## Friday 24 May 2024

### On This Day in Math - May 24

Hence no force, however great,
can stretch a cord, however fine,
into a horizontal line
which is accurately straight:
there will always be a bending downwards.

~William Whewell

The 144th day of the year; 144 is the only non-trivial square in the Fibonacci Sequence.In fact, there are only four  Fibonacci numbers that are  perfect powers, 0, 1, 8, and 144. And we haven't known that for so very long. Here is the story from , Professor Stewart's Incredible Numbers.

In 1913 R. D. Carmichael proved his conjecture that for any Fibonacci Number F(n), greater than F(12)=144, has at least one prime factor that is not a factor of any earlier Fibonacci number.

144 is the largest possible SP (sum times product) number. If you take the sum of the digits of a number, and also the product of the digits, and then multiply the two outcomes, there are only three positive numbers for which you will get the original value. One works, trivially. The other two are 135 and 144. 135-> (1+3+5)*(1*3*5) = 9*15=135. 144->(1+4+4)*(1*4*4)= 9 * 16 = 144.
.A Good exercise for students is to take the SP product in a iteration to find out if it goes to zero, or repeats some pattern, or lands eventually on one of these three fixed points(that's four fixed points if you count zero). (Try it with your students). 23->5*6 = 30. 30-> 3*0 = 0.... fixed point.

$144^5 = 27^5 + 84^5 + 110^5 + 133^5$ This counter-example disproved Euler's Conjecture that n nth powers are needed to sum to an nth power. It is also part of one of the shortest papers ever published in a math journal(two sentences)

(Squares are important in knowing if a number, n is Fibonacci or not. N is Fibonacci IFF one or both of $5n^2 \pm 4$ is a perfect square. )

And in a "Talking Numbers" post by John Golden I re-learned that 144 is the smallest perimeter of a primitive Pythagorean triple that is a square number. (see below)

144 is also the smallest square number which is also a square when its digits are reversed 144 = 12 2 while 441= 212

144 is a Sum-Product number, (n = (sum of digits of n)x(product of digits of n)  144=(1+4+4)x(1x4x4) . It is the largest such number.  There is one more multi-digit example that is slightly smaller.

The sum of the first 144 decimal digits of pi (don't use the 3.) is 666, "The Number of the Beast."  One person wrote that they thought that was gross!  (  sorry!   :-{  , bad pun)

144 is the only year day that is a square number that is the perimeter of a primitive Pythagorean Triangle.  (16, 63, 65) *Benjamin Vitale ‏@BenVitale

EVENTS

997 Al-Bırunı in Kath and Abul-Wafa in Baghdad simultaneously watch a lunar eclipse. The time obtained by this prearranged cooperation allowed them to determine the difference in longitude between the cities. **MacTutor

1032 The renowned "Arab" scientist Ibn Sina noted, “I saw Venus as a spot on the surface of the sun,” This is the first known record of witnessing the transit of Venus. The first recorded observation of a transit of Venus was made by Jeremiah Horrocks from his home at Carr House in Much Hoole, near Preston in England, on 4 December 1639. Kepler predicted the 1761 transit of Venus, the first such prediction in Western recorded history, and one that inspired several astronomical expeditions. *Sky and Telescope (A.M.S ‏@amoshaye pointed out that he was NOT Arabic, {Persian I believe}. It was once common to lump mid-eastern scholars who wrote in Arabic as Arabian. I write in English, but am not English, a fact that I, and all of England are equally grateful for.)
 Transit of Venus, 2004

1543 An advance copy of his work De revolutionibus orbium coelestium was presented to Copernicus. On the same day he died. *VFR

1547 Ferrari replied to Tartaglia’s letter of 21 April 1547 by sending 31 challenge problems of his own. Tartaglia solved all but the ﬁve dealing with cubic equations. *VFR
At the end of the 15th century Luca Pacioli, Leonardo’s math teacher, claimed in his Summa, the most influential maths textbook in the 16th century, that a general algebraic solution of the cubic equation was impossible. His timing was wonderful, his book was published in 1494 and a few years later the Bologna professor of mathematics Scipione del Ferro discovered the general solution of one class of cubics.
Tartaglia was convinced that he would easily defeat Ferrari but by the end of the first day it was obvious that Ferrari was his superior and so that night he slipped out of town thereby losing the contest and ruining his reputation. With the Ars magna Cardano become acknowledged as the greatest mathematician in Europe and the publication of his book is taken by many historians of mathematics to be the start of modern mathematics, a view that I don’t share.
Cardano and Ferrari’s triumph led, of course, to further speculation about a general solution to the quintic i.e. fifth order algebraic equations and Évariste Galois’ proof, in the 19th century, that none exists led to the development of group theory but that as they say is another story.
For more on this "math wars" story, see this Renaissance Mathematicus blog.

1626 Manhattan bought from the Indians for $24$ *VFR {In 1626 Peter Minuit bought Manhattan island from the local Indians for a load of cloth, beads, hatchets, and other odds and ends then worth 60 Dutch guilders. Estimated to have been worth  $24$ Dollars at a much later time. If we convert that to silver prices at the time, you could purchase about 18 Troy ounces of Silver.. Today the price of silver is about  $35$ per ounce, so .. not such a good investment in that sense... }

1683 The Ashmolean, Britain's first Museum, first opened to the public on 24 May 1683. *Ashmolean (in full the Ashmolean Museum of Art and Archaeology) on Beaumont Street, Oxford, England, is the world's first university museum. Its first building was built in 1678–1683 to house the cabinet of curiosities Elias Ashmole gave Oxford University in 1677. *Wik
G

1727 Euler arrived at St. Petersburg for the first time on 24 May 1727, only seven days after the Russian empress Katherine I, the wife of Peter I (an adopted daughter of Ernst Glilck, the provost of Marienburg, now Aliiksne) died, and he worked at the St. Petersburg Academy of Sciences until 1741 when he moved to
Berlin, there to stay until 1766. *Historia Mathematica

1844 Samuel F. B. Morse dispatched the first telegraphic message over an experimental line from Washington, D.C., to Baltimore. The message, taken from the Bible, Numbers 23:23 and recorded on a paper tape had been suggested to Morse by Annie Ellsworth, the young daughter of a friend. {Nice to have influential friends, she was the teenage daughter of the Commissioner of Patents. Congress appropriated $30,000 for a telegraph wire to be strung the 80 miles between Washington and Baltimore}..Morse sent the message from the chamber of the Supreme Court, then in the United States Capitol, to his assistant Albert Vail at the Mount Clair depot in Baltimore in 1844.*Library of Congress A photo of the actual paper tape with raised dots and dashes in the Library of Congress is here. Across the top of this artifact of his historic achievement Morse has given credit to Annie Ellsworth for suggesting the message. Wolfram alpha will let you convert any message to Morse code, or from Morse code by typing [Morse code "input"] to encode, or Morse Code message to decode. 1883 Brooklyn Bridge was opened over the East River, New York City, USA, of a breadth of 1,600 feet, navigable water with a single span. What was then regarded as the greatest engineering feat still stands in service today, and remains the world's only stone-towered, steel cabled bridge. Twice the size of the Niagara Suspension Bridge and four times the longest non-extension spans ever attempted, the total length of this colossal structure is 6,927 ft. The road bed is 80 feet wide, and at an elevation of 186 feet above high water. John Roebling, and after his death his son Washington Roebling, worked on its construction for 13 years. *TIS Contrary to the New York Times Magazine of 27 March 1983, the cables hang in the shape of parabolas, not catenaries. *VFR 1937 A temporary science exhibition called Le Palais de la Decouverte opened its doors in the west wing of the Grand Palais in time for the 1937 International Exposition of Art and Technology in Modern Life, which was to be held in Paris. It was primarily the inspiration of French physicist Jean Perrin, who won the Nobel prize in 1926 for his work on the atom. An interesting mathematical anecdote relates to the museum. The museum contains a circular room known as the "pi room". On its wall is inscribed 707 digits of the number π. The digits are large wooden characters attached to the dome-like ceiling. The digits were based on an 1853 calculation by English mathematician William Shanks, which included an error in the 528th digit. The error was detected in 1946 and corrected in 1949 Shank's 707 digits was the record at that time, taken from his teacher William Rutherford, whose record was 404 digits.His record, now adjusted to the 528 digits he had right, was the last record of hand computation. D. F. Ferguson's 1946 calculation of 606 digits was done with a desk top calculating machine. 1961 MIT's Clark Begins Work on LINC Computer : Wes Clark began his work on LINC, or the Laboratory Instrument Computer, at MIT's Lincoln Laboratory. His plan was to create a computer for biomedical research, that was easy to program and maintain, that could be communicated with while it operated, and that could process biotechnical signals directly. Building on his previous experience in developing the Whirlwind, TX-0, and other early computers, Clark set to work on one of the earliest examples of a "user friendly" machine -- setting the standard for personal computer design in the following decades. *CHM The first LINC included two oscilloscope displays. Twenty-one were sold by DEC at$43,600 (equivalent to \$390,600 in 2021),

"If you have to ask, you can't afford it!"

 *Wik

1965Hansard, the official record of all English parliamentary debates, recorded in an Appendix the statement by Mr Jay (President of the Board of Trade) saying he was impressed by the case for adoption of the metric system - by a long-term, gradual, voluntary process - and was arranging for the British Standards Institution to investigate. Eventually, as a member of the European Common Market, the transition to the metric system for trade and commerce became obligatory.*TIS

1973 French mathematicians Jean Guilloud and Mlle. Martine Bouyer computed π on a CDC 7600 computer to one million decimal places, the greatest accuracy to (that) date. The value was not veriﬁed until September 3, 1973. It is published in a 400 page book. *VFR
FYI: The millionth digt of Pi is one.
I came across a book on Amazon by Lee DeForest titled The Square Root of 4 to a Million Places.  (Hmmmm!)  Are you wondering what that digit is also?

1980  This poster is for the Rocket 150 event staged in Rainhill in the afternoons of 24, 25 and 26th May 1980.
Specially built replicas of the three locomotives and nine coaches were used to re-enact the Rainhill Trials of 1829. This was followed by a cavalcade of steam, diesel and electric locomotives, coaches and wagons to illustrate the history of railway development.
The Rainhill trials was an important competition run from the 6 to 14 October 1829, to test George Stephenson's argument that locomotives would have the best motive power for the then nearly-completed Liverpool and Manchester Railway (L&MR).[1] Ten locomotives were entered, of which five were able to compete, running along a 1 mile (1.6 km) length of level track at Rainhill, in Lancashire (now Merseyside).
Stephenson's Rocket was the only locomotive to complete the trials, and was declared the winner. (One contesting vehicle was a horse walking on a treadmill to propel it, but the floor broke before the vehicle would have been eliminated for failure to average 10 mph or more.)

1983 Marshall Stone received the National Medal of Science, the nation’s highest scientiﬁc honor, “for his original synthesis of analysis, algebra and topology into the new, vital area of functional analysis, in modern mathematics.” Notices AMS, v. 30, p. 485, contains more information. *VFR
Stone, who earned a doctorate in mathematics from Harvard, is credited with helping to modernize mathematics. He published numerous papers during his career and proved several theorems.
”Simply stated, Professor Stone was one of the greatest American mathematicians of this century,” Larry N. Mann, head of the Department of Mathematics and Statistics at the University of Massachusetts, where Stone was professor emeritus, told The New York Times in 1989 when Stone died.
In addition to his work at Harvard, Yale and Columbia universities, Stone was chairman of the mathematics department at the University of Chicago from 1945 to 1952. The university had lured him from Harvard with the request that he rebuild the once-proud mathematics department that had languished during World War II. The task, though monumental, proved no contest for Stone’s vision and tireless work.
Stone later took the helm of the International Mathematicians Union in 1952 and helped revitalized that group as well.

2014 NASA predicts a never before seen meteor shower. The shower is the May Camelopardalids (a faint constellation near the North Star), caused by dust from periodic comet 209P/LINEAR. No one has ever seen it before, but this year the Camelopardalids could put on a display that rivals the well-known Perseids of August.
"Some forecasters have predicted more than 200 meteors per hour," *science.nasa.gov

Returning tonight by popular demand!

BIRTHS

1544 William Gilbert (24 May 1544 – 30 November 1603)  English scientist, the "father of electrical studies" and a pioneer researcher into magnetism, who spent years investigating magnetic and electrical attractions. Gilbert coined the names of electric attraction, electric force, and magnetic pole. He became the most distinguished man of science in England during the reign of Queen Elizabeth I. Noting that a compass needle not only points north and south, but also dips downward, he thought the Earth acts like a bar magnet. Like Copernicus, he believed the Earth rotates on its axis, and that the fixed stars were not all at the same distance from the earth. Gilbert thought it was a form of magnetism that held planets in their orbits. *TIS  "Gilbert shall live, till Load-stones cease to draw,   Or British Fleets the boundless Ocean awe. ~ John Dryden
“At least, therefore, by Tychone Brahe, the diurnal motion of the earth is refuted by such arguments.”
Excerpt from the chapter titled, “The arguments of those who deny the motion of the Earth and their refutation” from ‘De Magnete’ [1600] by William Gilbert. *
@EponymousBreeze

1640 John Mayow (24 May 1640 - October 1679 ) English chemist and physiologist who, about a hundred years before Joseph Priestley and Antoine-Laurent Lavoisier, identified spiritus nitroaereus (oxygen) as a distinct atmospheric entity. He further recognized the role of oxygen in the combustion of metals. His medical writings include a remarkably correct anatomical description of respiration. *TIS  {It is said that he observed a mouse in a sealed jar with a candle, and as the candle flame was going out, the mouse fainted...}

1686 Daniel Gabriel Fahrenheit (24 May 1686 – 16 September 1736) German physicist and maker of scientific instruments. He is best known for inventing the alcohol thermometer (1709) and mercury thermometer (1714) and for developing the Fahrenheit temperature scale. He devoted himself to the study of physics and the manufacture of precision meteorological instruments. He discovered, among  other things, that water can remain liquid below its freezing point and that the boiling point of  liquids varies with atmospheric pressure.*TIS
From 1718 onward, he lectured in chemistry in Amsterdam. He visited England in 1724 and was elected into the Fellow of the Royal Society on May 5. In August of that year, he published five papers in Latin for the Royal Society's scientific journal, Philosophical Transactions, on various topics. In his second paper, Experimenta et observationes de congelatione aquae in value factae, he provides a description of his thermometers and the reference points he used for calibrating them. For two centuries, this document was the only description of Fahrenheit's process for making thermometers.:   In the 20th century, Ernst Cohen uncovered correspondences between Fahrenheit and Herman Boerhaave which cast considerable doubt on the veracity of Fahrenheit's article explaining the reference points for his scale and that, in fact, Fahrenheit's scale was largely derived from Rømer's scale.
Some people like to give the credit to Roemer and prefer to call it the Roemer scale, but that seems unwarranted, since Roemer never used his thermometer for anything, made only a few instruments in his lifetime, and never published on the subject.
Fahrenheit published 5 papers on his thermometer in the Philosophical Transactions of the Royal Society of London (all 5 papers in one busy year, 1724), and very explicitly gave us the zero, 32, 96 (later corrected to 98.6), and 212 degree benchmarks. Fahrenheit is buried in The Hague, and there is a large Fahrenheit thermometer on the outside of his tomb (third image above). One assumes that it works, and that it gives us the temperature of the ambient air, and not of Fahrenheit's mortal remains. There is no authentic known portrait of Fahrenheit.

1794 William Whewell,  (24 May 1794 – 6 March 1866)  British scientist, best known for his survey of the scientific method and for creating scientific words. He founded mathematical crystallography and developed Mohr's classification of minerals. He created the words scientist and physicist by analogy with the word artist. They soon replaced the older term natural philosopher. (actually the use of scientist was a very slow process often not well received.)  Other useful words were coined to help his friends: biometry for Lubbock; Eocine, Miocene and Pliocene for Lyell; and for Faraday, anode, cathode, diamagnetic, paramagnetic, and ion (whence the sundry other particle names ending -ion). In metereology, Whewell devised a self-recording anemometer. He was second only to Newton for work on tidal theory. He died as a result of being thrown from his horse*TIS
In a single letter to Faraday on 25 April, 1834;  he invented the terms cathode, anode and ion.  The letter is on display at the Wren Library at Trinity College, Cambridge, UK.

1820  William Chauvenet (24 May 1820, Milford, Pennsylvania - 13 December 1870, St. Paul, Minnesota) was born on a farm near Milford, Pennsylvania, in 1820 and was raised in Philadelphia. Early in life he exhibited a knack for mathematics and all things mechanical, and he attended Yale University. Entering Yale at age 16, he graduated in 1840 with high honors and soon after began his scholarly career by assisting a professor at Girard College in Philadelphia, Pennsylvania in a series of magnetic observations. In 1841 he was appointed professor of mathematics in the U.S. Navy and for a few months served on the U.S. steamer Mississippi, where he taught midshipmen. He later taught at and was instrumental in the establishment of the U.S. Naval Academy at Annapolis, Maryland.*Wik

1903  Władysław Roman Orlicz (May 24, 1903 in OkocimAustria-Hungary (now Poland) – August 9, 1990 in Poznań, Poland) was a Polish mathematician of Lwów School of Mathematics.  His main interest was topology. *Wik   Orlicz spaces are named after him.
Orlicz was awarded the Stefan Banach Prize by the Polish Mathematical Society in 1948.

1909 Karl Heinrich Weise (24 May 1909 in Gera, Thüringen, Germany, 15 April 1990 in Kiel, Germany) Weise's mathematical work was mainly on questions from differential geometry and topology. In 1951, jointly with Robert König, he published the book Mathematische Grundlagen der Höheren Geodäsie und Kartographie.
Weise acted as supervisor of PhD students from a wide range of mathematical fields, a dozen of them went on to become professors, among them Wolfgang Gaschütz (finite groups), Wolfgang Haken (knot theory and the solution of the four-colour-problem), Wilhelm Klingenberg (differential geometry) and Jens Mennicke (topology). Let us look in a little more detail at Weise's influence on one of these students, Wolfgang Haken, who studied mathematics, physics and philosophy at the University of Kiel. Haken attended Heinrich Heesch's talk on his contributions to the Four Colour Problem, but he was most enthused by Weise's lectures on topology. In these lectures, Weise described three long-standing unsolved problems - the Poincaré Conjecture, the Four Colour Problem, and a problem on knot theory. Haken decided to attempt to solve all three problems and began this challenge while studying for a doctorate at Kiel with Weise as his thesis advisor. His thesis, submitted in 1953, was Ein topologischer Satz über die Einbettung (d-1)-dimensionaler Mannigfaltigkeiten in d-dimensionale Mannigfaltigkeiten. He had solved the knot theory problem and this led to his appointment at the University of Illinois in the United States. Eventually, assisted by Kenneth Appel, he solved the Four Colour Problem in 1976 with the aid of computer techniques.
Weise was retired on 30 September1977, and in the following year the Christian Albrechts Universität conferred on him the title of 'Ehrensenator' (honorary senator). *SAU

1914 Federico Cafiero (24 May 1914 in Riposto, Catania, Sicily, 7 May 1980 in Naples, Italy) Cafiero played an important role in building a vigorous mathematical school at Naples which included (in alphabetical order) Luigi Albano, Ugo Barbuti, Antonio Chffi, Paolo De Lucia, Nicola Fedele, Renato Fiorenza, Francesco Guglielmino, Giuseppe Pulvirenti, Giuseppe Santagati and Antonio Zitarosa. We have already seen that Cafiero made contributions to the theory of ordinary differential equations and to the theory of measure and integration.
Two notable awards the Cafiero received for his mathematical contributions were the Tenore prize of the Accademia Pontaniana (awarded in 1953 for his monograph Funzioni additive d'insieme e integrazione negli spazi astratti) and the Golden medal 'Benemeriti della Scuola, della Cultura, dell'Arte' which he received from the President of the Italian Republic in 1976. *SAU

DEATHS

1543 Nicolaus Copernicus (19 February 1473 – 24 May 1543) Polish astronomer who proposed that the planets have the Sun as the fixed point to which their motions are to be referred; that the Earth is a planet which, besides orbiting the Sun annually, also turns once daily on its own axis; and that very slow, long-term changes in the direction of this axis account for the precession of the equinoxes    *TIS     An advance copy of his work De revolutionibus orbium coelestium was presented to Copernicus. On the same day he died. *VFR  Over 450 years after his death, Copernicus was reburied in the cathedral at Frombork on Poland’s Baltic coast. The astronomer whose ideas were once declared heresy by the Vatican—was reburied with full religious honors.

1734 Georg Ernst Stahl (22 October 1659 – 24 May 1734 [NS}) was a German chemist, physician and philosopher. He was a supporter of vitalism, and until the late 18th century his works on phlogiston were accepted as an explanation for chemical processes
Stahl used the works of Johann Joachim Becher to help him come up with explanations of chemical phenomena. The main theory that Stahl got from J. J. Becher was the theory of phlogiston. This theory did not have any experimental basis before Stahl. He was able to make the theory applicable to chemistry.[4] Becher's theories attempted in explaining chemistry as comprehensively as seemingly possible through classifying different earths according to specific reactions. Terra pinguis was a substance that escaped during combustion reactions, according to Becher.[10] Stahl, influenced by Becher's work, developed his theory of phlogiston.People who dismiss Phlogiston theory as early ignorance should read The Renaissance Mathematicus blog, The Phlogiston Theory – Wonderfully wrong but fantastically fruitful.

1843  Sylvestre François Lacroix (April 28, 1765, Paris – May 24, 1843) was the writer of important textbooks in mathematics and through these he made a major contribution to the teaching of mathematics throughout France and also in other countries. He published a two volume text Traité de calcul differéntiel et du calcul intégral (1797-1798) which is perhaps his most famous work. In the first of these volumes Lacroix introduces for the first time the expression "analyic geometry" writing:-
There exists a manner of viewing geometry that could be called géométrie analytique, and which would consist in deducing the properties of extension from the least possible number of principles, and by truly analytic methods.
He expanded his masterpiece to three volumes for the second edition published between 1810 and 1819.
Lacroix's texts had an influence beyond France,... and it was through English translations of Traité élémentaire de calcul differéntiel et du calcul intégral by Babbage, Peacock and Herschel that the 'new continental mathematics' entered universities in Britain. It is interesting that Lacroix held the view that algebra and geometry:-
... should be treated separately, as far apart as they can be, and that the results in each should serve for mutual clarification, corresponding, so to speak, to the text of a book and its translation.
His texts appeared in many editions for over 50 years .  *SAU

1896 Luigi Menabrea (4 Sept 1809 in Chambéry, Savoy, France - 24 May 1896 in St Cassin (near Chambéry), France) was a French-born soldier and engineer who made contributions to elasticity theory and became prime-minister of Italy. *SAU

1904 Cecil John Alvin Evelyn (25 August 1904 in London, England, 24 May 1976 in Deptford, Kent, England) He graduated with a B.A. in 1927. At Oxford he had become friendly with Hubert Linfoot who was one year older than Evelyn. Linfoot graduated in 1926 but had remained at Oxford undertaking research advised by G H Hardy. Both Evelyn and Linfoot were interested in number theory at this time and they worked together.

Between 1929 and 1933, Evelyn and Linfoot produced six joint papers, all with the title On a problem in the additive theory of numbers.
A book was to be Evelyn's final mathematical publication. He published the book (with G B Money-Coutts and J A Tyrrell) The seven circles theorem and other new theorems (1974) which was translated into French and published as (with G B Money-Coutts and J A Tyrrell) Le théorème des sept cercles (1975). R D Nelson, Ampleforth College, York, writes :-

This elegant book will please all geometers, amateur and professional, and deserves a place in every library. Using a variety of essentially elementary methods, the authors present and prove a number of new or little known theorems in plane geometry. To emphasise the aesthetic appeal of these results and to assist the argument in places, over forty of its pages carry diagrams of high quality. The book has three independent sections but the style of writing is uniform. The authors invite and sometimes require the co-operation of the reader as he works through the book and, in this way, they prepare him for the intricacies of the final and most difficult section. The first part re-introduces an algebra of vectors, due to Silberstein, in which the laws of addition and equivalence are such that few of the usual properties are obvious. For example, associativity of addition requires two applications of Desargues' theorem for its proof. No use is made of this algebra. The second section opens with a delightful new theorem concerning seven Pascal lines derived from a heptagon inscribed in a conic. This is followed by a number of extensions and generalisations of the theorems of Pascal and Brianchon. ... Finally there are four new theorems about closed chains of six circles ... In the first theorem each circle touches a seventh, in the second the circles alternately touch a pair of parallel lines, in the third each circle touches two of the sides of a triangle and in the fourth each circle touches two out of three fixed circles making a configuration of nine circles in all. The first theorem, beautifully proved by inversion, gives the book its title.

The remarkable thing about this book is that the theorem of the title is an elementary geometry theorem which appears to have been first discovered by the authors of this book. The theorem concerns six circles, all inside and touching a seventh circle. These six circles all touch each other. Join each of the six points on the outer circle where the six inner circles touch it, to the point of contact directly opposite it. The theorem states that these three lines are concurrent. *SAU

1968 Frances Sarnat Hugle (August 13, 1927 – May 24, 1968) was an American scientist, engineer, and inventor who contributed to the understanding of semiconductors, integrated circuitry, and the unique electrical principles of microscopic materials. She also invented techniques, processes, and equipment for practical (high volume) fabrication of microscopic circuitry, integrated circuits, and microprocessors which are still in use today.

In 1962, Hugle co-founded Siliconix, one of Silicon Valley's first semiconductor houses. She is the only woman included in the "Semiconductor Family Tree *Wik

*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