Hipparchus, Aedesia, Pappus, Theon, Hypatia, and Aristarchus of Samos. Galen recorded that all ships visiting Alexandria were searched and owners were obliged to surrender their books for immediate copying. The owners received a copy of their book (scroll) while the originals were stored in the Library. It is said that, at its height, the Library may have contained as many as half a million scrolls! I remember crying when I read that, in later years, the Library was burned.

Portrait of Eratosthenes

Eratosthenes of Cyrene (276 BC – c.195/194 BC) was one of the chief librarians there. He had access to records and maps that allowed him to calculate the circumference of the Earth in an ingenious way. He knew that on a particular day of the year, the sun shone directly down a well in Elephantine Island in the Nile River at Syene (near the current Aswan Dam). This means that it lies on the Tropic of Cancer. On the same day, he erected a measuring stick at Alexandria and, by measuring its shadow, concluded that Alexandria and Elephantine Island were separated by about 1/50th of a circle (just over 7° of latitude). Since the distances between various places in the Empire were fairly well established (marching soldiers counted out the number of paces, for example), he was able to find that the distance between the two locations was about 5,000 stadia. He concluded that the circumference of the Earth is about 50 x 5,000 stadia = 250,000 stadia.

A Diagram Showing Eratosthenes' Method for Determining the Size of the Earth

There is some dispute over which stade he used (there were more than one in existence). Even if he used the Attic stade (185 metres long), his calculation would yield a circumference of 46,620 km, about 16% greater than the actual figure of about 40,008 km for the polar circumference. His figure would have been almost exact (an error of less than 2%) if he had been using the Egyptian stade.  Here is Carl Sagan describing how he made this calculation.

I remember marvelling at the simple geometry that was used and the ingenious way in which he made this calculation. Perhaps, if you are a teacher, you might consider repeating this with your class using local measurements.

For example, there is a city called Rockhampton in Queensland, Australia which is almost due north of where I live in the Shoalhaven. Rockhampton lies on the Tropic of Capricorn which means that, on December 21/22 the sun is directly overhead at midday. If I set up a vertical stick here in the Shoalhaven on those days and measured its shadow, I could use simple trigonometry (or a protractor) to determine that the angle between the top portion of the stick and the sun's rays would be about 12.5°. The road distance between Rockhampton and the Shoalhaven is about 1560 km (so the direct distance would be a little shorter). Therefore, the circumference of the Earth would be a little less than 1560*(360/12.5) ≈ 45,990 km.

What an ingenious man Eratosthenes was!  Why not read the Wikipedia article about him?

Sofia Vasilyevna Kovalevskaya an Astoundingly Talented Russian Mathematician (1850-1891)

Image of Sofia Vasilyevna KovalevskayaLet me tell you a little about Sofia Vasilyevna Kovalevskaya.

If I tell you that she was born in Moscow on 15 January 1850 and died in Stockholm (Sweden) on 10 February 1891, you will gain some idea of the era in which she lived.

The story goes that, when she was about 11 years old, the walls of her room had been papered with lecture notes on differential and integral calculus.  Papering walls with any available paper was not uncommon in those days.

These notes piqued her interest in mathematics and she began studying more advanced mathematics with her tutor.  Her father tried to prevent her from studying mathematics but she took to studying algebra secretly in bed at night.  If only we encountered students with that passion today!

All her life she battled discrimination against her being allowed to study mathematics because she was a woman.  Despite this, she studied in Heidelberg, Germany (under such teachers as Hermann von Helmholtz, Gustav Kirchhoff and Robert Bunsen), attended George Eliot's Sunday salons in England (George Eliot appears to refer to her obliquely in Middlemarch) and took private lessons with Karl Weierstrass in Berlin (the university would not even allow her to sit in on lectures).

In 1874, as a doctoral dissertation, she presented three papers at the University of Göttingen (on partial differential equations, the dynamics of Saturn's rings, and on elliptic integrals).  Her paper on partial differential equations contained what is now known as the Cauchy–Kovalevskaya theorem.  This theorem establishes the conditions under which a certain class of differential equations has solutions.  With Weierstrass' support, she earned a doctorate in mathematics summa cum laude, all without the usually required lectures and examinations.  This made her the first woman in Europe to hold a doctorate in mathematics.  Sadly, despite all her achievements and Weierstrass' fervent support, her gender prevented her from obtaining an academic position.

About ten years later, she was able to obtain a position at the University of Stockholm, becoming the third female professor at a European university.  A couple of years later, in 1886, she won the Prix Bordin of the French Academy of Sciences for her work on the motion of rigid bodies.  Astoundingly, her work was so good that the prize money was raised from 3,000 to 5,000 Francs!  More prizes and honours followed.

Unfortunately, Sofia died of pneumonia in 1891 at the height of her career and at only 41 years of age.

Your method is perfect, I couldn’t imagine a better way to find derivatives with the chain rule. Thank you! Regards from France
CopainVG (on CCM YouTube video about the Chain Rule)

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