Pat'sBlog

Wednesday, 15 April 2026

Mathematical Induction, A Brief History of the Term

I have had an interest in the history and etymology of mathematical terms for many years, as witnessed by my MathWords web page. Recently I came across a couple of old journal articles, (1915-1918) related to the history of mathematical induction, and to the term itself. Most of this comes from an article by Cajori, and the very early dates above. Certainly those who know more about the current status of the usage could help by sharing their information.

The logical and scientific process called induction dates back as far as Cicero's translation of Aristotle. Cicero used the latin term "inductio", for the Greek "epagoge", which translates as "leading to." Levi ben Gerson wrote Art of Calculation (or Art of the Computer) in 1321. It deals with arithmetical operations, including extraction of square roots and cube roots. In this work he also gives formulas for the sum of squares and the sum of cubes of natural numbers as well as studying the binomial coefficients. In proofs, he uses induction making this one of the earliest texts to use this important technique.

Induction has always existed in mathematics, but the formal concept of mathematical induction did not appear until it was developed by Maurolycus in 1575 to prove that the sum of the first n odd numbers is n². While the roots of formal mathematical induction are nested in works from Fermat all the way back, one might say, to Euclid's proof of the infinity of the primes, the work of Maurolycus was unique in the formal use of attaching one term to the next in a general way.

The method of Maurolycus was repeated and extended in the works of Pascal to be a much more clear illustration of the present method but none of them used a particular name for their logical process. Then in his Arithmetica infinitorum in 1656 Wallis decided to name the term. On page 15 he creates the term "per modum inductionis" to prove that the limit of the ratio of the sum of the first n squares to n³ + n² was 1/3. His inductive method followed very much the unnamed method of Maurolycus.

Later Bernoulli gives an improvement to Wallis' method by showing the argument from n to n+1 as a general proof; this was the real foundation of modern mathematical induction. Bernoulli gives no specific name to his process, but uses his method as an improvement on the "incomplete induction" earlier used.

For the next 150 years, mathematicians used induction in both senses, to refer to the process of observing a relationship from a pattern , and in the method of Bernoulli to prove such an induced relationship by arguing from n to n+1. Then early in the 19th century, George Peacock uses the term "demonstrative induction" in his 1830 Treatise on Alebra. Then several years later, Augustus De Morgan proposes the name "successive induction" but then at the end of the article he talks about the method as "mathematical induction."

Isaac Todhunter used both names in his chapter on the method, but he used only Mathematical Induction in the chapter heading. When he defined and introduced the term "mathematical induction" (1838), he gave the process a rigorous basis and clarity that it had previously lacked.

Several popular textbook authors, Jevons and Ficklin, for example, used both terms. But among several others, Chrystal, Hall and Knight, used only the term mathematical induction. The same name seems to have been common in the early part of the 20th century in America and Europe, with Germany seemingly clinging to a single term for both "complete" and "incomplete" induction. Cajori, in 1918, says the Germans most commonly use the term, "vollstandige Induktion". I do not know if there is currently a more appropriate notation for the true mathematical induction of Bernoulli in Germany. If a reader is familiar with the current situation in German mathematics, please update me.

On This Day in Math - April 15

Duomo Santa Maria del Fiore, *Wik

For since the fabric of the universe is most perfect and the work of a most wise creator, nothing at all takes place in the universe in which some rule of the maximum or minimum does not appear.
~Leonhard Euler

The 105th day of the year, Paul Erdős conjectured that this is the largest number n such that the positive values of n - 2^k are all prime. *Prime Curios

105 is the first degree for which the cyclotomic polynomial factors are not all 1, 0 or -1.

105 is the sum of consecutive integers in seven distinct ways. 105 =
1 + 2 + 3 + … + 13 + 14 =
6 + 7 + 8 + … + 14 + 15 =
12 + 13 + … + 17 + 18 =
15 + 16 + 17 + 18 + 19 + 20 =
19 + 20 + 21 + 22 + 23 =
34 + 35 + 36 =
52 + 53

105 is the largest composite number for which all the odd numbers less than it either are prime, or share a factor with it.

The distinct prime factors of 105, (3,5,7) add up to 15. The same is true of the factors of 104, so they form a Ruth Aaron pair. Someone noticed the factor relation about these two shortly after Hank Aaron hit his 715th home run to break Ruth's record of 714 on April 8th, 1974. 104 and 105 form the fifth such pair in year days, and yet, there is only one more for the rest of the year.

As the sum of the first fourteen integers, 105 is a Triangular number.

105 is the middle number in a prime quadruplet (101, 103, 107, 109) all in the same decade of numbers so it is the only odd composite in that decade of numbers. 15 holds a similar position in the teens decade.

EVENTS

1566 Early Tycho Brahe in 1566 he left Denmark for the second time, and arrived at Wittenberg on the 15 th April. The University of Wittenberg had been founded in 1502, and had then for nearly fifty years been one of the most renowned in Europe. He Studied under Caspar Peucer, distinguished as a mathematician, a physician, and a historian. Tycho, however, did not profit very much from Peucer's instruction, as the plague broke out at Wittenberg, so that he was induced to leave it on the 16th September, after a stay of only five months. *TYCHO BRAHE, A PICTURE OF SCIENTIFIC LIFE AND WORK IN THE SIXTEENTH CENTURY BY J. L. E. DREYER

1726, writer William Stukeley held a conversation with Isaac Newton in Kensington during which Newton recalled “when formerly, the notion of gravitation came into his mind.” Later, Stukeley writing in his Memoirs of Sir Isaac Newton's Life, recorded that Newton said, “It was occasioned by the fall of an apple, as he sat in contemplative mood. Why should that apple always descend perpendicularly to the ground, thought he to himself. Why should it not go sideways or upwards, but constantly to the earth's centre.” *TIS The story was also related to John Conduitt who was Newton's assistant at the Royal mint, and the husband of Newton's niece. The idea that the apple hit Newton on the head seems to date from the early 20th Century. A copy of the page of Stukeley's manuscript where he tells this story is available on-line at the Royal Society.

1747 Euler, writing in response to a now lost letter from D'Alembert, that he opposed the suggestion that logarithms of negative numbers could exist and in particular that $e^1$ could have both a positive and a negative value. He adds that as soon as the value of e, in $ y = e^x $ is defined, then the logarithm of all values are also assigned.
In the same letter he continues his argument by giving a new definition, now popular, of $ e^x$ as $e^x = 1 + x + \frac{x^2}{1*2} \dots $ and hence the idea of a negative logarithm is impossible. *L E Dickson, History of the Exponential and Logarithmic Concept. Am Math Monthly Mar, 1913

1770, Dr. Joseph Priestley made the first mention in English that a piece of a rubber substance could erase marks from black-lead pencils. At the end of the Preface to his work, Familiar Introduction to the Theory and Practice of Perspective, he described it: "Since this Work was printed off, I have seen a substance excellently adapted to the purpose of wiping from paper the mark of a black-lead-pencil. It must, therefore, be of singular use to those who practice drawing. It is sold by Mr Nairne, Mathematical Instrument Maker, opposite the Royal Exchange. He sells a cubical piece of about half an inch for three shillings; and he says it will last several years." *TIS

It was not until 1770 that we found out that a natural rubber made from plants can be used as an eraser. That year, Edward Nairne, an English engineer, picked up a piece of rubber instead of breadcrumbs and discovered that rubber can erase pencil markings. Yes, you read that right, before gum rubber, the common pencil eraser was breadcrumbs.

Now, if only ..... Oh they did

In 1858 a Pencil with attached eraser patented. It has benefited generations of mathematics students. The first patent for attaching an eraser to a pencil was issued to a man from Philadelphia named Hyman Lipman. This patent was later held to be invalid because it was merely the combination of two things, without a new use.

1831 Gauss introduces the term "complex" for a+bi. Most of the 17th and 18th century writers spoke of a + bi as an imaginary quantity. Gauss saw the desirability of having different names for ai and a + bi, so he gave to the latter the Latin expression numeros integros complexos.

Gauss wrote:

...quando campus arithmeticae ad quantitates imaginarias extenditur, ita ut absque restrictione ipsius obiectum constituant numeri formae a + bi, denotantibus i pro more quantitatem imaginariam \/-1, atque a, b indefinite omnes numeros reales integros inter -oo et +oo. Tales numeros vocabimus numeros integros complexos, ita quidem, ut reales complexis non opponantur, sed tamquam species sub his contineri censeatur.

The citation above is from Gauss’s paper "Theoria Residuorum Biquadraticorum, Commentatio secunda," Societati Regiae Tradita, Apr. 15, 1831, published for the first time in Commentationes societatis regiae scientiarum Gottingensis recentiones, vol. VII, Gottingae, MDCCCXXXII (1832)]. [Julio González Cabillón]

The term complex number was used in English in 1856 by William Rowan Hamilton. The OED2 provides this citation: Notebook in Halberstam & Ingram Math. Papers Sir W. R. Hamilton (1967) III. 657: "a + ib is said to be a complex number, when a and b are integers, and i = [sqrt] -1; its norm is a^2 + b^2; and therefore the norm of a product is equal to the product of the norms of its factors."

*Jeff Miller, Earliest Known Uses of Some of the Words of Mathematics

1869 W.S. Gilman Jr (of the Naval Observatory, I think{help!}) to Prof Elias Loomis of Yale, sends an account of an Aurora viewed from Brooklyn, NY. He ranks the aurora "inferior in brightness to... one I Witnessed ... on 15th September" (1868) *American Journal of Science

1877, a steam-engine driven helicopter model built by Enrico Forlanini rose 40 ft (12 m). The machine weighed 3.5 kg (7.7 lbs). Its coaxial rotors were powered by a two-cylinder steam engine. Just before takeoff the spherical steam accumlator was charged with 10 atmospheres of pressure, enabling the craft to rise and remain aloft for 20 seconds. Forlanini (1848-1930) was an Italian pioneer of scientific aviation. He built a hydroplane, which could take off on water (1905) and a new type of semirigid aircraft in1914. He also invented the hydrofoil boat. Alexander Graham Bell secured the Italian's patents to pursue his own interest in hydrofoil development. TIS

In 1895, a mathematical relationship between the frequencies of the hydrogen light spectrum was reported by a Swiss school teacher, Johann Balmer, in Annalen der Physik. Its significance was overlooked until Niels Bohr realized this showed a structure of energy levels of the electron in the hydrogen atom. *TIS

1904 term "discrete mathematics was introduced in The Twelfth Annual Report of the Ohio State Academy of Science “The new mathematics...has triumphed for its own domain in cases where the continuity methods were wholly inapplicable, where arithmology, discrete mathematics was called for and victorious. *Jeff Miller, Earliest Known Uses of Some of the Words of Mathematics

In 1570 in Sir Henry Billingsley's translation of Euclid's Elements he described discrete numbers, but not a discrete mathematics : "Two contrary kynds of quantity; quantity discrete or number, and quantity continual or magnitude"

"Discrete Mathematics" is a biweekly peer-reviewed scientific journal in the broad area of discrete mathematics, combinatorics, graph theory, and their applications. It was established in 1971

The first modern Discrete Mathematics text, "Discrete Mathematics" by László Lovász, József Pelikán, and Katalin Vesztergombi, was published in 1975.

In 1912, the fourth dimension was spoken of by Albert Einstein as time. *TIS The great French mathematician d’Alembert, made the first published suggestion that time is the fourth dimension in his 1754 article on the dimensions of space in the Encyclopédie, edited by Diderot and himself. He attributed the idea there to “un homme d’esprit de ma connaissance,” who is thought to have been his fellow mathematician Lagrange, although the latter did not publish such a suggestion until 1797 in his Théorie des fonctions analytiques.

As well as discrediting Kant’s argument that space must be Euclidean, Poincaré declared that space need not even be three-dimensional. In an article in Nature, December, 1869,

Charles Howard Hinton (1853–1907) (Husband of Mary Boole, the daughter of George Boole) taught Uppingham School in Rutland, where Howard Candler, a friend of Edwin Abbott Abbott's, also taught.)

Rather than supporting scientists like Helmholtz who believed that the non-Euclidean geometries of Lobachevsky and Bolyai had discredited Kant’s contention that Euclidean geometry is true a priori, Hinton gave the philosopher credit for identifying space as the necessary means by which human beings cognise the world. However, instead of accepting the three-dimensionality of perception as an unalterable fact of life, Hinton proposed in his books A New Era of Thought (1888) and The Fourth Dimension (1904) that it was merely a temporary feature of man’s evolution. In 1888, Hinton coined the term Tesseract for a four dimensional cube. Earlier, W.I. Stringham had drawn and published in the American Journal of Mathematics an article containing one of the earliest known sets of illustrations of the projections on a plane of the six regular polyhedroids or polytopes — the four-dimensional counterparts of the five regular polyhedra: tetrahedron, octahedron, cube, icosahedron and dodecahedron.

Stringham's depiction of the four-dimensional cube, and current illustration of a tesseract

1949 Even though the paper on pp 1208 through 1226 of the 15 April 1949 issue of The Physical Review looks like any other, it is today seen as revolutionary. The entry for "Physical Principles Involved in Transistor Action" by John Bardeen (two-time Nobel in physics) and Walter Brattain (Nobel '72) was the defining technical publication on the transistor, which was the first massive step towards microminiaturization and the explosive new growth in the computer, *JF Ptak Science Books

1952 The ﬁrst bank credit card was issued by Franklin National Bank, Franklin Square, New York. Purchases were charged to the bank, which made the payments, and then billed the card holders. *FFF (Would love image of this if anyone has one?)

In 1966, the first X-ray three-dimensional stereo fluoroscopic system was installed for use in heart catherization by Richard J Kuhn. The $30,000 machine, developed by Joseph Quinn was put into use at the University of Oregon Medical Center, Portland, Oregon, U.S. The X-ray tube had one anode but two cathodes, an image intensifier with polarizers, and a synchronized analyzer. This produced a 3D image that could be seen through a viewing mirror without the use of special glasses. *TIS

Godfrey Hounsfield of EMI Laboratories created the first commercially available CT scanner in 1972. He co-invented the technology with physicist Dr. Allan Cormack and both researchers were later on jointly awarded the 1979 Nobel Prize in Physiology and Medicine.

The cross-sectional imaging, or “slices”, from CT scans made diagnosing health issues like heart disease, tumors, internal bleeding, and fractures simpler for doctors while also being easier on the patients. Through the following years, with how effective the CT scanners proved to be improvements on the design were quickly developed.

Allan Cormack

Godfrey Hounsfield

1977 First West Coast Computer Faire Begins:
The first West Coast Computer Faire begins, featuring the debut of the Apple II from Apple Computer. The new machine includes innovations such as built-in high-resolution color graphics. For about $1,300, buyers receive a machine and built-in keyboard, 16 kilobytes of memory, BASIC, and eight expansion slots.*CHM

The 1981 Pulitzer prize winner The Soul of a New Machine describes the development of their ECLIPSE computer. *VFR

BIRTHS

1452 Leonardo da Vinci (15 Apr 1452; 2 May 1519 at age 67) Italian painter, draftsman, sculptor, architect, and engineer. Da Vinci was a great engineer and inventor who designed buildings, bridges, canals, forts and war machines. He kept huge notebooks sketching his ideas. Among these, he was fascinated by birds and flying and his sketches include such fantastic designs as flying machines. These drawings demonstrate a genius for mechanical invention and insight into scientific inquiry, truly centuries ahead of their time. His greater fame lies in being one of the greatest painters of all times, best known for such paintings as the Mona Lisa and The Last Supper.*TIS In an interesting blog Thony Christie pointed out that "... Leonardo played absolutely no role what so ever in the history of science and or technology because none of his voluminous writings on those subjects saw the light of day before the 19th century when they were nothing more than a historic curiosity, admittedly a fascinating curiosity but nothing more than that.. " *Renaissance Mathematicus

This portrait attributed to Francesco Melzi, c. 1515–1518, is the only certain contemporary depiction of Leonardo

1548 Pietro Antonio Cataldi (15 April 1548, Bologna – 11 February 1626, Bologna) was an Italian mathematician. A citizen of Bologna, he taught mathematics and astronomy and also worked on military problems. His work included the development of continued fractions and a method for their representation. He was one of many mathematicians who attempted to prove Euclid's fifth postulate. Cataldi discovered (actually re-discovered, see bottom of article) the sixth and seventh primes later to acquire the designation Mersenne primes by 1588. His discovery of the 6th, that corresponding to p=17 in the formula Mp=2^p-1, exploded a many-times repeated number-theoretical myth (Until Cataldi, 19 authors going back to Nicomachus are reported to have made the claim in L.E.Dickson's History of the Theory of Numbers--with a few more repeating this afterward) that the perfect numbers had units digits that invariably alternated between 6 and 8; and that of the 7th (for p=19) held the record for the largest known prime for almost two centuries, until Leonhard Euler discovered that 2³¹ - 1 was the eighth Mersenne prime. Although Cataldi also claimed that p=23, 29, 31 and 37 all also generate Mersenne primes (and perfect numbers), his text's clear demonstration shows that he had genuinely established the fact through p=19.. *Wik In 1613 he published an important early work on continued fractions. The term “continued fraction” was coined by John Wallis in 1655. [DSB 3, 125]
(earlier discoverers of 5th-7th perfect numbers: Ismail ibn Ibrahim ibn Fallus (1194-1239) who wrote a treatise based on the Introduction to arithmetic by Nicomachus. Ibn Fallus gave, in his treatise, a table of ten numbers which were claimed to be perfect, the first seven are correct and are in fact the first seven perfect numbers, the remaining three numbers are incorrect.

The fifth perfect number has been discovered again (after the unknown results of the Arabs) and written down in a manuscript dated 1461. It is also in a manuscript which was written by Regiomontanus during his stay at the University of Vienna, which he left in 1461, see . It has also been found in a manuscript written around 1458, while both the fifth and sixth perfect numbers have been found in another manuscript written by the same author probably shortly after 1460. All that is known of this author is that he lived in Florence and was a student of Domenico d'Agostino Vaiaio.

In 1536, Hudalrichus Regius made the first breakthrough which was to become common knowledge to later mathematicians, when he published Utriusque Arithmetices in which he gave the factorisation 211 - 1 = 2047 = 23 . 89. With this he had found the first prime p such that (2^p-1)(2^p - 1) is not a perfect number. He also showed that 213 - 1 = 8191 is prime so he had discovered (and made his discovery known) the fifth perfect number $2^12(2^13 - 1) = 33550336. $

J Scheybl gave the sixth perfect number in 1555 in his commentary to a translation of Euclid's Elements. This was not noticed until 1977 and therefore did not influence progress on perfect numbers. *SAU

1541 "The discovery that comets are in fact supralunar entities has long been attributed to Tycho Brahe. Yet in a letter from Rheticus’ confidant Paul Eber to Melanchthon we learn that Copernicus and Rheticus had considered the matter long before Brahe:
Magister Rheticus wrote from Prussia, as he is expecting the completion of the work of his praeceptor he will not be able to return in the coming months, but rather in autumn. They have already discovered in those lands that Comets do not arise in the region of the elements, but rather in that of the ether above the lunar sphere. ..." April 15, 1541

Rheticus

1707 Leonhard Euler (15 Apr 1707, 18 Sep 1783) Swiss mathematician and physicist, one of the founders of pure mathematics. He not only made decisive and formative contributions to the subjects of geometry, calculus, mechanics, and number theory but also developed methods for solving problems in observational astronomy and demonstrated useful applications of mathematics in technology. At age 28, he blinded one eye by staring at the sun while working to invent a new way of measuring time. *TIS (Students who have not, should read Dunham's "Euler, The Master of us All")
He was the most productive mathematician of all times; his still only partly published collected works comprise over 75 large volumes. *VFR

1793 Friedrich Georg Wilhelm von Struve (15 Apr 1793, 23 Nov 1864) German-Russian astronomer, one of the greatest 19th-century astronomers and the first in a line of four generations of distinguished astronomers. He founded the modern study of binary (double) stars. In 1817, he became director of the Dorpat Observatory, which he equipped with a 9.5-inch (24-cm) refractor that he used in a massive survey of binary stars from the north celestial pole to 15°S. He measured 3112 binaries - discovering well over 2000 - and cataloged his results in Stellarum Duplicium Mensurae Micrometricae (1837). In 1835, Czar Nicholas I persuaded Struve to set up a new observatory at Pulkovo, near St. Petersburg. There in 1840 Struve became, with Friedrich Bessel and Thomas Henderson, one of the first astronomers to detect parallax. *TIS

1809 Hermann Günther Grassmann (15 Apr 1809, 26 Sep 1877) German mathematician chiefly remembered for his development of a general calculus of vectors in Die lineale Ausdehnungslehre, ein neuer Zweig der Mathematik (1844; "The Theory of Linear Extension, a New Branch of Mathematics"). *TIS One of the many examinations for which Grassmann sat, required that he submit an essay on the theory of the tides. In 1840, he did so, taking the basic theory from Laplace's Mécanique céleste and from Lagrange's Mécanique analytique, but expositing this theory making use of the vector methods he had been mulling over since 1832. This essay, first published in the Collected Works of 1894-1911, contains the first known appearance of what are now called linear algebra and the notion of a vector space. He went on to develop those methods in the book mentioned above. In spite of publishing the idea somewhat early in his career, it seems his work went largely unnoticed until the last decade of his life.*Wik

1874 Johannes Stark ( 15 April 1874 – 21 June 1957) was a German physicist who was awarded the Nobel Prize in Physics in 1919 "for his discovery of the Doppler effect in canal rays and the splitting of spectral lines in electric fields". This phenomenon is known as the Stark effect.

Stark received his Ph.D. in physics from the University of Munich in 1897 under the supervision of Eugen von Lommel, and served as Lommel's assistant until his appointment as a lecturer at the University of Göttingen in 1900. He was an extraordinary professor at Leibniz University Hannover from 1906 until he became a professor at RWTH Aachen University in 1909. In 1917, he became professor at the University of Greifswald, and he also worked at the University of Würzburg from 1920 to 1922.

A supporter of Adolf Hitler from 1924, Stark was one of the main figures, along with fellow Nobel laureate Philipp Lenard, in the anti-Semitic Deutsche Physik movement, which sought to remove Jewish scientists from German physics. He was appointed head of the German Research Foundation in 1933 and was president of the Reich Physical-Technical Institute from 1933 to 1939. In 1947 he was found guilty as a "Major Offender" by a denazification court. *Wik

1927 Robert L. Mills (15 Apr 1927; 27 Oct 1999 at age 72) American physicist who shared the 1980 Rumford Premium Prize with his colleague Chen Ning Yang for their “development of a generalized gauge invariant field theory” in 1954. They proposed a tensor equation for what are now called Yang-Mills fields. Their mathematical work was aimed at understanding the strong interaction holding together nucleons in atomic nuclei. They constructed a more generalized view of electromagnetism, thus Maxwell's Equations can be derived as a special case from their tensor equation. Quantum Yang-Mills theory is now the foundation of most of elementary particle theory, and its predictions have been tested at many experimental laboratories.*TIS

1929 Thomas Brooke Benjamin, FRS (15 April 1929 – 16 August 1995) was an English mathematical physicist and mathematician, best known for his work in mathematical analysis and fluid mechanics, especially in applications of nonlinear differential equations.

The Benjamin–Ono equation describes one-dimensional internal waves in deep water. It was introduced by Benjamin in 1967, and later studied also by Hiroaki Ono. Another equation named after Benjamin, the Benjamin–Bona–Mahony equation, models long surface gravity waves of small amplitude. Benjamin studied it with Jerry L. Bona and J. J. Mahony in a 1972 paper. *Wik

1931 Samaun Samadikun (15 April 1931 – 15 November 2006) was an Indonesian electrical engineer.

He was one of the founders of the Indonesian Academy of Sciences. He is known especially for his contributions in microelectronics research, but also worked on payload instrumentation for space programs. From 1978-1983, he was the Director General of Energy, Ministry of Mining and Energy for the Indonesian government. With co-inventor Kensall D Wise, he held a US Patent (No. 3,888,708, 10 Jun 1975) for his “Method for Forming Regions of Predetermined Thickness in Silicon” for pressure sensors. It was his vision to bring integrated chip (IC) fabrication to Indonesia. Though that was not accomplished before his death, he was active in planning Bandung High Tech Valley inspired by the success of California’s Silicon Valley. *TIS

1934 Professor James "Jim" Wiegold (15 April 1934 – 4 August 2009) was a Welsh mathematician. He earned a PhD at the University of Manchester in 1958, studying under Bernhard Neumann, and is most notable for his contributions to group theory.*Wik

DEATHS

1446 Filippo Brunelleschi (1377 in Florence, Italy - 15 April 1446 in Florence, Italy) Brunelleschi's most important achievement in mathematics came around 1415 when he rediscovered the principles of linear perspective using mirrors. He understood that there should be a single vanishing point to which all parallel lines in a plane, other than the plane of the canvas, converge. Also important was his understanding of scale, and he correctly computed the relation between the actual length of an object and its length in the picture depending on its distance behind the plane of the canvas. Using these mathematical principles, he drew various scenes of Florence with correct perspective. These perspective drawings by Brunelleschi have since been lost but a "Trinity" fresco by Masaccio still exists which uses Brunelleschi's mathematical principles. He is best known for best known for his construction of the dome of Florence's cathedral, the Duomo Santa Maria del Fiore.*SAU

The Santa Maria del Fiore cathedral in Florence possesses the largest brick dome in the world, and is considered a masterpiece of European architecture.

1704 Johan van Waveren Hudde (23 Apr 1628, 15 Apr 1704 at age 76) Dutch mathematician and statesman who, after an education in law, became interested in mathematics, though for a limited time (1654-63). He worked on improving the algebraic methods of René Descartes, seeking to extend them to the solution of equations of a higher degree by applying an algorithm. He also developed an algorithm based on Fermat's method to deal with the maxima, minima and tangents to curves of algebraic functions. Later, he served as burgomaster of Amsterdam for 30 years. During this time time he made a mathematical study of annuities. Hudde continued with an interest in physics and astronomy, producing lenses and microscopes. He collaborated with Baruch Spinoza, of Amsterdam, on telescopes. Hudde determine that in a telescope, a plano-convex lenses were better than concavo-convex . *TIS

1754 Jacopo Francesco Riccati (28 May 1676 in Venice, Venetian Republic (now Italy) - 15 April 1754 in Treviso, Venetian Republic (now Italy)) His work had a wide influence on leading mathematicians such as Daniel Bernoulli, who studied the equation in his Exercitationes quaedam mathematicae, and Leonard Euler who extended Riccati's ideas to integration of non-homogeneous linear differential equations of any order. Riccati also worked on cycloidal pendulums, the laws of resistance in a fluid and differential geometry. *SAU

1764 Peder [Nielsen] Horrebow (Horrebov) (14 May 1679; Løgstør, Jutland – 15 April 1764; Copenhagen) From 1703 to 1707, he served as an assistant to Ole Rømer and lived in Rømer's home. He worked as a household tutor from 1707 to 1711 to a Danish baron, and entered the governmental bureaucracy as an excise writer in 1711.
After repeatedly petitioning King Frederick IV, Horrebow became professor of mathematics at the University of Copenhagen in 1714. He also became director of the university's observatory (called the Rundetårn, "the Round Tower"). His son Christian succeeded him in this position. Horrebow and his wife, Anne Margrethe Rossing, had a total of 20 children.
In 1728, the great fire of Copenhagen destroyed all of the papers and observations made by Rømer, who had died in 1710. Horrebow wrote the Basis Astronomiae (1734–35), which describes the scientific achievements made by Rømer. Horrebow's own papers and instruments were destroyed in the same fire. Horrebow was given a special grant from the government to repair the observatory and instruments. Horrebow received further support from a wealthy patron.
Horrebow invented a way to determine a place's latitude from the stars. The method fixed latitude by observing differences of zenith distances of stars culminating within a short time of each other, and at nearly the same altitude, on opposite sides of the zenith. The method was soon forgotten despite its value until it was rediscovered by the American Andrew Talcott in 1833. It is now called the Horrebow-Talcott Method.
He wrote on navigation and determined the sun parallax, 9", an approximative solution to the Kepler equation. Horrebow also learned how to correct inherent flaws in instruments. This preceded Tobias Mayer's theory of correction of 1756.
Horrebow was a member of a number of scientific societies, including the Académie des Sciences (from 1746). He also worked as a medical doctor and as an academic notary (from 1720). *Wik

1873 Christopher Hansteen (26 Sep 1784, 15 Apr 1873 at age 88) Norwegian astronomer and physicist who is noted for his research in geomagnetism. In 1701, Edmond Halley had already published a map of magnetic declinations, and the subject was studied by Humboldt, de Borda, and Gay-Lussac, among others. Hansteen collected available data and also mounted an expedition to Siberia, where he took many measurements for an atlas of magnetic strength and declination. *TIS

From 1835 to 1838 he published textbooks on geometry and mechanics, largely a reaction to his former research assistant Bernt Michael Holmboe's textbooks. Compared to Holmboe's method of teaching, Hansteen's books were more practically oriented. After Holmboe wrote a review of the first textbook for the newspaper Morgenbladet, in which he advised schools not to use it, a public debate followed, with contributions from other mathematicians. It has been claimed that this was the first debate on the subject of school textbooks in Norway. Holmboe's textbooks proved more lasting, with Hansteen's textbook not being reprinted. In 1842 Hansteen wrote his Disquisitiones de mutationibus, quas patitur momentum acus magneticae. He also contributed various papers to different scientific journals, especially Magazin for Naturvidenskaberne.*WIK

1983 Vera Faddeeva ( 20 September 1906, 15 April 1983 (aged 76)) was a Soviet mathematician. Faddeeva published some of the earliest work in the field of numerical linear algebra. Her 1950 work, Computational methods of linear algebra was widely acclaimed and she won a USSR State Prize for it. Between 1962 and 1975, she wrote many research papers with her husband, Dmitry Konstantinovich Faddeev. She is remembered as an important Russian mathematician, specializing in linear algebra, who worked in the 20th century.

1993 John Tuzo Wilson, CC, OBE, FRS, FRSC, FRSE (October 24, 1908 – April 15, 1993) the world-renowned Canadian geophysicist, served as Director General of the Ontario Science Centre from 1974 to 1985. He was instrumental in developing the theory of Plate Tetonics in the 1960s. This theory describes the formation, motion and destruction of the Earth's crust, the origin of volcanic eruptions and earthquakes, and the growth of mountains. Dr. Wilson's signficant contributions to this theory revolutionized Earth Sciences. He proposed the existence of transform faults to explain the numerous narrow fracture zones and earthquakes along oceanic ridges. He also showed that rising magma plumes beneath the Earth's crust could create stationary hot spots, leading to the formation of mid-plate volcanic chains like the Hawaiian Islands.
The first graduate of geophysics from the University of Toronto in 1930, Dr. Wilson went on to study at Cambridge and Princeton, earning his doctorate in 1936. After spending two years with the Geological Survey of Canada and almost a decade with the Canadian Military Engineers, he accepted the position of Professor of Geophysics at the University of Toronto in 1946. Internationally recognized for his major contributions as a research scientist, educator and visionary, Dr. Wilson received many prestigious
awards, including the Vetlesen Prize, the Earth Sciences equivalent of the Nobel Prize.*THE HISTORICAL MARKER DATABASE

1999 Floyd Burton Jones (November 22, 1910, Cisco, Texas – April 15, 1999, Santa Barbara, California[1]) was an American mathematician, active mainly in topology.

Jones's father was a pharmacist and local politician in Shackelford County, Texas. As the valedictorian of his high school class, Jones earned a Regents' Scholarship to The University of Texas, intending to study law eventually. Jones soon discovered that he had a poor memory for dates and history, and thus changed his major to chemistry.

Jones had the extraordinary good fortune to be taught freshman calculus by Robert Lee Moore, a founder of topology in the US, a legendary mathematics teacher, and the inventor of the Moore method. Jones went on to take more mathematics courses than required to be a chemist. He displayed sufficient ability in those courses that when he graduated in 1932, Moore invited him to do a Ph.D. in mathematics and offered him a part-time job as a math instructor. Moore later supervised Jones's Ph.D. dissertation, completed in 1935.

Jones then taught at the University of Texas for the next 15 years except during 1942–44, when he was a research associate at the Harvard Underwater Sound Laboratory, helping develop scanning sonar for the Navy. In 1950, Jones moved to the University of North Carolina, where he eventually headed the Department of Mathematics. From 1962 until his 1978 retirement, he was at the University of California at Riverside, where he helped launch the doctoral program in mathematics. Over the course of his career, Jones published 67 articles and supervised 15 Ph.D. dissertations. In 1987, he endowed a Chair in Topology at the University of California at Riverside.

Jones taught using a modified version of the Moore method. He believed in "learning by doing" but unlike Moore, he incorporated textbooks into his courses. In 1969, Louis McAuley wrote of "the magical powers of Jones in the classroom—a master who breathes the very life of mathematics into his students." *Wik

2007 Leslie Colin Woods (6 December 1922 – 15 April 2007) was a New Zealand mathematician.

He was born on 6 December 1922 in Reporoa, New Zealand. Woods' father was a fisherman. His surname was originally Woodhead. His school education was completed in New Zealand, where he attended Seddon Memorial Technical College (where he was Head Boy). In his autobiography Against the Tide: An Autobiographical Account of a Professional Outsider, he gives credit to his school teachers, including Colin Maloy and G J Park, for kindling his interest in science and encouraging him to take up a career in academia.

In 1940 Woods went on to study at Auckland University College, but left to join the Royal New Zealand Air Force in December 1941. After completing his training as a pilot he was posted to the Pacific Area in 1943, serving three tours of duty in 1944 and 1945.

In 1943, Woods married Gladys Elizabeth (Betty) Bayley; they had five daughters.

After the war Woods returned to Auckland University College, taking his MSc in 1945 and a BE in 1947. The following year he matriculated at Merton College, Oxford on a Rhodes Scholarship; he studied there until 1951, taking a DPhil in 1950, and then, unusually, a first-class BA in Mathematics in 1951.

Woods was the Nuffield Research Professor of Mechanical Engineering at the University of Technology at Sydney. He was elected a Fellow of Balliol College, Oxford in 1961 where he researched the theory of magnetically-confined hot plasmas. Woods was professor of mathematics at the University of Oxford from 1970 until his retirement in 1990.

Woods died on 15 April 2007 in Oxford. *Wik

2018 George Frederick Oster NAS (April 20, 1940 – April 15, 2018) was an American mathematical biologist, and Professor of Cell and Developmental Biology at University of California, Berkeley. He made seminal contributions to several varied fields including chaos theory, population dynamics, membrane dynamics and molecular motors. He was a 1985 MacArthur Fellow.

He graduated from Columbia University, with a Ph.D., in Nuclear Engineering in 1967. He was appointed as an assistant professor in at UC Berkeley in 1970. In the early 1970s Oster collaborated with Aharon Katzir-Katchalsky on statistical mechanics.

Oster's work with E. O. Wilson on populations dynamics of social animals, particularly ants, is considered pioneering work in evolution in social insects. Oster was one of the first theoretical biologists to understand that a complex interplay between mechanical and chemical forces was at the root of most biological phenomena.

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

Tuesday, 14 April 2026

On This Day in Math - April 14

Can you guess what joke about Topology this was supposed to represent?

The 'control of nature' is a phrase conceived in arrogance, born of the Neanderthal age of biology and the convenience of man.
~Rachel Carson

The 104th day of the year; 104 is the smallest known number of unit line segments that can exist in the plane, 4 touching at every vertex. *What's Special About Number

HT Jesse Hammer

104 is the sum of eight consecutive even numbers, 104 = 6 + 8 + 10 + 12 + 14 + 16 + 18 + 20
Douglas W Boone noted that "_Every_ odd multiple of eight greater than 56 is the sum of eight consecutive positive even numbers. (The smallest sum of eight consecutive positive even numbers is 72 = 2+4+6+8+10+12+14+16.) Allowing zero and negative numbers, every odd multiple of eight, period, is the sum of eight consecutive even numbers. The _even_ multiples of eight (that is, multiples of sixteen) are the sum of eight consecutive _odd_ numbers."

The reversal of 104 is a prime, 401. It is the largest year day that has a prime reversal that is too large to be a year date

13 straight lines through an annulus can produce a maximum of 104 pieces (students might try to create the maximum for smaller numbers of lines, the sequence is 2, 5, 9, 14, 20,... https://oeis.org/A000096 the differences give a clue to the complete pattern.)

Douglas W Boone pointed out that The formula for the number of pieces can be stated as (n^2 + 3n)/2, or (1/2) × n × (n+3), which is an integer for integral n; exactly one of n and n+3 will be even, i.e. divisible by 2.(Students might like another way involving the counting numbers.)

There are 588939451 "left and right" truncatable primes (truncate the two outside digits at once) with an even number of digits. The largest is the 104-digited prime number 91617596742869619884432721391145374777686825634291523771171391111313737919133977331737137933773713713973.

Just introduced to these by a comment from William Gosnell, thanks. sum of number and cube of its digits is a square.(wonder if there are other power sums?)

104+1^3+0^3+4^3 = 13^2 . There is another smaller year day with this property, (Hint, it is prime),

but wait..104 + 1^2 + 0^2 + 4^2 = 11^2 Don't you wonder if there are more like this?

Japanese Route 104 ran from Hachinohe, near my former home in Misawa, Japan on the Pacific, to go across the mountains to Noshiro on the Sea of Japan in Akita prefecture. One of the better places to find the prized 36 inch green glass fishing floats washed up along the coast. (perhaps no more, almost all plastic in last two decades)

*************** Lots of additional math facts for days 91-120 at https://mathdaypballew.blogspot.com/

EVENTS

1129 Chinese accounts state “there was a Black spot within the Sun” on March 22, 1129, which “died away” on April 14th. This may well have been one of the sunspots John of Worcester had observed 104 days earlier (8 December, 1128), on the other side of the world. Worcester's observation prompted the earliest known drawing of sunspots, which appear in his Chronicle recorded in 1128. *Joe Hanson, itsokaytobesmart.com

The first sunspot drawing, John of Worcester around 1128 *Wik

1561 One of the earliest recorded citations of UFO's:
At sunrise on the 14th April 1561, the citizens of Nuremberg beheld "A very frightful spectacle." The sky appeared to fill with cylindrical objects from which red, black, orange and blue white disks and globes emerged. Crosses and tubes resembling cannon barrels also appeared whereupon the objects promptly "began to fight one another." This event is depicted in a famous 16th century woodcut by Hans Glaser.
*UFO Evidence Org

1611 Galileo (1564 1642) visited Rome at the height of his fame and was made the sixth member of the Accademia dei Lincei (Lynx Society) at a banquet on 14 Apr. The word 'telescopium' was first applied to his instrument at this dinner. He showed sunspots to several people. The term “telescope” was introduced by Prince Federico Cesi at a banquet given in Galileo’s honor. It derives from the Greek “tele” meaning “far away” and “skop´eo” meaning “to look intently.” For a change, a term which derives from the Greek was actually coined by a Greek, namely Ioannes Demisiani. [Willy Ley, Watchers of the Skies, p. 112]*VFR Thony Christie at the Renaissance Mathematicus blog has an enjoyable review of the telescope and how it got its name. This account of he events that evening by Girolamo Sirtori was published in his Telescopium, printed in 1618 but written in 1612. "I went to Rome... Galileo was there with his unforgettable telescope. By chance, on a certain day, Prince Federico Cesi,Marquis of Monticello, a learned man and benefactor of the sciences, had invited him [Galil eo] to dinner... Before sunset... they began to look through the telescope at the inscription of

Pope Sixtus Vii above the Lateran portal, which was about a mile distant. I took my turn and looked and read the inscription to my satisfaction. Later that night, after dinner, we observed Jupiter and the motion of his companion stars, after which, sufficiently invigorated by the sight of such brilliance and by the curiosity of the matter, they withdrew in order to examine the telescope. And Galileo himself, in order to satisfy their curiosity, took out the lens and the concave glass, and showed them openly."

1685 It's easy for students of Math History to get the impression that John Wallis was totally immersed in mathematics, but a perusal of his writing on religion, or his many varied contributions to the Royal Society paint the picture of a polymath.

“A Relation Concerning the Late Earthquake Neer Oxford: Together with Some Observations of the Sealed Weatherglass, and the Barometer Both upon That Phænomenon, and in General,” Phil Trans 1 (1665-1666):
166-171; Wallis, “A Discourse concerning the Air’s Gravity, Observd in the Baroscope, Occasioned by That of Dr. Garden: Presented to the Phil. Soc. of Oxford, by the Reverend Dr. Wallis, President of That Society. April, 14, 1685,” Phil Trans 15 (1685): 1002-1014; WC II, 282; WC III, 281-287.

This is almost certainly concerning the earthquake of 6 Oct, 1683 at Derbyshire. This earthquake also has the distinction of being the first British earthquake surveyed by the British Geological Survey.

1760 Four years after leaving the coal pits near Newcastle, 22 year old Charles Hutton advertises the opening of his private school.

"To Be Opened

On Monday, April 14, 1760, at the head of the Flesh Market, down the entry formerly known by the name of the Salutation Entry, Newcastle, A Writing and Mathematical School, where persons may be fully and expeditiously qualified for business, and where such as intend to go through a regular course of Arts and Sciences, may be completely grounded therein at large. " Four years later he would publish his first math text. It would still be in print 100 years later. *Gunpowder and Geometry, Benjamin Waedhaugh.

1790 Mathurin Jacques Brisson (1723–1806) proposed to the Paris Academy the establishment of a system of measurement resting on a natural unit of length. The general idea of decimal subdivision was obtained from a work of Thomas Williams, London, 1788. *F Cajori, History of Mathematics

1822 In a letter to Gauss, Bessell recommends his student, Heinrich Ferdinand Scherk. Gauss considered Scherk one of the best students he ever had. Scherk would go on to great educational success and Kummer was one of his students. * Dunnington, Gray, & Dohse , Carl Friedrich Gauss: Titan of Science

1845 First Light for Western Hemispheres Largest Refractor.

The Cincinnati Observatory is known as ‘The Birthplace of American Astronomy.’ It houses one of the oldest working telescopes in the world and was the first public observatory in the western hemisphere. Recently restored to its original beauty, the Observatory is a fully functioning 19th century observatory used daily by the public and amateur astronomers. The main telescopes are an 11-inch Merz and Mahler refractor from 1845 and a 16-inch Alvan Clark and Sons refractor from 1904. The historic buildings are designated as a National Historic Landmark, and the grounds provide a serene, park-like setting while still being centrally located in the city of Cincinnati.

The observatory originally sat on four acres of land at the top of Mt. Ida, which was donated to the Society by Nicholas Longworth. On the 9th of November, 1843, a crowd of thousands witnessed former president John Quincy Adams preside over the dedication of the observatory, “The Lighthouse of the Sky,” and the laying of the cornerstone. It was at the dedication that Adams gave his last public speech. Mt Ida was renamed Mt. Adams following this event.

When the great refractor saw first light on April 14, 1845 it was the largest refractor in the Western Hemisphere and third largest in the world. Mitchel, the first director, wrote and edited the first astronomical publication in the United States, The Sidereal Messenger. The second director, Cleveland Abbe, published the nation’s first weather forecasts and he later assisted in the founding of the National Weather Service. *Cincinnati Observatory

The 11-inch Merz & Mahler refractor at Cincinnati Observatory, installed in 1842, and still in operation, recent photograph, Observatories of Ohio *Linda Hall Org

1855 The ﬁrst chess problem of Sam Loyd, age fourteen, was published in the New York Saturday Courier. Within a few years he was recognized as the nation’s foremost composer of chess problems. Once he announced that he had discovered a way to mate a lone king in the center of the board with a knight and two rooks. Readers were ﬁrst furious, afterwards amused, by his preposterous solution: line them up in the order knight, rook, king, rook. [Mathematical Puzzles of Sam Loyd, edited by Martin Gardner, Dover 1959, p. xi-xii]

His first published puzzle is below:

1860 A printed article on the Four Color theorem (perhaps only the second public statement about it, see June 10, 1854) was printed on this date and spread knowledge of the problem to America. In the unusual form of an Atheaneum book review of The Philosophy of Discovery by William Whewell, the unsigned, but almost surely written by DeMorgan, review launched in to a discussion of the Four Color problem. The review treats the four color necessity as obvious to cartographers, and makes no mention of either Guthrie, since he most surely knew the mathematical community in England were aware of his contribution from DeMorgan's own letters.
The review of Whewell's book came to the attention of American Philosopher/Logician C. S. Peirce, son of Harvard Professor Benjamin Peirce, and became a lifelong fascination. He immediately crafted a proof, which is still unknown, to my knowledge. He wrote later that it had been the Atheanenum review which first ignited his interest, and that his own proof was never printed. Shortly before DeMorgan's death in 1871, he was visited by Peirce, but no record is known of what they talked.

See the story of DeMorgan and Guthrie at the origin of the four color problem, 1852.

Letter of De Morgan to Hamilton, 23 Oct. 1852

1894 The first ever commercial motion picture house opens in New York City, United States. It uses ten Kinetoscopes, devices for peep-show viewing of films.

*The Painter Flynn

1914 Ramanujan boarded the S.S. Nevasa on 17 March 1914, and at 10 o'clock in the morning, the ship departed from Madras. He arrived in London on 14 April, with E. H. Neville waiting for him with a car. I received a tweet from @amanicdroid who pointed out that, "this was significant for him culturally as a high-caste Hindu as crossing the ocean was taboo. "

Bust of Ramanujan in the garden of Birla Industrial & Technological Museum in Kolkata, India

1931 The ﬁrst issue of the review journal Zentralblatt f¨ur Mathematik was published by Springer. Otto Neugebauer, then a young professor at G¨ottingen, conceived the idea of a journal that would publish the reviews of articles as soon as possible after the papers had appeared and persuaded the publishing house of J. Springer to publish such a journal. The first issue of Zentralblatt f¨ur Mathematik und ihre Grenzgebiete, as the new journal was called, dated April 14, 1931, had Neugebauer as its editor. It also had a very distinguished and international editorial committee (consisting of P. Alexandroff, J. Bartels, W. Blaschke, R. Courant, H. Hahn, G. H. Hardy, F. Hund, G. Julia, O. Kellogg, H. Kienle, T.Levi-Civita, R. Nevanlinna, H. Thirring and B. L. van der Waerden). The first volume consisted of seven issues plus an index, in 466 pages. (The very first item reviewed was the second edition
of Methoden der mathematischen Physik, by Courant and Hilbert.) The classification system used was very similar to the scheme used by Jahrbuch.
Mathematical Reviews. Zentralblatt flourished under Neugebauer’s direction and became the primary reviewing journal in mathematics. Jahrbuch valiantly continued until issue number 4 of its Volume 68, for the year 1942, ceasing publication in mid-1944, but it had already lost its prominence in the research community. But, just as WorldWar I damaged Jahrbuch, serious harm was done to Zentralblatt soon after its founding by political conditions beyond its control. The anti-Semitic and anti-Soviet policies of the Nazi regime generated pressures on the editorial policies of Zentralblatt concerning the use of Jewish and Russian reviewers. Although Neugebauer left G¨ottingen for the University of Copenhagen in 1934, he had continued to edit Zentralblatt. But by 1938 the intrusion of politics had become intolerable and he and other members of the editorial board resigned. Despite these difficulties Zentralblatt continued its operation and, except for a brief suspension of publication from November 1944 until June 1948, has continued to publish to the present day.

1932, the atom was split by a proton beam on a lithium target. Two physicists, Englishman Sir John Douglas Cockcroft and Irishman Errnest Walton had developed the first nuclear particle accelerator (the Cockcroft-Walton generator for which they shared 1951 Nobel Prize for Physics. The accelerator was built in a disused room in the Cavendish Laboratory. With this equipment, Walton succeeded in being the first to split the atom (its nucleus). When a proton from the beam supplied by the accelerator struck a lithium nucleus, their unstable combination disintegrated into two alpha particles (helium nuclei). Walton observed the scintillations characteristic of alpha particles on a zinc sulphide screen.

Ernest Rutherford (centre) encouraged Ernest Walton (left) and John Cockcroft (right) to build a high-voltage accelerator to split the atom. Their success marked the beginning of a new field of subatomic research. Image credit: AIP Emilio Segrè Visual Archives.

1943 a proposal for an electronic computer was submitted to colleagues at the U.S. Army's Ballistics Research Laboratory by John Grist Brainerd, director of research at the University of Pennsylvania's Moore School, where the proposal was written by John Mauchly. In May 1943, the Army contracted the Moore School to build ENIAC, the first electronic computer. Although ENIAC was not finished until after the war had ended, it nevertheless marked a major step forward in computing. *TIS

1995 Chinese Government Works to Purge Its Agencies of Illegal Software:
The Chinese government launches widespread efforts to purge governmental agencies of illegally copied software, a practice that had been costing U.S. software publishers millions of dollars. The plan calls for allotting more money to purchase software while giving an enforcement agency the power to prosecute anyone bootlegging software. The announcement follows a March meeting at which China had signed an accord with the United States vowing to crackdown on piracy.*CHM

2014 Almost exactly a year after Yitang Zhang announced a proof (see April 17) that there are infinitely many pairs of prime numbers which differ by 70 million or less Terrance Tao's online group attack on the problem reduced the number to 243. Zhang's proof is the first to establish the existence of a finite bound for prime gaps, resolving a weak form of the twin prime conjecture.

2014 A total Lunar eclipse visible in most of North and South America occurred on this night. The total eclipse began around 3am EDT and last for about 80 minutes. More information is here. *Michael Zeiler

Another lunar eclipse took place on Wednesday 8 October 2014. The April eclipse was the first of two total lunar eclipses in 2014, and the second in a tetrad (four total lunar eclipses in series). Other eclipses in the tetrad are those of 4 April 2015, and 28 September 2015.

*Wik

BIRTHS

1527 Abraham Ortelius, (?4 or 14 Apr 1527, 28 June 1598) a Flemish cartographer. In 1570, Ortelius published Theatrum Orbis Terrarum, or Theater of the World. This was the first modern world atlas. It contained 53 maps, and its novelty lay in the fact that the maps were uniform in style, size, and lettering; had been engraved especially for this work; had descriptive text on the back of each map; and covered the entire world, region by region. Most of the maps were not original with Ortelius—he borrowed freely from previous cartographers and he fully credited all his sources—but many of the maps, such as the world map, are brand new.

The Theatrum was an immediate publishing success, and it went through 23 editions and translations in Ortelius’ own lifetime (he died in 1598). *Linda Hall Library

*Ortelius by Peter Paul Rubens

1629 Christiaan Huygens (14 Apr 1629; 8 Jul 1695 at age 66) Dutch physicist and astronomer who founded the wave theory of light, discovered the true shape of the rings of Saturn, and contributed to the science of dynamics - the study of the action of forces on bodies. Using a lens he ground for himself, on 25 Mar 1655, he discovered the first moon of Saturn, later named Titan. In 1656, he patented the first pendulum clock, which he developed to enable exact time measurement while observing the heavens. Cristiaan Huygens studied the relation of the length of a pendulum to its period of oscillation (1673) and stated theories on centrifugal force in circular motion which influenced Sir Isaac Newton in formulating his Law of Gravity. Huygens also studied and drew the first maps of Mars. On 14 Jan 2005, a NASA space probe, named after Huygens, landed on Titan. *TIS

Amazon has the Kindle version of his Treatise on Light for $2.99.

1868 Annie Scott Dill Maunder (née Russell) FRAS (14 April 1868 – 15 September 1947)

Edward Walter Maunder (12 Apr 1851, 21 Mar 1928 at age 76) English astronomer who was the first to take the British Civil Service Commission examination for the post of photographic and spectroscopic assistant at the Royal Observatory, Greenwich. For the next forty years that he worked there, he made extensive measurements of sunspots. Checking historical records, he found a period from 1645 to 1715 that had a remarkable lack of reports on sunspots. Although he might have questioned the accuracy of the reporting, he instead attributed the shortage of report to an actual dearth of sunspots during that period. Although his suggestion was not generally accepted at first, accumulating research has since indicated there are indeed decades-long times when the sun has notably few sunspots. These periods are now known as Maunder minima.*TIS

On the far side of the Moon lies the Maunder crater, named after two British astronomers - Annie and Walter Maunder.

Annie worked alongside her husband at the end of the 19th Century, recording the dark spots that pepper the Sun.

The name Maunder is still known in scientific circles, yet Annie has somehow slipped from history.

"I think the name Maunder is there and we have all rather forgotten that that's two people," says Dr Sue Bowler, editor of the Royal Astronomical Society magazine, Astronomy and Geophysics.

"She was acknowledged on papers, she published in her own name as well as with her husband, she wrote books, she was clearly doing a lot of work but she also clearly kept to the conventions of the day, I think." *By Helen Briggs BBC News

She is known to have worked closely with her husband on the study of sunspots, and she is often credited with discovering the butterfly pattern. *LH

1898 Harold Stephen Black (14 Apr 1898; 11 Dec 1983 at age 85) American electrical engineer who discovered and developed the negative-feedback principle, in which amplification output is fed back into the input, thus producing nearly distortionless and steady amplification. In 1921, Black joined the forerunner of Bell Labs, in New York City, working on elimination of distortion. After six years of persistence, Black conceived his negative feedback amplifier in a flash commuting to work aboard the ferry. Basically, the concept involved feeding systems output back to the input as a method of system control. The principle has found widespread applications in electronics, including industrial, military, and consumer electronics, weaponry, analog computers, and such biomechanical devices as pacemakers. *TIS

1927 Marcel Berger (14 April 1927 – 15 October 2016) was a French mathematician, doyen of French differential geometry, and a former director of the Institut des Hautes Études Scientifiques (IHÉS), France. Formerly residing in Le Castera in Lasseube, Berger was instrumental in Mikhail Gromov's accepting positions both at the University of Paris and at the IHÉS. His contributions to geometry were both broad and deep. The classification of Riemannian holonomy groups provided by his thesis has had a lasting impact on areas ranging from theoretical physics to algebraic
geometry. *Wik *AMS

1922 Betty Shannon (née Mary Elizabeth Moore) (April 14, 1922 – May 1, 2017) was a mathematician and the main research collaborator of Claude Shannon. Betty inspired and assisted Claude in building some of his most famous inventions.She was awarded a full scholarship to the New Jersey College for Women, where she graduated Phi Beta Kappa after studying mathematics.

She worked as a numerical analyst at Bell Labs, where as a computer she supported work on microwaves, and then on radar. She published her own research on "Composing Music by a Stochastic Process"; an "exceptional" accomplishment in an era when it was a "significant and unusual achievement for a woman to get her name on a research report".

While at Bell Labs she met the shy and insular Claude Shannon. Claude "didn’t have much patience with people who weren’t as smart as he was" and the two of them got on well. In 1948 he asked her on a date and they ended up dining each night together; they were married in 1949.

Many of his papers were written in her hand, and at her inspiration. Shannon's mind had little patience with filling in the gaps between his leaps of intuitive brilliance.

She also was the partner and co creator of many of his unusual creations.

In 1950, pioneering information theorist Claude Shannon engineered a mechanical mouse, theseus, that navigated a maze to find a hunk of metal “cheese.” The mouse was made from an erector set given to Claude by Betty, and the final electrical connections were by her hand.

In addition to her research, Shannon was a member of the Weavers' Guild of Boston, served as Dean of the Guild from 1976 to 1978 and received the Guild's Distinguished Achievement Award.

Shannon had three children, Robert James Shannon, Andrew Moore Shannon, and Margarita Shannon, and raised their family in Winchester, Massachusetts. Her oldest son, Robert Shannon, died in 1998 at the age of 45. Betty died on May 1, 2017 at her home at Brookhaven in Lexington, Massachusetts.

1937 Charles Coffin Sims (April 14, 1937 – October 23, 2017) was an American mathematician best known for his work in group theory. Together with Donald G. Higman he discovered the Higman–Sims group, one of the sporadic groups. The permutation group software developed by Sims also led to the proof of existence of the Lyons group (also known as the Lyons–Sims group) and the O'Nan group (also known as the O'Nan–Sims group).

Sims was born and raised in Elkhart, Indiana, and received his B.S. from the University of Michigan. He did his graduate studies at Harvard University, where he was a student of John G. Thompson and received his Ph.D. degree in 1963. In his thesis, he enumerated p-groups, giving sharp asymptotic upper and lower bounds. Sims is one of the founders of computational group theory and is the eponym of the Schreier–Sims algorithm. He was a faculty member at the Department of Mathematics at Rutgers University from 1965 to 2007. During that period he served, in particular, as Department Chair (1982–84) and Associate Provost for Computer Planning (1984–87). Sims retired from Rutgers in 2007 and moved to St. Petersburg, Florida.

In 2012, he became a fellow of the American Mathematical Society. *Wik

DEATHS

1792 Maximilian Hell (May 15, 1720 – April 14, 1792) was a Slovak astronomer and an ordained Jesuit priest from the Kingdom of Hungary.
Born as Rudolf Maximilian Höll in Selmecbánya, Kingdom of Hungary (present-day Banská Štiavnica, Slovakia)., but later changed his surname to Hell. He was the third son from the second marriage of his father Matthias Cornelius Hell (Matthäus Kornelius Hell) and his mother Julianna Staindl. The couple had a total of 22 children. Registry entries indicate that the family was of German descent, while Maximilian Hell later in life (ca 1750) is known to declare himself as Hungarian.
Hell became the director of the Vienna Observatory in 1756. He published the astronomical tables Ephemerides astronomicae ad meridianum Vindobonemsem ("Ephemerides for the Meridian of Vienna"). He and his assistant János Sajnovics went to Vardø in the far north of Norway (then part of Denmark-Norway) to observe the 1769 transit of Venus. He was elected as a foreign member of the Royal Danish Academy of Sciences and Letters on October 13, 1769. This society also funded the publication of his 1770 account of the Venus passage Observatio transitus Veneris ante discum Solis die 3. Junii anno 1769 (Copenhagen, 1770).
There was some controversy about Hell's observations of the transit of Venus because he stayed in Norway for eight months, collecting non-astronomical scientific data about the arctic regions for a planned encyclopedia (which never appeared, in part due to the suppression of the Jesuit order). The publication of his results was delayed, and some (notably Joseph Johann Littrow) accused Hell posthumously of falsifying his results. However, Simon Newcomb carefully studied Hell's notebooks and exonerated him a century after his death in Vienna.
Besides astronomy, Hell also had an interest in magnet therapy (the alleged healing power of magnets), although it was Franz Anton Mesmer who went further with this and received most of the credit.
In 1771, Hell was elected a foreign member of the Royal Swedish Academy of Sciences.
The crater Hell on the Moon is named after him. *Wik

1935 Amalie Emmy Noether (23 Mar 1882, 14 Apr 1935 at age 53) was a German mathematician best known for her contributions to abstract algebra, in particular, her study of chain conditions on ideals of rings. In theoretical physics, she produced Noether's Theorem, which proves a relationship between symmetries in physics and conservation principles. This basic result in the general theory of relativity was praised by Einstein. It was her work in the theory of invariants which led to formulations for several concepts of Einstein's general theory of relativity. For her obituary in The New York Times, Albert Einstein wrote: “Fraulein Noether was the most significant mathematical genius thus far produced since the higher education of women began.”*TIS Emmy Noether’s house in Erlangen

1948 Dr Clara Latimer Bacon (13 August 1866 – 14 April 1948) was a mathematician and Professor of Mathematics at Goucher College. She was the first woman to earn a PhD in mathematics from Johns Hopkins University.

Dr Clara Latimer Bacon (13 August 1866 – 14 April 1948) was a mathematician and Professor of Mathematics at Goucher College. She was the first woman to earn a PhD in mathematics from Johns Hopkins University.

In October 1907 she began graduate work at Johns Hopkins University in mathematics, education and philosophy. A fellowship from the Baltimore Association for Promotion of University Education of Women allowed her to spend the 1910-1911 academic year at the university. In 1911 she became the first woman to receive a Ph.D. in mathematics from Johns Hopkins University. Her dissertation was on "The Cartesian oval and the elliptic functions p and σ," later published in the American Journal of Mathematics, Vol. 35, No. 3. (July, 1913), pp. 261-280.

Bacon was promoted to associate professor at Goucher in 1905 and to full professor in 1914. She continued to teach at Goucher College until her retirement in 1934 as Professor Emeritus of Mathematics. She was by all accounts an outstanding teacher. One student wrote of her [4]:

She believed in us so simply and so deeply that we could not disappoint her. When she felt that circumstances prevented us from doing all she hoped, she tried to change the circumstances. It was her support that made graduate study possible for me. Her patience and understanding as a teacher opened up the beauty of mathematics. For many years her faith in all of us made life seem good.

At least eight of her students went on to earn the Ph.D. degree in mathematics,

1964 Tatyana Alexeyevna Afanasyeva (Kiev, 19 November 1876 – Leiden, 14 April 1964) (also known as Tatiana Ehrenfest-Afanaseva) was a Russian/Dutch mathematician. On 21 December 1904 she was married to Paul Ehrenfest (1880–1933) an Austrian physicist. They had two daughters and two sons: one daughter, Tatyana Pavlovna Ehrenfest, also became a mathematician.
Afanasyeva was born in Kiev, Ukraine, then part of the Russian Empire. After her father died she was brought up by an uncle in St Petersburg, Russia, where she attended a women's pedagogical school and a Women's College. In 1902 she transferred to Göttingen, where she met Ehrenfest. The couple got married in 1904, and in 1907 they returned to St Petersburg. In 1912 they moved to Leiden, where Paul Ehrenfest was appointed to succeed H.A. Lorentz as professor at the University of Leiden.
Tatyana collaborated closely with her husband, most famously on their classic review of the statistical mechanics of Boltzmann. She published many papers on various topics such as randomness and entropy, and teaching geometry to children. *Wik

1964 Rachel Louise Carson (27 May 1907, 14 Apr 1964 at age 56) was an American marine biologist, conservationist and writer well known for her writings on environmental pollution and the natural history of the sea. Embedded within all of Carson's writing was the view that human beings were but one part of nature distinguished primarily by their power to alter it, in some cases irreversibly. Disturbed by the profligate use of synthetic chemical pesticides after World War II, Carson reluctantly changed her focus in order to warn the public about the long term effects of misusing these chemicals.

1928 Errett Albert Bishop (July 10, 1928 – April 14, 1983) (His) work is so wide ranging that it is difficult to give an overview in a biography such as this. Let us look at the book Selected papers which was published in 1986 and reprints some of Bishop's most significant contributions. The book divided Bishop's papers into five categories:
(1) Polynomial and rational approximation. Examples are extensions of Mergelyan's approximation theorem and the theorem of Frigyes Riesz and Marcel Riesz concerning measures on the unit circle orthogonal to polynomials. Bishop found new methods in dealing with these problems;
(2) The general theory of function algebras. Here Bishop worked on uniform algebras (commutative Banach algebras with unit whose norms are the spectral norms) proving results such as antisymmetric decomposition of a uniform algebra, the Bishop-DeLeeuw theorem, and the proof of existence of Jensen measures. In 1965 Bishop wrote an excellent survey Uniform algebras examining the interaction between the theory of uniform algebras and that of several complex variables.
(3) Banach spaces and operator theory. An examples of a paper by Bishop on this topic is Spectral theory for operators on a Banach space (1957). He introduced the condition now called the Bishop condition which turned out to be very useful in the theory of decomposable operators.
(4) Several complex variables. Examples of Bishop's papers in this area are Analyticity in certain Banach spaces (1962). He proved important results in this area such as the biholomorphic embedding theorem for a Stein manifold as a closed submanifold in Cn, and a new proof of Remmert's proper mapping theorem.
(5) Constructive mathematics. Bishop become interested in foundational issues around 1964, about the time he was at the Miller Institute. He wrote a famous text Foundations of constructive analysis (1967) which aimed to show that a constructive treatment of analysis is feasible.*SAU

2005 Saunders Mac Lane (4 August 1909 – 14 April 2005) was an American mathematician who co-founded category theory with Samuel Eilenberg.

After a thesis in mathematical logic, his early work was in field theory and valuation theory. He wrote on valuation rings and Witt vectors, and separability in infinite field extensions. He started writing on group extensions in 1942, and in 1943 began his research on what are now called Eilenberg–MacLane spaces K(G,n), having a single non-trivial homotopy group G in dimension n. This work opened the way to group cohomology in general.

After introducing, via the Eilenberg–Steenrod axioms, the abstract approach to homology theory, he and Eilenberg originated category theory in 1945. He is especially known for his work on coherence theorems. A recurring feature of category theory, abstract algebra, and of some other mathematics as well, is the use of diagrams, consisting of arrows (morphisms) linking objects, such as products and coproducts. According to McLarty (2005), this diagrammatic approach to contemporary mathematics largely stems from Mac Lane (1948). Mac Lane also coined the term Yoneda lemma for a lemma which is an essential background to many central concepts of category theory and which was discovered by Nobuo Yoneda.

Mac Lane had an exemplary devotion to writing approachable texts, starting with his very influential A Survey of Modern Algebra, coauthored in 1941 with Garrett Birkhoff. From then on, it was possible to teach elementary modern algebra to undergraduates using an English text. His Categories for the Working Mathematician remains the definitive introduction to category theory.

Mac Lane supervised the Ph.Ds of, among many others, David Eisenbud, William Howard, Irving Kaplansky, Michael Morley, Anil Nerode, Robert Solovay, and John G. Thompson.

Mac Lane and Samuel Eilenberg at a conference in July 1992

2011 William Nunn Lipscomb (December 9, 1919 – April 14, 2011) was an American physical chemist who won the Nobel Prize for Chemistry in 1976 for his research on the structure of boranes (boron hydride compounds), work which also answered general questions about chemical bonding. Boranes became important in chemical research in the 1940s and ‘50s because of the need to find volatile uranium compounds (borohydrides) for isotope separation, as well as the need to develop high-energy fuels for rockets and jet aircraft. To map the molecular structures of boranes, Lipscomb also developed x-ray techniques that later found application in many other areas of chemical research. Lipscomb's research interests included the relationship of three-dimensional structure and mechanisms of enzymes and other proteins. *TIS

2015 Gordon Bamford Preston (28 April 1925 – 14 April 2015) was an English mathematician best known for his work on semigroups. He received his D.Phil. in mathematics in 1954 from Magdalen College, Oxford.

He was born in Workington and brought up in Carlisle. During World War II, he left his undergraduate studies at Oxford University for Bletchley Park, to help crack German codes with a small group of mathematicians, which included Alan Turing. At Bletchley Park he persuaded Max Newman (who thought that the women would not care for the "intellectual effort") to authorise talks to the Wrens to explain their work mathematically, and the talks were very popular.

After graduation, he was a teacher at Westminster School, London and then the Royal Military College of Science. In 1954 he wrote three highly influential papers in the Journal of the London Mathematical Society, laying the foundations of inverse semigroup theory. Before Preston and Alfred H. Clifford's book, The Algebraic Theory of Semigroups (Vol 1 1961) (Vol 2 1967) and the Russian, Evgenii S. Lyapin's, Semigroups (1960) there was no systematic treatment of semigroups. The Algebraic Theory of Semigroups was hailed as an excellent achievement that greatly influenced the future development of the subject.

In 1963, Preston moved to Australia to take up the chair of mathematics at Monash University, Melbourne. Preston was an important contributor to algebraic semigroup theory and a respected head of school during his various Monash appointments from 1963 until his retirement in 1990.

He subsequently spent six months each year in both Oxford, UK, and Melbourne, Australia, dying on 14 April 2015 in Oxford at age 89. *Wik