Pat'sBlog

Monday, 9 March 2026

Islands in the Mist, ----- of Polynomials, and Pretty Geometry

I once read a description of math as like seeing islands in a great ocean covered by a mist. As you learn the subject you work around on an island and clear away some of the mist. Often your education jumps from one island to another at the direction of a teacher and eventually you have mental maps of parts of many separate islands. But at some point, you clear away a fog on part of an island and see it connects off to another island you had partially explored, and now you know something deeper about both islands and the connectedness of math.

I was recently reminded of one of those kinds of connections that ties together several varied topics from the high school education of most good math students. It starts with that over-criticized (and under-appreciated, i) Algebra I technique of factoring.
Almost ever student in introductory algebra is introduced to a "sum and product" rule that relates the factors of a simple quadratic (with quadratic coefficient of one) to the coefficients. The rule says that if the roots are at p and q, then the linear coefficient will be the negative of p+q, and the constant term will be their product, pq. So for example, the simple quadratic with roots at x=2 and x=3 will be x² - 5x + 6.

I know from experience that if you take a cross section of 100 students who enter calculus classes after two+ year of algebra, very few will know that you can extend that idea out to cubics and higher power polynomials. An example for a polynomial with four roots will probably suffice for most to understand. Because the constant terms in linear factors are always the opposite of the roots, {if 3 is a root, (x-3) is a factor} it is easiest to negate all the roots before doing the math involved (at least for me it always was).
So if we wanted to find the simple polynomial with roots at -1, -2, -3, and -4 (chosen so all the multipliers are +) we would find that the fourth degree polynomial will have 10 for the coefficient of x³ because 1+2+3+4 = 10, just as it works in the second term of a quadratic. After that, the method starts to combine sets of them. The next coefficient will be the sum of the products of each pair of factor coefficients. In the example I created we would add 1x2+1x3+1x4+2x3+2x4+3x4 to get 35x². The next term sums all triple products of the numbers, 1x2x3 + 1x2x4 + 1x3x4 + 2x3x4 = 50 for the linear coefficient. And in the constant term, we simply multiply all of them together to get 24.

After you've carried that around for a while and maybe forgotten how to get all the other terms, the easy part may remain; the second term is the sum of the opposite of the roots, and the constant term is their product. Then you get to calculus and you learn how easy it is to take the derivative of a polynomial. Then maybe you are playing around with some simple derivatives and you realize that a function f(x) = xⁿ + Ax^n-1 + ... will have a derivative that is f'(x)=nx^n-1 + A(n-1)x^n-2. You realize that if f(x) has roots that sum to A, then f'(x) has roots that will sum to (n-1)A/n [If your younger and this seems unclear, note that the roots of f(x) are the same as the roots of n*f(x), for example, y= x² - 1 has the same roots (+/-1) as 2x²-2 or 3x²-3 etc].

Much later, you come back across this thought, but because you are at a different place in your understanding of math, you realize that means that the average of the zeros of f(x) is A/n, because there are n of them. So the average of f'(x) must also be A/n because there are n-1 of them... and since f"(x) is related to f'(x) by this same method, A/n must be the average of all the zeros of derivatives of f(x) that do not descend to a constant value.
Because that seems to glib to pass muster with most of my students, an example of these last two paragraphs, to show how interrelated they are. Take the example f(x) = $x^4 + 3x^3 + 7x^2 + 2x + 4 $. We simply inspect to see that the roots have a sum of 3, and since there are four of them, their average is -3/4. Without knowing the derivatives, we know the roots of f' will sum to $ \frac{3(-3)}{4} = \frac{-9}{4} $ and since there are three of them, their average is ...yeah... -3/4. We can find f" and the rest by continuing this, but the big flashing light here is that the average stays the same, so the sum of the roots is just the average root times the highest power of the derivative.

You smugly nest that away in your mind and go on about your business, occasionally refreshing it by relating it to a friend or colleague in the coffee shop or at a conference.

Someday down the line you wonder, or someone you relate it to asks, will that work with numbers that have complex roots, and you quickly convince yourself that it will, and feel pretty smug for knowing all this. Then you stumble across an old copy of Professor Dan Kalman's paper on Marden's Theorem (at least you will if you are as lucky as I was). (Professor Kalman was awarded the 2009 Lester R. Ford Award of the MAA for his 2008 paper on this theorem. Jörg Siebeck discovered this theorem 81 years before Morris Marden wrote about it (1965). However, Prof Kalman writes, "I call this Marden’s Theorem because I first read it in M. Marden’s wonderful book". The theorem says that if you take a trinomial with complex roots (even if the coefficients are complex numbers) there is a really beautiful geometric tie in to the average idea, and more.

I will illustrate with an example that is easy to picture. Suppose we take a trinomial with roots of 2+5i, 2-5i and 6, f(x) =x³ - 10 x² + 53 x - 174. The derivative will be 3x² - 20x + 53, with zeros at the complex conjugates x = 1/3 (10-i sqrt(59)) and x = 1/3 (10+i sqrt(59)). Both of these we can see quickly have averages of 10/3 for the zeros, but these first derivative zeros will play a special geometric role a little later in Marden's theorem.

The second derivative of the original cubic gives us 6x-20, with a zero which agrees with the average of the zeros above.

All those little islands with a common algebraic truth seem somehow connected, but then a little more of the mist clears, and the geometry is revealed.

But if we examine these zeros on a complex plane, the three zeros of the original function can form the vertices of a triangle. And the two zeros of the first derivative fall inside that triangle, with the zero of f" bisecting the segment joining them.

So the vertices of the triangle are the roots of f(x), the two red points are f' zeros, and they are the foci of the ellipse shown inscribed in the triangle. And f" has a zero at the center of the ellipse. The ellipse passes through the midpoints of the vertices, and it turns out it is the maximal area ellipse you can inscribe in that triangle, called the Steiner inellipse. (A little algebra, a little calculus, a little geometry, a little trig... maybe there are really no islands, just one math land mass. )

I backed it all up one level by integrating f(x) but the four roots did not appear to relate to the three vertices of the trinomial in any pretty way. They do obey the Gauss-Lucas Theorem. The Gauss–Lucas theorem gives a geometrical relation between the roots of a polynomial P and the roots of its derivative P'. The set of roots of a real or complex polynomial is a set of points in the complex plane. The theorem states that the roots of P' all lie within the convex hull of the roots of P, that is the smallest convex polygon containing the roots of P. When P has a single root then this convex hull is a single point and when the roots lie on a line then the convex hull is a segment of this line. The Gauss–Lucas theorem, named after Carl Friedrich Gauss and Félix Lucas is similar in spirit to Rolle's theorem, another high school calculus basic.

And here is a tie-in for the stats students, the line containing the foci and centroid is the least squares regression line for the three vertices.

If you have only three roots to a higher degree polynomial (one with some or all the roots multiple, such as f(x) = (x-a)^J (x-b)^K(x-c)^J then the ellipse will be tangent at points that divide the segments in ratios of J/K, K/L, and L/J. This is due to Linfield who published it in 1920.

And if you have an n-sided polygon which is tangent to an ellipse at all four midpoints, it seems that there is a complex polynomial with those roots whose derivative has zeros at the foci of the ellipse. I managed to create an easy example by using the idea of a rhombus centered at the origin. The polynomial f(x)=x⁴+3x² - 4 has zeros at 2i, -2i, 1 and -1. The derivative, 4x²+6 has zeros at +sqrt(3/2) and -sqrt(3/2). Using this focus and the point (1/2,1) which is the midpoint of one side of the rhombus I get the ellipse 4x² + y²=2 which seems to work.

I don't have a clear easy way to recognize what fourth power polynomials would have that property, so if you want to be next to teach me some math, send me what you know.

I communicated several times in 2007 with Professor Kalman when we shared some information about the history of a problem we were both working on. He went on to include that material, and Marden's Theorem in his wonderful book Uncommon Mathematical Excursions: Polynomia and Related Realms. If you pick it up, check the acknowledgements. There is actually a hat-tip from the professor to yours truly for (a very tiny bit of) assistance with the material for Lill's graphic method of solving for the roots of a polynomial. Still, I'm grateful for any recognition.

On This Day in Math - March 9

Waldsemuller 1507 first map to use the name "America" *Wik

Don't worry about people stealing your ideas. If your ideas are any good, you'll have to ram them down people's throats.
~Howard Aiken

The 68th day of the year; if you searched through pi for all the two digit numbers, the last one you would find is 68. The string 68 begins at position 605 counting from the first digit after the decimal point. (What is the last single digit numeral to appear? One might wonder how far out the string would you have to go to find all possible three digit numbers? )

68 is the largest known number to be the sum of two primes in exactly two different ways: 68=7+61=31+37.

68 is a stobogrammatic number, rotated it is 89. Some consider only invertible numbers (rotated they form the same value, like 181) as strobograms. HT to Paul O'Malley

There are exactly 68 ten digit binary numbers in which each digit is the same as one of it's adjacent digits.

The numbers on the two diagonals of Durer's Melencolia add up to 68. All the numbers not on the two main diagonals also add up to 68, but that's just the teaser. The sum of the squares of the numbers in each group sum to 748.....Not impressed? Try the sum of the cubes of each group...Yep also equal, their sums are each 9248....Now That's Magic...Math Magic!

$2^{68} = 295147905179352825856$, notice anything? Every digit 0 through 9 is included. There is no smaller power of two for which this is true.*@fermatslibrary

68 is the smallest composite number that can be read as a prime number when it is rotated 180^o HT Jim Wilder @wilderlab.

And a historical oddity, in 46 BCE, as a result of Julius Caesar's Calendar adjustment, there were 68 days inserted between November and December.

EVENTS

1497 Copernicus, then a student of canon law at the University of Bologna, made his ﬁrst recorded astronomical observation. Working with Domenica Maria Novara, a professor of astronomy at the university, from whom he rented a room, they observed an occultation of the star aldebaran by the moon. He will later mention this as one of the influential experiences in shaping his new theory.

In 1611, (Mar 9 (NS),Feb 27 (OS)) Johannes Fabricius, a Dutch astronomer, observed the rising sun through his telescope, and observed several dark spots on it. This was perhaps the first ever observation of sunspots. (This is not true, see below) He called his father to investigate this new phenomenon with him. The brightness of the Sun's center was very painful, and the two quickly switched to a projection method by means of a camera obscura. Johannes was the first to publish information on such observations. He did so in his Narratio de maculis in sole observatis et apparente earum cum sole conversione. ("Narration on Spots Observed on the Sun and their Apparent Rotation with the Sun"), the dedication of which was dated 13 Jun 1611. *TIS

In 1611 Fabricius’ son Johannes brought home a telescope from the University of Leiden where he was studying medicine. With this instrument the father and son, with the son this time in the leading role, discovered the sunspots. Although they were not the first European astronomers to make this discovery, this honor goes to Thomas Harriot, Johannes Fabricius was the first to publish it in his De Maculis in Sole in 1611. Unfortunately his publication went largely unnoticed and is not mentioned at all by Galileo and Christoph Scheiner in their monumental argument as to who first discovered the sunspots. *RMAT (correct answer, neither of them)

(though unclear statements in East Asian annals suggest that Chinese and Korean astronomers may have discovered them with the naked eye previously, and Fabricius may have noticed them himself without a telescope a few years before).

The first known drawing of sunspots dates back to John of Worcester in 1128.

John of Wooster drawing

1671 Hooke demonstrates vibration due to sound. For the benefit of two visiting Italian noblemen, Hooke shows how flour "moves like a liquid" when placed in a broad shallow glass when it is struck or vibrated. The flour would rise up the edge of the glass and run over. *Stephen Inwood, The Forgotten Genius

1736 Euler receives a letter challenging him to solve the Konigsburg Bridge Problem.

"Carl Leonhard Gottlieb Ehler was the mayor of Danzig in Prussia (now Gdansk in Poland), some 80 miles west of Kinigsberg. He corresponded with Euler from 1735 to 1742, acting as intermediary for Heinrich Kuhn, a local mathematics professor. Their initial communication has not been recovered, but a letter of 9 March 1736 indicates they had discussed the problem and its relation to the 'calculus of position':
You would render to me and our friend Kiihn a most valuable service, putting us greatly in your debt, most learned Sir, if you would send us the solution, which you know well, to the problem of the seven Kinigsberg bridges, together with a proof. It would prove to be an outstanding example of the calculus of position [Calculi Situs], worthy of your great genius. I have added a sketch of the said bridges ... "

*Brian Hopkins, Robin Wilson; The Truth About Konigsberg

1832 Wolfgang Bolyai made a corresponding member of the mathematics section of the Magyar Academy. *Bonola, Non-Euclidean Geometry, Appendix 1, p. xxv

In 1893, Professor James Dewar communicated to the meeting of the Royal Society that he had succeeded in freezing air into a clear, transparent solid. The precise nature of this solid was not known, and needed further research. It was speculated that it may be “a jelly of solid nitrogen containing liquid oxygen, much as calves'foot jelly contains water diffused in solid gelatine. Or it may be a true ice of liquid air, in which both oxygen and nitrogen exist in the solid form.” At this time, Dewar had not been able to solidify pure oxygen, although nitrogen had been frozen with comparative ease. It also had already been proved that in the evaporation of liquid air, nitrogen boils off first.*TIS

1914 The Mining and Metallurgical Society held a dinner to bestow its first gold medal on future President Herbert Hoover, and his wife Lou Henry Hoover for their joint translation of Georgius Agricola's De Re Metallica.(1556) Both Hovers had earned bachelor degrees in geology from Stanford. It is said that the future First Lady bore most of the "heavy lifting" in the translation. The President was poor at languages.

1951 Edward Teller and Stanislaw Ulam submit a classified paper at the Los Alamos lab, in which they proposed their revolutionary new design, staged implosion, for a practical megaton-range hydrogen bomb.

1961 Korabl-Sputnik 4 or Vostok-3KA No.1, also known as Sputnik 9 in the West, was a Soviet spacecraft which was launched on 9 March 1961. Carrying the mannequin Ivan Ivanovich, a dog named Chernushka, some mice and the first guinea pig in space, it was a test flight of the Vostok spacecraft. *Wik

1993 PowerOpen Association Formed: Apple Computer Inc., Motorola Inc., IBM Corp. and four other computer companies form the PowerOpen Association Inc., intended to promote new computer chip technology in preparation for the release of the next generation of personal computers. The association also tested conformance to the PowerOpen environment, which led to computers such as Apple's Power PC. *CHM

BIRTHS

1451 Amerigo Vespucci (9 Mar 1451; died 22 Feb 1512 at age 60. ), Italian navigator, who claimed to have reached North and South America in 1498. It is after him that the continents are named. *VFR

Waldsemuller 1507 first map to use the name "America" *Wik

1564 David Fabricius(9 Mar 1564; died 7 May 1617 at age 53) A German astronomer, friend of Tycho Brahe and Kepler, and one of the first to follow Galileo in telescope observation of the skies. He is best known for a naked-eye observation of a star in Aug 1596, subsequently named Omicron Ceti, the first variable star to be discovered, and now known as Mira. Its existence with variable brightness contradicted the Aristotelian dogma that the heavens were both perfect and constant. With his son, Johannes Fabricius, he observed the sun and noted sunspots. For further observations they used a camera obscura and recorded sun-spot motion indicating the rotation of the Sun. (David Fabricius wrote to Michael Maestlin (Kepler's old teacher) that he did not believe the spots were on the Sun's body, although the center of their motions clearly lay in the Sun. *Galileo Project)

Kepler had studied the stars and planets with a camera obscura that had a lens to sharpen the view in 1600.(he coined the term camera obscura in 1604 from the Latin for dark chamber or dark room. Before the term camera obscura was first used, other terms were used to refer to the devices: cubiculum obscurum, cubiculum tenebricosum, conclave obscurum, and locus obscurus.)
Fabricius, a Protestant minister, was killed by a parishioner angered upon being accused by him as a thief. *TIS (after denouncing a local goose thief from the pulpit, the accused man struck him in the head with a shovel and killed him.. *Wik)

*Wik

[re: invented, The Camera Obscura (Latin for dark room) was a dark box or room with a hole in one end. If the hole was small enough, an inverted image would be seen on the opposite wall. Such a principle was known by thinkers as early as Aristotle (c. 300 BC). It is said that Roger Bacon invented the camera obscura just before the year 1300, but this has never been accepted by scholars; more plausible is the claim that he used one to observe solar eclipses. In fact, the Arabian scholar Hassan ibn Hassan (also known as Ibn al Haitam), in the 10th century, described what can be called a camera obscura in his writings..] The image is the first published picture of camera obscura in Gemma Frisius' 1545 book De Radio Astronomica et Geometrica

1818 Ferdinand Joachimsthal (9 March 1818 in Goldberg, Prussian Silesia (now Złotoryja, Poland) - 5 April 1861 in Breslau, Germany (now Wrocław, Poland)) Influenced by the work of Jacobi, Dirichlet and Steiner, Joachimsthal wrote on the theory of surfaces where he made substantial contributions, particularly to the problem of normals to conic sections and second degree surfaces.
Joachimsthal applied the theory of determinants to geometry. He made the important step of introducing oblique coordinates. Joachimsthal surfaces are named after him, these have a family of plane lines of curvature within the plane of a pencil. He has a theorem named after him which concerns the intersection of surfaces. He is also remembered for another theorem on the four normals to an ellipse from a point inside it. *SAU His name is derived from the region in Silesia which was rich in Silver. Coins made with the silver were called "daler", from German T(h)aler, short for Joachimsthaler, a coin from the silver mine of Joachimsthal (‘Joachim's valley’), now Jáchymov in the Czech Republic. The term was later applied to a coin used in the Spanish American colonies, which was also widely used in the British North American colonies at the time of the American War of Independence, hence adopted as the name of the US monetary unit in the late 18th century.

1824 Birthdate of Leland Stanford, the American railroad builder and capitalist who founded Stanford University in 1885. *VFR

1852 Constantin Marie Le Paige (9 March 1852 in Liège, Belgium - 26 Jan 1929 in Liège, Belgium) worked on the theory of algebraic forms, a topic whose study was initiated by Boole in 1841 and then developed by Cayley, Sylvester, Hermite, Clebsch and Aronhold. In particular Le Paige studied the geometry of algebraic curves and surfaces, building on this earlier work. He is best known for his construction of a cubic surface given by 19 points.
Le Paige studied the generation of plane cubic and quartic curves, developing further Chasles's work on plane algebraic curves and Steiner's results on the intersection of two projective pencils.
The history of mathematics was another topic which interested Le Paige. He published Sluze's correspondence with Pascal, Huygens, Oldenburg and Wallis. *SAU

1900 Howard Hathaway Aiken (9 Mar 1900; 14 Mar 1973 at age 72) American mathematician who invented the Harvard Mark I, forerunner of the modern electronic digital computer. While a graduate student and instructor Harvard University, Aiken's research had led to a system of differential equations which could only be solved using numerical techniques, for which he began planning large computer. His idea was to use an adaptation of Hollerith's punched card machine. When eventually built, (1943) it weighed 35 tons, had 500 miles of wire and could compute to 23 significant figures. There were 72 storage registers and central units to perform multiplication and division. It was controlled by a sequence of instructions on punched paper tapes, and used punched cards to enter data and give output from the machine. *TIS

1923 Walter Kohn (March 9, 1923 – April 19, 2016) Austrian-American physicist who shared (with John A. Pople) the 1998 Nobel Prize in Chemistry. The award recognized their individual work on computations in quantum chemistry. Kohn's share of the prize acknowledged his development of the density-functional theory, which made it possible to apply the complicated mathematics of quantum mechanics to the description and analysis of the chemical bonding between atoms. *TIS
"Paris somehow lends itself to conceptual new ideas. There is a certain magic to that city." (Thanks to Arjen Dijksman)

1948 László Lovász (March 9, 1948 - ) is a Hungarian mathematician, best known for his work in combinatorics, for which he was awarded the Wolf Prize and the Knuth Prize in 1999, and the Kyoto Prize in 2010.*Wik

DEATHS

886 Abu masar (10 Aug 787, 9 Mar 886 at age 98)Persian astrologer, a.k.a. Abu Ma'shar al-Balkhi, or Ja'far ibn Muhammad, who was the leading astrologer of the Muslim world. He is known primarily for his theory that the world, created when the seven planets were in conjunction in the first degree of Aries, will come to an end at a like conjunction in the last degree of Pisces. *TIS

His discourses incorporated and expanded upon the studies of earlier scholars of Islamic, Persian, Greek, and Mesopotamian origin. His works were translated into Latin in the 12th century and, through their wide circulation in manuscript form, had a great influence on Western scholars. Kitab al-Mudkhal al-Kabīr (Great introduction) is his most important work and the one most frequently cited by scholars in the West. It contains an astrological theory on the nature of the moon's influence on the tides and was the key work on the subject during the Middle Ages. This edition is the 1140 translation into Latin by Hermann of Carinthia, first printed by Erhard Ratdolt in Augsburg, Germany, in 1489. The woodcut title vignette of a black-faced astronomer reading the stars with an astrolabe and dividers is one of the best-known Renaissance representations of an astronomer. *Library of Congress

1833 Jacques Frédéric Français (20 June 1775 in Saverne, Bas-Rhin, France - 9 March 1833 in Metz, France) In September 1813 Français published a work in which he gave a geometric representation of complex numbers with interesting applications. This was based on Argand's paper which had been sent, without disclosing the name of the author, by Legendre to François Français. Although Wessel had published an account of the geometric representation of complex numbers in 1799, and then Argand had done so again in 1806, the idea was still little known among mathematicians. This changed after Français' paper since a vigorous discussion between Français, Argand and Servois took place in Gergonne's Journal. In this argument Français and Argand believed in the validity of the geometric representation, while Servois argued that complex numbers must be handled using pure algebra. *SAU

Argand diagram

1851 Hans Christian Oersted (14 Aug 1777, 9 Mar 1851 at age 73) Danish physicist and chemist whose discovery (1820) that an electric current in a wire causes a nearby magnetized compass needle to deflect, indicating the electric current in a wire induces a magnetic field around it, marks the starting point for the development of electromagnetic theory. For this, he can be called “the father of electromagnetism,” for which his name was adopted for the magnetic field strength in the CGS system of units (for which the SI system now uses the henry unit). Philosophically, he had believed nature's forces had a common origin. Oersted was the first to isolate aluminum as a metal (1825). He also made the first accurate determination of the compressibility of water (1822). Late in his career, he researched diamagnetism. In his final years, he turned back to philosophy, and started writing The Soul in Nature. *TIS

Statue of Oerstead at Oxford

1866 Edmond Bour (19 May 1832 in Gray, Haute-Saône, France - 9 March 1866 in Paris, France)Bour made many significant contributions to analysis, algebra, geometry and applied mechanics despite his early death from an incurable disease. His remarkable achievements were cut short at the age of 33 and as a consequence Bour is hardly known in the history of mathematics whereas one feels that if he had been given the chance to continue his outstanding work he would today be remembered as one of the major figures in the subject. *SAU

1917 Agnes Sime Baxter (Hill) (18 March 1870 – 9 March 1917) was a Canadian-born mathematician. She studied at Dalhousie University, receiving her BA in 1891, and her MA in 1892. She received her Ph.D. from Cornell University in 1895; her dissertation was “On Abelian integrals, a resume of Neumann’s ‘Abelsche Integrele’ with comments and applications." *Wik

It must have been a difficult time for a woman such as Baxter taking what were considered at that time to be men's subjects. Although few women studied mathematics, Baxter was not the only one studying mathematics at Dalhousie; for example there were two other women in the class of 24 that studied second level mathematics with her. Her performance at university was outstanding and she was awarded a distinction and received the Sir William Young Gold Medal. With the award of her B.A. degree Baxter became the first ever woman to graduate with honours from Dalhousie University. But more than this, she had been clearly the best student in both mathematics and mathematical physics.

When she graduated in 1895, Baxter became only the second Canadian woman to be awarded Ph.D. in Mathematics. On a wider scale, she was only the fourth woman to receive such a degree in the whole of North America.*SAU

1923 Johannes Diederik van der Waals (23 Nov 1837; 9 Mar 1923) Dutch physicist, winner of the 1910 Nobel Prize for Physics for his research on the gaseous and liquid states of matter. He was largely self-taught in science and he originally worked as a school teacher. His main work was to develop an equation (the van der Waals equation) that - unlike the laws of Boyle and Charles - applied to real gases. Since the molecules do have attractive forces and volume (however small), van der Waals introduced into the theory two further constants to take these properties into account. The weak electrostatic attractive forces between molecules and between atoms are called van der Waals forces in his honour. His valuable results enabled James Dewar and Heike Kamerlingh-Onnes to work out methods of liquefying the permanent gases. *TIS

1931 Ivan Vladislavovich Sleszynski (23 July 1854 in Lysianka, Cherkasy, Kiev gubernia, Ukraine - 9 March 1931 in Kraków, Poland)Sleszynski's main work was on continued fractions, least squares and axiomatic proof theory based on mathematical logic. In a paper of 1892, based on his doctoral dissertation, he examined Cauchy's version of the Central Limit Theorem using characteristic function methods, and made several significant improvements and corrections. Because of the work, he is recognised as giving the first rigorous proof of a restricted form of the Central Limit Theorem. *SAU

1942 Mykhailo Pilipovich Krawtchouk (27 Sept 1892 in Chovnitsy, (now Kivertsi) Ukraine - 9 March 1942 in Kolyma, Siberia, USSR) In 1929 Krawtchouk published his most famous work, Sur une généralisation des polynômes d'Hermite. In this paper he introduced a new system of orthogonal polynomials now known as the Krawtchouk polynomials, which are polynomials associated with the binomial distribution.
However his mathematical work was very wide and, despite his early death, he was the author of around 180 articles on mathematics. He wrote papers on differential and integral equations, studying both their theory and applications. Other areas he wrote on included algebra (where among other topics he studied the theory of permutation matrices), geometry, mathematical and numerical analysis, probability theory and mathematical statistics. He was also interested in the philosophy of mathematics, the history of mathematics and mathematical education. Krawtchouk edited the first three-volume dictionary of Ukrainian mathematical terminology. *SAU

Mykhailo Krawtchouk was arrested on February 21, 1938. He was charged with a

membership in an underground Ukrainian nationalist and terrorist organization and spying.

The accusation of a membership in such fictitious organizations was a typical charge

against the country’s intellectuals during the Great Terror. The allegations of stemmed

from the fact that Krawtchouk closely identified with Ukrainian culture, playing a major

role in developing Ukrainian mathematical terminology and mathematical education.

On September 23, 1938, in a trial lasting one half-hour, the Military Collegium of the Supreme Court of the USSR sentenced Krawtchouk to 20 years in prison and 5 years in exile. Mykhailo Krawtchouk

was assigned to perform heavy manual work in a gold mine in Kolyma region, one of the coldest and most uninhabitable places on the planet. Like a large proportion of political prisoners, Krawtchouk did not survive the Kolyma camps.

1954 V(agn) Walfrid Ekman (3 May 1874, 9 Mar 1954 at age 79) Swedish physical oceanographer and mathematical physicist whose research into the dynamics of ocean currents led to his name remaining associated with terms for particular phenomena of the ocean or atmosphere, including Ekman spiral, Ekman transport and Ekman layer. Fridtjof Nansen pointed out to Ekman that he had noticed that icebergs drift at an angle of 20°-40° to the prevailing wind, rather than directly with the wind. In 1902, Ekman published an explanation, known now as the Ekman spiral, describing movement of ocean currents influenced by the Earth's rotation. He also developed experimental techniques and instruments such as the Ekman current meter and Ekman water bottle.*TIS

1962 Dr. Howard T. Engstrom (23 Jun 1902, 9 Mar 1962 at age 59) American computer designer who promoted the first commercially available digital computer, the Univac. As a Yale professor he had written a paper on the mathematical basis for cryptanalysis techniques. During WW II he was called to the Navy and placed in command of the OP-20-G automated machines "Research Section" for message decryption. After the war, he was a co-founder of Engineering Research Associates, a private company to work on electronic digital circuit technology for the Navy on a contract basis, with former Navy researchers. ERA delivered its first Atlas computer to the National Security Agency in Dec 1950. As vice president for research, Engstrom took the initiative to make a commercial version, renamed Univac. *TIS

1981 Max Ludwig Henning Delbrück (September 4, 1906 – March 9, 1981)
Delbrück was a German-American biophysicist and Nobel laureate.
Delbrück studied astrophysics, shifting towards theoretical physics, at the University of Göttingen. After receiving his Ph.D. in 1930, he traveled through England, Denmark, and Switzerland. He met Wolfgang Pauli and Niels Bohr, who got him interested in biology.
In 1937, he moved to the United States to pursue his interests in biology, taking up research in the Biology Division at Caltech on genetics of the fruit fly Drosophila melanogaster.
Delbrück was one of the most influential people in the movement of physical scientists into biology during the 20th century. Delbrück's thinking about the physical basis of life stimulated Erwin Schrödinger to write the highly influential book, What Is Life?. Schrödinger's book was an important influence on Francis Crick, James D. Watson and Maurice Wilkins who won a Nobel prize for the discovery of the DNA double helix. *TIA

1993 Max August Zorn (6 June 1906 in Krefeld, Germany - 9 March 1993 in Bloomington, Indiana, USA) To his chagrin, he is most famous for discovering something yellow and equivalent to the Axiom of Choice. *VFR (with a smile, I'm sure) He was an algebraist, group theorist, and numerical analyst. He is best known for Zorn's lemma, a powerful tool in set theory that is applicable to a wide range of mathematical constructs such as vector spaces, ordered sets, etc. Zorn's lemma was first discovered by K. Kuratowski (see June 18) in 1922, and then independently by Zorn in 1935.*Wik Today we know that the Axiom of Choice, the well-ordering principle, and Zorn's Lemma (the name now given to Zorn's maximum principle by Tukey and now the standard name) are equivalent. *SAU

2020 Richard Kenneth Guy (born September 30, 1916, Nuneaton, Warwickshire - March 9, 2020 ) is a British mathematician, and Professor Emeritus in the Department of Mathematics at the University of Calgary.
He is best known for co-authorship (with John Conway and Elwyn Berlekamp) of Winning Ways for your Mathematical Plays and authorship of Unsolved Problems in Number Theory, but he has also published over 100 papers and books covering combinatorial game theory, number theory and graph theory.
He is said to have developed the partially tongue-in-cheek "Strong Law of Small Numbers," which says there are not enough small integers available for the many tasks assigned to them — thus explaining many coincidences and patterns found among numerous cultures.
Additionally, around 1959, Guy discovered a unistable polyhedron having only 19 faces; no such construct with fewer faces has yet been found. Guy also discovered the glider in Conway's Game of Life.
Guy is also a notable figure in the field of chess endgame studies. He composed around 200 studies, and was co-inventor of the Guy-Blandford-Roycroft code for classifying studies. He also served as the endgame study editor for the British Chess Magazine from 1948 to 1951.
Guy wrote four papers with Paul Erdős, giving him an Erdős number of 1. He also solved one of Erdős problems.

Many number theorists got their start trying to solve problems from Guy's book Unsolved problems in number theory.

His son, Michael Guy, is also a computer scientist and mathematician.

Guy died on 9 March 2020 at the age of 103.

*Wik

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

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

Sunday, 8 March 2026

Heron and his Formula(s)

Ok, you probably know Heron's Formula (if your teacher calls it Hero's formula, it's the same)... Heron of Alexandria, sometimes called Hero, lived around the year 100 AD and is most often remembered for a formula for the area of a triangle. The formula gives a method of computing the area from the lengths of the three sides...no angles required. If we call the sides a, b, and c; then the area is given by $A= \sqrt {s(s-a)(s-b)(s-c)}$ where the "s" stands for the semi-perimeter, $s=\frac{a + b + c}{2}$. You can find a nice geometric proof of Heron's formula at this link to the Dr. Math site. The proof was done by Dr. Floor, who credits the method to Paul Yiu of Florida Atlantic University. Documents from the Arabic writers indicate Archimedes may well have known this formula 300 years before Heron. In 1896, a copy of Heron's Metrica was recovered in Constantinople (now Istanbul) that had been copied around 1100 AD. It contains the oldest known demonstration of the formula. Wikipedia also has several nice proofs of the theorem, including one derived from the Pythagorean Thm.

I recently (2024) found a very similar formula for the square of the volume of a Tetrahedron in a paper in Letters to the editor of The Mathematical Intellingencer by Martin Lukarevski. The Formula applies to a tetrahedron with all four faces congruent with edges a, b, c. I did a little simple algebra to enhance the similarity to Heron's formula.

$V(T)= \sqrt(\frac{( -a^2 + b^2 + c^2)( +a^2 - b^2 + c^2)( +a^2 + b^2 - c^2)}{72})$

Heron is also remembered for his invention of a primitive steam engine and many early automatons, and a coin operated vending machine, and one of the earliest forerunners of the thermometer. The image at right shows a picture of a reconstruction of Heron's steam engine. The image is from the Smith College museum of Ancient Inventions where you can find more about Heron's, and many other's, interesting creations. An extension of Heron's area formula for cyclic

Heron's Metrica also contains one of the earliest examples of a method of finding square roots that is called the divide and average model. To find an approximate square root of a number, N, think of any number smaller than N, which we will call M. Then find a new approximation by letting E = (M + N/M)/2. Another approximation can be found by repeating the method with this new approximation. For example, beginning with N=20 and M= 2, we get E= (2 + 20/2) / 2 or E= (2+10)/2 = 6. Repeating with M= 6 we get E= (6+ 20/6)/2 = ( 6 + 3 1/3 )/2 = 14/3 or 4 2/3. After only two iterations from a very bad starting guess the approximation is within .2 of the correct value.

Heron is also remembered for a problem he solved in Catoprica; Given two points, A and B, on the same side of a line, find a point X on the line so that the total distance AX+XB is a minimum. The solution may come quickly if you know that the translation of Catoprica is "About Mirrors". The solution given by Heron is to find the mirror reflection of point B in the line, B', and draw a straight line from A to B'. Where it intersects the line is the choice of point X.

Ok, so much for the old news... but recently I was going through some old journals that Dave Refro sends me from time to time to keep me out of mischief, and I came across an article in the 1885 Annals of Mathematics which listed 105 different formulas for the area of a triangle ( things to do on a rainy afternoon, list 110 different formulae for the area of a triangle). One was the well known Heron's formula above and then there was another that looked strikingly similar. If we let M_A be the length of the median to vertex A, and similarly for M_B and M_C . The we can call sigma 1/2 the sum of M_A + M_B+ M_C. Then we can write.

$A= \frac{4}{3}\sqrt {\sigma(\sigma - M_A)(\sigma - M_B)(\sigma - M_C)}$Now that is a new one to me..

Here is one more I only learned recently. The triangle at right has the lengths s-a, etc shown, and you realize that they are the radii of Soddy "kissing circles".(below) If we think of the side a, as opposite angle A (as we like to do in geometry) the a= (s-b )+(s-c), and b=(s-a)+(s-c), and you've figured out already that c=(s-a)+(s-b). Now if we use a' for s-a, and b' for s-b... then we have a= b' + c', and b=a'+c' and c=a' + b'.

"But Why?", you ask. Because the the formula can be written

as $ \sqrt{a' b' c' (a'+b'+c'}$, which looks much simpler.

On This Day in Math - March 8

Sheet from Kepler's Harmonices Mundi *Alamy.com

The teaching of Algebra in the early stages ought to consist of a gradual generalisation of Arithmetic; in other words, Algebra ought, in the first instance to be taught as Arithmetica Universalis in the strictest sense.
~George Chrystal

The 67th day of the year; 67 is the largest prime which is not the sum of distinct squares. It is the 19th prime number and the sum of five consecutive primes ending in 19 (7 + 11 + 13 + 17 + 19)

The maximum number of internal pieces possible if a circle is cut with eleven lines. These are sometimes called "lazy caterer's numbers."
$ 67 = \binom{11}{0} + \binom {11} {1} + \binom {11}{2} $

67 is the largest prime which is not the sum of distinct squares. It is also the smallest prime which contains all ten digits when raised to the tenth power. *Prime Curios

and Jim Wilder ‏@wilderlab sent 67 = 2⁶ + 2¹+ 2⁰ = 26 + 21 + 20 = 67

And one smoot is equal to 67 inches. The long and short of it is that a smoot is a unit of measurement that measures exactly 5 feet 7 inches (or 67 inches or 1.7018 meters – sorry, surveyors tend to get carried away with conversions). The smoot was created in 1958 when Lambda Chi Alpha fraternity members at MIT decided to use a pledge, Oliver R. Smoot, Jr., to calculate the length of the Massachusetts Avenue Bridge. Smoot lay down on the bridge, his fraternity brothers marked his head and feet, then he moved down one length and the process was repeated until the entire length of the bridge had been measured. The fraternity painted markings every ten smoots. The length of the bridge was calculated at 364.4 smoots, plus one ear. Succeeding pledge classes repainted the markings; it is a tradition that continues to this day.

EVENTS

1618 Kepler, On how he discovered his Third law:
...and if you want the exact moment in time, it was conceived mentally on 8th March in this year one thousand six hundred an eighteen, but submitted to calculation in an unlucky way, and therefore rejected as false, and finally returning on the 15th of May and adopting a new line of attack, stormed the darkness of my mind. So strong was the support from the combination of my labor of seventeen years on the observations of Brahe and the present study, which conspired together, that at first I believed I was dreaming, and assuming my conclusion among my basic premises. But it is absolutely certain and exact that the proportion between the periodic times of any two planets is precisely the sesquialterate proportion of their mean distances ...
* Harmonice mundi (Linz, 1619) Book 5, Chapter 3, trans. Aiton, Duncan and Field, p. 411.

", as Kepler later recalled, on the 8th of March in the year 1618, something marvelous "appeared in my head". He suddenly realized that

III. The proportion between the periodic times of any two planets is precisely one and a half times the proportion of the mean distances.

Presumably he used the word “proportion” here to signify the logarithm of the ratio, so he is asserting that log(T1/T2) = (3/2)log(r1/r2), where Tj are the periods and rj are the mean radii of the orbits of any two planets. In the form of a diagram, his insight looks like this:

At first it may seem surprising that it took a mathematically insightful man like Kepler over twelve years of intensive study to notice this simple linear relationship between the logarithms of the orbital periods and radii. In modern data analysis the log-log plot is a standard format for analyzing physical data. However, we should remember that logarithmic scales had not yet been invented in 1605. A more interesting question is why, after twelve years of struggle, this way of viewing the data suddenly "appeared in his head" early in 1618. (Kepler made some errors in the calculations in March, and decided the data didn't fit, but two months later, on May 15 the idea "came into his head" again, and this time he got the computations right, and thought he was dreaming because the fit is so exact.)

Is it just coincidental that John Napier's "Mirifici Logarithmorum Canonis Descripto" (published in 1614) was first seen by Kepler towards the end of the year 1616? We know that Kepler was immediately enthusiastic about logarithms, which is not surprising, considering the masses of computation involved in preparing the Rudolphine Tables. Indeed, he even wrote a book of his own on the subject in 1621. It's also interesting that Kepler initially described his "Third Law" in terms of a 1.5 ratio of proportions, exactly as it would appear in a log-log plot, rather than in the more familiar terms of squared periods and cubed distances. It seems as if a purely mathematical invention, namely logarithms, whose intent was simply to ease the burden of manual arithmetical computations, may have led directly to the discovery/formulation of an important physical law, i.e., Kepler's third law of planetary motion. (Ironically, Kepler's academic mentor, Michael Maestlin, chided him − perhaps in jest? − for even taking an interest in logarithms, remarking that "it is not seemly for a professor of mathematics to be childishly pleased about any shortening of the calculations".) By the 18th of May, 1618, Kepler had fully grasped the logarithmic pattern in the planetary orbits: 'Now, because 18 months ago the first dawn, three months ago the broad daylight, but a very few days ago the full Sun of a most highly remarkable spectacle has risen, nothing holds me back.' "

*mathpages.com

1758 Euler's paper on the game of Rencontre,(A type of solitaire card game, although it was sometimes played in a variation with two players... Rencontre takes two players, whom Euler names A and B. (Their descendents still populate mathematics problems worldwide. )The players have identical decks of cards. They both turn over cards, one at a time and at the same time. If they turn over the same card at the same time, there is a coincidence, and A wins. If they go all the way through the deck without a coincidence, then B wins. published in 1753, is E201, "Calcul de la probabilité dans le jeu de rencontre," Mémoires de l'académie de Berlin (1751), 1753, p. 255-270. Regarding this work, the editor says that a memoir entitled "Calcul des probabilités dans les jeux de hasard" was presented to the Academy of Berlin 8 March 1758. He asserts that it is probably memoir 201: "Calcul de la probabilité dans le jeu de rencontre." An analysis of it appeared in the Nova Acta eruditorum, Leipzig 1754, p, 179. Euler's paper can be found here Euler showed that the probability that A wins (there is a match in first n cards) is 1/n!, which rapidly converges to 1/e, or about 37%.

The function is called a derangement or subfactorial. A classic form of the problem is how many different can a clerk put n letters in n addressed envelopes so that no letter is in the correct address.

The symbol for subfactorial n is !n, a reversal of the usual factorial notation of today.

The problem of counting derangements was first considered by Pierre Raymond de Montmort in his Essay d'analyse sur les jeux de hazard in 1708; he solved it in 1713, as did Nicholas Bernoulli at about the same time.

The 9 derangements of the order of numbers one to four (from 24 permutations) are highlighted the probability of randomly picking a derangement is 9/24.

In 1775, Joseph Priestley, having discovered oxygen on 1 Aug 1774, experimented with mice in his home laboratory on whether it is necessary to support life. *TIS

Over the past experiments, Joseph Priestley was looking for different ‘airs’ and trying to observe their properties. In one of the experiments, he noticed that when a burning candle was placed in a jar, it was put out. In such a jar, a mouse would also die because of the lack of air. However, putting a green plant in the same jar and exposing it to sunlight would bring the air back, which would permit the flame to burn and the mouse to breathe.

On August 1, Priestley took a lump of reddish solid substance, which was mercury oxide, and put it inside an inverted container, which was placed in a pool of mercury. Then he took a ‘burning lens’ and focussed the sunlight on the reddish lump hoping the substance to burn and collect the air that was produced.

The produced ‘air,’ he wrote, was “five or six times as good as common air," and it allowed the mouse to breathe and the candle to burn for four times longer than earlier. Priestley had discovered what he called “dephlogisticated air," and which was later named by Antoine Lavoisier as Oxygen.

1838 US mint in New Orleans begins operation (producing dimes). “Dime” is based on the Latin word “decimus,” meaning “one tenth.” The French used the word “disme” in the 1500s when they came up with the idea of money divided into ten parts. In America, the spelling changed from “disme” to “dime.”

1896 James Dewar responds in answer to questions about his cryogenic experiments and safety precautions from Heike Kamerlingh Onnes, the Dutch Physicist whose laboratory had been shut down in Leiden for being to dangerous. "I may say that I have made all my experiments with high pressure apparatus before the Prince of Wales and the Sister of your Queen Dowager the Duchess of Albany without the slightest hesitation and no suggestions of danger were even suggested." *archive of the Kamerlingh Onnes Laboratory.

1945 A Patent is Filed for the Harvard Mark I: C.D Lake, H.H. Aiken, F.E. Hamilton, and B.M. Durfee file a calculator patent for the Automatic Sequence Control Calculator, commonly known as the Harvard Mark I. The Mark I was a large automatic digital computer that could perform the four basic arithmetic functions and handle 23 decimal places. A multiplication took about five seconds. *CHM

In 1976, the largest recovered single stony meteorite (1,774 kg) fell in Jilin, China, during a meteor shower that dropped more than 4,000 kg of extra-terrestrial rock. *TIS One piece weighed 1.77 tons, produced an impact pit 6 m deep (only a couple of hundred meters from the nearest house), and is the largest single fragment of stony meteorite ever found.

At about 3:00 pm on March 8, 1976 a red fireball moving southwest was sighted by townspeople of Hsinglung, Kirin Province. During flight there were several explosions and in the last stages of flight three distinct fireballs were observed.

2016 Ralph Bohun's, A Discourse Concerning the Origine and Properties of the Wind (1671), was Sold for $£562 (US$ 734)$ at auction by Bonhams. The Book is mentioned by John Wallis in a letter to Oldenburg of 24 January, 1672(NS) because the book's printing had been temporarily suspended over some wording that appeared "too favourable to the Royal Society" (*Beeley's correspondence of Wallis)

BIRTHS

1804 Alvan Clark (8 Mar 1804, 19 Aug 1887) American astronomer whose family became the first significant manufacturers of astronomical instruments in the U.S. His company manufactured apparatus for most American observatories of the era, including Lick and Pulkovo, and others in Europe. In 1862, while testing a telescope, Clark discovered the companion star to Sirius, which had previously been predicted but until then never sighted. The 18½-in objective telescope he used was subsequently delivered to the Dearborn Observatory, Chicago. His sons, Alvan Graham Clark and George Bassett Clark, continued the business. The unexcelled 40-in refractor telescopes for the 40-in Yerkes observatory was made by Alvan Graham Clark*TIS

Clark's telescopes at Lowell and Yerkes observatories

1851 George Chrystal (8 March 1851 in Old Meldrum (near Aberdeen), Scotland
- 3 Nov 1911 in Edinburgh, Scotland)is best remembered today for Algebra: a two volume work which was completed by 1889. He was also involved in educational reform throughout his career and was a major figure in setting up an educational system in Scotland. He became one of the first honorary members of the EMS in 1883. *SAU Chrystal was (one of?) the first to use the inverted exclamation mark for the subfactorial notation. Prior, and for sometime after, the Whitworth symbol was used. The name subfactorial was created by W A Whitworth around 1877. The symbol for the subractorial is !n, a simple reversal of the use of the exclamation for n-factorial n!, although both symbols are relatively newer than the word. Whitworth himself used a symbol something like || n which is still used in some places.

My "Notes on the History of the Factorial" are here.

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

1866 Pyotr Nikolayevich Lebedev (8 Mar 1866; 1 Apr 1912 at age 46) Russian physicist who, in experiments with William Crookes' radiometer, proved (1910) that light exerts a minute pressure on bodies (as predicted by James Clerk Maxwell's theory of electromagnetism), and furthermore that this effect is twice as great for reflecting surfaces than for absorbent surfaces. He had proposed that light pressure on small particles of cosmic dust could be greater than gravitational attraction, thus explaining why a comet's tail points away from the Sun (though it is now understood the solar wind has a greater influence). He built an extremely small vibrator source capable of generating 4-6 mm waves, which he used to demonstrate the first observation of douible refraction of electromagnetic waves in crystals of rhombic sulphur.*TIS

1879 Otto Hahn (8 Mar 1879; 28 Jul 1968 at age 89) German physical chemist who, with the radiochemist Fritz Strassmann, is credited with the discovery of nuclear fission. He was awarded the Nobel Prize for Chemistry in 1944 and shared the Enrico Fermi Award in 1966 with Strassmann and Lise Meitner. Element 105 carries the name hahnium in recognition of his work.*TIS

"For the rest of his life, Hahn provided a standard explanation: fission was a discovery that relied on chemistry only and took place after Meitner left Berlin; she and physics had nothing to do with it, except to prevent it from happening sooner." *Lise Meitner by Ruth Lewin Sime

The prize-winning science-fiction writer, Frederik Pohl, talking about Szilard's epiphany in Chasing Science (pg 25), ".. we know the exact spot where Leo Szilard got the idea that led to the atomic bomb. There isn't even a plaque to mark it, but it happened in 1938, while he was waiting for a traffic light to change on London's Southampton Row. Szilard had been remembering H. G. Well's old science-fiction novel about atomic power, The World Set Free and had been reading about the nuclear-fission experiment of Otto Hahn and Lise Meitner, and the lightbulb went on over his head." (Maybe she had a little idea?)

in 1939 during the Fifth Washington Conference on Theoretical Physics at the George Washington University, Nobel Laureate Niels Bohr publicly announced the splitting of the uranium atom. The resulting “fission,” with its release of two hundred million electron volts of energy, heralded the beginning of the atomic age.

The announcement came just weeks after Otto Hahn and Fritz Strassmann, two of Bohr’s colleagues at Copenhagen, reported that they had discovered the element barium after bombarding uranium with neutrons. After receiving the news in a letter, physicist Lise Meitner and her cousin, Otto Frisch, correctly interpreted the results as evidence of nuclear fission. Frisch confirmed this experimentally on January 13, 1939. *atomicheritage.org

Niels Bohr was planning a trip to America to discuss other problems with Einstein who had found a haven at Princeton's Institute for Advanced Studies. Bohr came to America, but the principal item he discussed with Einstein was the report of Meitner and Frisch. Bohr arrived at Princeton on January 16, 1939. He talked to Einstein and J. A. Wheeler who had once been his student. From Princeton the news spread by word of mouth to neighboring physicists, including Enrico Fermi at Columbia. Fermi and his associates immediately began work to find the heavy pulse of ionization which could be expected from the fission and consequent release of energy. *Atomic Archive

1920 George Keith Batchelor FRS (8 March 1920 – 30 March 2000) was an Australian applied mathematician and fluid dynamicist. He was for many years the Professor of Applied Mathematics in the University of Cambridge, and was founding head of the Department of Applied Mathematics and Theoretical Physics (DAMTP). In 1956 he founded the influential Journal of Fluid Mechanics which he edited for some forty years. Prior to Cambridge he studied in Melbourne High School.
As an applied mathematician (and for some years at Cambridge a co-worker with Sir Geoffrey Taylor in the field of turbulent flow), he was a keen advocate of the need for physical understanding and sound experimental basis.
His An Introduction to Fluid Dynamics (CUP, 1967) is still considered a classic of the subject, and has been re-issued in the Cambridge Mathematical Library series, following strong current demand. Unusual for an 'elementary' textbook of that era, it presented a treatment in which the properties of a real viscous fluid were fully emphasized. He was elected a Foreign Honorary Member of the American Academy of Arts and Sciences in 1959.*Wik

1928 Martin David Davis (March 8, 1928 – January 1, 2023) was an American mathematician and computer scientist who contributed to the fields of computability theory and mathematical logic. His work on Hilbert's tenth problem led to the MRDP theorem. He also advanced the Post–Turing model and co-developed the Davis–Putnam–Logemann–Loveland (DPLL) algorithm, which is foundational for Boolean satisfiability solvers.

Davis won the Leroy P. Steele Prize, the Chauvenet Prize (with Reuben Hersh), and the Lester R. Ford Award. He was a fellow of the American Academy of Arts and Sciences and a fellow of the American Mathematical Society.

Davis first worked on Hilbert's tenth problem during his PhD dissertation, working with Alonzo Church. The theorem, as posed by the German mathematician David Hilbert, asks a question: given a Diophantine equation, is there an algorithm that can decide if the equation is solvable?[1] Davis's dissertation put forward a conjecture that the problem was unsolvable. In the 1950s and 1960s, Davis, along with American mathematicians Hilary Putnam and Julia Robinson, made progress toward solving this conjecture. The proof of the conjecture was finally completed in 1970 with the work of Russian mathematician Yuri Matiyasevich. This resulted in the MRDP or the DPRM theorem, named for Davis, Putnam, Robinson, and Matiyasevich.[1] Describing the problem, Davis had earlier mentioned that he found the problem "irresistibly seductive" when he was an undergraduate and later had progressively become his "lifelong obsession".*Wik

DEATHS

1688 Honoré Fabri (8 April 1608 in Le Grand Abergement, Ain, France - 8 March 1688 in Rome, Italy) was a French Jesuit who worked on astronomy, physics and mathematics. His lectures on natural philosophy were published in 1646 as Tractatus physicus de motu locali. In this work he uses the parallelogram law for forces, correctly applying it to deduce the law of reflection and the motion of a body acted on simultaneously by two forces.*SAU (This seems to be one of the earlier statements of the law)

1974 Olive Clio Hazlett (October 27, 1890 - March 8, 1974) was an American mathematician who spent most of her career working for the University of Illinois. She mainly researched algebra, and wrote seventeen research papers on subjects such as nilpotent algebras, division algebras, modular invariants, and the arithmetic of algebras.*Wik

2002 George Francis Carrier (May 4, 1918 – March 8, 2002) was an engineer and physicist, and the T. Jefferson Coolidge Professor of Applied Mathematics Emeritus of Harvard University. He was particularly noted for his ability to intuitively model a physical system and then deduce an analytical solution. He worked especially in the modeling of fluid mechanics, combustion, and tsunamis.

Born in Millinocket, Maine, he received a master's in engineering degree in 1939 and a Ph.D. in 1944 from Cornell University with a dissertation in applied mechanics entitled Investigations in the Field of Aeolotropic Elasticity and the Bending of the Sectorial-Plate under the supervision of J. Norman Goodier. He was co-author of a number of mathematical textbooks and over 100 journal papers.

Carrier was elected to the American Academy of Arts and Sciences in 1953, the United States National Academy of Sciences in 1967, and the American Philosophical Society in 1976. In 1990, he received the National Medal of Science, the United States' highest scientific award, presented by President Bush, for his contributions to the natural sciences.

He died from esophageal cancer on March 8, 2002. *Wik

George Carrier was considered to be one of the best applied mathematicians the United States ever produced. He loved applied mathematical problems developing complex mathematical models, which he solved with ingenious approximations and asymptotic results.*SAU

2017 George Andrew Olah (born Oláh András György; May 22, 1927 – March 8, 2017) was a Hungarian-American chemist. His research involved the generation and reactivity of carbocations via superacids. For this research, Olah was awarded a Nobel Prize in Chemistry in 1994 "for his contribution to carbocation chemistry." He was also awarded the Priestley Medal, the highest honor granted by the American Chemical Society and F.A. Cotton Medal for Excellence in Chemical Research of the American Chemical Society in 1996.

After the Hungarian Revolution of 1956, he emigrated to the United Kingdom, which he left for Canada in 1964, finally resettling in the United States in 1965. According to György Marx, he was one of The Martians.