A catenary curve is the shape that a perfectly uniform rope would form when suspended between two points. The word is from the Latin catena for chain. The name was applied by Christen Huygens while studying the form of suspended chains. Galileo thought the shape would be a parabola. As can be seen from the image below, near the vertex a parabola and a catenary look very similar. When x is slightly greater than three however, the catenary begins to rapidly outgrow the value of the parabola. The two shapes are related by another relationship. If a parabola is rolled along a straight line, the focus of the parabola will move along a catenary curve. In the figure the parabola y=x2 + 1 is on the inside and the catanary whose equation is y= (ex+e-x)/2, is on the outside.
For students of American History, it may be interesting that the first use of "catenary", rather than the longer, more formal "catenaria", may have been in a letter from Thomas Jefferson to Thomas Paine. Jeff Miller's wonderful web-site on the first use of mathematical words has
In a letter to Thomas Jefferson dated Sept. 15, 1788, Thomas Paine, discussing the design of a bridge, used the term catenarian arch:
Whether I shall set off a catenarian Arch or an Arch of a Circle I have not yet determined, but I mean to set off both and take my choice. There is one objection against a Catenarian Arch, which is, that the Iron tubes being all cast in one form will not exactly fit every part of it. An Arch of a Circle may be sett off to any extent by calculating the Ordinates, at equal distances on the diameter. In this case, the Radius will always be the Hypothenuse, the portion of the diameter be the Base, and the Ordinate the perpendicular or the Ordinate may be found by Trigonometry in which the Base, the Hypothenuse and right angle will be always given.
In a reply to Paine dated Dec. 23, 1788, Thomas Jefferson used the word catenary:
You hesitate between the catenary, and portion of a circle. I have lately received from Italy a treatise on the equilibrium of arches by the Abbé Mascheroni. It appears to be a very scientifical work. I have not yet had time to engage in it, but I find that the conclusions of his demonstrations are that 'every part of the Catenary is in perfect equilibrium.'
The earliest citation for catenary in the OED2 is from the above letter.
Yesterday I mentioned LaGrange's theorem that every number can be written as the sum of four integral squares. Some can be written as the sum of four squares in many different ways which include the use of 02. The number 26 can be partitioned as the sum of four squares in 336 different ways. If that sounds too trivial, tell folks 336 is a Lipschitz integer quaternion.
EVENTS
1697 St Paul's Cathedral, reconstructed after the Great Fire of 1666 as redesigned by Christopher Wren, was officially opened on December 2nd, 1697.*History Today In 1895, James Dewar exhibited his new apparatus for the production of liquid air at the Royal Institution.*TIS In 1934, the molten glass was poured in the Corning, N.Y. for the first 200-inch diameter telescope mirror. Pyrex glass at 2,700 degrees Fahrenheit was poured into a ceramic mold. The mold had been constructed over a period of several months. The temperature of the glass was lowered during 11 months, a degree or two a day. It was then allowed to cool to room temperature. The 20-ton disk was shipped 26 Mar 1936 for grinding and polishing at the California Institute of Technology, which spanned 11 years, completed on 3 Oct 1947. It was installed in a telescope at the Mount Palomar Observatory on Palomar Mountain, San Diego County, California, which was named the Hale telescope in honour of Dr George Hale who had conceived and promoted it.*TIS 1942 At 3:36 p.m. in a squash court (Actually it was a Racketball court *James Zug Squash, A History of the Game pgs. 135–136.) under the West Stands of Stagg Field (the abandoned football stadium) at the University of Chicago, the first self-sustaining nuclear (fission) reaction took place. Enrico Fermi (1901–1954) was leader of the Manhattan Project. [DSB 4, 582]. *VFR This first run of the nuclear pile produced a single watt of power, Just enough to show that the process was feasible. One of the “about 40” people who watched was Leo Szilard who had conceived the idea of a chain reaction leading to power while stopped at a red light on Southampton Row in London only four years before. *Frederik Pohl, Chasing Science, Pg 20 One of Fermi's assistants ran from the test to the telephone to notify Openheimer, "The Italian navigator has just landed in the New World." *Brody & Brody, The Science Class You Wished You Had 1954 The U.S. Navy dedicates its Naval Ordnance Research Calculator (NORC) at the Naval Surface Weapons Center in Dahlgren, Virginia. John von Neumann was the keynote speaker. The machine was built at the Watson Scientific Computing Laboratory under the direction of Wallace Eckert. This computer was in demand by many organizations, including two different Navy facilities and Lawrence Livermore National Laboratory in California. Physicist Edward Teller had been trying to receive NORC arguing that the LLNL's nuclear calculations were more important than Dahlgren's ballistic calculations. The Navy won and NORC was delivered to Dahlgren, following the Mark II (1948) and the Mark III (1951).*CHM
1967 Italy issued a postage stamp to commemorate the 25th anniversary of the first atomic chain reaction. Pictured is Enrico Fermi at Los Alamos and a model of the first Atomic Reactor. *VFR 1978 Science News reports, p. 390, that 221,701 − 1 is prime.
BIRTHS
1831 Paul David Gustav du Bois-Reymond (2 Dec 1831 in Berlin, Germany - 7 April 1889 in Freiburg, Germany) Du Bois-Reymond's work is almost exclusively on calculus, in particular partial differential equations and functions of a real variable. The standard technique to solve partial differential equations used Fourier series but Cauchy, Abel and Dirichlet had all pointed out problems associated with the convergence of the Fourier series of an arbitrary function. In 1873 du Bois-Reymond was the first person to give an example of a continuous function whose Fourier series diverges at a point. Perhaps what was even more surprising, the Fourier series of du Bois-Reymond function diverged at a dense set of points. The important work Eine neue Theorie der Convergenz und Divergenz von Reihen mit positiven Gliedern ("A new theory of convergence and divergence of series with positive terms") led to an increasing understanding of the whole concept of a function. Du Bois-Reymond published an example of a continuous function which is nowhere differentiable in 1875. It was inspired by a similar function found by Weierstrass in 1872 but not published by him until much later. This example contradicted most mathematicians' intuition, for it was generally believed that a continuous function was differentiable everywhere except in special points. *SAU 1865 Niels Nielsen (2 Dec 1865 in Orslev, Denmark - 16 Sept 1931 in Copenhagen, Denmark) was a Danish mathematician who worked on special functions and number theory. *SAU 1901 Dom George Frederick James Temple FRS(born 2 December 1901, London; died 30 January 1992, Isle of Wight) was an English mathematician, recipient of the Sylvester Medal in 1969. He was President of the London Mathematical Society in the years 1951-1953.[2] Temple took his first degree as an evening student at Birkbeck College, London, between 1918 and 1922, and also worked there as a research assistant. In 1924 he moved to Imperial College as a demonstrator, where he worked with Professor Sydney Chapman. After a period spent with Eddington at Cambridge, he returned to Imperial as reader in mathematics. He was appointed professor of mathematics at King's College London in 1932, where he returned after war service with the Royal Aircraft Establishment at Farnborough. In 1953 he was appointed Sedleian Professor of Natural Philosophy at the University of Oxford, a chair which he held until 1968, and in which he succeeded Chapman. He was also an honorary Fellow of Queen's College, Oxford. After the death of his wife in 1980, Temple, a devout Christian, took monastic vows in the Benedictine order and entered Quarr Abbey on the Isle of Wight, where he remained until his death. *Wik 1914 Robert Palmer Dilworth (December 2, 1914 – October 29, 1993) was an American mathematician. His primary research area was lattice theory; his biography at the MacTutor History of Mathematics archive states "it would not be an exaggeration to say that he was one of the main factors in the subject moving from being merely a tool of other disciplines to an important subject in its own right". He is best known for Dilworth's theorem (Dilworth 1950) relating chains and antichains in partial orders; he was also the first to study antimatroids (Dilworth 1940). Dilworth advised 17 Ph.D. students and as of 2010 has 373 academic descendants listed at the Mathematics Genealogy Project, many through his student Juris Hartmanis, a noted complexity theorist.*Wik
DEATHS
1594 Gerardus Mercator (5 Mar 1512- 2 Dec 1594) Flemish cartographer whose most important innovation was a map, embodying what was later known as the Mercator projection, on which parallels and meridians are rendered as straight lines spaced so as to produce at any point an accurate ratio of latitude to longitude. He also introduced the term atlas for a collection of maps. *TIS 1873 Karl Gräffe Gräffe (7 Nov 1799 in Brunswick, Germany - 2 Dec 1873 in Zurich, Switzerland) is best remembered for his method of numerical solution of algebraic equations, developed to answer a prize question of the Berlin Academy of Sciences. It is particularly suitable for methods developed for using computers to solve mathematical problems. This method is today called the Dandelin-Gräffe method after the two mathematicians who independently investigated it. The history of the Dandelin-Gräffe method is discussed in and . Lobachevsky is also credited with the independent discovery of the method which appears in his little-known book on algebra.*SAU 1966 L(uitzen) E(gbertus) J(an) Brouwer (27 Feb 1881, 2 Dec 1966) was a Dutch mathematician who founded mathematical Intuitionism (a doctrine that views the nature of mathematics as mental constructions governed by self-evident laws). He founded modern topology by establishing, for example, the topological invariance of dimension and the fixpoint theorem. (Topology is the study of the most basic properties of geometric surfaces and configurations.) The Brouwer fixed point theorem is named in his honor. He proved the simplicial approximation theorem in the foundations of algebraic topology, which justifies the reduction to combinatorial terms, after sufficient subdivision of simplicial complexes, the treatment of general continuous mappings. *TIS 1982 Geoffrey Timms studied at Leeds and Cambridge and then took up a post at St Andrews. During World War II he served at Bletchley Park and Cheltenham and joined the Foreign Office afterwards. He became President of the EMS in 1941.*SAU 2006 Dikran "Dick" Tahta (7 August 1928 – 2 December 2006) was a British-Armenian mathematician, teacher and author. Dikran Tahta is a descendant of an Ottoman Armenian family who settled in Manchester after the First World War. Much of his childhood, and the influence of his Armenian religious upbringing, is reflected upon in his penultimate book Ararat Associations, in which he notes how his parents were keen for their children to have an English education, yet made sure that they spoke Armenian at home. He was christened by Bishop Tourian in the Armenian Church in Manchester, and his name Dikran was shortened to Dick, but he never forgot his Armenian roots. From Rossall School, in Fleetwood, Lancashire, he gained a scholarship to Christ Church, Oxford, in 1946. His main subject was Mathematics, but he also read widely in English literature, philosophy and history. In the 1970s he was involved in the ATV television programme of mathematics for schools entitled 'Leapfrogs' (produced and directed by Paul Martin) and promoted visual approaches to mathematics. His paper "On Geometry" argued that geometrical approaches to mathematics could not be reduced to algebraic approaches. In line with this thinking, he produced the ATM book Geometric Images, and co-authored Images of Infinity with Ray Hemmings. The Leapfrogs group of Tahta and Hemmings, together with David Sturgess, Leo Rogers and Derick Last also produced hands-on teaching materials including workbooks for the polycube. He also drew upon insights into pedagogy in the writings of Mary Boole on mathematics education. After retirement, he went to teach in the United States and South Africa, and became a tutor for the Open University. His last book was The Fifteen Schoolgirls about Thomas Kirkman, known for the Kirkman's schoolgirl problem, a problem in combinatorics, which also delved into the byways of Victorian amateur mathematics. In his obituary, The Guardian newspaper described Dick as "one of the outstanding mathematics teachers of his generation", who was notable for having inspired physicist Stephen Hawking. The Guardian commented on his death that "He was a wise and generous man who inspired love and an increase of intellectual energy in everyone who came within his ambit." *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
Sometimes blogs start when some kind of reoccurring theme pops up over several days. In this case the theme was (loosely) drawing things using parametric functions. I saw the image above which is the Logo for the MIT Lincoln Library. It reminded me of something called Bowditch (or Lissajous) curves which were a common amusement I would use to introduce my students to parametric equations after graphing calculators. (more about these later) And I mused that someday I would have to look up the history of mechanical methods of producing parametric functions.
Then within a short period of time I read that André Cassagnes, the French inventor of the Etch A Sketch, had died near Paris on January 16, 2013, at the age of 86. If you haven't heard of the Etch A Sketch, (right) it was a mechanical toy that was used to draw on a screen with an internal stylus that was moved right or left by one twist knob, and up or down by the other, sort of a mechanical x=f(t) and y=f(t).
It reminded me of my earlier intention a few days earlier, and so I decided to begin filling out my knowledge about that history.
From my own notes I knew that Nathaniel Bowditch, an under-appreciated American self-taught mathematician had drawn curves like this. He first drew these parametric curves in 1815 with a compound pendulum." Like most others, in school I had learned about them as Lissajous figures, images we drew on oscilloscopes using signal generators for the two inputs. But then,shortly after I first read about Bowditch I happened to be in Tokyo Visiting the Edo Museum for an exhibit named Worlds Revealed - The Dawn of Japanese and American Exchange. Like others, I had always had the misconception that Commodore Perry opened trade with Japan in 1853, so I was surprised to find that a number of American ships from Salem, Massachusetts, sailing under Dutch charters had traded with the Japanese as early as 1800. The company was called the East India Marine Society, and in 1802 the First Secretary was Bowditch. On exhibit was a much more popular mathematical creation of Bowditch; his book, The New American Practical Navigator, that Bowditch, and the Marine Society had published in 1802. The book was a compilation of the most accurate measures of the period giving the positions of major astronomical objects at numerous longitude and latitude coordinates. The book was, literally, a mariner's bible until an accurate sea clock would become commonly available that allowed sailors to conquer the longitude problem. Bowditch's position and accomplishments seem even greater in light of the fact that he was almost totally self educated in mathematics. Then, in August of 2008 I read a post by Milo Gardner on the almost unheard of Wilkes Expedition, which explored the western Americas and the Pacific, and Milo added that "... mathematicians during the early 1800's were assigned to working on Manifest Destiny issues and projects. On the Wilkes Expedition you'll find Bowditch as one of its navigators. An island in the Pacific is named for Bowditch, since it had not been on any US or European map prior to the expedition's visit." The island, I found out, is sometimes called Fakaofu, and is located in the Stork Archipelago in the South Pacific.
I decided to go back a little farther by looking for any historical references I could find for the history of mechanical curve drawing and hit a jackpot with an on-line article by Daina Taimina, of Cornell University titled Historical Mechanisms for Drawing Curves. It seems to be from the book,Hands on History: A Resource for Teaching Mathematics.
She stated that "Mechanical devices in ancient Greece for constructing different curves were invented mainly to solve three famous problems: doubling the cube, squaring the circle and trisecting the angle." She went on to give several examples, "There can be found references that Meneachmus (~380-~320 B.C.) had a mechanical device to construct conics which he used to solve problem of doubling the cube. One method to solve problems of trisecting an angle and squaring the circle was to use quadratrix of Hippias (~460-~400 B.C) {this was the first named curve other than circle and line – it is also the first example of a curve that is defined by means of motion and can not be constructed using only a straightedge and a compass.} Proclus (418-485) also mentions some Isidorus from Miletus who had an instrument for drawing a parabola.[ Dyck,p.58]. We can not say that those mechanical devices consisted purely of linkages, but it is important to understand that Greek geometers were looking for and finding solutions to geometrical problems by mechanical means. These solutions mostly were needed for practical purposes."
From her description it would seem that none of these still existed in physical or drawn form.
While her focus was on the use of linkages to create mechanical movement and drawings, I was searching for something closer to the idea of a parametric curve.
Certainly the early trammel which dates to Proclus or Archimedes (indeed it is sometimes called the trammel of Archimedes) but again, it is more of a mechanical linkage than parametric. And no offense intended to my neighbors here in Kentucky, but the instrument is often sold as a novelty made of wood with a crank knob on the end of the trammel bar that traces out the ellipse, and is referred to as a "Kentucky do-nothing".
Students may have also been shown how to draw an ellipse by taking a loop of string looped around two thumb tacks. By holding a pencil pulled against the string to keep it taut, and sliding it around the two thumb tacks as you keep the string taut, the pencil will trace out an ellipse. The first written description of this method of construction an ellipse by means with string was by Abud ben Muhamad, in the 9th century.
Then I came across an article in Wikipedia about the harmonograph, a mechanical platform that employs one or more pendulums to create a geometric image. Interestingly, they give credit for the first harmonograph to Scottish mathematician Hugh Blackburn. Trouble is, Blackburn was born in 1823; almost a full decade after Bowditch had written about his use of such a device.
I am beginning to accept that Bowditch may have been the first person to create the parametric images which sometimes, and should more often, bare his name. If someone has an example of an earlier non-linkage apparatus that suggests parametric input to draw figures, I would love to be notified.
Jules Antoine Lissajous, for whom the figures are more often called, invented a different type of device to create the images. He used a beam of light bounced off a mirror attached to a vibrating tuning fork, which then reflected off a second mirror attached to another vibrating tuning fork which was perpendicularly orientated (usually of a different pitch, creating a specific harmonic interval), which was then reflected onto a wall, tracing the figure.With frequency produced by audible frequencies the curve traced out by the light appeared as a complete image due to visual persistence. Lissajous device is sometimes credited with inspiring the two pendulum device, but he too was born after Bowditch had written of his device. None of this should be seen to diminish Lissajous mathematical stature. His experiments with waves, his novel method of creating the waves, and his dramatic lectures and demonstrations, including one at the Royal Society in London, exposed them to a much wider audience. These lectures were so impressive that he was awarded the Lacaze Prize in 1873 for his optical observation of vibration and, in particular, "for his beautiful experiments". Almost certainly he was completely unaware of Bowditch's work.
When I introduced these to my students I often used one similar to the Lincoln Library Logo at top and I called it the Chinese finger cuff curve (I am still waiting for the rest of the mathematical world to adopt this term, fall into line people) As I neared retirement it seemed that many of the students had never heard of finger cuffs, but there were always a few who knew of them, and often at least one student who would produce one from home over the next few days.
Beauty is the first test: there is no permanent place in the world for ugly mathematics.
~Godfrey Harold Hardy
The 335th day of the year; 2335 is the smallest power of two which equals the sum of four consecutive primes *Prime Curios This seems astounding to me, that such a huge number would be the first.
There are 67 primes smaller than 335, and so 335 is divisible by the number of primes less than itself.. How common is that for integers.
Lagrange's theorem tells us that each positive integer can be written as a sum of four squares (perhaps including zero), but many can be written as the sum of only one or two non-zero squares. 335 is one of the numbers that can not be written with less than four non-zero squares. The smallest examples are 7, 15, and 23. If you take any number in this sequence, and raise it to an odd positive power, you get another number in the sequence (Why teachers?), so now you know that 73 = 343 is also not expressible as the sum of less than four non-zero squares.
EVENTS
1729 Euler/Goldbach correspondence begins: Goldbach was also a kind of mentor to Leonhard Euler. For over 25 years they exchanged letters, 196 of which survive. These letters give us a window into Euler’s scientific and personal life. In Goldbach’s very first letter to Euler, dated December 1, 1729, Goldbach got Euler interested in number theory. Goldbach added note at the end of the letter: “P. S. Have you noticed the observation of Fermat that all numbers of the form 22x+1, that is 3, 5, 17, etc., are prime numbers, but he did not dare to claim he could demonstrate it, nor, as far as I know, has anyone else been able to prove it.” Three years later, in a five-page paper that now bears the index number E26, Euler shows that the F(5) = 4,294,967,297 = 641× 6,700,417 . That is, Fermat was wrong. *Ed Sandifer, How Euler Did It (Like Fermat before him, Euler found most mathematicians less than excited about problems in Number Theory, but for most of his life, Goldbach would be his mentor, and his student in this area.) 1764 Alexander Small writes Benjamin Franklin from England, "My Namesake the Virginian Professor (William Small) is here; and desires to be most particularly remembered to you." Small is known for being Thomas Jefferson's professor of Natural Philosophy at William and Mary, and for having an influence on the young Jefferson. (I could not determine if Alexander and William Small are related) *Natl. Archives 1783 J. A. C. Charles was the first man to see the sun set twice in one day. He did it by making a flight (to 9000 feet) in a hydrogen balloon. *VFR (Charles is often considered the inventor of the hydrogen balloon.) The first manned voyage of a hydrogen balloon left Paris carrying Professor Jacques Alexander Cesar Charles and Marie-Noel Robert to about 600 m and landed 43 km away after 2 hours in the air. Robert then left the balloon, and Charles continued the flight briefly to 2700 m altitude, measured by a barometer. This hydrogen-filled balloon was generally spherical and used a net, load ring, valve, open appendix and sand ballast, all of which were to be universally adopted later. His hydrogen generator mixed huge quantities of sulfuric acid with iron filings. On 27 Aug 1783, Charles had launched an unmanned hydrogen balloon, just before the Montgolfiers' flight. *TIS (One of these altitudes is obviously wrong. ) 1851 On December the first, Louis-Napoleon Bonaparte, who had been instrumental in supporting Foucault in the demonstration of his pendulum, ordered that the pendulum demonstration cease and the Pantheon return to being used as a church (Louis Philippe had secularized the Pantheon in 1830 and stopped burials in the crypt). Why did he stop the popular demonstrations? We do not know, but on the next day citizens of France awoke to find notices posted on the major buildings, “The National Assembly is dissolved… “ Louis-Napoleon had taken his first step to becoming Emperor of France. *Amir D Aczel, Pendulum, pg 174 1890, after regular competition, Peano was named extraordinary professor of infinitesimal calculus at the University of Turin. *Hubert Kennedy Eight Mathematical Biographies Pg 23 1896 Frank Broaker of New York City received certificate No. 1 from the New York State Board of Certified Public Account Examiners thus becoming the first CPA in the US. *JN Kane, Famous First Facts,
In 1997 eight planets from our Solar System lined up from West to East beginning with Pluto, followed by Mercury, Mars, Venus, Neptune, Uranus, Jupiter, and Saturn, with a crescent moon alongside, in a rare alignment visible from Earth that lasted until Dec 8. Mercury, Mars, Venus, Jupiter and Saturn were visible to the naked eye, with Venus and Jupiter by far the brightest. A good pair of binoculars is needed to see the small blue dots that are Uranus and Neptune. Pluto is visible only by telescope. The planets also aligned in May 2000, but too close to the sun to be visible from Earth. It will be at least another 100 years before so many planets will be so close and so visible.*TIS
BIRTHS
1671 John Keill (1 Dec 1671; 31 Aug 1721) Scottish mathematician and natural philosopher, who was a major proponent of Newton’s theories. He began his university education at Edinburgh under David Gregory, whom he followed to Oxford, where Keill lectured on Newton's work, and eventually became professor of astronomy. In his book, An Examination of Dr. Burnett's Theory of the Earth (1698), Keill applied Newtonian principles challenging Burnett's unsupportable speculations on Earth's formation. In 1701, Keill published Introductio ad Veram Physicam, which was the first series of experimental lectures and provided a clear and influential introduction to Isaac Newton’s Principia. He supported Newton against priority claims by Leibnitz for the invention of calculus. *TIS 1792 Nikolay Ivanovich Lobachevsky (1 Dec 1792; 24 Feb 1856) Russian mathematician who, with János Bolyai of Hungary, is considered the founder of non-Euclidean geometry. Lobachevsky constructed and studied a type of geometry in which Euclid's parallel postulate is false (the postulate states that through a point not on a certain line only one line can be drawn not meeting the first line). This was not well received at first, but his greatest vindication came with the advent of Einstein's theory of relativity when it was demonstrated experimentally that the geometry of space is not described by Euclid's geometry. Apart from geometry, Lobachevsky also did important work in the theory of infinite series, algebraic equations, integral calculus, and probabilty. *TIS William Kingdon Clifford called Lobachevsky the "Copernicus of Geometry" due to the revolutionary character of his work. Lobachevsky is the subject of songwriter/mathematician Tom Lehrer's humorous song "Lobachevsky" from his Songs by Tom Lehrer album. In the song, Lehrer portrays a Russian mathematician who sings about how Lobachevsky influenced him: "And who made me a big success / and brought me wealth and fame? / Nikolai Ivanovich Lobachevsky is his name." Lobachevsky's secret to mathematical success is given as "Plagiarize!", as long as one is always careful to call it "research". According to Lehrer, the song is "not intended as a slur on [Lobachevsky's] character" and the name was chosen "solely for prosodic reasons".*Wik (The lyrics are here) 1847 Christine Ladd-Franklin (1 Dec 1847; 5 Mar 1930) American scientist and logician known for contributions to the theory of colour vision accounting for the development of man's color sense which countered the established views of Helmholtz, Young, and Hering. Her position was that color-sense developed in stages. Ladd- Franklin's conclusions were particularly useful in accounting for color-blindness in some individuals. In logic, she published an original method for reducing all syllogisms to a single formula *TIS Ladd-Franklin was the first woman to have a published paper in the Analyst (at this time, 1877, it was more of a recreational mathematics publication still edited by the self-educated Ohio farmboy, Joel E Hendricks. The article was simply titled "Quaternions." ). She was also the first woman to receive a Ph.D. in mathematics and logic. The majority of her publications were based on visual processes and logic. Her views on logic influenced Charles S. Peirce’s logic and she was highly praised by Prior. *Wik 1892 Krishnaswami Ayyangar (1 Dec 1892 in Attipattu, Chingleput district, Tamil Nadu, India - June 1953 in Mysore, India) was an Indian mathematician who worked in Mysore. He produced important work on the history of Hindu mathematics. *SAU 1913 Colossus' Team Member Chandler is Born W.W. Chandler was born in Bridport, England. He obtained his B.Sc. from London University in 1938 by private study while working as a telephone engineer at the British Post Office Research Department. During the war he was responsible for the installation and maintenance of the Colossus at Bletchley Park. The Colossus represented the first electronic computer, however it was programmed by a mechanical switchboard. Its was used to crack the German Fish codes which guarded the highest levels of German communication. Winston Churchill characterized the Bletchley Park team as the geese who laid the golden eggs but never cackled. After the war Chandler participated in development and installation of the MOSAIC computer and worked on optical character recognition. He died on September 11, 1989. *CHM 1941 Stephen A. Benton (1 Dec 1941; 9 Nov 2003.) American physicist who was a pioneer in medical imaging and fine-arts holography. His fascination with optical phenomena began with the 3-D glasses he used as an 11-year-old to watch te 1953 movie "House of Wax." In 1968, he invented the "rainbow holograms" as seen on credit cards while working for Polaroid Corporation. He turned to academia as an assistant professor at Harvard (1968) and later a professor at Massachusetts Institute of Technology from 1985 where he helped set up the Spatial Imaging Group and headed the M.I.T. media art and sciences program. Benton was a pioneer in natural light holography as a artistic medium, and was a curator at the Museum of Holography in Manhattan until it closed in 1992.*TIS
DEATHS
1750 Johann Doppelmayr (27 Sept 1677 in Nuremberg, Germany - 1 Dec 1750 in Nuremberg, Germany)was a German mathematician who wrote on astronomy, spherical trigonometry, sundials and mathematical instruments. Doppelmayr also wrote a book of tremendous value giving biographical details of 360 mathematicians and instrument makers of Nuremberg from the 15th to the 18th century. This had the lengthy title Historische Nachricht von den Nürnbergischen Mathematicis und Künstlern, welche fast von dreyen Seculis her durch ihre Schriften und Kunst-Bemühungen die Mathematic und mehrere Künste in Nürnberg vor andern trefflich befördert und sich um solche sehr wohl verdient gemacht zu einem guten Exempel, und zur weitern rühmlichen Nachahmung and was published in 1730. *SAU 1866 Sir George Everest (1790, 1 Dec 1866) British military engineer and geodesist, born in Gwernvale, Powys, Wales, UK. He worked on the trigonometrical survey of India (1818-43), providing the accurate mapping of the subcontinent. For more than twenty-five years and despite numerous hardships, he surveyed the longest arc of the meridian ever accomplished at the time. Everest was relentless in his pursuit of accuracy. He made countless adaptations to the surveying equipment, methods, and mathematics in order to minimize problems specific to the Great Survey: immense size and scope, the terrain, weather conditions, and the desired accuracy. Mount Everest, formerly called Peak XV, was renamed in his honour in 1865. *TIS (Mary Boole, self-taught mathematician and wife of George Boole was his niece) 1935 Bernhard Voldemar Schmidt (30 Mar 1879, 1 Dec 1935) Astronomer and optical instrument maker who invented the telescope named for him. In 1929, he devised a new mirror system for reflecting telescopes which overcame previous problems of aberration of the image. He used a vacuum to suck the glass into a mold, polishing it flat, then allowing in to spring back into shape. The Schmidt telescope is now widely used in astronomy to photograph large sections of the sky because of its large field of view and its fine image definition. He lost his arm as a child while experimenting with explosives. Schmidt spent the last year of his life in a mental hospital.*TIS 1947 Godfrey Harold Hardy (1877, 1 Dec 1947)English mathematician known for his work in number theory and mathematical analysis. Hardy's interests covered many topics of pure mathematics - Diophantine analysis, summation of divergent series, Fourier series, the Riemann zeta function, and the distribution of primes. Although Hardy considered himself a pure mathematician, early in his career, he nevertheless worked in applied mathematics when he formulated a law that describes how proportions of dominant and recessive genetic traits will propagate in a large population (1908). Hardy considered it unimportant but it has proved of major importance in blood group distribution. As it was also independently discovered by Weinberg, it is known as the Hardy-Weinberg principle. *TIS G. H. Hardy died—on the same day that the Copley Medal was to be presented to him by the Royal Society of London. [Collected Papers of G. H. Hardy, vol. 1, p. 8]. 1964 J.B.S.(John Burdon Sanderson) Haldane (5 Nov 1892, 1 Dec 1964) was a British geneticist and biometrician who opened new paths of research in population genetics and evolution. He began studying science at the age of eight, as assistant to his father (the noted physiologist John Scott Haldane). J.B.S. Haldane also worked in biochemistry, and on the effects of diving on human physiology. A Marxist from the 1930s, Haldane was well known for his outspoken Marxist views.He resigned from the Communist Party c. 1950 on the issue of Lysenko's claims to have manipulated the genetic structure of plants and "Stalin's interference with science". He became known to a large public as a witty popularizer of science with such works as Daedalus (1924), and Possible Worlds (1927).*TIS 1977 Kenneth O. May (July 8, 1915, Portland, Or. – December 1,1977) was an American mathematician and historian of mathematics, who developed May's theorem. The Kenneth O. May Prize is awarded for outstanding contributions to the history of mathematics. Ken May established Historia Mathematica, and preserved it by separating it from its creator, "The distinguished predecessors of HM were associated with their founders and died with them. If HM is to avoid this fate, we must prepare and carry through a prompt transfer of editorial responsibility to younger hands." His list of publications numbers above 300. *Henry S. Tropp, E'loge, Isis 70, Sept 1979, Pgs 419-422 1983 Leon Mirsky (19 Dec 1918 in Russia - 1 Dec 1983 in Sheffield, England)worked in Number Theory, Linear Algebra and Combinatorics.*SAU 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
A catenary curve is the shape that a perfectly uniform rope would form when suspended between two points. The word is from the Latin catena for chain. The name was applied by Christen Huygens while studying the form of suspended chains. Galileo thought the shape would be a parabola. As can be seen from the image below, near the vertex a parabola and a catenary look very similar. When x is slightly greater than three however, the catenary begins to rapidly outgrow the value of the parabola. 
