Pat'sBlog

Thursday 17 October 2024

A Graphic Approach to Pythagorean Triangles using the Unit Circle

I wrote a couple of posts a while back on the Barning Tree method of finding Pythagorean triples using matrices, and then a followup(now included in previous post). I recently came across another approach to Pythagorean triples that involves a clever relationship between points on the positive y-axis, and points on the unit circle. (ok, maybe I should have known this, but I didn't.. and I think it is really a neat idea)

On the unit circle, x² + y ² = 1, if we draw a secant from the point (-1,0) through (0,t) on the y-axis, it turns out that if t is a rational number, then the coordinates of P=(x,y) where the secant intersects the circle, will also be rational. Since the slope of the line is also t, the equation is y=tx+t ... and so and t²= (1-x²)/(1+x)².

That means t = y/(x+1) which leads to x= (1-t^2)/(1+t^2) and y= ^(2t)/_(1+t^2) If we pick some rational number to be t, say t=2/7,

then x= ⁴⁵/₅₃ and y= ²⁸/₅₃.... Then by similar triangles, there must be a circle with radius 53 and a point on the circle would be (28, 45) and in fact 28²+45²=53²... and any such rational point will produce another.

On This Day in Math - October 17

To parents who despair because their children are unable to master the first problems in arithmetic I can dedicate my examples. For, in arithmetic, until the seventh grade I was last or nearly last.
~Jacques Hadamard

The 290th day of the year, 290 is a sphenic (wedge) number, the product of three distinct primes (290 = 2*5*29).

It is also the sum of four consecutive primes (67 + 71 + 73 + 79) [Students might try to construct and examine a list of numbers that can be written as the sum of two or more consecutive primes]

290 is conjectured to be the smallest number such that the Reverse and Add! algorithm in base 4 does not lead to a palindrome.

290 is the tenth prime(29) times ten

290^3 - 290^2 = 4930^2. 290 is only the 17th number for which n^3 - n^2 = y^2. There will only be two more year days that meet that relation and each is one more than a square. And, the sum of the squares of the divisors of 17 is 290.

EVENTS

1604 In Prague, Kepler first observes the supernova now known as supernova 1604 and Kepler's Star. The first recorded observation of this supernova was in northern Italy on October 9, 1604. It was named after Kepler because his observations tracked the object for an entire year and because of his book on the subject, entitled De Stella nova in pede Serpentarii ("On the new star in Ophiuchus's foot", Prague 1606). Here is an image of Kepler's De Stella Nova, open to the foldout star map placing the supernova of 1604, from the twitter feed of @Libroantiguo . It was the second supernova to be observed in a generation (after SN 1572 seen by Tycho Brahe in Cassiopeia). No further supernovae have since been observed with certainty in the Milky Way, though many others outside our galaxy have been seen since S Andromedae. *Wik

1776 Euler read a paper to the St. Petersburg Academy of Science entitled “De quadratis magicis,” in which he gave a method of constructing magic squares by means of two orthogonal Latin squares. *Peter Ullrich, “An Eulerian square before Euler and an experimental design before R. A. Fisher: On the early history of Latin squares,” Chance, vol. 12, no. 1, Winter 1999, pp. 22–26.

1831 After discovering induced current on October 1st using two electrified coils, on the 17th of October Michael Faraday observers the same effect on the galvanometer when he inserts a permanent steel magnet into the electrified coil. *A history of physics in its elementary branches By Florian Cajori

1843 Hamilton Writes to his friend, John Graves, with a description of Quaternions. By December, Graves will have extended the idea to an eight dimensional algebra which will become "octonians".

Observatory, October 17, 1843
My dear Graves,|A very curious train of mathematical speculation occurred to me
yesterday, which I cannot but hope will prove of interest to you. You know that I have long
wished, and I believe that you have felt the same desire, to possess a Theory of Triplets,
analogous to my published Theory of Couplets, and also to Mr. Warren's geometrical representation
of imaginary quantities. Now I think that I discovered yesterday a theory of
quaternions which includes such a theory of triplets.

The complete letter is available at this site. *David R. Wilkins, *John Derbyshire, Unkown Quantity
In his preface to the ‘Lectures on Quaternions’ and in a prefatory letter to a communication to the Philosophical Magazine for December 1844 are acknowledgments of his indebtedness to Graves for stimulus and suggestion. *Wik

1858 DeMorgan writes a letter about Euler’s prodigious output. *W W Rouse Ball, from The genius of Euler: reflections on his life and work, By William Dunham, pg 89

1933 Albert Einstein seeks asylum in the US, one of many Jewish/left-wing intellectuals fleeing the Nazi govt in Germany and Europe. The Nazi government put a bounty now worth £50,000 on his head while a German magazine included him in a list of the Nazis’ enemies who were 'not yet hanged'.

Professor Albert Einstein, who has taken up residence in England as a refugee from Nazi threats, was among the prominent speakers who addressed a great gathering at the Royal Albert Hall in London, recently, to aid the Jewish Refugee Fund. Commander Locker Lampson, M.P., Lord Rutherford and Sir Austen Chamberlain were among the principal speakers of the meeting. Photo shows Professor Einstein during the delivery of his speech. October 1933.

1952 D. H. Lehmer, University of California, announced that 2ⁿ − 1 for n = 2203 and 2281 are Mersenne primes. He was aided by a SWAC computing machine, the ﬁrst result taking 59 minutes. *VFR This may have been predated by Raphael Mitchel Robinson (November 2, 1911 – January 27, 1995) at Berkeley may have beaten him by a week or so on October 7th of the same year.
D. H. Lehmer continued his fathers interest in combinatorial computing and in fact wrote the article "Machine tools of Computation," which is chapter one in the book "Applied Combinatorial Mathematics," by Edwin Beckenbach, 1964. It describes methods for producing permutations, combinations etc. This was a uniquely valuable resource and has only been rivaled recently by Volume 4 of Donald Knuth's series. In 1950, Lehmer was one of 31 University of California faculty fired after refusing to sign a loyalty oath, a policy initiated by the Board of Regents of the State of California in 1950 during the Communist scare personified by Senator Joseph McCarthy. (see below)*Wik

1952 The California Supreme Court declared the state loyalty oath unconstitutional and declared that the eighteen faculty members who had refused to sign the oath be reinstated.*VFR

1978 James Burke's history of science series Connections first airs on BBC Television in the United Kingdom (with accompanying book). *Wik

1983 Gerard Debreu, who holds a joint appointment in Mathematics and Economics at Berkeley, won a Nobel Prize for his work in mathematical economics. For a non-technical description of his work see The Mathematical Intelligencer, 6(1984), no. 2, pp. 61–62. *VFR

1994 IBM Corp. announced it would be cutting back its line of personal computers from nine models to four. It also notified the public of several new models and said it would bring back the brand name many connected to the company, the IBM PC. The plan for consolidating IBM’s personal computer production was to have four divisions: IBM PC for commercial desktop machines, IBM PC Server for larger computers used on networks, Thinkpad for portables, and Aptiva for the home market.

2012 Car size pieces of Halley's Comet lit up the skies over the Bay Area in California. Hundreds of residents from Oakland, San Francisco and Santa Cruz called ABC News station KGO-TV, reporting a loud boom, explosions and streaks of light around 7:45 p.m. local time. The Orionids are one of two annual meteor showers produced by icy pieces of Halley's Comet. The other shower, called the Eta Aquarids, peaks each year in early May, according to NASA. Video *ABC News

BIRTHS
1759 Jakob II Bernoulli (17 October 1759, Basel – 3 July 1789, Saint Petersburg), younger brother of Johann III Bernoulli. Having finished his literary studies, he was, according to custom, sent to Neuchâtel to learn French. On his return he graduated in law. This study, however, did not check his hereditary taste for geometry. The early lessons which he had received from his father were continued by his uncle Daniel, and such was his progress that at the age of twenty-one he was called to undertake the duties of the chair of experimental physics, which his uncle’s advanced years rendered him unable to discharge. He afterwards accepted the situation of secretary to count de Brenner, which afforded him an opportunity of seeing Germany and Italy. In Italy he formed a friendship with Lorgna, professor of mathematics at Verona, and one of the founders of the Società Italiana for the encouragement of the sciences. He was also made corresponding member of the royal society of Turin; and, while residing at Venice, he was, through the friendly representation of Nicolaus von Fuss, admitted into the academy of St Petersburg. In 1788 he was named one of its mathematical professors. *Wik
He drowned while bathing in the Neva in July 1789, a few months after his marriage with a granddaughter of Leonhard Euler. (Can't tell your Bernoulli's without a scorecard? Check out "A Confusion of Bernoulli's" by the Renaissance Mathematicus.)

1788 Paul Isaak Bernays (17 Oct 1888; 18 Sep 1977) Swiss mathematician and logician who is known for his attempts to develop a unified theory of mathematics. Bernays, influenced by Hilbert's thinking, believed that the whole structure of mathematics could be unified as a single coherent entity. In order to start this process it was necessary to devise a set of axioms on which such a complete theory could be based. He therefore attempted to put set theory on an axiomatic basis to avoid the paradoxes. Between 1937 and 1954 Bernays wrote a whole series of articles in the Journal of Symbolic Logic which attempted to achieve this goal. In 1958 Bernays published Axiomatic Set Theory in which he combined together his work on the axiomatisation of set theory. *TIS

1927 Friedrich Ernst Peter Hirzebruch (17 October 1927 – 27 May 2012) was a German mathematician, working in the fields of topology, complex manifolds and algebraic geometry, and a leading figure in his generation. He has been described as "the most important mathematician in the Germany of the postwar period.
Amongst many other honours, Hirzebruch was awarded a Wolf Prize in Mathematics in 1988 and a Lobachevsky Medal in 1989. The government of Japan awarded him the Order of the Sacred Treasure in 1996. He also won an Einstein Medal in 1999, and received the Cantor medal in 2004.*Wik

DEATHS
1817 John West (10 April 1756 in Logie (near St Andrews), Scotland - 17 Oct 1817 in Morant Bay, Jamaica) The achievements of the little-known Scottish mathematician, John West (1756–1817), deserve recognition: his Elements of Mathematics(1784) shows him to be a skilled expositor and innovative geometer while his manuscript,Mathematical Treatises,unpublished until 1838, reveal him also to be an accomplished exponent of “continental” analysis, familiar with works of Lagrange, Laplace, and Arbogast then little studied in Britain.
First an assistant at St. Andrews University in Scotland, West then worked in isolation in Jamaica, combining mathematics with the duties of an Anglican rector. His life and his pastoral and mathematical works are here described. *abstract for Geometry, Analysis, and the Baptism of Slaves: John West in Scotland and Jamaica, Alex D.D. Craik

1877 Gustav Robert Kirchhoff (12 Mar 1824, 17 Oct 1887) German physicist who, with Robert Bunsen, established the theory of spectrum analysis (a technique for chemical analysis by analyzing the light emitted by a heated material), which Kirchhoff applied to determine the composition of the Sun. He found that when light passes through a gas, the gas absorbs those wavelengths that it would emit if heated, which explained the numerous dark lines (Fraunhofer lines) in the Sun's spectrum. In his Kirchhoff's laws (1845) he generalized the equations describing current flow to the case of electrical conductors in three dimensions, extending Ohm's law to calculation of the currents, voltages, and resistances of electrical networks. He demonstrated that current flows in a zero-resistance conductor at the speed of light. *TIS

Postage stamp honoring Gustav Kirchhoff, issued by the German Democratic Republic (East Germany), 1974 (mathshistory.st-andrews.ac.uk)

1923 August Adler (24 Jan 1863 in Opava, Austrian Silesia (now Czech Republic)-17 Oct 1923 in Vienna, Austria) In 1906 Adler applied the theory of inversion to solve Mascheroni construction problems in his book Theorie der geometrischen Konstruktionen published in Leipzig. In 1797 Mascheroni had shown that all plane construction problems which could be made with ruler and compass could in fact be made with compasses alone. His theoretical solution involved giving specific constructions, such as bisecting a circular arc, using only a compass.
Since he was using inversion Adler now had a symmetry between lines and circles which in some sense showed why the constructions needed only compasses. However Adler did not simplify Mascheroni proof. On the contrary, his new methods were not as elegant, either in simplicity or length, as the original proof by Mascheroni.
This 1906 publication was not the first by Adler studying this problem. He had published a paper on the theory of Mascheroni's constructions in 1890, another on the theory of geometrical constructions in 1895, and one on the theory of drawing instruments in 1902. As well as his interest in descriptive geometry, Adler was also interested in mathematical education, particularly in teaching mathematics in secondary schools. His publications on this topic began around 1901 and by the end of his career he was publishing more on mathematical education than on geometry. Most of his papers on mathematical education were directed towards teaching geometry in schools, but in 1907 he wrote on modern methods in mathematical instruction in Austrian middle schools. He produced various teaching materials for teaching geometry in the sixth-form in Austrian schools such as an exercise book which he published in 1908. *SAU

1937 Frank Morley (9 Sept 1860 in Woodbridge, Suffolk, England-17 Oct 1937 in Baltimore, Maryland, USA) wrote mainly on geometry but also on algebra.*SAU Morley is remembered most today for a singular theorem which bears his name in recreational literature. Simply stated, Morley's Theorem says that if the angles at the vertices of any triangle (A, B, and C in the figure) are trisected, then the points where the trisectors from adjacent vertices intersect (D, E, and F) will form an equilateral triangle. In 1899 he observed the relationship described above, but could find no proof. It spread from discussions with his friends to become an item of mathematical gossip. Finally in 1909 a trigonometric solution was discovered by M. Satyanarayana. Later an elementary proof was developed. Today the preferred proof is to begin with the result and work backward. Start with an equilateral triangle and show that the vertices are the intersection of the trisectors of a triangle with any given set of angles. For those interested in seeing the proof, check Coxeter's Introduction to Geometry, Vol 2, pages 24-25. Find more about this unusual man here. *PB

1941 John Stanley Plaskett (17 Nov 1865, 17 Oct 1941) Canadian astronomer known for his expert design of instruments and his extensive spectroscopic observations. He designed an exceptionally efficient spectrograph for the 15-inch refractor and measured radial velocities and found orbits of spectroscopic binary stars. He designed and supervised construction of the 72-inch reflector built for the new Dominion Astrophysical Observatory in Victoria and was appointed its first director in 1917. There he extended the work on radial velocities and spectroscopic binaries and studied spectra of O and B-type stars. In the 1930s he published the first detailed analysis of the rotation of the Milky Way, demonstrating that the sun is two-thirds out from the center of our galaxy about which it revolves once in 220 million years.*TIS

1952 Ernest Vessiot (8 March 1865 in Marseilles, France-17 Oct 1952 in La Bauche, Savoie, France) applied continuous groups to the study of differential equations. He extended results of Drach (1902) and Cartan (1907) and also extended Fredholm integrals to partial differential equations. Vessiot was assigned to ballistics during World War I and made important discoveries in this area. He was honoured by election to the Académie des Sciences in 1943. *SAU

1963 Jacques-Salomon Hadamard (8 Dec 1865, 17 Oct 1963) French mathematician who proved the prime-number theorem (as n approaches infinity, the limit of the ratio of (n) and n/ln(n) is 1, where (n) is the number of positive prime numbers not greater than n). Conjectured in the 18th century, this theorem was not proved until 1896, when Hadamard and also Charles de la Vallée Poussin, used complex analysis. Hadamard's work includes the theory of integral functions and singularities of functions represented by Taylor series. His work on the partial differential equations of mathematical physics is important. He introduced the concept of a well-posed initial value and boundary value problem. In considering boundary value problems he introduced a generalization of Green's functions (1932). *TIS

Quote : To parents who despair because their children are unable to master the first problems in arithmetic I can dedicate my examples. For, in arithmetic, until the seventh grade I was last or nearly last.

~Jacques Hadamard

1978 Gertrude Mary Cox (January 13, 1900 – October 17, 1978) was an influential American statistician and founder of the department of Experimental Statistics at North Carolina State University. She was later appointed director of both the Institute of Statistics of the Consolidated University of North Carolina and the Statistics Research Division of North Carolina State University. Her most important and influential research dealt with experimental design; she wrote an important book on the subject with W. G. Cochran. In 1949 Cox became the first female elected into the International Statistical Institute and in 1956 she was president of the American Statistical Association.*Wik

2008 Andrew Mattei Gleason (November 4, 1921 – October 17, 2008) was an American mathematician and the eponym of Gleason's theorem and the Greenwood–Gleason graph. After briefly attending Berkeley High School (Berkeley, California) he graduated from Roosevelt High School in Yonkers, then Yale University in 1942, where he became a Putnam Fellow. He subsequently joined the United States Navy, where he was part of a team responsible for breaking Japanese codes during World War II. He was appointed a Junior Fellow at Harvard in 1946, and later joined the faculty there where he was the Hollis Professor of Mathematicks and Natural Philosophy. He had the rare distinction among Harvard professors of having never obtained a doctorate. (In graph theory, the Greenwood–Gleason graph (Image at top of page) is also known as the Clebsch graph. It is an undirected graph with 16 vertices and 40 edges. It is named after Alfred Clebsch, a German mathematician who discovered it in 1868. It is also known as the Greenwood–Gleason graph after the work of Robert M. Greenwood and Andrew M. Gleason (1955), who used it to evaluate the Ramsey number R(3,3,3) = 17 *Wik

Credits :
*CHM=Computer History Museum
*FFF=Kane, Famous First Facts
*NSEC= NASA Solar Eclipse Calendar
*RMAT= The Renaissance Mathematicus, Thony Christie
*SAU=St Andrews Univ. Math History
*TIA = Today in Astronomy
*TIS= Today in Science History
*VFR = V Frederick Rickey, USMA
*Wik = Wikipedia
*WM = Women of Mathematics, Grinstein & Campbell

Wednesday 16 October 2024

A Curious Property of Vulgar Fractions

Another from 2008, hope you enjoy:

John Farey was a geologist, not a mathematician, but he is better remembered today for a single short (four paragraphs) paper he wrote in 1816 than for all his good works in geology. In that year he sent a paper to the Philosophical Magazine (or at least it was published in that year) called On a curious property of vulgar fractions and described a pattern that appears in sequences of what we would today call common (vulgar is the Latin term for common) fractions, like 3/8 etc, which are in simplest terms. The observation has almost NO practical use, not even to prove other things mathematically, and yet, it seems to have all kinds of interesting properties that tend to keep us fascinated with it. If you have never been introduced to it, here is a brief description, and some novel relationships that I think are interesting, with some links to places where they are made clearer than I could do in this brief space.

So First... What is it we are talking about? If you take ALL the fractions that could be written in simplest terms with a denominator less than some number n, say n=5 (since that is the one Farey used in his paper), and put them in order from lowest to highest... you get

The "curious" thing that Farey noticed is that if you ignore the way your fifth grade teacher taught you to add fractions, and do it the way YOU would have added them, "add the tops, add the bottoms", then each number in the sequence is the sum of the terms on each side of it... for example ¹/₄ and ²/₅ are on each side of ¹/₃, and if you add by this approach you get ⁽¹⁺²⁾/₍₄₊₅₎ = ³/₉ and that simplifies to ¹/₃ . The number obtained by adding two fractions in this fashion is often called the mediant .

OK, so that is how you make them. Our first question might be, how many of them are there? F(5) obviously has eleven terms (I counted). If we picked a value of N, what would be the number of fractions in the set F(N). A little investigation would show that F(1) = 2 (0 and 1); and F(2) = 3 (0, 1/2, and 1). So how many would be added to the next set... and the next... it turns out that each new set will have all the values of the previous set (of course) and will add one for every value of one through n that is co-prime (has no common divisors) to n. So the set for N=6 will have the eleven terms of F(5) plus 1/6 (one has no common factor with six), and 5/6. (notice that 2/6, 3/6, and 4/6 are already in the sequence in F(5) in simplified form)... thus F(6) has thirteen terms.. and in general we get a recursive formula that say the Order of F(N) = Order of F(n-1) + φ(n). Euler's $\phi$ function,(sometimes called the totient, like quotient) he number of integers k in the range 1 ≤ k ≤ n for which the greatest common divisor gcd(n, k) is equal to 1.( and no, Euler never used either term.)

A second nice curiosity related to Farey Sequences are the Ford Circles. "Ford circles are named after American mathsedematician s the R. Ford, Sr., who described them in an article in American Mathematical Monthly in 1938, volume 45, number 9, pages 586-601" (from Wikipedia). In fact, the wikipedia article is a nice place to see how they work, and I need say no more as it shows the circles for F(5). What is amazing is that each circle is tangent to every other circle for a fraction it will be adjacent to in ANY sequence.... ahhh, go on..say "cool".

I decided to mention this when I came across another curious property of Farey sequences that relates them to lines on the plane and Pick's Theorem. If you treated each fraction a/b as a point (b,a) then none of the lines cross. If you make a triangle with the origin and any two adjacent Farey fractions, since each of the triangles have a determinant of one (meaning the area is 1/2) and therefore, by PIck's theorem, they cannot contain any other lattice points in their interior. A nice explanation of this, including the photo below, was at the Cut-The-Knot web site

On This Day in Math - October 16

I have often pondered over the roles of knowledge or experience, on the one hand, and imagination or intuition, on the other, in the process of discovery. I believe that there is a certain fundamental conflict between the two, and knowledge, by advocating caution, tends to inhibit the flight of imagination. Therefore, a certain naivete, unburdened by conventional wisdom, can sometimes be a positive asset.
~Harish-Chandra

The 289th day of the year; 289 is a Friedman number since (8 + 9)² = 289 (A Friedman number is an integer which, in a given base, is the result of an expression using all its own digits in combination with any of the four basic arithmetic operators (+, −, ×, ÷) and sometimes exponentiation.)Students might try to find the first few multi-digit Friedman numbers.

289 = 17^2, and it is too little known by students that for every prime p > 3 , 24 divides p^2-1. Since every prime is =1, 3, 5 or 7 Mod 8, their squares mod 8 are 1, 9 25, and 49 and all of those simplify to 1 mod 8. Also all primes are == 1 or 2 mod 3, and their squares are 1 or 4 (=1 mod 3) . so p^2 mod 24 = p^2 mod 3 times p^2 mod 8 is always one. Subtract 1 and P62 -1 is divisible by 8.

289 is the square of the sum of the first four primes, 289 = (2 + 3 + 5 + 7)²

There are only two square year days that are neither the sum, or the difference of two primes, 289 is the second.

289 is the largest 3-digit square with increasing digits.

289 is the hypotenuse of a primitive Pythagorean triple. Find the legs students!

EVENTS
1707 Roger Cotes elected ﬁrst Plumian Professor of Astronomy and Experimental Philosophy at Cambridge at age 26. He is best known for his meticulous and creative editing of the second edition (1713) of Newton’s Principia. He was also an important developer of the integral calculus. *Ronald Gowing, Roger Cotes, Natural Philosopher, p. 14

He also found Euler's formula,

cosx + isinx = eix

before Euler, and expressed it in its logarithmic form.

1784 Jean-Pierre Blanchard, a French balloonist, was born July 4, 1753. The Golden Age of Ballooning began on Nov. 21, 1783, when Pilâtre de Rozier and François d'Arlandes soared aloft in a hot-air balloon made by the Montgolfier brothers. They launched from the Château de la Muette just outside Paris and floated for some 5 miles. Just over a week later, Jacques-Alexandre Charles and Nicolas Robert ascended to 3000 feet from the Tuileries in Paris, this time in a hydrogen balloon. Blanchard was caught up immediately in balloon frenzy, designed his own hydrogen balloon, complete with "oars" to swim through the air and an always-open parachute to slow descent should the gas bag spring a leak, and headed for the skies . He made his first ascent in a hydrogen balloon on Mar. 2, 1784, lifting off from the Champ de Mars. If there is a surviving contemporary image of that ascent, I have not seen it.

The difference between Blanchard and the Montgolfier brothers and Jacques Charles is that Blanchard was in it for the money. He was the first barnstorming balloonist who charged admission for his ascents and seems to have given the public (who showed up by the thousands) their money's worth, especially on the first ascent, when a military student demanded to come along and attacked Blanchard and the balloon with a sword when he was refused. The somewhat bloodied Blanchard proceeded with the flight anyway, which I am sure delighted the crowd.

Seeking larger paydays, Blanchard travelled with his balloon to England in August of 1784 and began to organize public ascents there. He made one ascent from Chelsea, for which (so it is recorded) 400,000 people showed up. He made the ascent with an English physician, who was added to the gondola to increase local interest. An engraving recorded the event, which took place on Oct. 16, 1784. Blanchard then ascended with another physician, John Jeffries (an ex-American, actually), on Nov. 30, 1784, and this time they wafted all the way from London to Kent.

This set the stage for Blanchard's goal all along, to balloon across the English Channel. Pilâtre de Rozier had the same idea; he was sitting on the other side of the channel with his hydrogen balloon, waiting for favorable winds to take him westward to Dover. Blanchard won the battle of the winds. He and Jeffries took off from Dover on Jan. 7, 1785. They almost ended up in the sea, as their bag of hydrogen was providing insufficient lift, and they threw nearly everything overboard, including most of their clothes, to maintain altitude. But the balloon for some reason recovered its buoyancy, and they made it to Calais and beyond, landing at Guines, to the great excitement of the local populace.

1797 Gauss records in his diary that he has discovered a new proof of the Pythagorean Theorem. See Gray, Expositiones Mathematicae, 2(1984), 97–130. *VFR I have had this here for several years, and no one seems to know the nature of his proof. Anyone???

1819 Thomas Young writes to Fresnel to thank him for a copy of his memoirs (sent to Young by Arago). "I return a thousand thanks, Monsieur, for the gift of your admirable memoir, which surely merits a very high rank amongst the papers which have contributed most to the progress of optics." *A history of physics in its elementary branches By Florian Cajori

Thomas Young

1843 Hamilton discovered quaternions while walking along the Royal Canal in Dublin and immediately scratches the multiplication formulas on a bridge. Today a plaque on the bridge reads, "Here as he walked by on the 16th of October 1843 Sir William Rowan Hamilton in a ﬂash of genius discovered the fundamental formula for quaternion multiplication i² = j² = k² = ijk = −1 & cut it in a stone on this bridge." Since 1989, the Department of Mathematics of the National University of Ireland, Maynooth has organized a pilgrimage, where scientists (including the physicists Murray Gell-Mann in 2002, Steven Weinberg in 2005, and the mathematician Andrew Wiles in 2003) take a walk from Dunsink Observatory to the Royal Canal bridge where no trace of Hamilton's carving remains, unfortunately.
Here is how Hamilton described his memory of the discovery of the Quaternions to his son, "Every morning in the early part of the above-cited month, on my coming down to breakfast, your (then) little brother, William Edwin, and yourself, used to ask me, `Well, papa, can you multiply triplets?' Whereto I was always obliged to reply, with a sad shake of the head: `No, I can only add and subtract them. But on the 16th day of the same month (Oct) - which happened to be Monday, and a Council day of the Royal Irish Academy - I was walking in to attend and preside, and your mother was walking with me along the Royal Canal, to which she had perhaps driven; and although she talked with me now and then, yet an undercurrent of thought was going on in my mind which gave at last a result, whereof it is not too much to say that I felt at once the importance. An electric circuit seemed to close; and a spark flashed forth the herald (as I foresaw immediately) of many long years to come of definitely directed thought and work by myself, if spared, and, at all events, on the part of others if I should even be allowed to live long enough distinctly to communicate the discovery. Nor could I resist the impulse - unphilosophical as it may have been - to cut with a knife on a stone of Brougham Bridge, as we passed it, the fundamental formula which contains the Solution of the Problem, but, of course, the inscription has long since mouldered away. A more durable notice remains, however, on the Council Books of the Academy for that day (October 16, 1843), which records the fact that I then asked for and obtained leave to read a Paper on `Quaternions,' at the First General Meeting of the Session; which reading took place accordingly, on Monday, the 13th of November following.'' *from Hamilton By Sir Robert Stawell Ball.

The plaque says:
Here as he walked by
on the 16th of October 1843
Sir William Rowan Hamilton
in a flash of genius discovered
the fundamental formula for
quaternion multiplication
i² = j² = k² = i j k = −1
& cut it on a stone of this bridge
(Quatenion was a Latin term before Hamilton used it. Milton uses it in Paradise Lost to refer to the four elements of antiquity: air, earth, water, and fire. The last three are “the eldest birth of nature’s womb” because they are mentioned in Genesis before air is mentioned. *John Cook )

In 1982, Halley's Comet was observed on its 30th recorded visit to Earth, first detected using the 5-m (200-in) Hale Telescope at the Mount Palomar Observatory by a team of astronomers led by David Jewett and G. Edward Danielson. They found the comet, beyond the orbit of Saturn, about 11 AU (1.6 billion km) from the Sun. While 50 million times fainter than the faintest objects our eyes can see, they needed to use not only the largest American telescope but also special electronic equipment developed for the Space Telescope. In 1705, Halley used Newton's theories to compute the orbit and correctly predicted the return of this comet about every 76 years. After his death, for correctly predicting its reappearance, it was named after Halley. *TIS (The next predicted perihelion of Halley's Comet is 28 July 2061)
In 1982 the first image of the returning Halley's Comet was recorded with the 200-inch Hale telescope at Palomar Mountain. Caltech astronomers David Jewitt and G. Edward Danielson found the comet when it was still beyond the orbit of Saturn, more than 1.6 billion kilometers (960 million miles) from the Sun. *National Air and Space Museum

1988 Connect Four Solved first by James D. Allen (Oct 1, 1988), and independently by Victor Allis (Oct 16, 1988). First player can force a win. Strongly solved by John Tromp's 8-ply database (Feb 4, 1995). Weakly solved for all boardsizes where width+height is at most 15 (Feb 18, 2006). *Wik

2016 The Hamilton walk takes place each year on this day. Students, professors, and math lovers in general will gather at the Dunsink Observatory around 3:30 pm and proceed to Broombridge in Cabra where he had his Eureka moment about Quaternions. (see 1843 in Events above) The annual event is part of Irish Math week.

The Hamilton Walk from Dunsink Observatory to Broom Bridge on the Royal Canal in Dublin takes place on 16 October each year. This is the anniversary of the day in 1843 when William Rowan Hamilton discovered the non-commutative algebraic system known as quaternions, while walking with his wife along the banks of the Royal Canal.(Tip to Mathematicians, take walks with your wife.)

It is famous for being the location where Sir William Rowan Hamilton first wrote down the fundamental formula for quaternions on 16 October 1843, which is to this day commemorated by a stone plaque on the northwest corner of the underside of the bridge. After being spoiled by the action of vandals and some visitors,^[2] the plaque was moved to a different place, higher, under the railing of the bridge.

The text on the plaque reads:

Here as he walked by
on the 16th of October 1843
Sir William Rowan Hamilton
in a flash of genius discovered
the fundamental formula for
quaternion multiplication
$i^{2}=j^{2}=k^{2}=ijk=-1$
& cut it on a stone of this bridge.

BIRTHS1689 Robert Smith (16 October,1689 – 2 February, 1768) was an English mathematician and Master of Trinity College.
Smith was probably born at Lea near Gainsborough, the son of the rector of Gate Burton, Lincolnshire. He entered Trinity College, Cambridge, in 1708, and becoming minor fellow in 1714, major fellow in 1715 and senior fellow in 1739. From 1716 to 1760 he was Plumian Professor of Astronomy,and was chosen Master in 1742, in succession to Richard Bentley.
Besides editing two works by his cousin, Roger Cotes, who was his predecessor in the Plumian chair, he published A Compleat System of Opticks in 1738, (which was the principal textbook on Optics in the 18th Century) , and Harmonics, or the Philosophy of Musical Sounds in 1749.
Smith never married but lived with his unmarried sister Elzimar (1683–1758) in the lodge at Trinity College. Although he is often portrayed as a rather reclusive character, John Byrom's journal shows that in the 1720s and 1730s Smith could be quite sociable. Yet ill health, particularly gout, took its toll and severely inhibited his academic work and social activities. He died at the lodge on 2 February 1768, and on 8 February he was buried in Trinity College Chapel.
In his will Smith left £3500 South Sea stock to the University of Cambridge. The net income on the fund is annually divided equally between the Smith's Prize and the stipend of the Plumian Professor. *Wik

1879 Philip Edward Bertrand Jourdain (16 October 1879 – 1 October 1919) was a British logician and follower of Bertrand Russell. He corresponded with Georg Cantor and Gottlob Frege, and took a close interest in the paradoxes related to Russell's paradox, formulating the card paradox version of the liar paradox. He also worked on algebraic logic, and the history of science with Isaac Newton as a particular study. He was London editor for The Monist. *Wik

1882 Ernst Erich Jacobsthal (16 October 1882, Berlin – 6 February 1965, Überlingen) was a German mathematician, and brother to the archaeologist Paul Jacobsthal.
In 1906, he earned his PhD at the University of Berlin, where he was a student of Georg Frobenius, Hermann Schwarz and Issai Schur; his dissertation, Anwendung einer Formel aus der Theorie der quadratischen Reste (Application of a Formula from the Theory of Quadratic Remainders), provided a proof that prime numbers of the form 4n + 1 are the sum of two square numbers. *Wik

1930 John Charlton Polkinghorne KBE FRS (16 October 1930 – 9 March 2021) was an English theoretical physicist, theologian, writer, and Anglican priest. He was professor of Mathematical physics at the University of Cambridge from 1968 to 1979, when he resigned his chair to study for the priesthood, becoming an ordained Anglican priest in 1982. He served as the president of Queens' College, Cambridge from 1988 until 1996.*Wik

DEATHS
1937 William Sealy Gosset (13 June 1876 in Canterbury, England - 16 October 1937 in Beaconsfield, England) Gosset was the eldest son of Agnes Sealy Vidal and Colonel Frederic Gosset who came from Watlington in Oxfordshire. William was educated at Winchester, where his favourite hobby was shooting, then entered New College Oxford where he studied chemistry and mathematics. While there he studied under Airy. He obtained a First Class degree in both subjects, being awarded his mathematics degree in 1897 and his chemistry degree two years later.

Gosset obtained a post as a chemist with Arthur Guinness Son and Company in 1899. Working in the Guinness brewery in Dublin he did important work on statistics. In 1905 he contacted Karl Pearson and arranged to go to London to study at Pearson's laboratory, the Galton Eugenics Laboratory, at University College in session 1906-07. At this time he worked on the Poisson limit to the binomial and the sampling distribution of the mean, standard deviation, and correlation coefficient. He later published three important papers on the work he had undertaken during this year working in Pearson's laboratory.
Many people are familiar with the name "Student" but not with the name Gosset. In fact Gosset wrote under the name "Student" which explains why his name may be less well known than his important results in statistics. He invented the t-test to handle small samples for quality control in brewing. Gosset discovered the form of the t distribution by a combination of mathematical and empirical work with random numbers, an early application of the Monte-Carlo method.

McMullen says:-

To many in the statistical world "Student" was regarded as a statistical advisor to Guinness's brewery, to others he appeared to be a brewer devoting his spare time to statistics. ... though there is some truth in both these ideas they miss the central point, which was the intimate connection between his statistical research and the practical problems on which he was engaged. ... "Student" did a very large quantity of ordinary routine as well as his statistical work in the brewery, and all that in addition to consultative statistical work and to preparing his various published papers.

From 1922 he acquired a statistical assistant at the brewery, and he slowly built up a small statistics department which he ran until 1934.
Gosset certainly did not work in isolation. He corresponded with a large number of statisticians and he often visited his father in Watlington in England and on these occasions he would visit University College, London, and the Rothamsted Agricultural Experiment Station. He would discuss statistical problems with Fisher, Neyman and Pearson. *SAU

In probability and statistics, Student's t-distribution (or simply the t-distribution)

t_nu is a continuous probability distribution that generalizes the standard normal distribution. Like the latter, it is symmetric around zero and bell-shaped.

However, t_nu has heavier tails and the amount of probability mass in the tails is controlled by the parameter nu . For nu = 1 the Student's t distribution

t_nu becomes the standard Cauchy distribution, whereas for nu→∞} it becomes the standard normal distribution

1983 Harish-Chandra (11 October 1923 – 16 October 1983) was an Indian mathematician, who did fundamental work in representation theory, especially Harmonic analysis on semisimple Lie groups.*Wik

1998 Jonathan Bruce Postel (6 Aug 1943, 16 Oct 1998) American computer scientist who played a pivotal role in creating and administering the Internet. In the late 1960s, Postel was a graduate student developing the ARPANET, a forerunner of the Internet for use by the U.S. Dept. of Defense. As director of the Internet Assigned Numbers Authority (IANA), which he formed, Postel was a creator of the Internet's address system. The Internet grew rapidly in the 1990s, and there was concern about its lack of regulation. Shortly before his death, Postel submitted a proposal to the U.S. government for an international nonprofit organization that would oversee the Internet and its assigned names and numbers. He died at age 55, from complications after heart surgery.*TIS