Thursday 31 January 2019

Mechanical Drawing with Harmony, a Brief History

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=g(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.

If you want to create you own, you can find on-line parametric graphers and even an ipad app for a harmonograph.

Several nice examples, with their equations, are given at this Wikipedia link. Enjoy

On This Day in Math - January 31

Joost Bürgi nich at Kepler  monument
on the market-place in the city Weil der Stadt in  Baden-Württemberg

The capacity to blunder slightly is the real marvel of DNA. 
Without this special attribute, we would still be anaerobic bacteria
and there would be no music.
~Antoine-Thomson d' Abbadie

The 31st day of the year; 31 = 22 + 33, i.e., The eleventh prime, and third Mersenne prime, it is also the sum of the first two primes raised to themselves. *Number Gossip  (Is there another prime which is the sum of consecutive primes raised to themselves?  A note from Andy Pepperdine of Bath who informed me that \(2^2 + 3^3 +5^5 + 7^7 = 826699 \), a prime.)

There are exactly 31 positive integers which cannot be written as the sum of two distinct squares; and one of them is the number 31. Finding the 31 is not so hard, they all occur in the first six months of the year.  Proving there are no more is a little tougher. 

Jim Wilder ‏@wilderlab offered, The sum of digits of the 31st Fibonacci number (1346269) is 31.

If you like unusual speed limits, the speed limit in downtown Trenton, a small city in northwestern Tennessee, is 31 miles per hour.
And the little teapot on the sign? Well, Trenton also bills itself as the teapot capital of the nation. The 31 mph road sign seems to come from a conflict between Trenton and a neighboring town which I will not name ,...but I will tell you they think of themselves as the white squirrel capital.

31 is also the smallest integer that can be written as the sum of four positive squares in two ways 1+1+4+25; 4+9+9+9.

31 is an evil math teacher number. The sequence of  the maximum number of regions obtained by joining n points around a circle by straight lines begins 2, 4, 8, 16... but for five points, it is 31.

And 31 is also the minimal number of moves to solve the Towers of Hanoi problem with five disks.  (now wondering if there is a mathematical connection between these two ideas other than coincidence)

@JamesTanton posted a mathematical fact and query regarding 31.  31 =111(base 5) =11111(base 2) and 8191 =111(base 90) = 111111111111(base 2) are the only two integers known to be repunits at least 3 digits long in two different bases.
Is there an integer with representations 10101010..., ,at least three digits, in each of two different bases?

Which made me wonder, are there other pairs that are repdigits (all alike, but not all units) in two (or more) different bases?

1599 During an observation of the lunar eclipse, Tycho Brahe discovers that his predictive theory about the movement of the Moon is wrong since the eclipse started 24 minutes before his calculations predicted: he improves on his theory. On March 21 he sent a letter to Longomontanus, in which he reports his revised theory.*Wik

1802 Gauss elected a corresponding member of the St. Petersburg Academy of Science. *VFR

1834 Felix Klein declines to be the successor of J. J. Sylvester at John's Hopkins.  Klein had been offered the position on December 13th of the previous year, but had demanded a salary equal to the departing Sylvester and some form of security for his family which Johns Hopkins did not meet. By October he would send notes to his family, "Gottingen is beginning to make noises."  In the spring of 1836 he took over as Professor at Gottingen (he had been their second choice). *Constance Reid, The Road Not Taken, Mathematical Intelligencer, 1978

1839, Fox Talbot read a paper before the Royal Society, London, to describe his photographic process using solar light, with an exposure time of about 20 minutes: Some Account of the Art of Photogenic Drawing or the Process by which Natural Objects may be made to Delineate Themselves without the Aid of the Artist's Pencil. He had heard that Daguerre of Paris was working on a similar process. To establish his own priority, Fox Talbot had exhibited "such specimens of my process as I had with me in town," the previous week at a meeting of the Royal Institution, before he had this more detailed paper ready to present.*TIS

1939 Hewlett-Packard founded. Their calculators use the “reverse Polish notation” devised by Jan L Lukasiewicz (see here, 1878). *VFR

1939 Joseph Ehrenfried Hofmann began his academic career as a professor of the history of mathematics at the University of Berlin. He is noted for his work on Leibniz, especially the book Leibniz in Paris, 1672–1676: His Growth to Mathematical Maturity. *VFR Leibniz in Paris 1672-1676: His Growth to Mathematical Maturity

1958 Explorer 1 was launched on January 31, 1958 at 22:48 Eastern Time (equal to February 1, 03:48 UTC because the time change goes past midnight). It was the first spacecraft to detect the Van Allen radiation belt, returning data until its batteries were exhausted after nearly four months. It remained in orbit until 1970, and has been followed by more than 90 scientific spacecraft in the Explorer series. *Wik 
Actually the Van Allen radiation was detectable by the Russian’s first satellite, Sputnik.  Because the signals were sent in a secret code, it’s signal could not be received by the Russians when it was detecting the radiation of the belt.  *Frederich Pohl, Chasing Science, pg 85

1995 AT&T Bell Laboratories and VLSI Technology announce plans to develop strategies for protecting communications devices from eavesdroppers. The goal would be to prevent problems such as insecure cellular phone lines and Internet transmissions by including security chips in devices. *CHM

2016, Since the year is a leap year beginning on a Friday, the typical calendar page for January takes six lines.  Such months are called perverse months. (The same months will be perverse in a year starting on Saturday.  2016 has three such months, Jan, July and October. 2012 had only two. It is possible for there to be four in a single year. When will that year be?  Is it possible for there to be a year with no perverse months?
There is an inverse relationship between Friday-the-thirteenths and perverse months; so what is good for the calendar makers is bad for the superstitious., so 2016 has only one Friday the 13th

Image credit: NASA/JPL-Caltech/MSSS/TAMU
2014 The Mars rover's view of its original home planet even includes our moon, just below Earth.
The images, taken about 80 minutes after sunset during the rover's 529th Martian day (Jan. 31, 2014) are available for a broad scene of the evening sky, and a zoomed-in view of Earth and the moon.
The distance between Earth and Mars when Curiosity took the photo was about 99 million miles (160 million kilometers). * NASA

2018 The rare combination of a blue moon (generally the second full moon of a month), a Supermoon (the full moon occurring nearest to perigee when moon is closer to earth), and a total lunar eclipse occurs early in the morning (8:37 am EST). Unfortunately it was only total in the western US. It is the first such triple treat in the US since 1866. *USA Today

1715 Giovanni Francesco Fagnano dei Toschi (31 Jan 1715 in Sinigaglia, Italy - 14 May 1797 in Sinigaglia, Italy) He proved that the triangle which has as its vertices the bases of the altitudes of any triangle has those altitudes as its bisectors. *VFR  Of all the triangles that could be inscribed in a given triangle, the one with the smallest perimeter is the orthic triangle. This has sometimes been called Fagnano's Problem since it was first posed and answered by Giovanni Francesco Fagnano dei Toschi. Fagnano also was the first to show that the altitudes of the original triangle are the angle bisectors of the orhtic triangle, so the incenter of the orthic triangle is the orthocenter of the original triangle.*pb
He was the son of the mathematician
Giulio Carlo Fagnano. He calculated the integral of the tangent and also proved the reduction  formula \( \int x^n \sin {x} dx = -x^n \cos{x}+n\int x^{n-1} \cos{x} dx \)


1763 The Rt. Rev. John Mortimer Brinkley D.D. (ca. 1763 (Baptized 31 Jan,1763, Woodbridge, Suffolk – 14 September 1835, Dublin) was the first Royal Astronomer of Ireland and later Bishop of Cloyne.
He graduated B.A. in 1788 as senior wrangler and Smith's Prizeman, was elected a fellow of the college and was awarded M.A. in 1791. He was ordained at Lincoln Cathedral in the same year, and in 1792 became the second Andrews Professor of Astronomy in the University of Dublin, which carried the new title of Royal Astronomer of Ireland. Together with John Law, Bishop of Elphin, he drafted the chapter on "Astronomy" in William Paley's Natural Theology. His main work concerned stellar astronomy and he published his Elements of Plane Astronomy in 1808. In 1822 he was elected a Foreign Honorary Member of the American Academy of Arts and Sciences. He was awarded the Copley Medal by the Royal Society in 1824. Brinkley's observations that several stars shifted their apparent place in the sky in the course of a year were disproved at Greenwich by his contemporary John Pond, the Astronomer Royal. In 1826, he was appointed Bishop of Cloyne in County Cork, a position he held for the remaining nine years of his life. Brinkley was elected President of the Royal Astronomical Society in 1831, serving in that position for two years.
He died in 1835 at Leeson Street, Dublin and was buried in Trinity College chapel. He was succeeded at Dunsink Observatory by Sir William Rowan Hamilton. *Wik

1841 Samuel Loyd (31 Jan 1841 ; died 10 Apr 1911)  was an American puzzlemaker who was best known for composing chess problems and games, including Parcheesi, in addition to other mathematically based games and puzzles. He studied engineering and intended to become a steam and mechanical engineer but he soon made his living from his puzzles and chess problems. Loyd's most famous puzzle was the 14-15 Puzzle which he produced in 1878. The craze swept America where employers put up notices prohibiting playing the puzzle during office hours. Loyd's 15 puzzle is the familiar 4x4 arrangement of 15 square numbered tiles in a tray that must be reordered by sliding one tile at a time into the vacant space. *TIS When he offered a cash prize to anyone who could solve the puzzle with 14&15 reversed, it swept the country.  To show it impossible requires only a little group theory; see W. E. Story, “Note on the ‘15’ puzzle,” American Journal of Mathematics, 2, 399–404. For samples of Loyd’s many puzzles, see Mathematical Puzzles of Sam Loyd, edited by Martin Gardner, Dover 1959 [p. xi]. *VFR 
Although Lloyd popularized the puzzle in his books and articles, he most certainly did not invent it. Loyd's first article about the puzzle was published in 1886 and it wasn't until 1891 that he first claimed to have been the inventor.  The article mentioned by Story(1878) was dated prior to Loyd's first mention of the puzzle) Here is the history of the puzzle as related by Wikipedia:The puzzle was "invented" by Noyes Palmer Chapman, a postmaster in Canastota, New York, who is said to have shown friends, as early as 1874, a precursor puzzle consisting of 16 numbered blocks that were to be put together in rows of four, each summing to 34. Copies of the improved Fifteen Puzzle made their way to Syracuse, New York by way of Noyes' son, Frank, and from there, via sundry connections, to Watch Hill, RI, and finally to Hartford (Connecticut), where students in the American School for the Deaf started manufacturing the puzzle and, by December 1879, selling them both locally and in Boston, Massachusetts. Shown one of these, Matthias Rice, who ran a fancy woodworking business in Boston, started manufacturing the puzzle sometime in December 1879 and convinced a "Yankee Notions" fancy goods dealer to sell them under the name of "Gem Puzzle". In late-January 1880, Dr. Charles Pevey, a dentist in Worcester, Massachusetts, garnered some attention by offering a cash reward for a solution to the Fifteen Puzzle.
The game became a craze in the U.S. in February 1880, Canada in March, Europe in April, but that craze had pretty much dissipated by July. Apparently the puzzle was not introduced to Japan until 1889.
Noyes Chapman had applied for a patent on his "Block Solitaire Puzzle" on February 21, 1880. However, that patent was rejected, likely because it was not sufficiently different from the August 20, 1878 "Puzzle-Blocks" patent (US 207124) granted to Ernest U. Kinsey.*Wik
Play with an online version here.

1886 George Neville Watson (31 Jan 1886 in Westward Ho!, Devon, England - 2 Feb 1965 in Leamington Spa, Warwickshire, England) studied at Cambridge, and then taught at Cambridge and University College London before becoming Professor at Birmingham. He is best known as the joint author with Whittaker of one of the standard text-books on Analysis. Titchmarsh wrote of Watson's books, "Here one felt was mathematics really happening before one's eyes. ... the older mathematical books were full of mystery and wonder. With Professor Watson we reached the period when the mystery is dispelled though the wonder remains." *SAU

1914 Lev Arkad'evich Kaluznin (31 Jan 1914 in Moscow, Russia - 6 Dec 1990 in Moscow, Russia) Kaluznin is best known for his work in group theory and in particular permutation groups. He studied the Sylow p-subgroups of symmetric groups and their generalisations. In the case of symmetric groups of degree pn, these subgroups were constructed from cyclic groups of order p by taking their wreath product. His work allowed computations in groups to be replaced by computations in certain polynomial algebras over the field of p elements. Despite the fact that the earliest applications of wreath products of permutation groups was due to C Jordan, W Specht and G Polya, it was Kaluznin who first developed special computational tools for this purpose. Using his techniques, he was able to describe the characteristic subgroups of the Sylow p-subgroups, their derived series, their upper and lower central series, and more. These results have been included in many textbooks on group theory. *SAU

1928 Heinz Bauer (31 January 1928 – 15 August 2002) was a German mathematician.
Bauer studied at the University of Erlangen-Nuremberg and received his PhD there in 1953 under the supervision of Otto Haupt and finished his habilitation in 1956, both for work with Otto Haupt. After a short time from 1961 to 1965 as professor at the University of Hamburg he stayed his whole career at the University of Erlangen-Nuremberg. His research focus was the Potential theory, Probability theory and Functional analysis
Bauer received the Chauvenet Prize in 1980 and became a member of the German Academy of Sciences Leopoldina in 1986. Bauer died in Erlangen. *Wik

1929 Rudolf Ludwig Mössbauer (31 Jan 1929 -  14 September 2011) German physicist and co-winner (with American Robert Hofstadter) of the Nobel Prize for Physics in 1961 for his researches concerning the resonance absorption of gamma-rays and his discovery in this connection of the Mössbauer effect. The Mössbauer effect occurs when gamma rays emitted from nuclei of radioactive isotopes have an unvarying wavelength and frequency. This occurs if the emitting nuclei are tightly held in a crystal. Normally, the energy of the gamma rays would be changed because of the recoil of the radiating nucleus. Mössbauer's discoveries helped to prove Einstein's general theory of relativity. His discoveries are also used to measure the magnetic field of atomic nuclei and to study other properties of solid materials. *TIS
Rudolf Mössbauer was an excellent teacher. He gave highly specialized lectures on numerous courses, including Neutrino Physics, Neutrino Oscillations, The Unification of the Electromagnetic and Weak Interactions and The Interaction of Photons and Neutrons With Matter. In 1984, he gave undergraduate lectures to 350 people taking the physics course. He told his students: “Explain it! The most important thing is, that you are able to explain it! You will have exams, there you have to explain it. Eventually, you pass them, you get your diploma and you think, that's it! – No, the whole life is an exam, you'll have to write applications, you'll have to discuss with peers... So learn to explain it! You can train this by explaining to another student, a colleague. If they are not available, explain it to your mother – or to your cat!” *Wik 

1945 Persi Warren Diaconis (January 31, 1945;  ) is an American mathematician and former professional magician. He is the Mary V. Sunseri Professor of Statistics and Mathematics at Stanford University. He is particularly known for tackling mathematical problems involving randomness and randomization, such as coin flipping and shuffling playing cards.
Diaconis left home at 14 to travel with sleight-of-hand legend Dai Vernon, and dropped out of high school, promising himself that he would return one day so that he could learn all of the math necessary to read William Feller's famous two-volume treatise on probability theory, An Introduction to Probability Theory and Its Applications. He returned to school (City College of New York for his undergraduate work graduating in 1971 and then a Ph.D. in Mathematical Statistics from Harvard University in 1974), and became a mathematical probabilist.
According to Martin Gardner, at school Diaconis supported himself by playing poker on ships between New York and South America. Gardner recalls that Diaconis had "fantastic second deal and bottom deal".
Diaconis is married to Stanford statistics professor Susan Holmes. *Wik

1632 Joost Bürgi (28 Feb 1552, 31 Jan 1632) Swiss watchmaker and mathematician who invented logarithms independently of the Scottish mathematician John Napier. He was the most skilful, and the most famous, clockmaker of his day. He also made astronomical and practical geometry instruments (notably the proportional compass and a triangulation instrument useful in surveying). This led to becoming an assistant to the German astronomer Johannes Kepler. Bürgi was a major contributor to the development of decimal fractions and exponential notation, but his most notable contribution was published in 1620 as a table of antilogarithms. Napier published his table of logarithms in 1614, but Bürgi had already compiled his table of logarithms at least 10 years before that, and perhaps as early as 1588.
*TIS  I posted about Burgi and his work w/ "proto" logarithms here if you would like more detail.

1903 Norman Macleod Ferrers; (11 Aug 1829 in Prinknash Park, Upton St Leonards, Gloucestershire, England - 31 Jan 1903 in Cambridge, England)  John Venn wrote of him,.. ,
the Master, Dr Edwin Guest, invited Ferrers, who was by far the best mathematician amongst the fellows, to supply the place. His career was thus determined for the rest of his life. For many years head mathematical lecturer, he was one of the two tutors of the college from 1865. As lecturer he was extremely successful. Besides great natural powers in mathematics, he possessed an unusual capacity for vivid exposition. He was probably the best lecturer, in his subject, in the university of his day.
It was as a mathematician that Ferrers acquired fame outside the university. He made many contributions of importance to mathematical literature. His first book was "Solutions of the Cambridge Senate House Problems, 1848 - 51". In 1861 he published a treatise on "Trilinear Co-ordinates," of which subsequent editions appeared in 1866 and 1876. One of his early memoirs was on Sylvester's development of Poinsot's representation of the motion of a rigid body about a fixed point. The paper was read before the Royal Society in 1869, and published in their Transactions. In 1871 he edited at the request of the college the "Mathematical Writings of George Green" ... Ferrers's treatise on "Spherical Harmonics," published in 1877, presented many original features. His contributions to the "Quarterly Journal of Mathematics," of which he was an editor from 1855 to 1891, were numerous ... They range over such subjects as quadriplanar co-ordinates, Lagrange's equations and hydrodynamics. In 1881 he applied himself to study Kelvin's investigation of the law of distribution of electricity in equilibrium on an uninfluenced spherical bowl. In this he made the important addition of finding the potential at any point of space in zonal harmonics (1881).
Ferrers proved the proposition by Adams that "The number of modes of partitioning (n) into (m) parts is equal to the number of modes of partitioning (n) into parts, one of which is always m, and the others (m) or less than (m). " with a graphic transformation that is named for him. *SAU

1934 Duncan MacLaren Young Sommerville (24 Nov 1879 in Beawar, Rajasthan, India - 31 Jan 1934 in Wellington, New Zealand) Sommerville studied at St Andrews and then had a post as a lecturer there. He left to become Professor of Pure and Applied mathematics at Victoria College, Wellington New Zealand. He worked on non-Euclidean geometry and the History of Mathematics. He became President of the EMS in 1911. *SAU

1966 Dirk Brouwer (1 Sep 1902; 31 Jan 1966) Dutch-born U.S. astronomer and geophysicist known for his achievements in celestial mechanics, especially for his pioneering application of high-speed digital computers for astronomical computations. While still a student he determined the mass of Titan from its influence on other Saturnian moons. Brouwer developed general methods for finding orbits and computing errors and applied these methods to comets, asteroids, and planets. He computed the orbits of the first artificial satellites and from them obtained increased knowledge of the figure of the earth. His book, Methods of Celestial Mechanics, taught a generation of celestial mechanicians. He also redetermined astronomical constants.*TIS

1973 Noel Bryan Slater, often cited NB Slater, (29 July 1912 in Blackburn, Lancashire, England - January 31 1973 in Hull, England) was a British mathematician and physicist who worked on including statistical mechanics and physical chemistry, and probability theory.*Wik

1995 George Robert Stibitz (30 Apr 1904, 31 Jan 1995) U.S. mathematician who was regarded by many as the "father of the modern digital computer." While serving as a research mathematician at Bell Telephone Laboratories in New York City, Stibitz worked on relay switching equipment used in telephone networks. In 1937, Stibitz, a scientist at Bell Laboratories built a digital machine based on relays, flashlight bulbs, and metal strips cut from tin-cans. He called it the "Model K" because most of it was constructed on his kitchen table. It worked on the principle that if two relays were activated they caused a third relay to become active, where this third relay represented the sum of the operation. Also, in 1940, he gave a demonstration of the first remote operation of a computer.*TIS

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 30 January 2019

Lies, Damned Lies, and Something About Statistics

In a long passed discussion about quotations on the AP Statistics news group the quotation, “There are three kinds of lies; lies, damned lies, and statistics.” came up. The quote is usually attributed to either Mark Twain or Disraili, and several nice notes regarding the veracity of the quote, and its origin, were contributed. Here are snips and direct quotes from the ones that seemed most interesting…

Chris Olsen waded in with this:
“I was reading something or other recently -- I don't remember what it was, but do remember it was a statistician writing -- and he alluded to this quote. In the following the [] are my interjections. What the statistician said was that Disraeli was arguing for or against [I kind of think against] the repeal of the Corn Laws in the English Parliament [possibly in 1846]. An individual [Robert Peel?] on the other side pointed out some sort of statistic in arguing the other side of the issue, and that is when Disraeli is alleged to have made the damn remark.“

David Bee added a source for a slightly different version of the quote:
“…on Page 242 of their compilation 'Statistically Speaking' (1996), compilers CC Gaither and AE Cavazos-Gaither have the following, attributed to Disraeli in George Seldes's 1960 book The Great Quotations: There are lies, damn lies, and church statistics.”

Rex Bogg’s contributed a link to HYPERLINK "" Quote-Unquote , a web site with the radio articles of Nigel Rees. About the topic in question, he writes:
“Although sometimes attributed to Mark Twain – because it appears in his posthumously-published Autobiography (1924) – this should more properly be ascribed to Disraeli, as indeed Twain took trouble to do: his exact words being, ‘The remark attributed to Disraeli would often apply with justice and force: “There are three kinds of lies: lies, damned lies, and statistics”.’
On the other hand, the remark remains untraced among Disraeli’s writings and sayings and Lord Blake, Disraeli’s biographer, does not know of any evidence that Disraeli said any such thing and thinks it most unlikely that he did. So why did Twain make the attribution? A suggestion: Leonard Henry Courtney, the British economist and politician (1832-1918), later Lord Courtney, gave a speech on proportional representation ‘To My Fellow-Disciples at Saratoga Springs’, New York, in August 1895, in which this sentence appeared: ‘After all, facts are facts, and although we may quote one to another with a chuckle the words of the Wise Statesman, “Lies - damn lies - and statistics,” still there are some easy figures the simplest must understand, and the astutest cannot wriggle out of.’
It is conceivable that Twain acquired the quotation from this - and also its veiled attribution to a ‘Wise Statesman’, whom he understood to be Disraeli. The speech was reproduced in the (British) National Review, No. 26, in the same year. Subsequently, Courtney’s comment was reproduced in an article by J.A. Baines on ‘Parliamentary Representation in England illustrated by the Elections of 1892 and 1895’ in the Journal of the Royal Statistical Society, No. 59 (1896): ‘We may quote to one another with a chuckle the words of the Wise Statesman, lies, damn lies, and statistics, still there are some easy figures which the simplest must understand but the astutest cannot wriggle out of.’ It would be a reasonable assumption that Courtney was referring to Disraeli by his use of the phrase ‘Wise Statesman’, though the context in which the phrase is used is somewhat complicated. For some reason, at this time, allusions to rather than outright quotations of Disraeli were the order of the day (he had died in 1881). Compare the fact that the remark to an author who had sent Disraeli an unsolicited manuscript – ‘Many thanks; I shall lose no time in reading it’ – is merely ascribed to ‘an eminent man on this side of the Atlantic’ by G.W.E. Russell in Collections and Recollections, Chap. 31 (1898).
Comparable sayings: Dr Halliday Sutherland’s autobiographical A Time to Keep (1934) has an account of Sir Henry Littlejohn, ‘Police Surgeon, Medical Officer of Health and Professor of Forensic Medicine at the University [Edinburgh] ... Sir Henry’s class at 9 a.m. was always crowded, and he told us of the murder trials of the last century in which he had played his part. It was Lord Young [judge] who said, “There are four classes of witnesses - liars, damned liars, expert witnesses, and Sir Henry Littlejohn”.’ Lies, Damn Lies, and Some Exclusives was the title of a book about British newspapers (1984) by Henry Porter. ‘There are lies, damned lies ... and Fianna Fáil party political broadcasts’ - Barry Desmond MEP, (Irish) Labour Party director of elections, in November 1992.

On This Day in Math - January 30

God may not play dice with the universe,
but something strange is going on with the prime numbers.
~Paul Erdos **

(Not realy,Referencing Albert Einstein's famous remark that "God does not play dice with the universe", this is attributed to Erdős in "Mathematics : Homage to an Itinerant Master" by D. Mackenzie, in Science 275:759 (1997), but has also been stated to be a comment originating in a talk given by Carl Pomerance on the Erdős-Kac theorem, in San Diego in January 1997, a few months after Erdős's death. Confirmation of this by Pomerance is reported in a statement posted to the School of Engineering, Computer Science & Mathematics, University of Exeter, where he states it was a paraphrase of something he imagined Erdős and Mark Kac might have said, and presented in a slide-show, which subsequently became reported in a newspaper as a genuine quote of Erdős the next day. In his slide show he had them both reply to Einstein's assertion: "Maybe so, but something is going on with the primes.")

The 30th day of the year; both the dodecahedron and the icosahedron have 30 edges. They may be positioned at a common center so that in the center of each of the 12 faces of the dodecahedron is one of the 12 vertices of the icosahedron, in the center of each of the 20 faces of the icosahedron is one of the 20 vertices of the dodecahedron, and the 30 edges of the dodecahedron and the 30 edges of the icosahedron cross each other at right angles at their midpoints. (I find this incredibly wonderful)

astounding to me, but 11+22+33...+3030 = 208492413443704093346554910065262730566475781 is prime Republic of Math ‏@republicofmath If there is another prime of this type, it will have over 20025 digits.

7! hours is 30 weeks

and from *@MathYearRound
30 = 2*3*5 (first 3 primes).
30 =\( 1^2+2^2+3^2+4^2 \)(first 4 perfect squares).
30 = 1*1*2*3*5 (first 5 Fibonacci #).


10 BCE Solar eclipse in Iran, as reported on a cuneiform tablet from the British Museum *History of Astronomy ‏@hist_astro  (I have to admit that every time I look at this image it reminds me of the map of Michigan's lower peninsula, and that little chip up by the little finger is Grand Traverse Bay, which I see from my dining room)

1610 Galileo writes to Belisario Vinta, with notes on his long observation of the moon with a new twenty-power scope. A letter containing much of what was to appear about the Moon in Sidereus Nuncius, two months later. *Drake, Galileo at Work; 1978

1830 In a letter to Laplace, Gauss writes about a "curious problem" that he had been working on for twelve years.  He gives the limiting value of  the frequency of distribution of positive integers in the continued fraction of a random number (now called the Gauss-Kuzmin Distribution) as \( log_2(1+x) \) . He then asks if Laplace can offer help in finding the error term. *Math World

1897 Mary Frances Winston elected to membership in the American Mathematical Society. The previous year she received her PhD at G¨ottingen, being the first American woman to receive a PhD in mathematics at a German university. *G. B. Price, History of the Department of Mathematics of the University of Kansas, 1866–1970, p. 70

1884 Sonja Kovelevskiaya gives her first university lecturer. This was the first regular lecture by a woman at a research institution in any field in modern times. [The Mathematical Intelligencer, 6(1984), no. 1, p. 29] *VFR

1925 The U.S. History of Science Society was incorporated under the laws of the District of Columbia. The first president was Lawrence Joseph Henderson (1878–1924). The movement to form the society was begun by David Eugene Smith and today is the most important historical society in the world. *VFR

1952 Two New Primes Found with SWAC. Using the Standards Western Automatic Computer (SWAC), researchers found two new prime numbers the first time they attempted a prime-searching program on the computer. Within the year, three other primes had been found. The National Bureau of Standards funded construction of the SWAC in Los Angeles in 1950 and it ran, in one form or another, until 1967.
*CHM {The first two primes found with SWAC were M521, M607. In 1951 Ferrier used a mechanical desk calculator to find the 44 digit prime (2148+1)/17 = 20988936657440586486151264256610222593863921.
The first primes found with an electronic computer were by Miller and Wheeler (Nature, 168 (1951) 838) in 1951 when they found several new primes, including the 79 digit 180(2127-1)2+1 }

1982 First computer virus, the Elk Cloner, written by 15-year old Rich Skrenta, is found in the wild. It infects Apple II computers via floppy disk. *Wik

1988 Science News reports that Noam D. Elkies, age 21, of Harvard found four fourth-powers whose sum is another fourth-power, thereby providing a counterexample to a conjecture of Euler in 1769. (Euler's conjecture was that the sum of the first n integers each raised to the nth power can not be an nth power.) The smallest number in his counterexample had eight digits. Later Roger Frye of Thinking Machines Corporation, Cambridge, MA, found the smallest counterexample:
95,8004 + 217,5194 + 414,5604 + 414 5604 = 422,4184 .
This took some 100 hours on a Connection Machine. Can you figure out how to verify this example using your calculator (which only displays 8 or 10 digits)? [Mathematics Magazine 61 (1988), p 130; Science 239 (1988), p 464]. *VFR
(Euler's general conjecture had been proven false by L. J. Lander and T. R. Parkin in 1966 when they found a counterexample for fifth powers. Elkies had suggested the computer approach that provided the minimal solution. It is still unknown if there are counterexamples above n=5)

1990 Ruth Lawrence sends a paper on homological representations of the Hecke algebra, introducing, among other things, certain novel linear representations of the braid group, the Lawrence–Krammer representation to the journal, Communications in Mathematical Physics

1996 Yuji Hyakutake in Japan discovered a new comet using 25x150 binoculars. The comet was designated Comet C/1996 B2 (Hyakutake). As subsequent observations of the new comet were obtained, Brian Marsden from the IAU Central Bureau was able to compute the comet's orbital elements, and these computations indicated that the comet will pass as close as 0.10 AU (9.3 million miles) from the Earth on March 25, 1996! The comet has become a bright naked-eye object and remained so in March, April and May in 1996. The comet had exceeded expectations, becoming the brighest comet since Comet West in 1976. A long tail of up to 100 degrees was reported, and small fragments have been observed to break off the main nucleus. Comet Hyakutake is indeed the Great Comet of 1996. *jpl.nasa
Hyakutake discovered C/1996 B2 while looking for C/1995 Y1, a comet he had discovered a few weeks before. He died in Kokubu, Kagoshima, in 2002 at age 51 of an aneurysm which had led to internal bleeding. *Wik


1619 Michelangelo Ricci  (30 Jan 1619 in Rome, Italy - : 12 May 1682 in Rome) In 1666, he found the tangent lines to the parabolas of Fermat. *VFR Michelangelo Ricci was a friend of Torricelli; in fact both were taught by Benedetti Castelli. He studied theology and law in Rome and at this time he became friends with René de Sluze. It is clear that Sluze, Torricelli and Ricci had a considerable influence on each other in the mathematics which they studied.
Ricci made his career in the Church. His income came from the Church, certainly from 1650 he received such funds, but perhaps surprisingly he was never ordained. Ricci served the Pope in several different roles before being made a cardinal by Pope Innocent XI in 1681.
Ricci's main work was Exercitatio geometrica, De maximis et minimis (1666) which was later reprinted as an appendix to Nicolaus Mercator's Logarithmo-technia (1668). It only consisted of 19 pages and it is remarkable that his high reputation rests solely on such a short publication.
In this work Ricci finds the maximum of xm(a - x)n and the tangents to ym = kxn. The methods are early examples of induction. He also studied spirals (1644), generalised cycloids (1674) and states explicitly that finding tangents and finding areas are inverse operations (1668). *SAU

1755 Nikolai Fuss (30 Jan 1755 in Basel, Switzerland - 4 Jan 1826 in St Petersburg, Russia) was a Swiss mathematician whose most important contribution was as amanuensis to Euler after he lost his sight. Most of Fuss's papers are solutions to problems posed by Euler on spherical geometry, trigonometry, series, differential geometry and differential equations. His best papers are in spherical trigonometry, a topic he worked on with A J Lexell and F T Schubert. Fuss also worked on geometrical problems of Apollonius and Pappus. He made contributions to differential geometry and won a prize from the French Academy in 1778 for a paper on the motion of comets near some planet Recherche sur le dérangement d'une comète qui passe près d'une planète. Fuss won other prizes from Sweden and Denmark. He contributed much in the field of education, writing many fine textbooks. *SAU

1805 Edward Sang,(30 Jan 1805 in Kirkcaldy, Fife, Scotland - 23 Dec 1890) A native of Fife, Sang wrote extensively on mathematical, mechanical, optical and actuarial topics. *SAU

1865 Georg Landsberg (30 Jan 1865 , 14 Sept 1912) studied the theory of functions of two variables and also the theory of higher dimensional curves. In particular he studied the role of these curves in the calculus of variations and in mechanics.
He worked with ideas related to those of Weierstrass, Riemann and Heinrich Weber on theta functions and Gaussian sums. His most important work, however was his contribution to the development of the theory of algebraic functions of a single variable. Here he studied the Riemann-Roch theorem.
He was able to combine Riemann's function theoretic approach with the Italian geometric approach and with the Weierstrass arithmetical approach. His arithmetic setting of this result led eventually to the modern abstract theory of algebraic functions.
One of his most important works was Theorie der algebraischen Funktionen einer Varaiblen (Leipzig, 1902) which he wrote jointly with Kurt Hensel. This work remained the standard text on the subject for many years. *SAU

1918 Heinz Rutishauser (30 January 1918 in Weinfelden, Switzerland; 10 November 1970 in Zürich) was a Swiss mathematician and a pioneer of modern numerical mathematics and computer science. *Wik

1925 Douglas Engelbart is Born, best known for inventing the computer mouse. Engelbart publically demonstrated the mouse at a computer conference in 1968, where he also showed off work his group had done in hypermedia and on-screen video teleconferencing. The founder of the Bootstrap Institute, Engelbart has 20 patents to his name.*CHM


1954 Gino Benedetto Loria (19 May 1862 in Mantua, Italy - 30 Jan 1954 in Genoa, Italy) In his day, Loria was arguably the pre-eminent historian of mathematics in Italy. A full professor of higher geometry at the University of Genoa beginning in 1891, Loria wrote the history of mathematics as a mathematician writing for other mathematicians. He emphasised this approach repeatedly in his works. For instance, in the introduction to his 'Storia delle matematiche dall'alba della civilità al tramonto del secolo XIX' (History of Mathematics from the Dawn of Civilisation to the End of the 19th Century), he stated that general history of mathematics was written "by a mathematician for mathematicians". *SAU

1977 Harry Clyde Carver (December 4, 1890 – January 30, 1977) was an American mathematician and academic, primarily associated with the University of Michigan. He was a major influence in the development of mathematical statistics as an academic discipline.
Born in Waterbury, Connecticut, Carver was educated at the University of Michigan, earning his B.S. degree in 1915, and the next year becoming an instructor in mathematics; he taught statistics in actuarial applications. At the time, the University of Michigan was only the second such institution in the United States to offer this type of course, after the pioneering Iowa State University. Carver was appointed assistant professor at Michigan in 1918, then associate professor (1921) and full professor (1936); during this period the University's program in mathematical statistics and probability underwent significant expansion.
In 1930 Carver founded the journal Annals of Mathematical Statistics, which over time became an important periodical in the field. Financial support, however, was lacking in the midst of the Great Depression; in January 1934 Carver undertook financial responsibility for the Annals and maintained the existence of the journal at his own expense. In 1935 he helped to start the Institute of Mathematical Statistics, which in 1938 assumed control over the journal; Samuel S. Wilks succeeded Carver as editor in the same year. The Institute has named its Harry C. Carver Medal after him.
With the coming of World War II, Carver devoted his energies to solving problems in aerial navigation, an interest he maintained for the remainder of his life. *Wik

1991 John Bardeen (23 May 1908, 30 Jan 1991) American physicist who was cowinner of the Nobel Prize for Physics in both 1956 and 1972. He shared the 1956 prize with William B. Shockley and Walter H. Brattain for their joint invention of the transistor. With Leon N. Cooper and John R. Schrieffer he was awarded the 1972 prize for development of the theory of superconductors, usually called the BCS-theory (after the initials of their names). *TIS

1992 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

1998 Samuel Eilenberg (September 30, 1913 – January 30, 1998) was a Polish and American mathematician born in Warsaw, Russian Empire (now in Poland) and died in New York City, USA, where he had spent much of his career as a professor at Columbia University.
He earned his Ph.D. from University of Warsaw in 1936. His thesis advisor was Karol Borsuk. His main interest was algebraic topology. He worked on the axiomatic treatment of homology theory with Norman Steenrod (whose names the Eilenberg–Steenrod axioms bear), and on homological algebra with Saunders Mac Lane. In the process, Eilenberg and Mac Lane created category theory.
Eilenberg was a member of Bourbaki and with Henri Cartan, wrote the 1956 book Homological Algebra, which became a classic.
Later in life he worked mainly in pure category theory, being one of the founders of the field. The Eilenberg swindle (or telescope) is a construction applying the telescoping cancellation idea to projective modules.
Eilenberg also wrote an important book on automata theory. The X-machine, a form of automaton, was introduced by Eilenberg in 1974. *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

Tuesday 29 January 2019

On This Day in Math - January 29

There is no philosophy which is not founded upon knowledge of the phenomena, but to get any profit from this knowledge it is absolutely necessary to be a mathematician.
~Daniel Bernoulli

The 29th day of the year; 229 = 536870912 a nine-digit number with no digit repeated. Is it possible to create a power of a single digit number that has ten distinct digits?

The digits of 29 appear in one of the most unusual un-mistake I can imagine in numbers, if you inadvertently wrote the exponents of \(2^5 9^2 \) on the same line as the bases you get 2592 which is = \(2^5 9^2 \). It is believed that this is the only such example.

28 is a perfect number, and I think it is interesting that we have proven that if there is ever an odd perfect number it will have at least 29 prime factors, the largest will be greater than 108

And I like this "Euler-like" function from Legendre in 1798, 2n2 + 29 is prime for all  n from 0 through 28.


1697 (o.s.) Newton received two challenge problems from Johann Bernoulli, one being the Brachistochrone problem published in Acta eruditorum the previous June and addressed “to the shrewdest mathematicians in the world.” The next day Newton posted his solution to the Royal Society. When Bernoulli saw the anonymous solution he recognized it as “ex ungue leonem” (as the lion is recognized by his paw). *Westfall, Never at Rest, pg 581

1769 "On the morning of the 29 January 1769, seven ‘transit’ astronomers went to Catherine the Great’s Winter Palace in St Petersburg because the Empress had requested to meet her astronomical army before they set out to their destinations across the Russian empire. The German Georg Moritz Lowitz and his assistant, the Russian Pjotr Inochodcev were going to Guryev, Russia (modern Atyrau, Kazakhstan), the Russian Stepan Rumovsky and the Swiss Jacques André Mallet and Jean-Louis Pictet were all travelling to different locations on the Kola peninsula, the Germans Christoph Euler was ordered to Orsk and Wolfgang Ludwig Krafft to Orenburg. *Andrea Wulf, Transit of Venus Web Site

1824 Even right at the end of his life, former President Thomas Jefferson was still reporting on the current news in mathematics. On this day he writes to Patrick K. Rogers concerning the abandonment of fluxional calculus at Cambridge in favour of the Leibnizian notation , "The English generally have been very stationary in later times, and the French, on the contrary, so active and successful, particularly in preparing elementary books, in mathematics and natural sciences, that those who wish for instruction without caring from what nation they get it, resort universally to the latter language. Besides the earlier and invaluable works of Euler and Bezout, we have latterly that of Lacroix in mathematics, of Legendre in geometry, . . . to say nothing of the many detached essays of Monge and others, and the transcendent labours of Laplace, and I am informed by a highly instructed person recently from Cambridge, that the mathematicians of that institution, sensible of being in the rear of those of the continent, and ascribing the cause much to their long-continued preference of the geometrical over the analytical methods, which the French have so long cultivated and improved, have now adopted the latter; and that they have also given up the fluxionary, for the differential calculus. " *John Fauval, Lecture at Univ of Va.

1939 J. Robert Oppenheimer hears about the discovery of fission. Within a few minutes, he realizes that excess neutrons must be emitted, and that it might be possible to build a bomb. Fission was discovered on December 17, 1938 by German Otto Hahn and his assistant Fritz Strassmann, but Oppenheimer probably hear about it through the publications which explained it (and named it) theoretically in January 1939 by Lise Meitner and her nephew Otto Robert Frisch. Frisch named the process by analogy with biological fission of living cells. *Wik

1957 SRI and GE Meet to Choose a Place for ERMA's MICR Encoding
ERMA (Electronic Recording Machine - Accounting), developed by SRI and General Electric for the Bank of America in California, employed Magnetic Ink Character Recognition (MICR) as a tool that captures data from checks. IBM was making a strong case to place the encoding at the top of a check. SRI and GE conducted a series of tests that clearly demonstrated the advantage of the bottom-of-the-check encoding. *CHM

1970 Yuri Matiyasevich presents proof of Hilbert's 10th Problem.  Having been frustrated  by the problem, he had given up hope of solving it. In December of the previous year after having been asked to review an article by Julia Robinson, he was inspired by the novelty of her approach and went back to work on H10.  By Jan 3, 1970 he had a proof.  He would present the proof on January 29, 1970


1688 Emanuel Swedenborg (29 Jan 1688; 29 Mar 1772) Swedish scientist, philosopher and theologian. While young, he studied mathematics and the natural sciences in England and Europe. From Swedenborg's inventive and mechanical genius came his method of finding terrestrial longitude by the Moon, new methods of constructing docks and even tentative suggestions for the submarine and the airplane. Back in Sweden, he started (1715) that country's first scientific journal, Daedalus Hyperboreus. His book on algebra was the first in the Swedish language, and in 1721 he published a work on chemistry and physics. Swedenborg devoted 30 years to improving Sweden's metal-mining industries, while still publishing on cosmology, corpuscular philosophy, mathematics, and human sensory perceptions. *TIS

1700 Daniel Bernoulli (29 January 1700 (8 Feb new style), 8 March 1782) was a Dutch-Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family. He is particularly remembered for his applications of mathematics to mechanics, especially fluid mechanics, and for his pioneering work in probability and statistics. Bernoulli's work is still studied at length by many schools of science throughout the world. The son of Johann Bernoulli (one of the "early developers" of calculus), nephew of Jakob Bernoulli (who "was the first to discover the theory of probability"), and older brother of Johann II, He is said to have had a bad relationship with his father. Upon both of them entering and tying for first place in a scientific contest at the University of Paris, Johann, unable to bear the "shame" of being compared as Daniel's equal, banned Daniel from his house. Johann Bernoulli also plagiarized some key ideas from Daniel's book Hydrodynamica in his own book Hydraulica which he backdated to before Hydrodynamica. Despite Daniel's attempts at reconciliation, his father carried the grudge until his death.
He was a contemporary and close friend of Leonhard Euler. He went to St. Petersburg in 1724 as professor of mathematics, but was unhappy there, and a temporary illness in 1733 gave him an excuse for leaving. He returned to the University of Basel, where he successively held the chairs of medicine, metaphysics and natural philosophy until his death.
In May, 1750 he was elected a Fellow of the Royal Society. He was also the author in 1738 of Specimen theoriae novae de mensura sortis (Exposition of a New Theory on the Measurement of Risk), in which the St. Petersburg paradox was the base of the economic theory of risk aversion, risk premium and utility.
One of the earliest attempts to analyze a statistical problem involving censored data was Bernoulli's 1766 analysis of smallpox morbidity and mortality data to demonstrate the efficacy of vaccination. He is the earliest writer who attempted to formulate a kinetic theory of gases, and he applied the idea to explain Boyle's law. He worked with Euler on elasticity and the development of the Euler-Bernoulli beam equation. *Wik

1761 Josef (also José or Joseph) de Mendoza y Ríos (29 January 1761; Sevilla, Spain - 4 March 1816 Brighton, England) was a Spanish astronomer and mathematician of the 18th century, famous for his work on navigation. The first work of Mendoza y Ríos was published in 1787: his treatise, Tratado de Navegación, about the science and technique of navigation in two tomes. He also published several tables for facilitating the calculations of nautical astronomy and useful in navigation to calculate the latitude of a ship at sea from two altitudes of the sun, and the longitude from the distances of the moon from a celestial body.
In the field of the nautical instruments, he improved the reflecting circle.
In 1816, he was elected a foreign member of the Royal Swedish Academy of Sciences. @Wik

1810 Ernst Eduard Kummer (29 Jan 1810; 14 May 1893) He was professor at the University of Breslau(now Wroclaw, Poland) in 1842-1855 and developed his theory of ideals here. Kronecker studied with him. Later he replaced Dirichlet at The University of Berlin. He died at age 83, after a short attack of influenza. German mathematician whose introduction of ideal numbers, which are defined as a special subgroup of a ring, extended the fundamental theorem of arithmetic to complex number fields. He worked on Function theory, and extended Gauss's work on hypergeometric series, giving developments that are useful in the theory of differential equations. He was the first to compute the monodromy groups of these series. Later. Kummer devoted himself to the study of the ray systems, but treated these geometrical problems algebraically. He also discovered the fourth order surface based on the singular surface of the quadratic line complex. This Kummer surface has 16 isolated conical double points and 16 singular tangent planes. *TIS and others An oft told, and almost certianly untrue anecdote is told about Kummer: Kummer was so inept at simple arithmetic that he often asked students to help him in class. On one occasion, Kummer sought the result of a simple multiplication. "Seven times nine," he began. "Seven times nine is er - ah - ah - seven times nine is..." "Sixty-one," a mischievous student suggested and Kummer wrote the "answer" on the blackboard. "Sir," another one interjected, "it should be sixty-seven." "Come, gentlemen, it can't be both," Kummer exclaimed. "It must be one or the other!" According to Erdos, Kumer reasoned out the answer as follows, -It can't be 61 as that is prime, as is 67, and 65 is a multiple of five, and 69 is too big, so it must be 63.

1817 William Ferrel (29 Jan 1817; 18 Sep 1891) American meteorologist who was an important contributor to the understanding of oceanic and atmospheric circulation. He was able to show the interrelation of the various forces upon the Earth's surface, such as gravity, rotation and friction. Ferrel was first to mathematically demonstrate the influence of the Earth's rotation on the presence of high and low pressure belts encircling the Earth, and on the deflection of air and water currents. The latter was a derivative of the effect theorized by Gustave de Coriolis in 1835, and became known as Ferrel's law. Ferrel also considered the effect that the gravitational pull of the Sun and Moon might have on the Earth's rotation and concluded (without proof, but correctly) that the Earth's axis wobbles a bit. *TIS (A more complete biography is here)

1838 Edward Williams Morley (29 Jan 1838; 24 Feb 1923) American chemist who is best known for his collaboration with the physicist A.A. Michelson in an attempt to measure the relative motion of the Earth through a hypothetical ether (1887). He also studied the variations of atmospheric oxygen content. He specialized in accurate quantitative measurements, such as those of the vapour tension of mercury, thermal expansion of gases, or the combining weights of hydrogen and oxygen. Morley assisted Michelson in the latter's persuit of measurements of the greatest possible accuracy to detect a difference in the speed of light through an omnipresent ether. Yet the ether could not be detected and the physicists had seriously to consider that the ether did not exist, even questioning much orthodox physical theory. *TIS

1888 Sydney Chapman (29 Jan 1888; 16 Jun 1970) English mathematician and physicist noted for his research in geophysics. After graduation (1910) he worked at the Greenwich Observatory, but returned to Cambridge upon the outbreak of WW I. Between 1915 and 1917 he completed a series of important papers on thermal diffusion and the fundamentals of gas dynamics. He developed systematic approximations to the Maxwell-Boltzmann formulation for the velocity distribution function for interacting particles under general force laws. During WW II he worked on military operational research and incendiary bomb problems. Chapman's main area of research was geomagnetism, beginning in 1913 and extending to terrestrial and interplanetary magnetism, the ionosphere and the aurora borealis.*TIS

1894 Miss Helen Almira Shaffer, A. M., LL. D., President of Welleslev College,
died of pneumonia at the college, on January 29, aged 54 years. She was chief teacher
of Mathematics for ten years in the St. Louis High School. In 1877 she accepted the
professorship of Mathematics in Wellesley, which she filled until 1888, when she became
president of that institution. *The American Mathematical Monthly Vol. 1, No. 2, Feb., 1894

1926 Abdus Salam (29 Jan 1926; 21 Nov 1996) Pakistani-British nuclear physicist who shared the 1979 Nobel Prize for Physics with Steven Weinberg and Sheldon Lee Glashow. Each had independently formulated a theory explaining the underlying unity of the weak nuclear force and the electromagnetic force. His hypothetical equations, which demonstrated an underlying relationship between the electromagnetic force and the weak nuclear force, postulated that the weak force must be transmitted by hitherto-undiscovered particles known as weak vector bosons, or W and Z bosons. Weinberg and Glashow reached a similar conclusion using a different line of reasoning. The existence of the W and Z bosons was eventually verified in 1983 by researchers using particle accelerators at CERN. *TIS

1928 O. Timothy O’Meara born in South Africa. This expert in quadratic forms is now Provost at the University of Notre Dame. *VFR On October 8, 2008, the Mathematics Library at Notre Dame was rededicated and named for Prof. O. Timothy O’Meara. Prof. O’Meara is a noted Mathematician, who has been on the faculty of the Mathematics Department since 1962, and twice served as its chairman. In 1976 he was named to the Kenna Endowed Chair in Mathematics. He is noted for serving as the first lay Provost of the University, 1978-1996. He is now an emeritus faculty member, but still very active and interested in the library *ND Web Site

1928 Joseph Bernard Kruskal, Jr. (January 29, 1928 – September 19, 2010) was an American mathematician, statistician, computer scientist and psychometrician. He was a student at the University of Chicago and at Princeton University, where he completed his Ph.D. in 1954, nominally under Albert W. Tucker and Roger Lyndon, but de facto under Paul Erdős with whom he had two very short conversations.Kruskal has worked on well-quasi-orderings and multidimensional scaling.
He was a Fellow of the American Statistical Association, former president of the Psychometric Society, and former president of the Classification Society of North America.
In statistics, Kruskal's most influential work is his seminal contribution to the formulation of multidimensional scaling. In computer science, his best known work is Kruskal's algorithm for computing the minimal spanning tree (MST) of a weighted graph. In combinatorics, he is known for Kruskal's tree theorem (1960), which is also interesting from a mathematical logic perspective since it can only be proved nonconstructively. Kruskal also applied his work in linguistics, in an experimental lexicostatistical study of Indo-European languages, together with the linguists Isidore Dyen and Paul Black.
Kruskal was born in New York City to a successful fur wholesaler, Joseph B. Kruskal, Sr. His mother, Lillian Rose Vorhaus Kruskal Oppenheimer, became a noted promoter of Origami during the early era of television. He died in Princeton. *Wik


1707 Otto Mencke (22 March (OS) April 2, 1644 – 18 Jan (OS) 29 Jan 1707) was a 17th-century German philosopher and scientist. He obtained his doctorate at the University of Leipzig in 1666 with a thesis entitled: Ex Theologia naturali — De Absoluta Dei Simplicitate, Micropolitiam, id est Rempublicam In Microcosmo Conspicuam.
He is notable as being the founder of the very first scientific journal in Germany, established 1682, entitled: Acta Eruditorum. *Wik

1715 Bernard Lamy (15 June 1640, in Le Mans, France – 29 January 1715, in Rouen, France) was a French Oratorian mathematician and theologian. He wrote on geometry and mechanics and developed the idea of a parallelogram of forces at about the same time as Newton and Verignon. The Law of Sines as applied to three static forces in mechanics is sometimes called Lamy's Rule. (Would provide an interesting variation for Pre-calc classes)

1859 William Cranch Bond (9 Sep 1789, 29 Jan 1859) American astronomer who, with his son, George Phillips Bond (1825-65), discovered Hyperion, the eighth satellite of Saturn, and an inner ring called Ring C, or the Crepe Ring. While W.C. Bond was a young clockmaker in Boston, he spent his free time in the amateur observatory he built in part of his home. In 1815 he was sent by Harvard College to Europe to visit existing observatories and gather data preliminary to the building of an observatory at Harvard. In 1839 the observatory was founded. He supervised its construction, then became its first director. Together with his son he developed the chronograph for automatically recording the position of stars. They also took some of the first recognizable photographs of celestial objects.*TIS

1864 Benoît "Claudius" Crozet (December 31, 1789; Villefranche, France – January 29, 1864) was an educator and civil engineer.
After serving in the French military, in 1816, he immigrated to the United States. He taught at the U.S. Military Academy at West Point, New York, and helped found the Virginia Military Institute at Lexington, Virginia. He was Principal Engineer for the Virginia Board of Public Works and oversaw the planning and construction of canals, turnpikes, bridges and railroads in Virginia, including the area which is now West Virginia. He became widely known as the "Pathfinder of the Blue Ridge."
On June 7, 1816, in Paris, Crozet married Agathe Decamp.
Late in fall of 1816, Crozet and his bride headed for the United States. Almost immediately after arriving, Crozet began work as a professor of engineering at the U.S. Military Academy at West Point, New York.
While at West Point, Crozet is credited by some as being the first to use the chalkboard as an instructional tool. He also designed several of the buildings at West Point. Thomas Jefferson referred to Claudius Crozet as "by far the best mathematician in the United States." He also published A Treatise on Descriptive Geometry while at West Point, a copy of which was sent to Jefferson. Jefferson's response on Nov 23, 1821 began, "I thank you, Sir, for your kind attention in sending me a copy of your valuable treatise on Descriptive geometry." He continued the messsage with praise for the work, and the instructor both. The dining hall at the Virginia Military Institute is named in his honor. It has been affectionately nicknamed "Club Crozet" by the Cadets. * Wik & Natl. Archives

1905 Robert Tucker (26 April 1832 in Walworth, Surrey, England - 29 Jan 1905 in Worthing, England) A major mathematical contribution made by Tucker was his work as editor of William Kingdon Clifford's papers. Fifty-seven of Clifford's papers were collected and edited by Tucker and published in 1882 as Mathematical Papers. Tucker also wrote many biographies including those of Gauss, Sylvester, Chasles, Spottiswoode, and Hirst, all of which appeared in Nature. But, like a number of schoolmaster's at this time, he also made a contribution to research in geometry. He wrote over 40 research papers which were published in leading journals. These papers, although sometimes not of the highest quality, do contain a number of interesting ideas. Hill specially singles out for special mention his work on the Triplicate-Ratio Circle, the group of circles sometimes known as Tucker Circles, and the Harmonic Quadrilateral. *SAU

1984 John Macnaghten Whittaker I(7 March 1905 in Cambridge, England - 29 Jan 1984 in Sheffield, England) was the son of Edmund Whittaker. He studied at Edinburgh University and Cambridge. After posts at Edinburgh and Cambridge he became Professor at Liverpool though his tenure was interrupted by service in World War II. He left Liverpool to become Vice-Chancellor of Sheffield University. He worked in Quantum Mechanics and Complex Analysis. *SAU

1999 Viktor Aleksandrovich Gorbunov (17 Feb 1950 in Russia - 29 Jan 1999 in Novosibirsk, Russia) He published his first paper in 1973 being a joint work with A I Budkin entitled Implicative classes of algebras (Russian). The implicative class of algebras is a generalisation of quasivarieties. The structural characteristics of the implicative class are studied in this paper. A second join paper with Budkin On the theory of quasivarieties of algebraic systems (Russian) appeared in 1975. In the same year he published Filters of lattices of quasivarieties of algebraic systems (Russian), this time written with V P Belkin. In fact he had written six papers before his doctoral thesis On the Theory of Quasivarieties of Algebraic Systems was submitted. He received the degree in 1978. Gorbunov continued working at Novosibirsk State University, being promoted to professor. He also worked as a researcher in the Mathematics Institute of the Siberian Branch of the Russian Academy of Sciences. *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

Monday 28 January 2019

On Circles and Equilateral Triangles

One blog I follow regularly is Antonio Gutierrez's gogeometry. If you teach/study/like plane geometry he should be one of your regular references.

Recently among his posts have been a couple with a related theme, circles inscribed or circumscribed about an equilateral triangle. I'm listing these because they are each a wonderful relationship, and together give these otherwise somewhat mundane seeming triangles a luster students?teachers/others might miss.
I will post the problems, but not the proofs, which (if you can't/won't work them out yourself you can find at the links provided to Antonio's site.

So on we go...
1) draw a circle and inscribe an equilateral triangle. Now pick any point on the circumference and construct segments from this point to the three vertices. The sum of the lengths of the two shorter segments will equal the third.  If you want to work on this solution yourself, there is a hint a little below the image, and then a link to the solution as well. 

Here is the Hint for the first problem,   Think about what you know about the quadrilateral ABDC?      The problem, and solution is here.

2) OK:
Same triangle, same circle, but now we sum the square of the three distances ...????? and they sum to twice the square of a side of the equilateral triangle. That proof is here.  and the hint???????? it's something you might know about Cosines. 

3) And now one with the circle on the inside. Again, from any point on the circle construct segments to the three vertices of the equilateral triangle. Again the sum of the squares is related to a side length, but I'll let you chase that down for yourself. No hint this time, work that out on your own, or you can go to the site here.

Addendum: John Golden sent a comment with a link to a GeoGebra sketch showing all three.