Tuesday, 10 February 2026

Trisecting an Angle, an Interesting Historical Comparison and a Mystery

   A correspondent, who chooses to remain nameless recently wrote me with some notes on Jost Burgi's (written in Pitiscus' Trigonometria, 1595)& and Viete's (Supplementum Geometriae, 1593 as described by Robin Hartshorne)  approaches to trisecting a general angle. I was first struck by how Burgi's approach seemed to be very algebraic, while Viete, father of algebra, uses a ancient geometric approach with the neusis, (sort of a marked straight edge approach) .


In addition, even though I own Smith's 'Sourcebook', and  had probably read the Pitiscus/Burgi article several times, I had never worked through his actual math.  Doing so left me somewhat confused, confounded, and perhaps enlightened.


Links to both documents on the internet are below.

First, I will explain Viete's use, somewhat amplified to make it clearer to younger readers, and a bit about why he might have chosen this method.  Then I will do somewhat the same with Burgi's method as presented by Pitiscus.
Viete explains clearly that he will extend the classical compass and straightedge approach by allowing, "from any point to any two straight lines, to draw a straight line cutting off between them any segment fixed in advance."

The Neusis Approach of Viete (and Archimedes)

Students who may hear about Viete's introduction of letters as variables in mathematical operations may not realize he was an excellent geometer. In fact he chose to follow a homogeneity of dimension restriction that kept him from addressing simple quadratics like x^2 -2x = 3 since it involved adding the area of a square to the length of a line. His ability to adjust such problems to become problems of proportions, so that x^2 + Zx = B^2 where the S and B were considered as known values (or pronumerals) allowed him to effect the same results. But in this case, he was seeking a solution to cubics, the so called "casus irreducibilis", and trying to avoid the use of Cardano's imaginaries (complex numbers) which were still unexplained geometrically. Caspar Wessel's paper on the geometric meaning of the complex numbers would be printed within a few years (1597) but even then, it would make hardly a ripple in the world of math outside his home country. Wessel was as much unknown as Cardano was known.
The relation of trisecting to the cubic equation \(x^3 - 3x = 2a)\) was well known, and Viete knew that with a suitable change of variables, it could be used to solve any cubic with all real roots.The method he would use dates back to the early Greeks, and was known and used by Archimedes. 
I have tried to illustrate the "neusis" approach with static pictures from a very dynamic program, but teachers who are interested in making this clearer to their students can easily construct an interactive model for their students.


We begin with an isosceles triangle with a base angle at the origin and the angle to be trisected and the vertex on a circle centered at the origin.  The marked ruler is represented here by the ray EG, and for our purpose has G fixed on the circle passing through the top vertex of the triangle.  In use, the mathematician would lay his straightedge along one of the two congruent sides of the triangle, and mark EG so that it was the length of a side.  Then holding E on the x-axis, and G on the circle, he would move the points along the respective curves until the straight edge passed through the vertex angle at C .  The resulting at \(\angle \) AEG is the trisector.


For ease in seeing a simple proof, I add one drawing with a second isosceles triangle shown:


There are actually three isosceles triangles in the figure, the two shaded ones, \(\Delta \)ACH , and \(\Delta \)AGE and the one between them \(\Delta \)CAG.   If we label the \( \angle \) CAH to be trisected as y,  then it's supplementary angle, \( \angle \) CAE is 180 - y.  Now if we call the equal angles \( \angle \)AEG and \( \angle \)GAE each x, then the exterior angle \( \angle \)AGC to this isosceles triangle must be 2x, as must it's base angle partner \( \angle \)ACG.  This leaves \( \angle \) CAG as 180 - 4x.  Adding back the value x at \( \angle \)GAE means that \( \angle \)CAE is equal to 180 - 3x, but we already know it is equal to 180-y, so y=3x as we sought.

Burgi   Using Brahmagupta's theorem and double false position. 
Writing at the same time, Pituscus describes Jost Burgi's approach with a much more algebraic or arithmetical approach.  They will create a complex construction involving Brahmagupta's theorem for inscribed quadrilaterals, then make an two educated guess of the trisecting angle's chord, then use the errors and a linear approximation technique called the rule of double false position to find the chord of the trisected angle.  The idea of double false position as used by Piticus/Burgi  is almost certainly worth showing to students for the historical value alone. Fibonacci described double false position in his well know “Liber Abaci” , using it's Arabic name, and says the method is the one “by which the solutions of nearly all problems are found."  I would think a simple illustration to students is essential before exposing them to the approach used by Burgi and is well within the grasp of any student who has worked with linear equations.

 Double False Position
Here is an example using a linear equation for which the method is exact.  For non-linear functions, like the trigonometric functions, it serves as a method of approximation.
Suppose we take a simple expression like 5x-3 = 7 and have no idea how to solve it.  We might begin with a guess of x=1.  Trying that out we get 5(1)-3 = 2, an error that is five too low, or -5.  We decide to guess larger, and try x=4.  This time 5(4)-3  gives us 17, which is 10 above our target.  The method of double false position was to take the cross products of each trial value, and the opposite error, and find their difference, then divide the result by the difference in the errors.  If we label the x values x1 and x2 and the errors e1  and e2, then the equation can be written as \( \frac{x_1 * e_2 - x_2 * e_1}{e_2 - e_1}\).  If we apply this to the trials above we get \( \frac{1 * 10 - 4 * (-5)}{10 - (-5)}\), and to none of our surprise, it tells us the correct value of x was 2.

As to why Burgi would have used this approach, when he might have used the much simpler neusis method of Archimedes and Viete, I think it was a matter of focus.  Burgi was an astronomer and precision clock maker who had just before this writing, made an astronomical clock which was based on the Copernican system, and along the way created a table of sines (chord lengths) for astronomical use, his Canon Sinuum which seems not to have been publishedBurgi had independently invented logarithms, almost certainly prior to Napier, as an aide to his computations.  I believe that Burgi, and Pitiscus use of Burgi in his Trigonometria, was for the purpose of computing chord lengths for Sine Tables.  To obtain more accurate chord lengths, astronomers increased the radius of their calculating circles to seven or more digits.  In the time before decimals were common (although Viete supposedly, "... wrote decimal fractions with the fractional part printed in smaller type than the integral and separated from the latter by a vertical line." )  the use of a large radius allows the use of a chord to be expressed with greater accuracy and in integer values.  When Pitiscus uses 5176381 for the chord length of a 30o arc, he is giving what we would now write as .5176381 for 2 sin(15o). 

Burghi's Use of Brahmagupta's Thm
Burgi realized that if you start with the chord of an angle (in this case, the chord of AB in \( \angle \)ADB) and double the angle by adding another chord of the same length, (BC, thus making  \( \angle \) ADB congruent to \( \angle \)BDC, and the chord of  \( \angle \) ADC would be AC). Then with one more congruent chord (CD) we have the chord of triple the arc of AB, and thus an angle AD is the chord for an angle three times \( \angle \) ABD (you can pick the point P for this APD anywhere on the arc AD).



With this groundwork, Burgi recognized that the four points form an inscribed quadrilateral, and so Brahmagupta's theorem would apply. That is, in this case, AC * BD = AB*CD + AD * BC. But since AB = BC = CD, and AC = BD we can rewrite that as \( AC^2 = AB^2 + AB * AD \).  And by a little algebra we arrive at \( \frac{AC^2 - AB^2}{AB} = AD \) 
Now to trisect the angle, we will be able to construct the chord AD from the given angle, and we want to find the length of the chord that would subtend 1/3 of that angle, that is, the length of AB.

The Mystery
As this point, Pitiscus says to, take a third of the chord, add something to it (the chords of the three equal angles are simply longer than the chord of the triple angle, so we must have something a little larger), and use the method of problem 3 (Brahmagupta's thm)  to compute the given chord.  He knows this will have some error, and so he will repeat it again to get a second error and use the method of double false position to find a close approximation.  But......   I can see no way he knew AC, or AC2 .  If we are building a table of chords,  and we knew AC or AC2 and hence the chord of 2/3 of the angle, we would have bisected this angle to begin with rather than trisecting the original angle since such a method was known at least as early as Hipparchus .  And Burgi was an expert in  prosthaphaeresis, the use of combining addition and subtraction of products to sums and is often credited with inventing at least two of them. How could we... how could HE carry out this operation?

Doubt and Suspicion (or guesses and wild conjectures)
 Immediately after the explanation of this problem, he gives "another method by algebra" and described the relation  which in modern language would be x^3-3x =2a.   Written in terms of chord lengths this would be crd3δ = 3crdδ−crd3 δ which might have been known as early as Ptolemy, and which Burgi and Pitiscus both well knew.

For the example of a 30o inscribed angle, we know that the chord length is about .5176381 (or 2 sin(15o) . Burgi's approach is to guess that the chord of 1/3 the arc is a little more than 1/3 this number, and he chooses .173 as his first guess.  Letting this be x in the expression \( f(x) = 3x - x^3 \) which I now suspect is what he did, we arrive at .5138223 (Burgi / Pitiscus use this number without decimal points, as explained before) .
Since he intends to use the method of double false position, he picks a second guess (.174) and gets a second estimate for his triple chord, .5167320
He arrives at an estimate using the double false position method of  .1743114.

You might compare this to the actual value of 2 sin (5o) to compare. The error depends on how close the original guesses were, since the chord lengths are not linear.

Testing Alternatives

What if we try to follow the approach using Brahmagupta's formula to get the values?  Beginning with .173.   We have to find a way to tease out a guess for AC from this.  Since \( \Delta \) ABC is isosceles, with Arcs AB and BC both being 10 o arcs, the inscribed angles \( \angle \) BAC and \( \angle \) BCA are both 5,sup>o
leaving 170 degrees for \( \angle \) ABC.  Using my very modern calculator I arrive at a length for AC2 of  .1206146401.  When this and .173 are used in the Brahmagupta formula, we get .5241944515; not miles away from the .5138... result he found, but obviously not the approach he used. 
I tried a few other things as well and all were as far, or farther away from Pitiscus' value than this, and none seem to get me very close to his result other than the cubic.  I would love to hear some other approaches to how he might have done this other than the cubic. 

And my special thanks to my unnamed correspondent for his note,  sending me on several interesting days of exploration and study.  Now on to write about the other ways of trisecting an angle. 

Robin Hartshorne Paper on Viete https://math.berkeley.edu/~robin/Viete/construction.html

D E Smith "Piticus on Bergi's trisection of an Arc \" https://books.google.com/books?id=awAfO7Ff_z0C&q=Pitiscus+trisection#v=snippet&q=Pitiscus%20trisection&f=false

On This Day in Math - February 10

 


Newton First Day Cover *BFDC.co.uk




It is impossible to be a mathematician without being a poet in soul.

~Sofya Kovalevskaya

The 41st day of the year; Euler (1772) observed that the polynomial f(x)= x2 + x + 41 will produce a prime for any integer value of x in the interval 0 to 39. In 1778 Legendre realized that x2 - x + 41 will give the same primes for interval (1-40).
 n^2 + n + 41 is prime for n = 0 ... 39 and Is prime for nearly half the values of n up to 10,000,000. *John D. Cook

If you multiply 41 by any three digit number to produce a five digit number, every cyclic representation of that number formed by moving the last digit to the front is also divisible by 41. (for example 41*378 = 15,498. 41 will also divide 81,549; 98154; 49815; and 54,981 *The Moscow Puzzles
41 can be expressed as the sum of consecutive primes in two ways, (2 + 3 + 5 + 7 + 11 + 13), and the  (11 + 13 + 17).

41 is the smallest integer whose reciprocal has a 5-digit repeating decimal.  \(\frac{1}{41} = \overline{.02439}\)

And even more from @Math Year-Round 41=1!+2!+3!+1¹+2²+3³

Found this on Twitter(X) by Abakcus




The Expanded Number Facts collection for year days 1-60 are now available here


EVENTS

1676/7 “The truth of it is, mathematical learning will not go off without a dowry; the booksellers have lost so much by the works of Drs. Wallis and ... Barrow ... that it is no easy task to persuade booksellers to undertake any thing but toys that are mathematical.” [Collins to Baker, The Correspondence of Isaac Newton, Vol. 2, p. 192]


In 1720, Edmund Halley was appointed second Astronomer Royal of England. Halley succeeded John Flamsteed as Astronomer Royal, a position he would hold until his death in 1742 at the age of 85. *Wik  On the 300th anniversary of his trip to St Helena, the island issued three stamps. The 5p value shows Halley’s comet as it appeared on the Bayeux Tapestry.

Flamsteed may still be spinning in his grave over the choice.  John Flamsteed revealed his irritation with the choice of Edmond Halley as successor to Wallis as the Savilian Professor of Geometry. In a letter to Abraham Sharpe he wrote:-

Dr Wallis is dead – Mr Halley expects his place – who now talks, swears and drinks brandy like a sea-captain.




1734 The Marquise de Chatelet writes to ask Maupertuis to explain his claim in a 1732 paper that "God preferred the inverse square law of attraction over all others." Apparently he failed to convince her. *J. B. Shank, The Newton Wars and the Beginning of the French Enlightenment



1747  "On the acceptance of trigonometry in wasan: Evidence from a text of Aida Yasuaki" [Born 10 February 1747 Yamagata, Japan ]  J. Marshall Unger; Historia Mathematica Volume 52, August 2020, Pages 51-65  .  Wasan, or wanska was a term used for great mathematicians of the 18th and 19th century.  their geometric methods avoided the use of trigonometry tables.  

The diagonals of a cyclic quadrilateral divide it into four non-overlapping quarter-triangles and its circumcircle into four skewed sectors.1 Each quarter-triangle and each skewed sector has an incircle. Japanese mathematicians of the 18th and 19th centuries (wasanka) discovered how to find the radius of the circumcircle given the radii of the incircles either of the four quarter-triangles or of the four skewed-sectors.


"Aida was hard-working and strong-willed and produced as many as fifty to sixty works a year. Nearly 2000 works survived him, including many on non-mathematical subjects. He was a distinguished teacher of traditional mathematics and a successful populariser of that discipline."

*mathshistory.st-andrews


1860 Joseph Toynbee (English otologist, whose career was dedicated to pathological and anatomical studies of the ear.) writes to ask Faraday for tickets to Thomas Henry Huxley's lecture.

18 Savile Row. Burlington Gardens. W | 10 Feb 1860

Dear Mr. Faraday,

Having been unsuccessful in my attempts to obtain a ticket for Mr. Huxley’s lecture tonight I shall esteem it a favour if you can give me one.

Believe me yours sincerely ; obliged,

Joseph Toynbee

*JAIPREET VIRDI, From the Hands of Quacks

WEEKLY EVENING MEETING,

Friday, February 10, 1860.


SIR HENRY HOLLAND, BART. M.D. F.R.S. Vice-President,

in the Chair.

PROFESSOR T. H. HUXLEY, F.R.S.

On Species and Races, and their origin.

The speaker opened his discourse by stating that its object was to place the fundamental propositions of Mr. Darwin's work "On the Origin of Species by Natural Selection," in a clear light, and to consider whether, as the question at present stands, the evidence adduced in their favour is, or is not, conclusive.


Faraday lecturing



1860  Published #OTD 1860 by the Linnean Society, Alfred Russel Wallace's  paper : "On the Zoological Geography of the Malay Archipelago". The paper having been communicated on his behalf by Charles Darwin to the Society on 3/11/1859.  



1880 On 10 February 1880, an article ran in the Daily Times (of Bethlehem, Pennsylvania) describing a remarkable invention recently demonstrated by a local inventor, Dr. H.E. Licks. The invention allowed images to be transmitted by telegraph. In other words, it resembled what people today would recognize as a television. However, Licks called his invention a "diaphote," from the Greek dia meaning "through" and photos meaning "light".
Despite the excitement it generated, the diaphote turned out to be a hoax. However, it wasn't until 1917 that "Dr. H.E. Licks" revealed the hoax in a book, Recreations in Mathematics. Licks himself is believed to have been the pseudonym of Mansfield Merriman (1848-1925), a professor of engineering at Lehigh University. *Museum of Hoaxes

He published Recreations in Mathematics in 1917 (a wonderful book) under the pseudonym H. E. Licks, which included a story, "The Diaphote Hoax", a republication of a detailed newspaper report from February 10, 1880 which purported to describe the scientific demonstration of a device that transmitted images by electricity. The report is peppered throughout with scientific jokes including mentions of "Dr. H. E. Licks" ('helix'), "Prof. M. E. Kannick" ('mechanic'), "Col. A. D. A. Biatic" ('adiabatic'), and "Prof. L. M. Niscate" ('lemniscate').



1883 Edith Clarke (February 10, 1883 – October 29, 1959) was an American electrical engineer. She was the first woman to be professionally employed as an electrical engineer in the United States, and the first female professor of electrical engineering in the country. She was the first woman to deliver a paper at the American Institute of Electrical Engineers; the first female engineer whose professional standing was recognized by Tau Beta Pi, the oldest engineering honor society and the second oldest collegiate honor society in the United States; and the first woman named as a Fellow of the American Institute of Electrical Engineers. She specialized in electrical power system analysis and wrote Circuit Analysis of A-C Power Systems.

After being orphaned at age 12, she was raised by an older sister. She used her inheritance to study mathematics and astronomy at Vassar College, where she graduated in 1908.


Unable to find work as an engineer, Clarke went to work for General Electric as a supervisor of computers in the Turbine Engineering Department. During this time, she invented the Clarke calculator, an early graphing calculator, a simple graphical device that solved equations involving electric current, voltage and impedance in power transmission lines. The device could solve line equations involving hyperbolic functions ten times faster than previous methods. She filed a patent for the calculator in 1921 and it was granted in 1925.

She was offered a job by GE as a salaried electrical engineer in the Central Station Engineering Department – the first professional female electrical engineer in the United States. She retired from General Electric in 1945.

Her background in mathematics helped her achieve fame in her field. On February 8, 1926, as the first woman to deliver a paper at the American Institute of Electrical Engineers' (AIEE) annual meeting, she showed the use of hyperbolic functions for calculating the maximum power that a line could carry without instability. The paper was of importance because transmission lines were getting longer, leading to greater loads and more chances for system instability, and Clarke's paper provided a model that applied to large systems.


In 1943, Clarke wrote an influential textbook in the field of power engineering, Circuit Analysis of A-C Power Systems, based on her notes for lectures to GE engineers. This two-volume textbook teaches about her adaption of the symmetrical components system, in which she became interested while working for the second time at GE.

In 1947, she joined the faculty of the Electrical Engineering Department at the University of Texas at Austin, making her the first female professor of electrical engineering in the country.

Ms. Clarke died in 1959 at the age of seventy-six.

Her calculator was a dynamic nomogram




1914 The Girl Scout’s patent the trefoil design for their membership badge. The badge was designed by the founder of Girl Guides (renamed in 1913), Juliette "Daisy" Gordon Low who began the program in Savannah, Ga. with a group of 18. This (2012) is the centennial year of the Girl Scouts. The organization has declared 2012 the "Year of the Girl", and the Smithsonian's National Portrait Gallery opened a year-long exhibit "Juliette Gordon Low and 100 Years of Girl Scouts" on January 13, 2012.*Wik

Juliette Gordon Low in 1887 *Wik



1946 War Department revealed the development of ENIAC, an electronic numerical integrator and computer. It is 1000 times faster than human computation. *VFR


In 1958, radar signals were bounced off the planet Venus by MIT engineers at Lincoln Laboratories in experiments conducted during an inferior conjunction with Venus. A maser installed at the Millstone Hill radar site, Westford, Mass., was used. The return echoes were distinguished from the background noise using digital signal processing. The results were confirmation by other researchers, and together led to a more precise determination of the value of the astronomical unit (AU), the Earth's mean distance to the sun. A new value of the AU, 149,600,000 km was adopted at a general meeting of the International Astronomical Union in Hamburg (1964). Radar was also bounced off the Sun (7 Apr 1959). Lincoln Laboratories had been set up to conduct military rather than astronomical research.*TIS


https://dai.ly/x9dw7c2


1996 Deep Blue defeats Kasparov
In the first game of a six game match, IBM's Deep Blue chess computer defeated world champion Garry Kasparov. No computer had ever won a game against a world champion in chess. Kasparov would eventually win the series 4-2, but would lose to Deep Blue in a re-match a year later.*CHM


2012 2/10/2012 is a palindrome, and even if it's written 2/10/12. Or even in Roman numerals II/X/MMXII. What other years may be expressed as palindromes in two of these three methods (or European date style, dd/mm/yy  4/10/2014 works with or without the 20, ... 41014.  And April 20, 2024 (4202024) or (42024) does as well.  



BIRTHS

1747 Aida Yasuaki (February 10, 1747 – October 26, 1817) Japanese mathematician who published about 2000 works. Aida compiled Sampo tensi shinan which appeared in 1788. It is a book of geometry problems, developing formulae for ellipses, spheres, circles etc. Aida explained the use of algebraic expressions and the construction of equations. He also worked on number theory and simplified continued fraction methods due to Seki. *SAU




1785 Claude Louis Marie Henri Navier (10 Feb 1785 in Dijon, France - 21 Aug 1836 in Paris, France) Claude-Louis Navier was a French mathematician best known for the Navier-Stokes equations describing the behaviour of a incompressible fluid. We should note, however, that Navier derived the Navier-Stokes equations despite not fully understanding the physics of the situation which he was modelling. He did not understand about shear stress in a fluid, but rather he based his work on modifying Euler's equations to take into account forces between the molecules in the fluid. Although his reasoning is unacceptable today. *SAU



1811 Jean-Laurent Palmer, a French metal worker and instrument maker ran a workshop in Paris that produced drawn wire and seamless metal tubes, and he apparently needed an instrument to measure wire diameter and the thickness of sheet stock and tube walls. In 1848, he produced the instrument illustrated in our first image: a screw micrometer. The idea for a micrometer – an instrument that measures distance or thickness by counting the turns of a screw – was 180 years old, and calipers were far older, but no one had ever combined the two into a single hand-held tool. Palmer's micrometer had two numerical scales--one to count the screw turns, and another to measure fractions of a turn. Since the threads were 1 mm apart, and since the fractional scale was divided into 20 parts, that means Palmers' instrument could measure thickness to an accuracy of .05 mm, and with a Vernier scale that could be reduced to .01 mm – one hundred-thousandth of a meter.  Impressive for such a simple instrument. *Linda Hall Org




1842 Agnes Mary Clerke (10 Feb 1842, 20 Jan 1907) Irish astronomical writer who was a diligent compiler of facts rather than a practicing scientist. Nevertheless, by 1885, her exhaustive treatise, A Popular History of Astronomy in the Nineteenth Century gained international recognition as an authoritative work. In 1903, with Lady Huggins, she was elected an honorary member of the Royal Astronomical Society, a rank previously held only by two other women, Caroline Herschel and Mary Somerville. Her publications included several books and 55 pieces in the Edinburgh Review. She contributed some astronomer biographies to the Dictionary of National Biography and some astronomical entries in the Encyclopaedia Britannica. *TIS More detail about her here




1885 Hardy Cross (10 Feb 1885; died 11 Feb 1959 at age 73) U.S. professor of civil and structural engineering whose outstanding contribution was a method of calculating tendencies to produce motion (moments) in the members of a continuous framework, such as the skeleton of a building. By the use of Cross's technique, known as the moment distribution method, or simply the Hardy Cross method, calculation can be carried to any required degree of accuracy by successive approximations, thus avoiding the immense labour of solving simultaneous equations that contain as many variables as there are rigid joints in a frame. He also successfully applied his mathematical methods to the solution of pipe network problems that arise in municipal water supply design; these methods have been extended to gas pipelines.
*TIS



1888 Selig Brodetsky (10 February 1888 - 20 May 1954)was educated at Cambridge and Leipzig. He became a lecturer at Bristol and later lecturer and professor at Leeds. He worked on fluid flow with particular emphasis on aerodynamics. He was President of the Hebrew University of Jerusalem for a short time. *SAU



1901 Richard Dagobert Brauer (10 Feb 1901; 17 Apr 1977 at age 76) German-American mathematician and educator, a pioneer in the development of algebra theory. He worked with Weyl on several projects including a famous joint paper on spinors (published in 1935 in the American Journal of Mathematics). This work provided a background for Paul Dirac's theory of the spinning electron within the framework of quantum mechanics. With Nesbitt, Brauer introduced the theory of blocks (1937). Brauer used this to obtain results on finite groups, particularly finite simple groups, and the theory of blocks would play a big part in much of Brauer's later work. Starting with his group-theoretical characterisation of the simple groups (1951), he spent the rest of his life formulating a method to classify all finite simple groups. *TIS

Richard and Ilse Brauer in 1970 *Wik



1908 Ida Winifred Busbridge (10 Feb, 1908– Dec 27, 1988) was a British mathematician who taught at the University of Oxford from 1935 until 1970 She was the first woman to be appointed to an Oxford fellowship in mathematics.
In 1926 she enrolled in Royal Holloway College, London, intending to specialize in physics, but switched to mathematics in 1928.She  graduated in 1929 with first class honors and was the awarded the Sir John Lubbock Prize for best first class honors of all London University Colleges. She continued her education at Royal Holloway, earning a masters with distinction in mathematics in 1933.
She began teaching as a demonstrator in mathematics at University College, London in 1933

She moved to St Hugh's College Oxford in 1935 to teach mathematics alongside Dorothy Wrinch to undergraduates of five women's colleges. Influenced by Madge Adam and Harry Plaskett, she shifted her interest to applications of maths in astronomy and physics.

During the Second World War, her workload increased to take on the education of physicists and engineers at Oxford. Her workload was especially great - not only because other mathematicians at the university were called up for special war service - but also because women formed a higher percentage of the undergraduate population during the war years. She was appointed to a Fellowship of St Hugh's College, Oxford, in 1946 – the first women to be appointed to a college fellowship in mathematics.
In 1962, she was awarded a Doctor of Science degree by Oxford. She was also a Fellow of the Royal Astronomical Society.
She was president of the Mathematical Association for 1964.
Busbridge's work included integral equations and radiative transfer. She was highly regarded as a lecturer and tutor, attending to her students' educational and personal needs. She retired from Oxford in 1970, and became one of the early tutors and developers of courses at the Open University, teaching Lebesgue integration and tutoring complex analysis.*Wik




1911 Mstislav Vsevolodovich Keldysh ( 10 February [O.S. 28 January] 1911 – 24 June 1978) was a Soviet mathematician who worked as an engineer in the Soviet space program.

He was the academician of the Academy of Sciences of the Soviet Union (1946), President of the Academy of Sciences of the Soviet Union (1961–1975), three-time Hero of Socialist Labour (1956, 1961, 1971), and fellow of the Royal Society of Edinburgh (1968). He was one of the key figures behind the Soviet space program. Among scientific circles of the USSR Keldysh was known by the epithet "the Chief Theoretician" in analogy with epithet "the Chief Designer" used for Sergei Korolev.
The strongest influence on Keldysh was his older sister, Lyudmila Keldysh (1904–1976), a mathematician and Keldysh's first teacher. Among her children are Leonid Keldysh, director of Lebedev Physical Institute and Sergei Novikov, the first Soviet mathematician to win a Fields Medal (1970).
Keldysh was a member of many foreign academies of sciences, including the Mongolian Academy of Sciences (1961), Polish Academy of Sciences (1962), Czechoslovak Academy of Sciences (1962), and Romanian Academy of Sciences (1965). He was also an honorary member of the American Academy of Arts and Sciences (1966), Bulgarian Academy of Sciences (1966), Hungarian Academy of Sciences (1970), and Royal Society of Edinburgh (1968), foreign corresponding member of the German Academy of Sciences (1966), and Saxon Academy of Sciences in Leipzig (1966).

Keldysh was awarded the Stalin Prize (1942, 1946), Lenin Prize (1957), six Orders of Lenin, three other orders, numerous medals and four foreign orders.*Wik 




1932  Vivienne Lucille Malone-Mayes (February 10, 1932 – June 9, 1995) was an American mathematician and professor. Malone-Mayes studied properties of functions, as well as methods of teaching mathematics. She was the fifth African-American woman to gain a PhD in mathematics in the United States, and the first African-American member of the faculty of Baylor University (which had rejected her application to study there five years earlier).
She decided to attend the University of Texas full-time as a graduate student when rejected entry at Baylor. In graduate school she was very much alone. In her first class, she was the only Black, the only woman. Her classmates ignored her completely, even terminating conversations if she came within earshot. She was denied a teaching assistantship, although she was an experienced and excellent teacher.
She wrote, "... it took a faith in scholarship almost beyond measure to endure the stress of earning a Ph.D. degree as a Black, female graduate student. I could not join my advisor and other classmates to discuss mathematics over coffee at Hilsberg's cafe .... Hilsberg's would not serve Blacks.
Some classes were closed to her despite the fact that the University of Texas was required to take Black students. For example R L Moore refused to have any Black students in his classes.
She was a member of the board of directors of the National Association of Mathematicians. She was elected Director-at-large for the Texas section of Mathematical Association of America and served as director of the High School Lecture Program for the Texas section.
She had a successful, lengthy career and served on several boards and committees of note, retiring in 1994 due to ill health.  She was the fifth African-American woman to be allowed in the White House.She was also active in her local community as a lifetime member of New Hope Baptist Church. She served on boards of directors for Cerebral Palsy, Goodwill Industries, and Family Counseling and Children. She was on the Texas State Advisory Council for Construction of Community Mental Health Centers and served on the board of the Heart of Texas Region Mental Health and Mental Retardation Center.
After Lillian K. Bradley in 1960, Malone-Mayes became one of the first African-American women to receive a PhD in Mathematics from University of Texas (and fifth African-American woman in the United States). She was the first African-American member of the faculty at Baylor University, and the first African-American person elected to Executive Committee of the Association of Women in Mathematics.
The student congress of Baylor voted her the "Outstanding Faculty Member of the Year" in 1971.  *Wik & *SAU





1944 Avraham Naumovich Trahtman (Trakhtman) ( 10 February 1944 – 17 July 2024) was a Soviet-born Israeli mathematician and academic at Bar-Ilan University (Israel). In 2007, Trahtman solved a problem in combinatorics that had been open for 37 years, the Road Coloring Conjecture posed in 1970. Trahtman died in Jerusalem on 17 July 2024, at the age of 80.
Trahtman's solution to the road coloring problem was accepted in 2007 and published in 2009 by the Israel Journal of Mathematics. The problem arose in the subfield of symbolic dynamics, an abstract part of the field of dynamical systems. The road coloring problem was raised by R. L. Adler and L. W. Goodwyn from the United States, and the Israeli mathematician B. Weiss. The proof used resulted from earlier work.
The finite basis problem for semigroups of order less than six in the theory of semigroups was posed by Alfred Tarski in 1966, and repeated by Anatoly Maltsev and L. N. Shevrin. In 1983, Trahtman solved this problem by proving that all semigroups of order less than six are finitely based. *Wik







DEATHS

1576 Wilhelm Xylander (born Wilhelm Holtzman, graecized to Xylander) (December 26, 1532 – February 10, 1576) was a German classical scholar and humanist.
Xylander was the author of a number of important works. He translated the first six books of Euclid into German with notes, the Arithmetica of Diophantus, and the De quattuor mathematicis scientiis of Michael Psellus into Latin. *Wik



1865 Heinrich Friedrich Emil Lenz (24 Feb 1804, 10 Feb 1865 at age 61) was the Russian physicist who framed Lenz's Law to describe the direction of flow of electric current generated by a wire moving through a magnetic field. Lenz worked on electrical conduction and electromagnetism. In 1833 he reported investigations into the way electrical resistance changes with temperature, showing that an increase in temperature increases the resistance (for a metal). He is best-known for Lenz's law, which he discovered in 1834 while investigating magnetic induction. It states that the current induced by a change flows so as to oppose the effect producing the change. Lenz's law is a consequence of the, more general, law of conservation of energy. *TIS




1868 Sir David Brewster (11 Dec 1781; 10 Feb 1868) Scottish physicist noted for his experimental work in optics and polarized light (light in which all waves lie in the same plane.) He is known for Brewster's Law, which relates the refractive index of a material to its polarizing angle (which is the incident angle at which reflected light becomes completely polarized. He patented the kaleidoscope in 1817. Later, he used lenses to improve three-dimensional images viewed with a stereoscope. Brewster also recommended the use of the lightweight, flat Fresnel lens in lighthouses.*TIS

The first biography of Newton was written by John Conduit, his step-nephew-in-law, if that is a viable term. Conduit inherited all of Newton's papers that the other heirs did not want, but if he used them to write his biography, it does not show, for the Newton who emerges from his pages is brilliant, devout, humble, friendly, caring, and devoted to physics and God. However, in the early 19th century, some of Newton’s correspondence was brought to light by a French physicist, Jean-Baptiste Biot, who published a short biography of Newton in 1822, in which he revealed that Newton had gone mad in 1693, suffering a mental breakdown from which he never recovered. Biot maintained that after his psychotic episode, Newton could no longer do physics, and that is when he found religion. His Christian faith, in other words, developed only when his mind had weakened and could no longer handle more important matters.

When Biot’s account was translated into English in 1829, British scientists, who worshipped Newton as the greatest genius of all time, got quite a jolt. The British had little use for the French, who had recently inflicted Napoleon on the world, but this was going too far. It was Brewster who took it upon himself to defend his countryman from Gallic slurs, and he published his own short biography, The Life of Isaac Newton in 1831 (first image). Brewster was not a historian, and he did not have access to Newton's papers (they were owned by the Portsmouth family, who refused all requests for access), and so Brewster’s Newton was a reincarnation of Conduit's, a perfect human being in nearly every way, and the psychotic episode of 1693 was treated as just a minor event, the kind of thing that happens to geniuses now and then, when they have to deal with ordinary mortals. Newton had been restored to the ranks of the superhuman.*Linda Hall Org







1891 Sonya Kovalevsky (Sofya Vasilyevna Kovalevskaya) (15 Jan 1850; 10 Feb 1891) professor of mathematics at Stockholm. In her youth, her bedroom was wallpapered with the pages of a text from her father’s schooldays, namely, Ostrogradsky’s lithographed lecture notes on the calculus. Study of the novel wallpaper introduced her to the calculus at age 11. She became the greatest woman mathematician prior to the twentieth century. *VFR a Russian mathematician and novelist who made valuable contributions to the theory of differential equations.*TIS She died of influenza in 1891 at age forty-one, after returning from a pleasure trip to Genoa. She is buried in Solna(Stockholm suburb), Sweden, at Norra begravningsplatsen (Northern Cemetery). Alfred Nobel is buried in the same cemetery.

Kovalevskaya at 18 years

*Wik



1907 Agnes Mary Clerke (10 Feb 1842, 20 Jan 1907) Irish astronomical writer who was a diligent compiler of facts rather than a practicing scientist. Nevertheless, by 1885, her exhaustive treatise, A Popular History of Astronomy in the Nineteenth Century gained international recognition as an authoritative work. In 1903, with Lady Huggins, she was elected an honorary member of the Royal Astronomical Society, a rank previously held only by two other women, Caroline Herschel and Mary Somerville. Her publications included several books and 55 pieces in the Edinburgh Review. She contributed some astronomer biographies to the Dictionary of National Biography and some astronomical entries in the Encyclopaedia Britannica. *TIS More detail about her here



1923 Wilhelm Conrad Röntgen (27 Mar 1845 - 10 Feb 1923 at age 77) was a German physicist who discovered the highly penetrating form of radiation that became known as X-rays on 8 Nov 1895. He received the first Nobel Prize for Physics (1901), “in recognition of the extraordinary services he has rendered by the discovery of the remarkable rays subsequently named after him.” This high-energy radiation, though first called Röngen rays, became known as X-rays. His discovery initiated revolutionary improvements in making medical diagnoses and enabled many new advances in modern physics. *TIS "In 1901 he became the first physicist to receive a Nobel prize." *VFR


Rontgen Family Grave, Giessen


1927 Alfred George Greenhill (29 Nov 1847 in London, England - 10 Feb 1927 in London, England) graduated from Cambridge and became Professor of Mathematics at the Royal Military Academy at Woolwich. His main work was on Elliptic Functions but he published widely on applications of mathematics to practical problems. He became an honorary member of the EMS in 1908. *SAU




1936 Eleanor Mildred (Balfour) Sidgwick, (11 March 1845 – 10 February 1936) was an activist for the higher education of women, Principal of Newnham College of the University of Cambridge and a leading figure in the Society for Psychical Research.
She was born in East Lothian, daughter of James Maitland Balfour and Lady Blanche Harriet. She was born into perhaps the most prominent political clan in nineteenth-century Britain, the 'Hotel Cecil': her brother Arthur would eventually himself become prime minister. Another brother, Frank, a biologist, died young in a climbing accident.
One of the first students at Newnham College in Cambridge, in 1876 she married (and became converted to feminism by) the philosopher Henry Sidgwick. In 1880 she became Vice-Principal of Newnham under the founding Principal Anne Clough, succeeding as Principal on Miss Clough's death in 1892. She and her husband resided there until 1900, the year of Henry Sidgwick's death. In 1894 Mrs Sidgwick was one of the first three women to serve on a royal commission, the Bryce commission on Secondary Education.
As a young woman, Eleanor had helped (John William Strutt, who was married to her sister, Evelyn) Lord Rayleigh improve the accuracy of experimental measurement of electrical resistance. She conducted several experiments in electricity and with him published three papers in the Philosophical Transactions of the Royal Society.
She subsequently turned her careful experimental mind to the question of testing the veracity of claims for psychical phenomena. She was elected President of the Society for Psychical Research in 1908 and named 'president of honour' in 1932. Her Husband, Henry, her brother and future Prime Minister, Arthur, and Lord Rayleigh all were also Presidents of the Society.)
She was a member of the Ladies Dining Society in Cambridge, with 11 other members.
In 1916 Mrs Sidgwick left Cambridge to live with one of her brothers near Woking; she remained there until her death in 1936.
She was awarded honorary degrees by the universities of Manchester, Edinburgh, St Andrews and Birmingham.Most of her writings related to Psychical Research, and are contained in the Proceedings of the Society for Psychical Research. However, some related to educational matters, and a couple of essays dealt with the morality of international affairs. *Wik & encyclopedia com

Portrait of Eleanor Mildred Sidgwick painted
by Sir 
James Jebusa Shannon, 1889 *Wik




1951 George Abram Miller (31 July 1863 – 10 February 1951) was an early group theorist whose many papers and texts were considered important by his contemporaries, but are now mostly considered only of historical importance.
Miller was born in Lynnville, Lehigh County, Pennsylvania, and died in Urbana, Illinois.*Wik




1994 Fritz John (14 June 1910 – 10 February 1994) was a German-born mathematician specializing in partial differential equations and ill-posed problems. His early work was on the Radon transform and he is remembered for John's equation.*Wik



1997 Jerome Namias (19 Mar 1910, 10 Feb 1997 at age 86) American meteorological researcher most noted for having pioneered the development of extended weather forecasts and who also studied the Dust Bowl of the 1930s and the El Niño phenomenon. *TIS In 1971 he joined the Scripps Institution and established the first Experimental Climate Research Center. His prognosis of warm weather during the Arab oil embargo of 1973 greatly aided domestic policy response.*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