Pat'sBlog

Monday, 21 September 2026

Imaginary Numbers, Imaginary Constant, ... History and Etymology of Math Terms

Imaginary Numbers The word imaginary was first applied to the square root of a negative number by Rene Descartes around 1635. He wrote that although one can imagine every Nth degree equation had N roots, there were no real numbers for some of those imagined roots. Around 1685 the English mathematician John Wallis wrote, "We have had occasion to make mention of Negative squares and Imaginary roots". Some mathematicians have suggested the name be changed to avoid the stigma that it seems to create in young students, "Why do we have to learn them if they aren't even real." (To be honest, in thirty years as an educator, I never heard the question.) Perhaps the weight of history is too much to support the change.

Imaginary and real are found in English in 1668 in Philosophical Transactions. The words are found in a review of the book Geometriæ Universalis: "And for the like reason a Cubick Æquation, having three reals roots, can never be reduced to a pure Æquation, which hath but one onely root, for in these Æquations, Reduction shall no wise profit, for as much as 'tis impossible, by aid thereof to change an Imaginary root into a real one, and the Converse." [Google print search by James A. Landau]

As a way of removing the stigma of the name, the American mathematician Arnold Dresden (1882-1954) suggested that imaginary numbers be called normal numbers, because the term "normal" is synonymous with perpendicular, and the y-axis is perpendicular to the x-axis (Kramer, p. 73). The suggestion appears in 1936 in his An Invitation to Mathematics.

Some other terms that have been used to refer to imaginary numbers include "sophistic" (Cardan), "nonsense" (Napier), "inexplicable" (Girard), "incomprehensible" (Huygens), and "impossible" (many authors).

The first person ever to write about employing the square roots of a negative number was Jerome Cardin (1501-1576). In his Ars Magna (great arts) he posed the problem of dividing ten into two parts whose product if forty. After pointing out that there could be no solution, he proceeded to solve the two equations, x+y = 10, and xy=40 to get the two solutions,

5 \pm \sqrt{15}

. He then points out that if you add the two solutions, you get ten, and if you multiply them then the product is indeed forty, and concluded by saying that the process was "as subtle as it was useless."

Cardin also left a seed to inspire future work int he mystery of roots of negative numbers. Cardin had published a method of finding soltuons to certain types of cubic functions of the form

x^{3} + a x + b

. His solution required finding the roots of a derived equation. For functions in which the value was negative, his method would not work, even if one of the three roots was a known real solution.
About thirty years later Rafael Bombelli found a way to use the approach to find a root to

x^{3} - 15 x - 4

with the known solution of four. He went on to develop a set of operations for these roots of negative numbers. By the eighteenth century the imaginary numbers were being used widely in applied mathematics and then in the nineteenth century mathematicians set about formalizing the imaginary number so that they were mathematically "real."
Some prefer the term "complex number" (a term first coined by Gauss) for combinations of a real and imaginary number like 2 + 3i, and reserve imaginary for numbers that have no real part.

Imaginary Unit The imaginary number with a magnitude of one used to represent the square root of

\sqrt{- 1}

, has been the letter i since it was adopted by Euler in 1777 in a memoir to the St Petersburg Academy, but it was not published until 1794 after his death. It seemed not to have gained much use until Gauss adopted it in 1801, and began to use it regularly. Gauss did not like the term "imaginary" in referring to complex numbers. In is Theoria residiorum biquadraticorum, (1832) he states that, "That the subject has hitherto been considered from the wrong point of view and surrounded by by a mysterious obscurity, is to be attributed largely to an ill adapted terminology. If for instance, +1, -1, and the square root of -1 had been called direct, inverse, and lateral units ...such an obscurity would have been out of the question."

The term, imaginary unit, was first created, it seems, by William Rowan Hamilton in writing about quaternions in 1843 to the Royal Irish Academy. For his three-dimensional algebra of quaternions, Hamilton added two more imaginary constants, j, and k, which were both considered perpendicular to the i, and to each other.

The term IMAGINARY PART appears in 1748 in A Treatise of Algebra: In Three Parts by Colin Maclaurin. [Google print search by James A. Landau]

On This Day in Math - September 22

Nothing is too wonderful to be true if it be consistent with the laws of nature.

~Michael Faraday in his Laboratory Notebook

The 265th day of the year; 265 is !6 (sub-factorial 6), the number of ways that six ordered objects can be mis-ordered so that each is in the wrong spot. See Subfactorial

The term "subfactorial "was introduced by Whitworth (1867 or 1878; Cajori 1993, p. 77). Euler (1809) calculated the first ten terms. For example, the only derangements of {1,2,3} are {2,3,1} and {3,1,2}, so !3=2 (Pssst, students... You can find !n by dividing n! by e, and rounding to nearest integer)

265 is the sum of two squares in two different ways, including one that is the sum of consecutive squares. \(265 = 3^2 + 16^2 = 11^2 + 12^2 \)

265² is also the sum of two squares in two different ways, making 265 the hypotenuse of two Pythagorean Triangles. One of them is 23, 264, 265.

265 = 16^2 + 3^2 and 29^2 - 24^2,

265 is a semi-prime, 5 x 53. The sum of the digits of 265 is the same as the sum of the digits of its factors, sometimes called Joke numbers.

To form a 5x5 magic square with a constant of 265, take the standard 5x5 using 1-25, and add forty to each term. You can also multiply all the digits from 1-25 by four, then add one, using the numbers 5, 9, 13,....., 101

EVENTS

1602 In a public address at T¨ubingen University, Michael Maestlin, Kepler’s teacher, on the basis of chronological research put Jesus’ birth more than four years before the conventional date of A.D. 1. This date is now generally accepted. *VFR

A student of Philipp Apian, Maestlin is recognized as the teacher who had the greatest influence on Kepler. He is regarded as one of the most significant astronomers of the period between Copernicus and Kepler.

The first known calculation of the (inverse) golden ratio as a decimal, approximately 0.6180340, was made by Maestlin in 1597. He included this calculation in a letter to Kepler about the Kepler triangle. a special right triangle with edge lengths in geometric progression.

1636 From a letter by Fermat to Roberval, it is clear that Fermat conceived the idea of analytic geometry as early as 1629, yet he published nothing on the subject. [Struik, Source Book, p. 397] *VFR In the same letter he found the first new pair of amicable numbers since the early Greeks found 220 & 284. Fermat's pair was 17296 & 18416. Two numbers are called amicable, if the sum of the aliquot divisors of each, sums to the other. *L E Dickson, History of Theory of Numbers

1792 This date is considered the beginning of the Republican Calendar of France, for on this date the Republic was proclaimed and this was also the date of the autumnal equinox in that year. The new calendar was not oﬃcially approved until 5 October 1793. *Cecil B. Read, “A book printed in the year VII,” The Mathematics Teacher, 59(1966), 138–140.

1822 “Jean-Francoise Champollion the younger wrote his famous Lettre `a Monsieur Dacier, secr´etaire perp´etuel de l’Acad´`ero¬emie royale des inscriptions et belles-lettres, relative a l’alphabet des hi´glyphes phon´etiques. On that day he opened the great book of Ancient Egypt, sealed for some two thousand years and now at last decipherable.” Quoted from p. 15 of Tutankhamen (1963) by Christiane Desroches-Noblecourt. *VFR

1986 In a decisive victory for the makers of a computer's insides, a federal judge ruled that code used to run computers and other electronic devices could be copyrighted like printed material.*CHM

2006 Britney Crystal Gallivan of Pomona, California was a keynote speaker at the September 22, 2006 National Council of Teachers of Mathematics convention. As a Junior in high school in 2002 she had disproved a commonly held mathematical myth that a piece of paper could not be folded more than eight times. Gallivan demonstrated that a single piece of toilet paper 4000 ft in length can be folded in half twelve times. Not only did she provide the empirical proof, but she also derived an equation that yielded the width of paper or length of paper necessary to fold a piece of paper of thickness t any n number of times. Wik

BIRTHS

1711 Thomas Wright (22 September 1711 – 25 February 1786) was an English astronomer, mathematician, instrument maker, architect and garden designer. He was the first to describe the shape of the Milky Way and speculate that faint nebulae were distant galaxies.
Wright is best known for his publication An original theory or new hypothesis of the Universe (1750), in which he explains the appearance of the Milky Way as "an optical effect due to our immersion in what locally approximates to a flat layer of stars."
Another of Thomas Wright's ideas, which is often attributed to Kant, was that many faint nebulae are actually incredibly distant galaxies.*Wik

1759 William Playfair (22 September 1759 – 11 February 1823), a Scottish engineer and political economist, served as a secret agent on behalf of Great Britain during its war with France. The founder of graphical methods of statistics, Playfair invented several types of diagrams: in 1786 the line, area and bar chart of economic data, and in 1801 the pie chart and circle graph, used to show part-whole relations. As a secret agent, Playfair reported on the French Revolution and organized a clandestine counterfeiting operation in 1793 to collapse the French currency.

*Linda Hall org

1765 Paolo Ruffini (22 September 1765 – 10 May 1822) He anticipated Abel by providing an almost correct proof of the insolubility of the quintic. *VFR Italian mathematician and physician who made studies of equations that anticipated the algebraic theory of groups. In 1799 Ruffini published a book on the theory of equations with his claim that quintics could not be solved by radicals, General theory of equations in which it is shown that the algebraic solution of the general equation of degree greater than four is impossible. Ruffini used group theory in his work but he had to invent the subject for himself. He also wrote on probability and the application of probability to evidence in court cases.*TIS (He also published the method now often called Horner's method ,,, see Horner below)

1769 Louis Puissant (22 Sept 1769, 10 Jan 1843) He is best remembered for his invention of a new map projection for a new map of France, and he was involved in the production of the map. The map was produced with considerable detail, the projection used spherical trigonometry, truncated power series and differential geometry. Puissant wrote on geodesy, the shape of the earth and spherical trigonometry. *SAU

1791 Michael Faraday born. (22 Sep 1791; 25 Aug 1867) English physicist and chemist whose many experiments contributed greatly to the understanding of electromagnetism. Although one of the greatest experimentalists, he was largely self-educated. Appointed by Sir Humphry Davy as his assistant at the Royal Institution, Faraday initially concentrated on analytical chemistry, and discovered benzene in 1825. His most important work was in electromagnetism, in which field he demonstrated electromagnetic rotation and discovered electromagnetic induction (the key to the development of the electric dynamo and motor). He also discovered diamagnetism and the laws of electrolysis. He published pioneering papers that led to the practical use of electricity, and he advocated the use of electric light in lighthouses. *TIS

1922 Chen Ning Yang (22 Sep 1922, )Chinese-American theoretical physicist who shared the 1957 Nobel Prize for Physics (with Tsung-Dao Lee) for a ground-breaking theory that the weak force between elementary particles did not conserve parity, thus violating a previously accepted law of physics. (Parity holds that the laws of physics are the same in a right-handed system of coordinates as in a left-handed system.) The theory was subsequently confirmed experimentally by Chien-Shiung Wu in observations of beta decay. Yang is also known for his collaboration with Robert L. Mills. They developed the Yang-Mills fields theory - a mathematical idea for describing interactions among elementary particles and fields*TIS

1959 Saul Perlmutter (born September 22, 1959) is a U.S. astrophysicist, a professor of physics at the University of California, Berkeley, where he holds the Franklin W. and Karen Weber Dabby Chair, and head of the International Supernova Cosmology Project at the Lawrence Berkeley National Laboratory.

He shared (with Brian P. Schmidt and Adam G. Riess) the 2011 Nobel Prize in Physics for “the discovery of the accelerating expansion of the Universe through observations of distant supernovae.” For this purpose, he had co-founded (1988) and led the international Supernova Cosmology Project based at the Lawrence Berkeley National Laboratory. *TiS

DEATHS

1703 Vincento Viviani died. His problem of cutting four congruent windows in a hemispherical cupolo so that the remainder was quadrable led to Euler’s development of the double integral. *VFR The leading geometer of his time, who founded the Accademia del Cimento. As one of the first important scientific societies, this organization came before England's Royal Society. In 1639, at age 17, he was a pupil of Torricelli and became the student, secretary and assistant of Galileo (now blind) in Arcetri, until Galileo died in 1642. During his long career, Viviani published a number of books on mathematical and scientific subjects. He edited the first edition of Galileo's collected works (1655-1656), and worked tirelessly to have his master's memory rehabilitated. In 1660, together with Borelli, he measured the velocity of sound by timing the difference between the flash and the sound of a cannon. They obtained the value of 350 metres per second. *TIS

1837 William George Horner (9 June 1786 – 22 September 1837) was a British mathematician and schoolmaster. The invention of the zoetrope, in 1834 and under a different name (Daedaleum), has been attributed to him. *Wik
Horner is largely remembered only for the method, Horner's method, of solving algebraic equations ascribed to him by Augustus De Morgan and others. He published on the subject in the Philosophical Transactions of the Royal Society of London in 1819, submitting his article on 1 July. But Fuller has pointed out that, contrary to De Morgan's assertion, this article does not contain the method, although one published by Horner in 1830 does. Fuller has found that Theophilus Holdred, a London watchmaker, did publish the method in 1820 and comments"At first sight, Horner's plagiarism seems like direct theft. However, he was apparently of an eccentric and obsessive nature ... Such a man could easily first persuade himself that a rival method was not greatly different from his own, and then, by degrees, come to believe that he himself had invented it. "
This discussion is somewhat moot because the method was anticipated in 19th century Europe by Paolo Ruffini (What a strange coincidence that he dies on Ruffini's birthdate) , but had, in any case, been considered by Zhu Shijie in China in the thirteenth century. In the 19th and early 20th centuries, Horner's method had a prominent place in English and American textbooks on algebra. It is not unreasonable to ask why that should be. The answer lies simply with De Morgan who gave Horner's name and method wide coverage in many articles which he wrote.
Horner made other mathematical contributions, however, publishing a series of papers on transforming and solving algebraic equations, and he also applied similar techniques to functional equations. It is also worth noting that he gave a solution to what has come to be known as the "butterfly problem" which appeared in The Gentleman's Diary for 1815. The problem is the following:-
Let M be the midpoint of a chord PQ of a circle, through which two other chords AB and CD are drawn. Suppose AD cuts PQ at X and BC cuts PQ at Y. Prove that M is also the midpoint of XY.

The butterfly problem, whose name becomes clear on looking at the figure, has led to a wide range of interesting solutions. Finally we mention that Horner published Natural magic, a familiar exposition of a forgotten fact in optics (1832). *SAU

1956 Frederick Soddy (2 Sep 1877, 22 Sep 1956). English chemist and physicist who received the Nobel Prize for Chemistry in 1921 for investigating radioactive substances. He suggested that different elements produced in different radioactive transformations were capable of occupying the same place on the Periodic Table, and on 18 Feb 1913 he named such species "isotopes" from Greek words meaning "same place." He is credited, along with others, with the discovery of the element protactinium in 1917 *TIS
Soddy is also the author of a mathematical poem about the solution to Descartes' four tangent circles theorem from the letter to Princess Elisabeth of Bohemia. The poem is called The Kiss Precise, and begins:

For pairs of lips to kiss maybe
Involves no trigonometry.
'Tis not so when four circles kiss
Each one the other three.
To bring this off the four must be
As three in one or one in three.
If one in three, beyond a doubt
Each gets three kisses from without.
If three in one, then is that one
Thrice kissed internally.

The complete poem and more about the history of the problem can be found here.

1970 Vojtěch Jarník (22 Dec 1897 , 22 Sept 1970) a Czech mathematician.

In 1929, Vojtěch Jarník was appointed an extraordinary professor of mathematics at Charles University and six years later a full professor. He worked at the university until 1967 , when he retired. After the establishment of the Czechoslovak Academy of Sciences, he was appointed academician in 1952 and became the first chairman of its mathematical-physical section.

Vojtěch Jarník was interested in the history of mathematics, he paid special attention to the study of Bernard Bolzan's work . The main content of his work was in the areas of number theory and mathematical analysis .

His main area of work was in number theory and mathematical analysis; he proved a number of results on lattice point problems. He also developed the graph theory algorithm known as Prim's algorithm. The Vojtěch Jarník International Mathematical Competition, held each year in Ostrava, is named in his honor. *Wik

1979 Charles Ehresmann (19 April 1905, 22 Sept 1979) He was one of the creators of differential topology. Beginning in 1941, Ehresmann made major contributions toward establishing the current view of fibre spaces, manifolds, foliations and jets. His work in the creation and development of fibre spaces followed on from the study of a special case made earlier by Seifert and Whitney.
After 1957 Ehresmann became a leader in category theory and he worked in this area for 20 years. His principal achievements in this area concern local categories and structures defined by atlases, and germs of categories. The article contains a list of 139 articles written by Ehresmann during his productive career as well as listing several volumes which he edited. *SAU

Charles Ehresmann (right) at the topology conference 1949 in Oberwolfach, together with Paul Vincensini (middle) and Georges Reeb (left)

1979 Otto Robert Frisch OBE FRS (1 October 1904 – 22 September 1979) was an Austrian-born British physicist who worked on nuclear physics. With Otto Stern and Immanuel Estermann he first measured the magnetic moment of the proton.

With his aunt Lise Meitner, they described the division of neutron-bombarded uranium into lighter elements. He named the process fission, borrowing a term from biology (1939). At the time, Meitner was working in Stockholm and Frisch (1934-39) at Copenhagen under Niels Bohr, who brought their observation to the attention of Albert Einstein and others in the United States. He did research with James Chadwick 1940-43, and was head of the Critical Assembly Group on the Los Alamos project 1943-46. After World War II, Frisch became a science writer on atomic physics for the layman.*TiS

2005 Hans Samelson (3 March 1916 – 22 September 2005) was a German-American mathematician who worked in differential geometry, topology and the theory of Lie groups and Lie algebras—important in describing the symmetry of analytical structures.

His family helped him leave Nazi Germany in 1936 for Zurich, Switzerland, where he studied with the geometer Heinz Hopf and received his doctorate in 1940 from the Swiss Federal Institute of Technology.

In 1941, he accepted a position at the Institute for Advanced Study in Princeton and immigrated to the United States; he arrived by ship six months before the United States entered World War II and acquired U.S. citizenship several years later. After leaving Princeton, he held faculty positions at the University of Wyoming (1942–1943), Syracuse University (1943–1946) and the University of Michigan (1946–1960) before coming to Stanford in 1960. He was recognized with the Dean's Award for Distinguished Teaching in 1977. He served as chair of the Mathematics Department from 1979 to 1982.

Though he became emeritus in 1986, he remained professionally active throughout his retirement, publishing articles on both contemporary and historical mathematical topics. One solved an architectural puzzle associated with the construction of the Brunelleschi Dome in Florence, Italy.

He was active in the Palo Alto Friends Meeting (Quakers) during his retirement, serving as treasurer for several years.

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

Degree, Gradient, Grads, Gons etc....History and Etymology of Math Terms

Degree is the union of the Latin roots de, down, and gradus, step. Gradus is actually derived from the Greek word for "to walk" or "go". Related words with the same root are congress (come together), regress (go back), and of course grade (the step you are on in school, or earned on an evaluation). Degree as the measure (or step) of an angle dates back at least to the writings of Chaucer who used the word both in his Canterbury Tails (Squires Tale) in 1386, and his more famous (then) book on the Astrolabe in 1400.

Degrees, why 360 in a circle I have seen many responses to why we use 360<sup>o</sup> in a circle, but the one that most impressed me was by the one below by the late Alexander Bogomolny. I have copied the entire thing from his response to a question on geometry discussion site, so here's why there are 360<sup>o</sup> in a circle,
"Babylonians used base 60 notation, which is convenient to divide a whole into 2, 3, 4, ... 30 parts. Early Greeks then probably divided the radius of a circle into 60 parts. Hence, the diameter must have 120 parts. As Pi was known to be close to 3, the circumference would have 360 parts.
This argument may be used to exonerate the Bible (I Kings. 7:23 and II Chronicles, 4:2) which is said to quote 3 as the value of Pi. Not being a geometry manual, the Bible just picked out a simple approximation to Pi to convey the order of magnitude of the measured quantity. ...
Some history of the sexagesimal (base 60) notations appear in D. E. Smith, History of Mathematics, v2, Dover"
DEGREE for angle measure is found in English in about 1386 in Chaucer's Canterbury Tales: "The yonge sonne That in the Ram is foure degrees vp ronne" [OED]. He again used the word in about 1391 in A Treatise on the Astrolabe: "9. Next this fole with the cercle of the daies, that ben figured in manere of degres, that contenen in nombre 365, dividid also with longe strikes fro 5 to 5, and the nombre in augrym writen under that cercle."

GRAD :Gradus is a Latin word equivalent to "degree."

Gradian/Grade/Grad/Gon In trigonometry, the gradian, also known as the gon (from the same Greek root for angle that gave us the gon in polygon) is a unit of measurement of an angle, defined as one hundredth of the right angle; in other words, there are 100 gradians in 90 degrees.

Grad or grade originally referred to one ninetieth of a right angle, but the term is now used primarily to refer to one hundredth of a right angle. Some early scientific calculators had a key labeled DRG for selecting between degrees, radians, and grads. The Sharp EL501X2BWH Engineering/Scientific Calculator shown has the key in the top row 2nd from left next to 2nd Function key.

The OED2 shows a use of grade in English in about 1511, referring to one-ninetieth of a right angle.

The OED2 shows a use of grade, meaning one-hundredth of a right angle, in 1801 in Dupré Neolog. Fr. Dict. 127: "Grade .. the grade, or decimal degree of the meridian." (being French and at this period when France had tried to decimalize clocks, calendars, and pretty much everything, it may be where the first use of grad as 1/100 of a degree began.

first use of grad as 1/100 of a degree began.

In France, it was originally called "grade" (or "grade nouveau" - new grade) Gradian - *HandWiki. Due to confusion with the existing term grad(e) in some northern European countries (meaning a standard degree, 1/360 of a turn), the name gon was later adopted, first in those regions, and later as the international standard

You may also hear about centesimal degrees. In the centesimal system, a right angle is divided into 100 centesimal degrees; each centesimal degree, into 100 centesimal minutes; and each centesimal minute into 100 centesimal seconds. (Centesimal degrees are also known as grads , grades , or gon .)

400 degree compass

If you see a sign on a US highway warning about a steep grade, it refers to the tangent of the angle the road makes with the horizontal.

A sign showing a 6% grade will go down (or up) by six feet per hundred feet of horizontal change. This seems a little confusing since the mile as recorded by your odometer will be slightly more than a mile to get this mile of horizontal change, think Pythagoras. Or just don't think about it at all since for most highway grades the difference is very small; as you travel one mile in horizontal change, for a vertical decline of 264 feet, you will actually have to travel about 6.5 feet more than a mile.

Sunday, 20 September 2026

On This Day in Math - September 21

To throw in a fair game at Hazards only three-spots, when something great is at stake, or some business is the hazard, is a natural occurrence and deserves to be so deemed; and even when they come up the same way for a second time if the throw be repeated. If the third and fourth plays are the same, surely there is occasion for suspicion on the part of a prudent man.
~Girolamo Cardano

The 264th Day of the Year
264 = 2³x 3 x 11 is a harshad number (a number divisible by the sum of its digits). The word "Harshad" comes from t h e Sanskrit harṣa (joy) + da (give), meaning joy-giver. The numbers were defined and named by the famous Indian Mathematician D. R. Kaprekar.

Jim Wilder @wilderlab pointed out that the sum of all 2-digit numbers you can make from 264 totals 264... 24 + 42 + 26 + 62 + 46 + 64.

264² = 69696, a palindrome; and 264 is the sum of ten consecutive primes. *Chaw points out that this is the only square less than 2,000,000,000 that is both a palindrome and a sum of twin primes, 34847 and 34849.

264 is a repdigit in base 11 (222).

264 = 66^2 - 64^2 = 35^2- 31^2,

264 is one more than a prime.

264 is the Area of two integer sided triangles, with sides 33, 34, 65 and 44, 37, 15.

264 = 2^8 + 2^3,

Events

1781 Writing to his friend and mentor d’Alembert, Lagrange expressed concern that mathematics was reaching its creative end. “It seems,” he wrote, “that the mine is almost too deep already, and unless new seams are discovered, it will be necessary to abandon it sooner or later. Physics and chemistry now offer riches that are more brilliant and easier to exploit.” *Amir R. Alexander , Tragic Mathematics, Isis, Vol. 97, No. 4, December 2006

Strangely, around 1900 Physicists were bemoaning that "there was nothing new to discover", and then the cornucopia exploded.

1784 The nation's first daily newspaper, the Pennsylvania Packet and Daily Advertiser, began publication on September 21, 1784. Many independent newspapers ran before that on a weekly or monthly basis. America's first independent newspaper, the New England Courant, was published by Benjamin Franklin's older brother in 1721. By the start of the Revolutionary War in 1775, there were 37 independent newspapers to keep the colonists informed. *Libray of Congress

The paper was founded by John Dunlap as a weekly paper in late 1771. It was based in Philadelphia, except during the British occupation of the city between 1777 and 1778, when Dunlap published the paper in Lancaster. David C. Claypoole eventually became a partner with Dunlap. As of September 21, 1784, the paper was issued as the Pennsylvania Packet, and Daily Advertiser, reflecting the paper's move to daily publication.

This newspaper subsequently underwent additional name changes, dropping the Pennsylvania Packet prefix in 1791 and becoming Dunlap's American Daily Advertiser (1791–1793), Dunlap and Claypoole's American Daily Advertiser (1793–1795), and Claypoole's American Daily Advertiser (1796-1800).

On September 21, 1796, it became the first to publish George Washington's Farewell Address.

In 1800, Zachariah Poulson purchased the paper and renamed it Poulson's American Daily Advertiser.

In 1825, the Marquis De Lafayette granted an interview to "Poulson's Advertiser" during his famous visit to the United States.

Poulson ran the paper for almost forty years; at the end of 1839, he sold the publication to the owners of the recently founded North American. The North American featured the 1771 founding of the Packet as its heritage.

To the extent it can honestly be traced past this point, the final successor of the Packet can be said to be The Philadelphia Inquirer.*Wik

1832 Mary Somerville's gender prevented her from attending university, or full membership in scientific societies. On 21 September 1832, The Naval and Military Library and Museum of London became the first society to list Somerville as an honorary member . Honorary memberships in several other scientific societies would follow, including Royal Astronomical Society (RAS) (With Caroline Herschel), in February 1835. *Mathematical Intelligencer

When she died in 1872, The Morning Post declared in her obituary that "Whatever difficulty we might experience in the middle of the nineteenth century in choosing a king of science, there could be no question whatever as to the queen of science".

Page 157 from Mechanism of the Heavens, Somerville discusses the law of universal gravity and Kepler's laws of planetary motion.

1908 On this day in 1908 Hermann Minkowski began his famous lecture at the University of Cologne with these words:-

The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.

*BYJUS

1925 Edith Clarke patent for the Clark Calculator is approved. The calculator was 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. Ms. Clarke is generally thought of as the first female electrical engineer in the U. S.

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. She taught for 10 years and retired in 1957.

1963 September 21, 1963: The front page of the New York Times reports President John F. Kennedy's proposal for a joint U.S.-Russian Moon mission. "Washington is Surprised By President's Proposal." * Chasing The Moon: The Book

1983 This set of four stamps honors some of America's top inventors. Charles Steinmetz devised methods of harnessing high voltage for the Niagara Falls Power Plant, Edwin Armstrong invented FM radio, Nikola Tesla developed the generator and other devices that made alternating current (AC) possible, and Philo T. Farnsworth invented the first television camera.

1984 Science reported (pp. 1379-1380) that Narendra Karmarkar of AT & T Bell Labs found a practical polynomial-time algorithm that is far faster than the simplex algorithm for linear programming problems. [Mathematics Magazine 58 (1985), p. 53]. *VFR

Karmarkar's algorithm falls within the class of interior-point methods: the current guess for the solution does not follow the boundary of the feasible set as in the simplex method, but moves through the interior of the feasible region, improving the approximation of the optimal solution by a definite fraction with every iteration and converging to an optimal solution with rational data.

2012 Clear skies allowed a clear view of the fire balls which were witnessed right across the UK, Ireland and even parts of north west Europe.Bright 'meteor-like' tails could be seen in the sky for approximately two minutes at 10.55pm on Friday night.
The fireball was also seen in the US, "Portsmouth, RI, USA 2245 EST 4 to 5 seconds South to North Initially white, turned orange, then turned greenish-blue as it got low on horizon Like a flare but definately not a flare. Yes, from the "ball" itself. The tail was approximately 5 times longer than the ball itself. The tail remained a bright white even though the ball changed color. Was definitely not the typical "shooting star"."
According to the modeling done by Finnish mathematician Esko Lyytinen, the big UK fireball of the 21st of September was captured by Earth`s gravity.
After one circle around the Earth one of the remnants seems to have re-entered the skies over North America.
"It looks now that the fireball witnessed 155 minutes later in US and Canada, may have been one fragment of the British fireball, most probably the biggest one. This was its second entry into the Earth`s atmosphere", Lyytinen says. "If so, this is historical scientific observation, but it needs to be confirmed." * LunarMeteorite*Hunter and other sources

BIRTHS
1623 Stefano degli Angeli (21 Sept 1623 , 11 Oct 1697) His many mathematical works were on infinitesimals and he used them to study spirals, parabolas and hyperbolas. While in Venice he published De infinitorum parabolis (1654), De infinitorum spiralium spatiorum mensura (1660) which contains a generalisation of Archimedes' spiral, and De infinitorum cochlearum mensuris ac centris gravitatis (1661) which carries out Torricelli's intention of finding the centre of gravity of a solid body called a cochlea. The approach followed by Angeli in all these works is that of his teacher Cavalieri and of Torricelli, so when Guldin and Tacquet attacked these methods and defended the approach of the ancient Greeks, Angeli disputed with them over indivisibles. One has to see both sides in this argument for although Angeli's methods were much more powerful, they were less rigorous than the method of exhaustion adopted by Archimedes. Angeli examined fluid statics based on Archimedes' principle and Torricelli's experiments. He published Della gravita dell aria e fluidi in 1671 while holding the chair at Padua. He also considered the motion bodies falling towards a rotating Earth. Of course Angeli held the chair at Padua which had been held earlier by Galileo and his work shows strong influences from his predecessor. For example Angeli often refers to Galileo in his writings on physics, showing clearly how he has been influenced, particularly in terms of ways of approaching problems via the experimental method. Also clearly influenced by Galileo is Angeli's writings on the two systems of Ptolemy and Copernicus which he writes in Galileo's dialogue style.*SAU

*Wik

1845 Elizaveta Fedorovna Litvinova (21 Sept, 1845–1919?) was a Russian mathematician and pedagogue. She is the author of over 70 articles about mathematics education.

Born in 1845 in czarist Russia as Elizaveta Fedorovna Ivashkina, she completed her early education at a women's high school in Saint Petersburg. In 1866 Elizaveta married Viktor Litvinov, who, unlike Vladimir Kovalevsky (Sofia Kovalevskaya's husband), would not allow her to travel to Europe to study at the universities there. Thus, Litvinova started to study with Strannoliubskii, who had also privately tutored Kovalevskaya.

In 1872, as soon as her husband died, Litvinova went to Zürich and enrolled at a polytechnic institute. In 1873 the Russian czar decreed all Russian women studying in Zürich had to return to Russia or face the consequences. Litvinova was one of the few to ignore the decree and she remained to continue her studies, earning her baccalaureate in Zürich in 1876. She completed her doctoral degree in 1878 from the University of Bern, as a student of Ludwig Schläfli, becoming the first woman to earn a doctorate in mathematics in Switzerland.

When Litvinova returned to Russia, she was denied university appointments because she had defied the 1873 recall. She taught at a women's high school and supplemented her meager income by writing biographies of more famous mathematicians such as Kovalevskaya and Aristotle. After retiring, Litvinova moved to the countryside in 1917. Although no subsequent records have been found, it is believed that she must have died soon after, possibly in the Russian famine of 1921–1922 or earlier.

1853 Heike Kamerlingh Onnes (21 Sep 1853; 21 Feb 1926) Dutch winner of the Nobel Prize for Physics in 1913 for his work on low-temperature physics and his production of liquid helium. He discovered superconductivity, the almost total lack of electrical resistance in certain materials when cooled to a temperature near absolute zero.*TIS

1884 Denes König (21 Sept 1884, 19 Oct 1944) His book, Theorie der endlichen und unendlichen Graphen, was published in 1936, and was a major factor in the growth of interest in graph theory worldwide. It was eventually translated into English under the title Theory of finite and infinite graphs (translated by R McCoart), Birkhauser, 1990; this also contains a biographical sketch by Tibor Gallai.
König's work on the factorisation of bipartite graphs relates closely to the marriage problem of Philip Hall. König's use of graphs to give a simpler proof of a determinant result of Frobenius seems to have led to some hostility between the two men.
After the Nazi occupation of Hungary, König worked to help persecuted mathematicians. This led to his death a few days after the Hungarian National Socialist Party took over the country. *SAU

1895 Joseph Leonard Walsh (21 Sept 1895, 6 Dec 1973) Walsh had a remarkable publication record. An obituary by Morris Marden (a student of Walsh) lists 279 articles, 7 books and 31 PhD students. He studied the relative location of the zeros of pairs of rational functions, zeros and topology of extremal polynomials, the critical points and level lines of Green's functions and other harmonic functions, conformal mappings, Padé approximation, and the interpolation and approximation of continuous, analytic or harmonic functions. Sewell writes "Polynomial approximation was neither discovered nor invented by J L Walsh (which may come as a surprise to some mathematicians). He is the one individual, however, who took a few scattered results on the subject and extended them, added mightily to them, and knit the whole together into a comprehensive, coherent theory." *SAU

1899 Juliusz Paweł Schauder (September 21, 1899, Lwów, Austria-Hungary – September 1943, Lwów, Occupied Poland) was a Polish mathematician of Jewish origin, known for his work in functional analysis, partial differential equations and mathematical physics.
He had to fight in World War I right after his graduation from school. He was captured and imprisoned in Italy. He entered the university in Lwów in 1919 and received his doctorate in 1923. He got no appointment at the university and continued his research while working as teacher at a secondary school. Due to his outstanding results, he obtained a scholarship in 1932 that allowed him to spend several years in Leipzig and, especially, Paris. In Paris he started a very successful collaboration with Jean Leray. Around 1935 Schauder obtained the position of a senior assistant in the University of Lwów.
Schauder was Jewish, and after the invasion of German troops in Lwów it was impossible for him to continue his work. In his letters to Swiss mathematicians, he wrote that he had important new results, but no paper to write them down. He was executed by the Gestapo, probably in October 1943.
Most of his mathematical work belongs to the field of functional analysis, being part of a large Polish group of mathematicians, i.e. Lwów School of Mathematics. They were pioneers in this area with wide applications in all parts of modern analysis. Schauder is best known for the Schauder fixed point theorem which is a major tool to prove the existence of solutions in various problems, the Schauder bases (a generalization of an orthonormal basis from Hilbert spaces to Banach spaces), and the Leray−Schauder principle, a way to establish solutions of partial differential equations from a priori estimates. *Wik

1907 Sir Edward Crisp Bullard (21 Sep 1907; 3 Apr 1980) English marine geophysicist noted for his work in geomagnetism who made the first satisfactory measurements of geothermal heat-flow through the oceanic crust. In early work, he measured minute gravitational variations by timing the swings of an invariant pendulum, which he used to study the East African Rift Valley. Bullard helped to develop the theory of continental drift. He made a computer analysis of the precise fit of the rifted continental borders along the two sides of the Atlantic Ocean, and presented his results to the Royal Society of London. He developed a "dynamo" theory of geomagnetism, which explained the Earth's magnetic field results from the convection of molten material within the Earth's core. He was knighted in 1953.*TIS

1917 Phyllis Nicolson (21 September 1917 – 6 October 1968) was a British mathematician and physicist best known for her work on the Crank–Nicolson method together with John Crank.

Nicolson was born Phyllis Lockett in Macclesfield and went to Stockport High School for Girls. She graduated from Manchester University with a B.Sc. in 1938, M.Sc. in 1939 and a Ph.D. on Three Problems in Theoretical Physics in 1946. Her Ph.D. thesis began with cosmic ray research conducted under Lajos Jánossy during 1939 and 1940.

Nicolson's Ph.D. was expected to be submitted in 1941 but was interrupted by wartime work with Douglas Hartree's research group at Manchester University from 1940 to 1945. During this time, Nicolson became a proficient numerical analyst and an expert user of Hartree's differential analyser. Nicolson, along with other members of the research group worked on defence-related problems for the Air Defence Research and Development Establishment (later the Radar Research and Development Establishment), both part of the Ministry of Supply. Nicolson's two significant bodies of wartime research, "Transient behaviour in the single anode magnetron" and "heat conduction", formed the basis of parts II and III of her 1946 PhD thesis Three Problems in Theoretical Physics.

Nicolson's research on heat conduction related to solutions of the heat equation, and with her colleague John Crank she investigated the numerical stability of several solution techniques. The algorithm now known as the Crank–Nicolson method emerged from this work and was published in 1947.

Nicolson was a research student in Cambridge from 1945 and completed her Ph.D. thesis completed at the Victoria University of Manchester (now Manchester University) in 1946. She was a Tucker-Price Research Fellow of Girton College, Cambridge from 1946 to 1949, working at the Cavendish Laboratory. Nicolson moved to Leeds around January 1950 with her husband Malcolm Nicolson, also a physicist, as he had been appointed to a lectureship in Physics at Leeds University. Phyllis Nicolson had married Malcolm in 1942 and they had two sons, Donald Macleod Nicolson (born 20 September 1947 in Cambridge) and Roderick Ian Nicolson (born 5 February 1950 in Leeds).

Malcolm Nicolson, aged 33, died in a train accident in December 1951, and Phyllis was appointed to take over his lectureship. In 1955, Nicolson married physicist Malcolm McCaig, who had a son Ian McCaig (born February 1946) from a previous marriage. In May 1957, Nicolson and McCaig had a son together, Andrew Malcolm McCaig. All three of Nicolson's sons ended up getting PhDs – in mathematics, psychology, and geology

1926 Donald A. Glaser (21 Sep 1926, 28 Feb 2013)American physicist, who was awarded the Nobel Prize for Physics in 1960 for his invention of the bubble chamber in which the behavior of subatomic particles can be observed by the tracks they leave. A flash photograph records the particle's path. Glaser's chamber contains a superheated liquid maintained in a superheated, unstable state without boiling. A piston causing a rapid decrease in pressure creates a tendency to boil at the slightest disturbance in the liquid. Then any atomic particle passing through the chamber leaves a track of small gas bubbles caused by an instantaneous boiling along its path where the ions it creates act as bubble-development centers.*TIS With the freedom that accompanies a Nobel Prize, he soon began to explore the new field of molecular biology, and in 1971 joined two friends, Ronald E. Cape and Peter Farley, to found the first biotechnology company, Cetus Corp., to exploit new discoveries for the benefit of medicine and agriculture. The company developed interleukin and interferon as cancer therapies, but was best known for producing a powerful genetic tool, the polymerase chain reaction, to amplify DNA. In 1991, Cetus was sold to Chiron Corp., now part of Novartis. Glaser died in his sleep Thursday morning, Feb. 28, at his home in Berkeley. He was 86. *Philosophy of Science Portal

1935 Yakov Grigorevich Sinai (Russian: Я́ков Григо́рьевич Сина́й; born September 21, 1935) is a Russian–American mathematician known for his work on dynamical systems. He contributed to the modern metric theory of dynamical systems and connected the world of deterministic (dynamical) systems with the world of probabilistic (stochastic) systems. He has also worked on mathematical physics and probability theory. His efforts have provided the groundwork for advances in the physical sciences.

Sinai has won several awards, including the Nemmers Prize, the Wolf Prize in Mathematics and the Abel Prize. He has served as professor of mathematics at Princeton University since 1993 and holds the position of Senior Researcher at the Landau Institute for Theoretical Physics in Moscow, Russia.

*Wik

DEATHS

1576 Girolamo Cardano (24 September 1501 – 21 September 1576) He wrote more than 200 works on medicine, mathematics, physics, philosophy, religion, and music.His gambling led him to formulate elementary rules in probability, making him one of the founders of the field.
One story says that it was by his own hand so as to fulﬁll his earlier astrological prediction of of his death on this date. *H. Eves, Introduction to the History of Mathematics, Pg 221...
He was the first to give a clinical description of typhus fever. His book, Ars magna ("Great Art," 1545) was one of the great achievements in the history of algebra, in which he published the solutions to the cubic and quartic equations.(Ars Magna was the first Latin treatise devoted solely to algebra. In it he gave the methods of solution of the cubic and quartic equations which he had learned from Tartaglia.*SAU) His mechanical inventions included the combination lock, the compass gimbal consisting of three concentric rings, and the universal joint to transmit rotary motion at various angles (as used in present-day vehicles). He contributed to hydrodynamics and held that perpetual motion is impossible, except in celestial bodies. He published two encyclopedias of natural science and introduced the Cardan grille, a cryptographic tool (1550).*TIS

And just for a bonus, the drive shaft in your automobile that allows rotary power to be transmitted at an angle, is called the Cardan joint, after Cardano.*PB

1842 Sir James Ivory (17 February 1765 – 21 September 1842) was a Scottish mathematician born in Dundee. He was essentially a self-trained mathematician, and was not only deeply versed in ancient and modern geometry, but also had a full knowledge of the analytical methods and discoveries of the continental mathematicians.
His earliest memoir, dealing with an analytical expression for the rectification of the ellipse, is published in the Transactions of the Royal Society of Edinburgh (1796); and this and his later papers on Cubic Equations (1799) and Kepler's Problem (1802) evince great facility in the handling of algebraic formulae. In 1804 after the dissolution of the flax-spinning company of which he was manager, he obtained one of the mathematical chairs in the Royal Military College at Marlow (afterwards removed to Sandhurst); and until the year 1816, when failing health obliged him to resign, he discharged his professional duties with remarkable success.*Wik

1859 Isidore Auguste Marie François Xavier Comte (28 January 1794 – 21 September 1859), better known as Auguste Comte (French: [oɡyst kɔ̃t]), was a French philosopher. He was a founder of the discipline of sociology and of the doctrine of positivism. He is sometimes regarded as the first philosopher of science in the modern sense of the term.
Strongly influenced by the utopian socialist Henri Saint-Simon, Comte developed the positive philosophy in an attempt to remedy the social malaise of the French Revolution, calling for a new social doctrine based on the sciences. Comte was a major influence on 19th-century thought, influencing the work of social thinkers such as Karl Marx, John Stuart Mill, and George Eliot. His concept of sociologie and social evolutionism, though now outdated, set the tone for early social theorists and anthropologists such as Harriet Martineau and Herbert Spencer, evolving into modern academic sociology presented by Émile Durkheim as practical and objective social research.
Comte's social theories culminated in the "Religion of Humanity", which influenced the development of religious humanist and secular humanist organizations in the 19th century. Comte likewise coined the word altruisme (altruism)*Wik

1924 Édouard Gaston (Daniel) Deville (21 Feb 1849; 21 Sep 1924 at age 75)
was a French-Canadian surveyor was a French-born Canadian surveyor of Canadian lands (1875-1924) who perfected the first practical method of photogrammetry, or the making of maps based on photography. His system used projective grids of images taken from photographs made with a camera and theodolite mounted on the same tripod. Photographs were taken from different locations, at precise predetermined angles, with measured elevations. Each photograph slightly overlapped the preceding one. With enough photographs and points of intersection, a map could be prepared, including contour lines. He also invented (1896) the first stereoscopic plotting instrument called the Stereo-Planigraph, though its complexity resulted in little use. *TIS

1936 Frank Hornby (15 May 1863, 21 Sep 1936) English inventor and toy manufacturer who patented the Meccano construction set in 1901. This toy used perforated metal strips, wheels, roods, brackets, clips and assembly nuts and bolts to build unlimited numbers of models. His original sets, marketed as "Mechanics Made Easy" produced in a rented room, were initially sold at only one Liverpool toy shop. By 1908, he had formed his company, Meccano Ltd., and within five more years had established manufacturing in France, Germany, Spain and the U.S. He introduced Hornby model trains in 1920, originally clockwork and eventually electrically powered with tracks and scale replicas of associated buildings. The "Dinky" range of miniature cars and other motor vehicles was added in 1933. *TIS

1937 Jessie Chrystal MacMillan (13 June 1872 in Edinburgh, Scotland - 21 September 1937 in Edinburgh, Scotland) was the first female science graduate at Edinburgh University and the first female honors graduate in Mathematics. She went on to study at Berlin. She was the first woman to plead a case before the House of Lords. She became active in the Women's Suffrage Movement and went on to become a lawyer.
A Millennial plaque is at Kings Buildings (West Mains Road), in Edinburgh. It reads:

In honour of
JESSIE CHRYSTAL MACMILLAN
1872-1937
Suffragist, founder of Women's International League for Peace and Freedom,
first woman science graduate of the University (1896).

*SAU

1950 Edward Arthur Milne (14 Feb 1896, 21 Sep 1950) English astrophysicist and cosmologist best known for his development of kinematic relativity. Poor eyesight prevented him from active service in WWI, he did important war service in research in ballistics and sound ranging, and problems related to the atmosphere of the earth.. From 1920-29, he studied problems of radiative equilibrium and the theory of stellar atmospheres. He extended work done earlier by Schuster and by Schwarzschild, which he combined in a mathematical interesting integral equation now known as Milne's integral equation. Later, he turned to the theory of stellar structure and cosmology. After 1932, he concentrated on a new form of relativity called kinematic relativity, an alternative to Einstein's general theory.*TIS (In Eurekas and Euphorias Walter Gratzer tells an interesting story about Milne's rejected offer to provide his services to the war effort in WWII. Milne had given important service (above) in WWI and wrote to offer his services for this war as well, but received a rather dismissive letter to which he took offense. He used his extensive connections to have his anger made known to the higher-ups at the War Office. Eventually he received a request from a Brigadier General to come to his office. Milne arrived and amidst his tirade advised the General that the War Office should know that this war would be a scientific one, and the way he was treated was not the way to make the best use of eminent scientists. The General waited out Milne's outburst, and then asked a single question, "Did you win the Adams Prize in your year?" When Milne said he had not, but asked what that had to do with anything, the General replied, "I did!"
I'm not sure what, if any, were Milne's contributions to the war effort in WWII.)

1981 Henry George Forder (27 Sept 1889, 21 Sept 1981) Forder spent much of his career in the Chair of Mathematics at Auckland. In fact he only once left New Zealand after settling there, this being in 1947 when he spent part of his leave in England. He spent 21 years building up the Mathematics Department at Auckland from a Department of a professor with one assistant when he arrived to one of six staff by the time he retired in 1955.
It is the books that Forder wrote which have given him a high reputation in the mathematical world. These are: The Foundations of Euclidean Geometry (1927), A School Geometry (1930), Higher Course Geometry (1931), The Calculus of Extension (1941), Geometry (1950), and Coordinates in Geometry (1953). In the Preface to the first of these, Forder writes, "Although the Euclidean geometry is the oldest of the sciences and has been studied critically for over two thousand years, it seems there is no textbook which gives a connected and rigorous account of that doctrine in the light of modern investigations. It is hoped that this book will fill the gap."
Geometry (1950) was reviewed by Donald Coxeter who was clearly fascinated by Forder's use of language, "The two-cusped epicycloid is described as the bright curve seen "when the sun shines on a cup of tea." ... The chapter on logical structure stresses the abstract nature of the order relation (ABC) by comparing it with the human relation "A prefers C to B." The possibility of coordinatising any descriptive geometry of three or more dimensions is epitomized in the statement that "we can create magnitudes from a mere muchness," and Archimedes' axiom in the statement that "you will always reach home, if you walk long enough." *SAU

*WM = Women of Mathematics, Grinstein & Campbell