Friday, 3 April 2026

Extending Some Basic Geometry - Pedal Triangles and More

  


Somewhere in geometry class they usually introduce the concurrency theorems. Often they are introduced, then dropped almost as quickly and seldom see the light of day again. This seems somewhat a shame to me as there are some really nice generalizations of most of them. For now I want to address some extensions of perpendiculars from a point in a triangle, and some more general properties of them.
The basic class often explains that if the student takes any triangle and draws the three altitudes, they will meet in a point, often called the orthocenter of the triangle.
I have covered several properties and generalizations of the orthocenter, but one I will repeat is the neat property that if you multiply the lengths of the two sections of any altitude created by the orthocenter, the products are equal. I think of this as similar to the theorem they will (or may have already) learned about the sections of intersecting chords in a circle (actually to prove the theorem it is very helpful to know that theorem and know that the reflection of the orthocenter over any side of the triangle will always land on the circumcircle)
The Orthic triangle is a name for the triangle joining the feet of the altitudes (where they intersect the opposite side) and students may be led to discover that 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.
A simple but pretty formula relates the lengths of the sides of the orthic triangle to the sides and angles of the original triangle. For example, in the Triangle ABC, the side of the orthic triangle nearest to vertex A is given by a*Cos (A), and likewise for the other two sides. The perimeter of the orthic triangle, then, is equal to a*Cos(A) + b*Cos(B) + c*Cos(C). I have days where I think that formula is right up there with the Pythagorean Theorem.
The Orthic triangle is also known as a pedal triangle, from the Latin word for foot, since it is the foot of the altitude. I wanted to talk today about a theorem that is very little known to students (and truth be told, to teachers of geometry) about the generalized Pedal Triangle.
In the figure above a random point P is selected in the triangle. The foot of the perpendiculars to each side from this point mark three points that are the vertices of the general pedal triangle.
Some special points and their associated pedal triangles are well known. For the orthocenter I have mentioned above some characteristics of the orthic triangle. If the incenter is chosen as the pedal point, the vertices of the pedal triangle will occur at the points where the incircle is tangent to the sides of the triangle. If the point P falls on the circumcircle of ABC, then the three feet of the perpendiculars will lie in a straight line, known as the Simson line.
There are two little known theorems about pedal triangles that are, in my mind, very beautiful. OK, actually there is only one theorem which is also generalizable to other polygons. I have to admit that I wasn't aware of the generalize version until I saw it recently on the Futility Closet blog of Greg Ross.
If you take the pedal triangle of the pedal triangle from that same point, then do that one more time (that should be ABC''' ) then the third pedal triangle will be similar to the original triangle you started with... and that happens with any random point you start with, inside or outside the original triangle.

Greg's post even had a catchy poem to describe the situation.


Begin with any point called P
(That all-too-common name for points),
Whence, on three-sided ABC
We drop, to make right-angled joints,
Three several plumb-lines, whence ’tis clear
A new triangle should appear.

A ghostly Phoenix on its nest
Brooding a chick among the ashes,
ABC bears within its breast
A younger ABC (with dashes):
A figure destined, not to burn,
But to be dropped on in its turn.

By going through these motions thrice
We fashion two triangles more,
And call them ABC (dashed twice)
And thrice bedashed, but now we score
A chick indeed! Cry gully, gully!
(One moment! I’ll explain more fully.)

The fourth triangle ABC,
Though decadently small in size,
Presents a form that perfectly
Resembles, e’en to casual eyes
Its first progenitor. They are
In strict proportion similar.

Greg credits the poem to the English Geographer, Mary (Tunstall) Pedoe in 1947, who was the wife of Dan Pedoe, the English-born mathematician and geometer who has authored such extraordinary books as The Gentle Art of Mathematics.  Together they authored a unit for The Minnesota School Mathematics and Science Teaching(MINNEMAST) Project on teaching addition and subtraction.

Pedoe Sangaku_of_Soddy's_hexlet_in_Samukawa_Shrine 


Dan Pedoe



The extension of this theorem, which was the new part to me, is that you can create a pedal rectangle or a pedal pentagon, etc by the same process and by the time you get to the n'th copy of the n-gon, "Eureka", it is similar to the original.
Simply an awesome idea, share it with your students.

I wanted to show a pedal quadrilateral progression.  They get a little messy, but notice that the nth reduction of each appears to have rotated about 180 degrees.  Which made me wonder...And after experimenting it seems that the rotation is 180 degrees.  That's from a sketch, not a geometric proof......but as the old Monkees song goes, "Now I'm a believer."



On This Day in Math - April 3

   

Alberti's Statue in the courtyard of the Uffizi Gallery, Florence; *Wik



Knowing what is big and what is small is more important than being able to solve partial differential equations.
~Stan Ulam


The 93rd day of the year; The first 93 digits of 93! form a prime number. *Prime Curios Can students find a smaller number n for which the first n digits of n! form a prime? Send results to me.)

93! = 1156772507081641574759205162306240436214753229576413535186142281213246807121467
315215203289516844845303838996289 ...


93 is the sum of three distinct squares, 93 = 22 + 52+ 82 )
and six consecutive integers 93= 13 + 14 + 15 + 16 + 17 + 18

There are 93 five-digit prime palindromes. The smallest (I think) is 10301

A potato can be cut into 93 pieces with just nine straight cuts.

and 93 in base 10 is 333 in base 5




EVENTS

1501  Friar Pietro de Novellara writes to inform Isabell d'Esta that Leonardo DaVinci is too preoccupied with geometry to do the portrait of her she desired, "he devotes much of his time to geometry and has no fondness at all for the paintbrush. " It seems that the great master had fallen prey to the sickness of circle squaring.  One wonders if his period working to produce the geometric studies for Pacioli's De Divina Proportione, (just before 1494) had enticed him into a geometric passion.  *David Richeson, Tales of Impossibility.



1736 Euler replied to Ehler on Konigsberg Bridge Challenge, "Not a Mathematical Problem..."

... Thus you see, most noble Sir, how this type of solution bears little relationship to mathematics, and I do not understand why you expect a mathematician to produce it, rather than anyone else, for the solution is based on reason alone, and its discovery does not depend on any mathematical principle. Because of this, I do not know why even questions which bear so little relationship to mathematics are solved more quickly by mathematicians than by others. In the meantime, most noble Sir, you have assigned this question to the geometry of position, but I am ignorant as to what this new discipline involves, and as to which types of problem Leibniz and Wolff expected to see expressed in this way ...

(See March 9th for Ehler's challenge to Euler) *Brian Hopkins, Robin Wilson; The Truth About Konigsberg

- See more at: http://www.maa.org/programs/maa-awards/writing-awards/the-truth-about-konigsberg#sthash.c6jO9L76.dpuf


Map of Königsberg in Euler's time showing the actual layout of the seven bridges, highlighting the river Pregel and the bridges




1753 Goldbach wrote Euler with a conjecture that every odd number greater than 3 is the sum of an odd number and twice a square (2n^2 + p, he allowed 02). Euler would reply on Dec 16 that it was true for the first 1000 odd numbers, and then again on April 3, 1753, to confirm it for the first 2500. A hundred years later, German mathematician Moritz Stern found two contradictions, 5777 and 5993. The story appears in Alfred S. Posamentier's Magnificent Mistakes in Mathematics, (but gloriously, has a mistake for the date, using 1852, but such a wonderful book can forgive a print error.)




1763 Easter Saturday 1763, Lalande recorded that the Astronomer Royal Nathaniel Bliss said ‘that Mr Lemonnier attached the wire to his quadrant with wax from his ears, that he went to Oxford with his sword broken, and that his observations agree less well with those of Mr Bevis than those of Caille.’ Pierre Charles Lemonnier or Le Monnier was a talented French astronomer 17 years Lalande’s senior who had a penchant for British instruments and astronomical methods and was a member of the Royal Society and the French Academy of Sciences. *Royal Museum Greenwich blog
His persistent recommendation of British methods and instruments contributed effectively to the reform of French practical astronomy, and constituted the most eminent of his services to science. He corresponded with James Bradley, was the first to represent the effects of nutation in the solar tables, and introduced, in 1741, the use of the transit-instrument at the Paris Observatory.  *Wik
Astronomical quarter-circle wall quadrant or mural quadrant (rotatable by 180 °) built by John Bird. Le Monnier adapted and used a version in 1774.






1769 A letter from Mr. Richard Price, F. R. S. to Benjamin Franklin, Esq; LL.D. discusses De Moivre's work on Population and survival rates. The paper runs to 38 pages. "Observations on the Expectations of Lives, the Increase of Mankind, the Influence of Great Towns on Population, and Particularly the State of London with Respect to Healthfulness and Number of Inhabitants." *Phil. Trans. January 1, 1769 59:89-125;
De Moivre's Law is a survival model applied in actuarial science, named for Abraham de Moivre.  It is a simple law of mortality based on a linear survival function.  De Moivre's law first appeared in his 1725 Annuities upon Lives, the earliest known example of an actuarial textbook. Despite the name now given to it, de Moivre himself did not consider his law (he called it a "hypothesis") to be a true description of the pattern of human mortality. Instead, he introduced it as a useful approximation when calculating the cost of annuities. 
A copy of de Moivre's Annuities upon Lives sold at Christie's in 1998 for USD 6325.

de Moivre's illustration of his piecewise linear approximation






1898 Jan Szczepanik, who with Ludwig Kleiberg obtained a British patent (patent nr. 5031) for his video transmission device in 1897. Szczepanik's telectroscope, although never actually exhibited and, as some claim, likely never existed, was covered in the New York Times on April 3, 1898, where it was described as "a scheme for the transmission of colored rays", and it was further developed and presented on the exhibition in Paris in 1900.
 Szczepanik's experiments fascinated Mark Twain, who wrote a fictional account of his work in his short story From The Times of 1904. Both the imagined "telectroscope" of 1877 and Mark Twain's fictional device (called a telectrophonoscope) had an important effect on the public. 
#Wik


In 1934, a British patent application for the first catseye road marker was recorded for inventor Percy Shaw (1889-1975), described as "Improvements relating to Blocks for Road Surface." These are the familiar reflectors which mark the lines that are lit up at night by the lights of passing vehicles. The raised surface in which the reflectors are mounted have a construction that "will yield when traveled over by a vehicle wheel and sink to the level of the road surface" such as a resilient white rubber cushion mounted in a metal holder sunk below the road surface. The patent No. 436,290 was accepted 3 Oct 1935. A revised design was patented the following year as No. 457,536. Shaw started Reflecting Roadstuds Ltd. to manufacture them. *TIS (While this is of little mathematical interest, before I learned what the term meant, I regularly wondered about a sign I passed on the way from Stoke Ferry to King's Lynn in Norfolk Shire which said, "Cat's Eyes Removed Ahead."

*Wik


1964 It is reported in the New York Times that the casinos in Las Vegas have changed their rules in blackjack so as to defeat the winning strategy devised by Edward O. Thorp. See 27 March 1964. *VFR
Thorp is described as an American mathematics professor, author, hedge fund manager, and blackjack player. He was a pioneer in modern applications of probability theory, including the harnessing of very small correlations for reliable financial gain.*Wik  Thorp's method was a modified card counting system.  He let the genie out of the bottle and published Beat the Dealer, in 1962, detailing his card-counting method to the general public. Now, everyone could be a card counter. 






1983 The Republic of China (Taiwan) marks the 400th anniversary of the arrival of Father Matteo Ricci (1552–1610) in China with a pair of stamps. [Scott #2359–2360] *VFR




2016 First day of the major league baseball season. The exact width of home plate is irrational: 12 times the square root of two.
History: The plate was originally a circle of diameter one, then a square of the same size(!), which, by mistake was a one-by-one square. Then the corners were filled in to make the current pentagonal plate. *VFR Home plate is an irregular pentagon. The front is 17 inches wide, faces the pitcher, and defines the width of the strike zone. Then parallel sides 8.5 inches long connect to the foul lines. Finally 12 inch sides run down the foul lines, connecting where the foul lines meet.
It can be thought of as a 17 inch square with the parts that would be in foul territory removed. The figure described in the official rules of MLB, as well as above, is technically impossible. One of two things must be true to make it possible:


the parallel sides of 8.5" are in reality approximately 8.5295" (the square root of 71.75)
or
the 12" sides that run along the foul lines are approximately 12.0208" (square root of 144.5)
The latter is more likely the case, as it would produce the angle measurements of 90º at the base and rear point and 135º at the sides.

Rick Pearce ‏@MrPearceMath has written to object: "The width is 17", not irrational. Other 4 sides are. But you're right about impossibility of defn." I'm not sure the difference of 17 and 12 square root (2) is a miniscule .0295" (Will wander down to nearest ball park with steel tape in hand, this challenge will not stand!)


BIRTHS

1529 – Michael Neander (April 3, 1529 – October 23, 1581) German mathematician and astronomer was born in Joachimsthal, Bohemia, and was educated at the University of Wittenberg, receiving his B.A. in 1549 and M.A. in 1550.
From 1551 until 1561 he taught mathematics and astronomy in Jena, Germany. He became a professor in 1558 when the school where he taught became a university. From 1560 until his death he was a professor of medicine at the University of Jena. He died in Jena, Germany. The crater Neander on the Moon is named after him. *Wik



1835 John Howard Van Amringe (3 April 1835 in Philadelphia, Pennsylvania, USA - 10 Sept 1915 in Morristown, New Jersey, USA) was a U.S. educator and mathematician. He was born in Philadelphia, and graduated from Columbia in 1860. Thereafter, he taught mathematics at Columbia, holding a professorship from 1865 to 1910 when he retired. Van Amringe was also the first Dean of Columbia College, the university's undergraduate school of arts and sciences, which he defended from dismemberment and incorporation into the larger university. During his long presence at the school, he made many addresses and enjoyed unrivaled popularity. He is memorialized with a bust enshrined in a column-supported cupola on "Van Am Quad" in the southeastern portion of the campus, surrounded by three College dormitories (John Jay Hall, Hartley Hall, and Wallach Hall) and by the main College academic building, Hamilton Hall. He is buried in Greenwood Cemetery in Brooklyn.
Van Amringe served as the first president of the American Mathematical Society between 1888 and 1890.
In honor of Van Amringe, Columbia University's Department of Mathematics has presented a "Van Amringe Mathematical Prize" each year (since 1911) to the best freshman or sophomore mathematics student, based on a very challenging examination. *Wik



1842 Hermann Karl Vogel (3 Apr 1842; died 13 Aug 1907 at age 65) German astronomer who discovered spectroscopic binaries (double-star systems that are too close for the individual stars to be discerned by any telescope but, through the analysis of their light, have been found to be two individual stars rapidly revolving around one another). He pioneered the study of light from distant stars, and introduced the use of photography in this field.*TIS



1859 Karl Heun (3 April 1859 in Wiesbaden, Germany - 10 Jan 1929 in Karlsruhe, Germany) was a German mathematician best known for the Heun differential equation which generalizes the hypergeometric differential equation. *SAU

1892 Hans Rademacher (3 April 1892 in Wandsbeck (part of Hamburg), Schleswig-Holstein, Germany - 7 Feb 1969 in Haverford, Pennsylvania, USA) It was philosophy that he intended to take as his main university subject when he entered the university of Göttingen in 1911, but he was persuaded to study mathematics by Courant after having enjoyed the excellent mathematics teaching of Hecke and Weyl. He is remembered for the system of orthogonal functions (now known as Rademacher functions) which he introduced in a paper published in 1922. Berndt writes "Since its discovery, Rademacher's orthonormal system has been utilised in many instances in several areas of analysis." Rademacher's early arithmetical work dealt with applications of Brun's sieve method and with the Goldbach problem in algebraic number fields. About 1928 he began research on the topics for which he is best known among mathematicians today, namely his work in connection with questions concerning modular forms and analytic number theory. Perhaps his most famous result, obtained in 1936 when he was in the United States, is his proof of the asymptotic formula for the growth of the partition function (the number of representations of a number as a sum of natural numbers). This answered questions of Leibniz and Euler and followed results obtained by Hardy and Ramanujan. Rademacher also wrote important papers on Dedekind sums and investigated many problems relating to algebraic number fields. *SAU



1900 Albert Edward Ingham (3 April 1900–6 September 1967) was an English mathematician. His book On the distribution of prime numbers published in 1932 was his only book and it is a classic. Many of the ideas here, as in other work of Ingham's, came from the joint work undertaken by Harald Bohr and Littlewood.






1907 Mark Grigorievich Krein (3 April 1907 – 17 October 1989) was a Soviet Jewish mathematician, one of the major figures of the Soviet school of functional analysis. He is known for works in operator theory (in close connection with concrete problems coming from mathematical physics), the problem of moments, classical analysis and representation theory.
He was born in Kiev, leaving home at age 17 to go to Odessa. He had a difficult academic career, not completing his first degree and constantly being troubled by anti-Semitic discrimination. His supervisor was Nikolai Chebotaryov.
He was awarded the Wolf Prize in Mathematics in 1982 (jointly with Hassler Whitney), but was not allowed to attend the ceremony.
He died in Odessa.
On 14 January 2008, the memorial plaque of Mark Krein was unveiled on the main administration building of I.I. Mechnikov Odessa National University. *Wik





1937  Enrique Aurelio Planchart Rotundo (3 April 1937 – 27 July 2021) was a Venezuelan mathematician and academic. He was rector of Simón Bolívar University in Caracas from 2009 until his death in 2021.

Planchart graduated as a Bachelor of Science from the Central University of Venezuela and obtained his Doctorate in Mathematics from the University of California, Berkeley, where he was also a visiting professor in its Department of Mathematics between 1986 and 1987. From 1973 he was part of the Department of Pure and Applied Mathematics of the Simón Bolívar University.

While at Simón Bolívar University, between 1989 and 1999 he directed the National Center for the Improvement of Science Education, and from 1999 he directed the Equal Opportunities Program (PIO). In 1989 he was awarded the National Council for Scientific and Technological Research Award.

Throughout his scientific career, Planchart published nine books and nine journal articles and gave thirty lectures *Wik






DEATHS

1472 Leone Battista Alberti (18 Feb 1404 in Genoa, French Empire (now Italy)- 3 April 1472 in Rome, Papal States (now Italy) The date of his death is given by Wikipedia as April 20, 1472) Italian mathematician who wrote the first general treatise on the laws of perspective and also wrote a book on cryptography containing the first example of a frequency table. Alberti died in Rome, but his ashes were brought from Rome and put in the family vault in the Santa Croce Cathedral (where Galileo is buried).




1717 Jacques Ozanam (16 June 1640, Sainte-Olive, Ain - 3 April 1718, Paris)In 1670, he published trigonometric and logarithmic tables more accurate than the then existing ones of Ulacq, Pitiscus, and Briggs. An act of kindness in lending money to two strangers secured for him the notice of M. d'Aguesseau, father of the chancellor, and an invitation to settle in Paris. There he enjoyed prosperity and contentment for many years. He married, had a large family, and derived an ample income from teaching mathematics to private pupils, chiefly foreigners. *Wik
He is remembered for his book on mathematical recreations. “He was wont to say that it was the business of the Sorbonne doctors to discuss, of the pope to decide, and of a mathematician to go straight to heaven in a perpendicular line.” [DSB 10, 264]. *VFR
On the flyleaf of J. E. Hofmann's copy of the 1696 edition of Ozanam's Recreations is a pencil portrait labelled Ozanam -- the only one I know of. This copy is at the Institut für Geschichte der Naturwissenschaft in Munich. *David Singmaster



1817 Friedrich Ludwig Wachter (1792–1817), a student of Gauss, called the geometry obtained by denying Euclid’s parallel postulate “anti-Euclidean geometry”. Had he returned from his cus­tomary evening walk on this date he might now be known as one of the founders of non-Euclidean geometry. [G. E. Martin, Foundations of Geometry and the Non-Euclidean Plane, p. 306] *VFR
(… In a letter to Gauss in 1816 he states that as a circle’s radius increases toward infinity as being identical to a plane, “even in the case of the fifth postulate being false, there would be geometry on this surface identical with that of the ordinary plane.)
Wachter was last seen on April 3 (Maundy Thursday) 1817 on his usual evening walk, during which he was alone. Despite an intensive search, his body was never found. In 1827 he was declared dead. The news of his death shocked Gauss deeply.  The fact that his body was not found  and there was no suicide note spoke against it. The authorities searched intensively for him and offered a considerable reward (200 thalers) for information about him. People still remembered the disappearance of the Austrian officer and mathematician Georg Freiherr von Vega , whose body was pulled from the Danube in 1802, where it was also initially suspected that he had committed suicide, but was later proven to be a robbery-murder. But no evidence of murder was found either, and similar incidents were not known in Danzig at the time. Some contemporaries suspected suicide due to Wachter's unstable character.  In the letter from Wachter's father to Gauss dated 10 May 1817, the latter also quotes extensively from a letter from his son in which he reports his high expectations of his work, which, according to Wachter, only Gauss could understand among contemporary mathematicians and with which he had achieved something that mathematicians had tried in vain to achieve since the time of Euclid. Kurt Biermann suspects that on 3 April a letter from Gauss reached Wachter, informing him of his negative opinion of Gauss, and that Wachter then committed suicide.*Wik



 


1827 Ernst Florenz Friedrich Chladni, (30 November 1756 – 3 April 1827) physicist and amateur musician, died. He is best remembered for the spectacular symmetrical patterns formed when a sand covered plate is vibrated with a violin bow. He was a Professor of Physics in Breslau when he developed Chladni figures c1800. He came to Paris in 1808 to present his work at the Institut and Laplace had him give a two hour demonstration to Napoleon, who gave him 6000 francs.
German physicist who is known as the “father of acoustics” for his mathematical investigations of sound waves. Chladni figures, seen when thin plates covered in sand at set in vibration, are complex patterns of vibration with nodal lines that remain stationary and retain sand. He demonstrated these to an audience of scientists in Paris in 1809. He measured the speed of sound in various gases by determining the pitch of the note of an organ pipe filled with different gases. To determine the speed of sound in solids, Chladni used analysis of the nodal pattern in standing-wave vibrations in long rods. He performed on the euphonium, an instrument he invented, made of glass and steel bars vibrated by rubbing with a moistened finger. He also investigated meteorites. *TIS  Chladni is also considered by many to be the father of scientific meteorite studies. In 1794, Ernst F. F. Chladni published a 63-page book, Über den Ursprung der von Pallas gefundenen und anderer ihr änlicher Eisenmassen und über einige damit in Verbindung stehende Naturerscheinungen, in which he proposed that meteor-stones and iron masses enter the atmosphere from cosmic space and form fireballs as they plunge to Earth. These ideas violated two strongly held contemporary beliefs: 1) fragments of rock and metal do not fall from the sky, and 2) no small bodies exist in space beyond the Moon.  On  April 26, 1803, thousands of L-chondrite fragments bombarded L'Aigle in Normandy, France, an event investigated by Jean-Baptiste Biot of the French Academy of Science. Until that time the Academy had been one of the staunchest holdouts against acceptance of Chladni's theory, but after Biot's analysis they, too, had to accept rocks falling from space.




1900 Dame Mary Lucy Cartwright (17 Dec 1900 in Aynho, Northamptonshire, England
- 3 April 1998 in Cambridge, England) In 1930 Cartwright was awarded a Yarrow Research Fellowship and she went to Girton College, Cambridge, to continue working on the topic of her doctoral thesis. Attending Littlewood's lectures, she solved one of the open problems which he posed. Her theorem, now known as Cartwright's Theorem, gave an estimate for the maximum modulus of an analytic function which takes the same value no more than p times in the unit disc. To prove the theorem she used a new approach, applying a technique introduced by Ahlfors for conformal mappings.
Cartwright was appointed, on the recommendation of both Hardy and Littlewood, to an assistant lectureship in mathematics in Cambridge in 1934, and she was appointed a part-time lecturer in mathematics the following year. In 1936 she became director of studies in mathematics at Girton College, and in 1938 she began work on a new project which had a major impact on the direction of her research. The Radio Research Board of the Department of Scientific and Industrial Research produced a memorandum regarding certain differential equations which came out of modelling radio and radar work. They asked the London Mathematical Society if they could help find a mathematician who could work on these problems and Cartwright became interested in their memorandum.
The dynamics which lay behind the problems was unfamiliar to Cartwright and so she approached Littlewood for help with this aspect. They began to collaborate studying the equations. Littlewood wrote, "For something to do we went on and on at the thing with no earthly prospect of "results"; suddenly the whole vista of the dramatic fine structure of solutions stared us in the face. "
The fine structure which Littlewood describes here is today seen to be a typical instance of the "butterfly effect". The collaboration led to important results, and these have greatly influenced the direction that the modern theory of dynamical systems has taken. In 1947, largely on the basis of her remarkable contributions in the collaboration with Littlewood, she was elected a Fellow of the Royal Society and, although she was not the first woman to be elected to that Society, she was the first woman mathematician. *SAU

 She was one of the pioneers of what would later become known as chaos theory.



2016 David James Foulis (July 26, 1930- April 3, 2018) was an American mathematician known for his research on the algebraic foundations of quantum mechanics. He spent much of his career at the University of Massachusetts Amherst, retiring in 1997 but continuing to be very active in mathematics as professor emeritus. He is the namesake of Foulis semigroups, an algebraic structure that he studied extensively under the alternative name of Baer *-semigroups.

After completing his doctorate, Foulis taught for one year in the mathematics department at Lehigh University, four years at Wayne State University, and two years at the University of Florida before moving to the University of Massachusetts as a professor of mathematics and statistics in 1965. He retired in 1997, but continued to be active as a researcher after retirement.

Foulis's doctoral students at Massachusetts have included DIMACS associate director Melvin Janowitz, graph theorist David Sumner, and mathematics and statistics educator and textbook author Patti Frazer Lock.*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



Thursday, 2 April 2026

On This Day in Math - May 8

 




A small circle is quite as infinite as a large circle.
G. K. Chesterton

The 128th day of the year; 128 is The largest known even number that can be expressed as the sum of two primes in exactly three ways. (Find them) *Prime Curios   How many smaller numbers (and which) are there that can be so expressed?

But, it can not be expressed as the sum of distinct squares, for any number of squares.
And it is the largest such number, ever.... no, I mean EVER.  The very last.  
 (Surprisingly, there are only 31 numbers that can not be expressed as the sum of distinct squares. )

128 can be expressed by a combination of its digits with mathematical operators thus 128 = 28 - 1, making it a Friedman number in base 10 (Friedman numbers are named after Erich Friedman, as of 2013 an Associate Professor of Mathematics and ex-chairman of the Mathematics and Computer Science Department at Stetson University, located in DeLand, Florida.)

128 the sum of the factorials of the first three prime numbers, 2! + 3! + 5! =128.

128 = 2^8, so in binary it is a 1 followed by 7 zeros, which makes it also 4^4, and in base 4 its a 2 with three zeros.  But it's also 8^2, so in base eight its a 2 with two zeros, 
128 is a power of two, and all of its digits are powers of two.  I don't know of any other.

 128 can be expressed by a combination of its digits with mathematical operators thus 128 = 28 - 1, making it a Friedman number in base 10 (Friedman numbers are named after Erich Friedman, as of 2013 an Associate Professor of Mathematics and ex-chairman of the Mathematics and Computer Science Department at Stetson University, located in DeLand, Florida.) 

 128 the sum of the factorials of the first three prime numbers, 2! + 3! + 5! =128.

Some nice relationships between 128 and its digits, 128 + (1+2+8) = 139, a prime number.  But 128 + (8 + 1) is 137, also prime, and 128 + (2 + 1)  is 131, a prime, AND 128 +( 8+2 ) is not prime, but 138 is between a twin prime pair.  ..... And 1*2*8 = 16 is a divisor of 128.  
And that pair of cousin primes, 127 and 131, are the largest  such pair with a power of two (128) between them.  

The name for a particular 7th dimensional Hyperplex with 128 vertices is a Hepteract.  Dazzle your friends.  

Oh, I told you 128 is the 7th power of two.... but there are no more three digit numbers that are 7th powers... 

And if you like to keep score, 128 is 6 score and 8.  In old commercial terminology, a schock was a lot of 60 items, so 128 is also two shock and 8, or 28 in sexigesimal (base sixty).   The number of  stalks of corn or wheat (supposedly)  gathered and stood on ends in the fields to dry, like in "When the frost is on the Pumpkin and the Fodders in the shock. "  


128  is divisible by four so it is the difference of two squares of numbers that differ by 2, and since  128 / 4 = 32, the numbers must straddle 32, 33² - 31² = 128. 
But it is also divisible by eight, so it is the difference of two squares of numbers that differ by four(there is a power of two relation working here, which students might find).  And since 128/8 = 16, 18² - 14² = 128

And in 1968 the 128 K Mac was the hottest desktop computer around.  







EVENTS

1654 Otto von Guericke demonstrates the Magdenburg hemispheres in front of the imperial Diet, and the Emperor Ferdinand IIII in Regensburg.
The Magdeburg hemispheres, around 50 cm (20 inches) in diameter, were designed to demonstrate the vacuum pump that Guericke had invented. One of them had a tube connection to attach the pump, with a valve to close it off. When the air was sucked out from inside the hemispheres, and the valve was closed, the hose from the pump could be detached, and they were held firmly together by the air pressure of the surrounding atmosphere.
Thirty horses, in two teams of fifteen, could not separate the hemispheres until the valve was opened to equalize the air pressure. In 1656 he repeated the demonstration with sixteen horses (two teams of eight) in his hometown of Magdeburg, where he was mayor. He also took the two spheres, hung the two hemispheres with a support, and removed the air from within. He then strapped weights to the spheres, but the spheres would not budge.Gaspar Schott was the first to describe the experiment in print in his Mechanica Hydraulico-Pneumatica (1657).
In 1663 (or, according to some sources, in 1661) the same demonstration was given in Berlin before Frederick William, Elector of Brandenburg with twenty-four horses. It is unclear how strong a vacuum Guericke's pump was able to achieve, but if it was able to evacuate all of the air from the inside, the hemispheres would have been held together with a force of around 20 000 N (4400 lbf, or 2.2 short tons), equivalent to lifting a car or small elephant; a dramatic demonstration of the pressure of the atmosphere. *Wik


1661 “On 8 May 1661 the Society’s Journal Book notes that ‘a motion was made for the erecting of a library’, and later in the same month ‘it was resolved that every member, who hath published or shall publish any work, give the Society one copy’.” (from Emma Davidson at RSI)

The Library and Archives of the Royal Society are open to researchers and members of the public. Access is free of charge.




1698  Henry Baker was born on May 8.  His book The Microscope Made Easy (1743) has been described as the first laboratory manual for microscopy. *RMAT



1774 The conjunction of the Planets Jupiter, Mars, Venus, Mercury and the Moon on this date would herald the apocalypse according to a treatise by Eelco Alta, a Frisian clergyman and theologian. However, the apocalypse did not occur, perhaps because the projected conjunction of the heavenly bodies never occurred. One good result attributed to the treatise was the creation of what is now the oldest continuously operating planetarium in the world, the Eise Eisinga Planetarium in the ceiling of his former home in the Netherlands. It is driven by a pendulum clock, which has 9 weights or ponds. The planets move around the model in real time, automatically. (A slight "re-setting" must be done by hand every four years to compensate for the February 29th of a leap year.) In addition to the basic orrery, there are displays of the phase of the moon and other astronomical phenomena. The planetarium includes a display for the current time and date. The plank that has the year numbers written on it has to be replaced every 22 years. To create the gears for the model, 10,000 handmade nails were used. *Wik *collected notes

*HT Erik K sent. "This UNESCO world heritage site (since september 2023) is a museum now and open to the public.  There you can actually sit inside that bedroom but also have insight giving views above the ceiling. 

This unique craftsmanship is located in the city of Franeker in the province of Friesland."



1790 The Assembly (French) ordered the Académie des Sciences to standardize weights and measures on 8 May 1790. The Académie appointed a Commission of Lagrange, Borda, Condorcet, Laplace and Tillet to compare the decimal and duodecimal systems. Another Commission, with Monge instead of Tillet, was to examine how to make a standard of length. The Commissions continued functioning through the Revolution.

The traditional French units of measurement prior to metrication were established under Charlemagne during the Carolingian Renaissance. Based on contemporary Byzantine and ancient Roman measures, the system established some consistency across his empire but, after his death, the empire fragmented and subsequent rulers and various localities introduced their own variants. Some of Charlemagne's units, such as the king's foot (French: pied du Roi) remained virtually unchanged for about a thousand years, while others important to commerce—such as the French ell (aune) used for cloth and the French pound (livre) used for amounts—varied dramatically from locality to locality. By the 18th century, the number of units of measure had grown to the extent that it was almost impossible to keep track of them and one of the major legacies of the French Revolution was the dramatic rationalization of measures as the new metric system. The change was extremely unpopular, however, and a metricized version of the traditional units—the mesures usuelles—had to be brought back into use for several decades.

Woodcut dated 1800 illustrating the new decimal units which became the legal norm across all France on 4 November 1800





1794 Lavoisier Guillotined along with twenty-seven other members of the Ferme Générale, including his father-in-law. See Deaths below

In September 1793 a law was passed ordering the arrest of all foreigners born in enemy countries and all their property to be confiscated. Lavoisier intervened on behalf of Lagrange, who certainly fell under the terms of the law, and he was granted an exception. On 8 May 1794, after a trial that lasted less than a day, a revolutionary tribunal condemned Lavoisier, who had saved Lagrange from arrest, and 27 others to death. Lagrange said on the death of Lavoisier, who was guillotined on the afternoon of the day of his trial:-

It took only a moment to cause this head to fall and a hundred years will not suffice to produce its like.

1795 French astronomer Jerome Lalande observes a "star". It is in fact the planet Neptune, which is not officially discovered until 1846. * Liz Suckow@LizMSuckow
A second recording, noting a possible error on the 10th was entered. Discovery of these recordings 1n 1947 by Sears C. Walker of the U.S. Naval Observatory led to a better calculation of the planet's orbit. *Wik




In 1886, Coca-Cola, the soft drink, was first sold to the public at the soda fountain in Jacob's Pharmacy in Atlanta, Georgia. It was invented by pharmacist, John Stith Pemberton, who mixed it in a 30-gal. brass kettle hung over a backyard fire. Until 1905, the drink, marketed as a "brain and nerve tonic," contained extracts of cocaine as well as the caffeine-rich kola nut. The name, using two C's from its ingredients, was suggested by his bookkeeper Frank Robinson, whose excellent penmanship provided the first scripted "Coca-Cola" letters as the famous logo. Asa Candler marketed Coke to world after buying the company from Pemberton. *TIS

John Pemberton, *Wik





1910 The New York Times Sunday Magazine publishes banner headline, "Fears Of The Comet Are Foolish And Ungrounded," only ten days before the Earth moves into the tail of Halley's Comet. The article featured the famous female astronomer, Mary Proctor, debunking horror stories such as :

Here is a gigantic monster in the sky with a head over two hundred thousand miles in width… and a train two million miles in length, rushing through space at the alarming rate of a thousand miles a minute.
On May 18 the earth will be plunged in this white hot mass of glowing gas, and, according to the report of the ignorant and superstitious, the world will be set on fire.
These sensation makers further say that the oceans on the side facing the comet will be boiled by the intense heat, and the land scorched and blistered as the dread wanderer passes by on its baneful way.

*SundayMagazine.org

Halley’s comet approached Earth and killed England’s King Edward VII, according to some superstitious folk. No one could definitively say how it did, but it certainly did. And that wasn’t its only offense. The Brits also figured it was an omen of a coming invasion by the Germans, while the French reckoned it was responsible for flooding the Seine. 

A French Scientist named Camille Flammarion, in typical French despair, reckoned that as we passed through the comet’s tail, “cyanogen gas would impregnate the atmosphere and possibly snuff out all life on the planet,” *Wired


1932 The USS Akron, an American dirigible and the world's first purpose-built flying aircraft carrier, flew mail from Lakehurst, New Jersey,  to San Diego, On This Day in 1932. The ship reached Camp Kearny in San Diego, on the morning of 11 May and attempted to moor. Since neither trained ground handlers nor specialized mooring equipment were present, the landing at Camp Kearny was fraught with danger. By the time the crew started the evaluation, the helium gas had been warmed by sunlight, increasing lift. Lightened by 40 short tons (36 t), the amount of fuel spent during the transcontinental trip, the Akron was now all but uncontrollable.

The mooring cable was cut to avert a catastrophic nose-stand by the errant airship which floated upward. Most of the mooring crew—predominantly "boot" seamen from the Naval Training Station San Diego—released their lines although four did not. One let go at about 15 ft (4.6 m) and suffered a broken arm while the three others were carried further aloft. Of these Aviation Carpenter's Mate 3rd Class Robert H. Edsall and Apprentice Seaman Nigel M. Henton soon plunged to their deaths while Apprentice Seaman C. M. "Bud" Cowart held on to his line until being hoisted on board the airship an hour later. The Akron moored at Camp Kearny later that day before proceeding to Sunnyvale, California. The deadly accident was recorded on newsreel film. *Postal Museum @PostalMuseum *Wik


1961 President J. F. Kennedy presented astronaut Alan B. Shepard the first National Aeronautics and Space Administration Distinguished Flying Medal for making America’s first space flight on May 5, 1961.*VFR

Shephard and Capsule recovered after splashdown





BIRTHS

1859 Johan Ludwig William Valdemar Jensen (8 May 1859 – 5 March 1925) contributed to the Riemann Hypothesis, proving a theorem which he sent to Mittag-Leffler who published it in 1899. The theorem is important, but does not lead to a solution of the Riemann Hypothesis as Jensen had hoped. It expresses
... the mean value of the logarithm of the absolute value of a holomorphic function on a circle by means of the distances of the zeros from the centre and the value at the centre.
He also studied infinite series, the gamma function and inequalities for convex functions.*SAU




1905 Karol Borsuk (May 8, 1905, Warsaw – January 24, 1982, Warsaw) Polish mathematician. His main interest was topology.
Borsuk introduced the theory of absolute retracts (ARs) and absolute neighborhood retracts (ANRs), and the cohomotopy groups, later called Borsuk-Spanier cohomotopy groups. He also founded the so called Shape theory. He has constructed various beautiful examples of topological spaces, e.g. an acyclic, 3-dimensional continuum which admits a fixed point free homeomorphism onto itself; also 2-dimensional, contractible polyhedra which have no free edge. His topological and geometric conjectures and themes stimulated research for more than half a century. *Wik



1905  Winifred Lydia Caunden Sargent (8 May 1905 – October 1979) was an English mathematician. She studied at Newnham College, Cambridge and carried out research into Lebesgue integration, fractional integration and differentiation and the properties of BK-spaces.

Sargent's first publication was in 1929, On Young's criteria for the convergence of Fourier series and their conjugates, published in the Mathematical Proceedings of the Cambridge Philosophical Society. In 1931 she was appointed an Assistant Lecturer at Westfield College and became a member of the London Mathematical Society in January 1932. in 1936 she moved to Royal Holloway, University of London, at the time both women's colleges. In 1939 she became a doctoral student of Lancelot Bosanquet, but World War II broke out, preventing his formal supervision from continuing. In 1941 Sargent was promoted to lecturer at Royal Holloway, moving to Bedford College in 1948. She served on the Mathematical Association teaching committee from 1950 to 1954. In 1954 she was awarded the degree of Sc.D. (Doctor of Science) by Cambridge and was given the title of Reader. While at the University of London she supervised Alan J. White in 1959.

Bosanquet started a weekly seminar in mathematics in 1947, which Sargent attended without absence for twenty years until her retirement in 1967. She rarely presented at it, and did not attend mathematical conferences, despite being a compelling speaker.

Much of Sargent's mathematical research involved studying types of integral, building on work done on Lebesgue integration and the Riemann integral. She produced results relating to the Perron and Denjoy integrals and Cesàro summation. Her final three papers consider BK-spaces or Banach coordinate spaces, proving a number of interesting results. *Wik


1923 Dionisio Gallarati (May 8, 1923 – May 13, 2019) was an Italian mathematician, who specialised in algebraic geometry. He was a major influence on the development of algebra and geometry at the University of Genova.

Gallarati published 64 papers between 1951 and 1996.

Important amongst his research was the study of surfaces in P3 with multiple isolated singularities. His lower bounds for maximal number of nodes of surfaces of degree n stood for a long time, and exact solutions for large n were still unknown in 2001.

In Grassmannian geometry he extended Segre's bound "for the number of linearly independent complexes containing the curve in the Grassmannian corresponding to the tangent lines of a nondegenerate projective curve."[3] He extended the results to arbitrarily dimensioned varieties' tangent spaces, to higher degree complexes, and to arbitrary curves in Grassmannians corresponding to degenerate scrolls. *Wik 




DEATHS

1794 Antoine Laurent Lavoisier (26 August 1743 – 8 May 1794) after a trial that lasted less than a day, a revolutionary tribunal condemned Antoine Laurent Lavoisier to death. He was 51 and guillotined on the same afternoon.  " It took only a moment to cause this head to fall and a hundred years will not suffice to produce its like." Joseph Louis Lagrange, the day of Lavoisier’s execution.
Lavoisier was guillotined in the terror following the French Revolution. In 1778, he found that air consists of a mixture of two gases which he called oxygen and nitrogen. By studying the role of oxygen in combustion, he replaced the phlogiston theory. Lavoisier also discovered the law of conservation of mass and devised the modern method of naming compounds, which replaced the older nonsystematic method. Under the Reign of Terror, despite his eminence and his services to science and France, he came under attack as a former Ferme Générale. In November 1793, all former members of the Ferme Générale including Lavoisier and his father-in-law, were arrested and imprisoned. After a trial that lasted less than a day, they were all found guilty of conspiracy against the people of France and condemned. When Lavoisier requested time to complete some scientific work, the presiding judge was said to have answered "The Republic has no need of scientists." He was guillotined and thrown in a common grave in the Cimetière de Picpus. Mathematician Joseph Louis Lagrange lamented the execution: "It took them only an instant to cut off that head, but France may not produce another like it in a century." About eighteen months following his death, Lavoisier was exonerated by the French government. When his belongings were delivered to his widow, a brief note was included reading "To the widow of Lavoisier, who was falsely convicted."
For more about Lavoisier see SomeBeans blog





1779 John Farrar (July 1, 1779 – May 8, 1853) born at Lincoln, Massachusetts. As Hollis professor of mathematics and natural philosophy at Harvard, he was responsible for a sweeping modernization of the science and mathematics curriculum, including the change from Newton’s to Leibniz’s notation for the calculus. *VFR
He was an American scholar. He first coined the concept of hurricanes as “a moving vortex and not the rushing forward of a great body of the atmosphere”, after the Great September Gale of 1815. Farrar remained Professor of Mathematics and Natural Philosophy at Harvard University between 1807 and 1836. During this time, he introduced modern mathematics into the curriculum. He was also a regular contributor to the scientific journals.

After attending Phillips Academy, Andover, and graduating from Harvard in 1803. In 1805, he was appointed Greek tutor at Harvard. Farrar was chosen Hollis Professor of Mathematics and Natural Philosophy in 1807. He retained the chair till 1836, when he resigned in consequence of a painful illness that finally caused his death. His second wife, Eliza Ware Farrar (née Rotch), was Flemish. She married him in 1828. She authored several children's books.

Farrar maintained weather records between 1807 and 1817 at Cambridge, Massachusetts. For the 23 September 1815 hurricane, he particularly noted the shape as "a moving vortex". He also observed the veering of the wind, and its different times of subsequent impacts on the cities of Boston and New York City.
This was the first hurricane, although the word had not been created yet, to hit New England in 180 yrs. In the aftermath of the Great Gale, the concept of a hurricane as a "moving vortex" was presented by John Farrar, Hollis Professor of Mathematics and Natural Philosophy at Harvard University. In an 1819 paper he concluded that the storm "appears to have been a moving vortex and not the rushing forward of a great body of the atmosphere". The word "hurricane" comes from Spanish huracán, from the Taino hurakán, “god of the storm.” While the Taino have been essentially wiped out by disease brought by the Spanish, there are still several words from the language remaining in English. Two of my favorites, Barbecue and Hammock. *Assorted sources (The Merriem Webster gives the first use of Hurricane in 1555, the same year as another Taino word, Yuca,  was first used in English.)Farrar was elected a Fellow of the American Academy of Arts and Sciences in 1808 and a member of the American Antiquarian Society in 1814.

 This was the first hurricane, although the word had not been created yet, to hit New England in 180 yrs. In the aftermath of the Great Gale, the concept of a hurricane as a "moving vortex" was presented by John Farrar, Hollis Professor of Mathematics and Natural Philosophy at Harvard University. In an 1819 paper he concluded that the storm "appears to have been a moving vortex and not the rushing forward of a great body of the atmosphere". The word "hurricane" comes from Spanish huracán, from the Taino hurakán, “god of the storm.” While the Taino have been essentially wiped out by disease brought by the Spanish, there are still several words from the language remaining in English. Two of my favorites, Barbecue and Hammock. *Assorted sources (The Merriem Webster gives the first use of Hurricane in 1555, the same year as another Taino word, Yuca,  was first used in English.)

Mathematical Treasure: Farrar’s Translation of Lacroix’s Algebra MAA




1904 Eadweard Muybridge English photographer important for his pioneering work in photographic studies of motion and in motion-picture projection. For his work on human and animal motion, he invented a superfast shutter. Leland Stanford, former governor of California, hired Muybridge to settle a hotly debated issue: Is there a moment in a horse’s gait when all four hooves are off the ground at once? In 1972, Muybridge took up the challenge. In 1878, he succeeded in taking a sequence of photographs with 12 cameras that captured the moment when the animal’s hooves were tucked under its belly. Publication of these photographs made Muybridge an international celebrity. Another noteworthy event in his life was that he was tried (but acquitted) for the murder of his wife's lover. *TIS




1951 Gilbert Ames Bliss, (9 May 1876, Chicago – 8 May 1951, Harvey, Illinois), was an American mathematician, known for his work on the calculus of variations. Bliss once headed a government commission that devised rules for apportioning seats in the U.S. House of Representatives among the several states.

After obtaining the B.Sc. in 1897, he began graduate studies at Chicago in mathematical astronomy (his first publication was in that field), switching in 1898 to mathematics. He discovered his life's work, the calculus of variations, via the lecture notes of Weierstrass's 1879 course, and Bolza's teaching. Bolza went on to supervise Bliss's Ph.D. thesis, The Geodesic Lines on the Anchor Ring, completed in 1900 and published in the Annals of Mathematics in 1902.

Bliss was elected to the National Academy of Sciences (United States) in 1916.[1] He was the American Mathematical Society's Colloquium Lecturer (1909), Vice President (1911), and President (1921–22). He received the Mathematical Association of America's first Chauvenet Prize, in 1925, for his article "Algebraic functions and their divisors," which culminated in his 1933 book Algebraic functions. He was also an elected member of the American Philosophical Society and the American Academy of Arts and Sciences. *Wikipedia




1959 Renato Caccioppoli (20 January 1904 – 8 May 1959) His most important works, out of a total of around eighty publications, relate to functional analysis and the calculus of variations. Beginning in 1930 he dedicated himself to the study of differential equations, the first to use a topological-functional approach. Proceeding in this way, in 1931 he extended the Brouwer fixed point theorem, applying the results obtained both from ordinary differential equations and partial differential equations.
In 1932 he introduced the general concept of inversion of functional correspondence, showing that a transformation between two Banach spaces is invertible only if it is locally invertible and if the only convergent sequences are the compact ones.
Between 1933 and 1938 he applied his results to elliptic equations, establishing the majorizing limits for their solutions, generalizing the two-dimensional case of Felix Bernstein. At the same time he studied analytic functions of several complex variables, i.e. analytic functions whose domain belongs to the vector space Cn, proving in 1933 the fundamental theorem on normal families of such functions: if a family is normal with respect to every complex variable, it is also normal with respect to the set of the variables. He also proved a logarithmic residue formula for functions of two complex variables in 1949.
In 1935 Caccioppoli proved the analyticity of class C2 solutions of elliptic equations with analytic coefficients.
The year 1952 saw the publication of his masterwork on the area of a surface and measure theory, the article Measure and integration of dimensionally oriented sets (Misura e integrazione degli insiemi dimensionalmente orientati, Rendiconti dell'Accademia Nazionale dei Lincei, s. VIII, v.12). The article is mainly concerned with the theory of dimensionally oriented sets; that is, an interpretation of surfaces as oriented boundaries of sets in space. Also in this paper, the family of sets approximable by polygonal domains of finite perimeter, known today as Caccioppoli sets or sets of finite perimeter, was introduced and studied.
His last works, produced between 1952 and 1953, deal about a class of pseudoanalytic functions, introduced by him to extend certain properties of analytic functions.
In his last years, the disappointments of politics and his wife's desertion, together perhaps with the weakening of his mathematical vein, pushed him into alcoholism. His growing instability had sharpened his "strangenesses", to the point that the news of his suicide on May 8, 1959 by a gunshot to the head did not surprise those who knew him. He died at his home in Palazzo Cellammare
In 1992 his tormented personality was remembered in a film directed by Mario Martone, The Death of a Neapolitan Mathematician (Morte di un matematico napoletano), in which he was portrayed by Carlo Cecchi. * Wik




1953 Benjamin Fedorovich Kagan (10 March 1869 in Shavli, Kovno (now Kaunas, Lithuania)
- 8 May 1953 in Moscow, USSR) Kagan worked on the foundations of geometry and his first work was on Lobachevsky's geometry. In 1902 he proposed axioms and definitions very different from Hilbert. Kagan studied tensor differential geometry after going to Moscow because of an interest in relativity.
Kagan wrote a history of non-euclidean geometry and also a detailed biography of Lobachevsky. He edited Lobachevsky's complete works which appeared in five volumes between 1946 and 1951. *SAU




1960 John Henry Constantine Whitehead FRS (11 November 1904 – 8 May 1960), known as "Henry", was a British mathematician and was one of the founders of homotopy theory. He was born in Chennai (then known as Madras), in India, and died in Princeton, New Jersey, in 1960.

During the Second World War he worked on operations research for submarine warfare. Later, he joined the codebreakers at Bletchley Park, and by 1945 was one of some fifteen mathematicians working in the "Newmanry", a section headed by Max Newman and responsible for breaking a German teleprinter cipher using machine methods.Those methods included the Colossus machines, early digital electronic computers.

From 1947 to 1960 he was the Waynflete Professor of Pure Mathematics at Magdalen College, Oxford.

He became president of the London Mathematical Society (LMS) in 1953, a post he held until 1955. The LMS established two prizes in memory of Whitehead. The first is the annually awarded, to multiple recipients, Whitehead Prize; the second a biennially awarded Senior Whitehead Prize *Wik



2016  Tom Mike Apostol (/əˈpɑːsəl/ ə-POSS-əl; August 20, 1923 – May 8, 2016) was an American analytic number theorist and professor at the California Institute of Technology, best known as the author of widely used mathematical textbooks.

Apostol received his Bachelor of Science in chemical engineering in 1944, Master's degree in mathematics from the University of Washington in 1946, and a PhD in mathematics from the University of California, Berkeley in 1948. Thereafter Apostol was a faculty member at UC Berkeley, MIT, and Caltech. He was the author of several influential graduate and undergraduate level textbooks.

Apostol was the creator and project director for Project MATHEMATICS! producing videos which explore basic topics in high school mathematics. He helped popularize the visual calculus devised by Mamikon Mnatsakanian with whom he also wrote a number of papers, many of which appeared in the American Mathematical Monthly. Apostol also provided academic content for an acclaimed video lecture series on introductory physics, The Mechanical Universe.

In 2001, Apostol was elected in the Academy of Athens. He received a Lester R. Ford Award in 2005, in 2008, and in 2010.  In 2012 he became a fellow of the American Mathematical Society

A favorite of mine...






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