We can only see a short distance ahead,

but we can see plenty there that needs to be done.

Alan Turing, From his paper on the Turing test

The 175th day of the year; 175 is the smallest number

*n*greater than 1 such that n

^{6}

\(\pm 6\) are both prime. *Prime Curios & Derek Orr

If S(n) = the sum of the proper divisors of n ( so S(8) = 1+2+4=7) then if S(n1) = n2 and s(n2)= n3... and s(n)=n1 we call the sequence a "sociable chain" of length n. There are, at this writing, 175 sociable chains with length greater than 2

(sociable chains of link two are called amicable numbers. The pair known to the ancients are 220 and 284 )

From Jim Wilder @wilderlab : \( 175 = 1^1 + 7^2 + 5^3 \)

1191 "In the month of June, the Vigil of the Nativity of St John the Baptist (June 23), the 9 th day before the Kalends of July, on the 27 th day of the Moon, at the 9 th hour of the day, the Sun was eclipsed and it lasted for three hours; the Sun was so obscured that the darkness arose over the Earth and stars appeared in the sky. And when the eclipse withdrew, the Sun returned to its

original beauty." This was an annular solar eclipse.

**1585**Thomas Harriot arrived off the coast of Virginia (actually Cape Lookout, NC). He was the ﬁrst substantial mathematician to visit North America. [John W. Shirley in Thomas Harriot: A Biography, 1983, p. 129; Thanks to Kullman] *VFR Thomas Harriot's name was once synonymous with a common method of solving quadratics taught in nearly every high school. Once commonly called Harriot's Method, today it is simply referred to as factoring.

And how did he come to be in the exploration of Virginia?? Here is the story from Encyclopedia Virginia, 2010:

Thomas Hariot (often spelled Harriot) was an English mathematician, astronomer, linguist, and experimental scientist. During the 1580s, he served as Sir Walter Raleigh's primary assistant in planning and attempting to establish the English colonies on Roanoke Island off the coast of present-day North Carolina. He taught Raleigh's sea captains to sail the Atlantic Ocean using sophisticated navigational methods not well understood in England at the time. He also learned the Algonquian language from two Virginia Indians, Wanchese and Manteo. In 1585, Hariot joined the expedition to Roanoke, which failed and returned to England the next year. During his stay in America, Hariot helped to explore the present-day Outer Banks region and, farther north, the Chesapeake Bay. He also collaborated with the artist John White in producing several maps notable at the time for their accuracy. Although Hariot left extensive papers, the only work published during his lifetime was "A Briefe and True Report of the New Found Land of Virginia", which evaluated the economic potential of Virginia. The report appeared most impressively in Theodor de Bry's 1590 edition that included etchings based on the White-Hariot maps and White's watercolors of Indian life. After a brief imprisonment in connection to the Gunpowder Plot (1605), Hariot calculated the orbit of Halley's Comet, sketched and mapped the moon, and observed sunspots. He died in 1621.

**1676**Newton, via Oldenburg, sent his famous Epistola prior to Leibniz. It contained the ﬁrst use of fractional exponents as well as the newly discovered binomial theorem.*VFR

**In 1775**, the first American-made book was advertised in Philadelphia, Penn. Titled Impenetrable Secret, the book was printed and sold by Story and Humphreys. Their advertisement in the Pennsylvania Mercury announced it was "printed with types, paper and ink manufactured in this Province."*TIS

**1835**Mobius receives a letter from Bellavitis with a method for adding and subtracting non-collinear vectors. (A history of vector analysis: the evolution of the idea of a vectorial system By Michael J. Crowe) A geometrical work by Bellavitis was published in 1832 which also contains vector type quantities. His basic objects are line segments AB and he considers AB and BA as two distinct objects. He defines two line segments as 'equipollent' if they are equal and parallel, so, in modern notation, two line segments are equipollent if they represent the same vector. Bellavitis then defines the 'equipollent sum of line segments' and obtains an 'equipollent calculus' which is essentially a vector space. *SAU

1868 Christopher Latham Sholes receives a patent for an invention he calls the "Type-Writer." *OnThisDay & Facts @NotableHistory (Sholes was an American inventor who invented the first practical typewriter and the QWERTY keyboard still in use today. He was also a newspaper publisher and Wisconsin politician.)

**1993**Over the course of three lectures delivered at Isaac Newton Institute for Mathematical Sciences on June 21, 22, and 23 of 1993, Wiles announced his proof of the Taniyama–Shimura conjecture, and hence of Fermat's Last Theorem. There was a relatively large amount of press coverage afterwards. After announcing his results, (Nick) Katz was a referee on his manuscript and he asked Wiles a series of questions that led Wiles to recognize that the proof contained a gap. There was an error in a critical portion of the proof which gave a bound for the order of a particular group: the Euler system used to extend Flach's method was incomplete. Wiles and his former student Richard Taylor spent almost a year resolving it. Wiles indicates that on the morning of September 19, 1994 he realized that the specific reason why the Flach approach would not work directly suggested a new approach with the Iwasawa theory which resolved all of the previous issues with the latter and resulted in a CNF that was valid for all of the required cases. On 6 October Wiles sent the new proof to three colleagues including Faltings. The new proof was published and, despite its size, widely accepted as likely correct in its major components. *Wik

**1988**Global warming became more widely popular after 23 June, 1988 when NASA climate scientist James Hansen used the term in a testimony to Congress. He said: "global warming has reached a level such that we can ascribe with a high degree of confidence a cause and effect relationship between the greenhouse effect and the observed warming." His testimony was widely reported and afterward global warming was commonly used by the press and in public discourse. *Wik

**2013**The Moon will make its closest approach to the Earth (at perigee) for the year on Sunday, 23 June, at 11:11 (UTC), and at this time the Moon will be 356,989 km from the Earth (that means 221,823 miles for us non-geeks). *Bob Mrotek

**1612 André Tacquet**(23 June 1612 Antwerp – 22 December 1660 Antwerp, also referred to by his Latinized name Andrea Tacquet) was a Flemish mathematician and Jesuit Priest. His work prepared ground for the eventual discovery of the calculus.

He was born in Antwerp, and entered the Jesuit Order in 1629. From 1631 to 1635, he studied mathematics, physics and logic at Leuven. Two of his teachers were Saint-Vincent and Francois d'Aguilon.

Tacquet became a brilliant mathematician of international fame and his works were often reprinted and translated (into Italian and English). He helped articulate some of the preliminary concepts necessary for Isaac Newton and Gottfried Leibniz to recognize the inverse nature of the quadrature and the tangent. He was one of the precursors of the infinitesimal calculus, developed by John Wallis. His most famous work, which influenced the thinking of Blaise Pascal and his contemporaries, is Cylindricorum et annularium (1651). In this book Tacquet presented how a moving point could generate a curve and the theories of area and volume. *Wik

**1756 Thomas Jones**(23 June 1756 – 18 July 1807) was Head Tutor at Trinity College, Cambridge for twenty years and an outstanding teacher of mathematics. He is notable as a mentor of Adam Sedgwick.

He was born at Berriew, Montgomeryshire, in Wales. On completing his studies at Shrewsbury School, Jones was admitted to St John's College, Cambridge on 28 May 1774, as a 'pensioner' (i.e. a fee-paying student, as opposed to a scholar or sizar). He was believed to be an illegitimate son of Mr Owen Owen, of Tyncoed, and his housekeeper, who afterwards married a Mr Jones, of Traffin, County Kerry, Thomas then being brought up as his son.

On 27 June 1776, Jones migrated from St John's College to Trinity College. He became a scholar in 1777 and obtained his BA in 1779, winning the First Smith's Prize and becoming Senior Wrangler. In 1782, he obtained his MA and became a Fellow of Trinity College in 1781. He became a Junior Dean, 1787–1789 and a Tutor, 1787-1807. He was ordained a deacon at the Peterborough parish on 18 June 1780. Then he was ordained priest, at the Ely parish on 6 June 1784, canon of Fen Ditton, Cambridgeshire, in 1784, and then canon of Swaffham Prior, also 1784. On 11 December 1791, he preached before the University, at Great St Mary's, a sermon against duelling (from Exodus XX. 13), which was prompted by a duel that had lately taken place near Newmarket between Henry Applewhaite and Richard Ryecroft, undergraduates of Pembroke, in which the latter was fatally wounded. Jones died on 18 July 1807, in lodgings in Edgware Road, London. He is buried in the cemetery of Dulwich College. A bust and a memorial tablet are in the ante-chapel of Trinity College. *Wik

**1775 Étienne-Louis Malus**(23 Jun 1775, 24 Feb 1812 at age 36)French physicist who discovered that light, when reflected, becomes partially plane polarized; i.e., its rays vibrate in the same plane. He served in Napoleon's corps of engineers, fought in Egypt, and contracted the plague during Napoleon's aborted campaign in Palestine. Posted to Europe after 1801, he began research in optics. In 1808, he discovered that light rays may be polarized by reflection, while looking through a crystal of Iceland spar at the windows of a building reflecting the rays of the Sun. He noticed that on rotating the crystal the light was extinguished in certain positions. Applying corpuscular theory, he argued that light particles have sides or poles and coined the word "polarization." *TIS

**1824 Johann Martin Zacharias Dase**(June 23, 1824, Hamburg – September 11, 1861, Hamburg) was a German mental calculator.

He used to spend a lot of time playing dominoes, and suggested that this played a significant role in developing his calculating skills. Dase suffered from epilepsy from early childhood throughout his life.

At age 15 he began to travel extensively, giving exhibitions in Germany, Austria and England. Among his most impressive feats, he multiplied 79532853 × 93758479 in 54 seconds. He multiplied two 20-digit numbers in 6 minutes; two 40-digit numbers in 40 minutes; and two 100-digit numbers in 8 hours 45 minutes. The famous mathematician Carl Friedrich Gauss commented that someone skilled in calculation could have done the 100-digit calculation in about half that time with pencil and paper.

These exhibitions however did not earn him enough money, so he tried to find other employments. In 1844 he obtained a position in the Railway Department of Vienna, but this didn't last long since in 1845 he was reported in Mannheim and in 1846 in Berlin.

In 1844, Dase calculated π to 200 decimal places over the course of approximately two months, a record for the time, from the Machin-like formula:

\( \frac{\pi}{4} = \arctan \frac{1}{2} + \arctan \frac{1}{5} + \arctan \frac{1}{8} \)

He also calculated a 7-digit logarithm table and extended a table of integer factorizations from 7,000,000 to 10,000,000.

Dase had very little knowledge of mathematical theory. The mathematician Julius Petersen tried to teach him some of Euclid's theorems, but gave up the task once he realized that their comprehension was beyond Dase's capabilities. Gauss however was very impressed with his calculating skill, and he recommended that the Hamburg Academy of Sciences should allow Dase to do mathematical work on a full-time basis, but Dase died shortly thereafter.

The book "Gödel, Escher, Bach" by Douglas Hofstadter mentions his calculating abilities. "... he also had an uncanny sense of quantity. That is, he could just 'tell', without counting, how many sheep were in a field, or words in a sentence, and so forth, up to about 30." *Wik

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

**1912 Alan Mathison Turing**(23 June 1912 – 7 June 1954) born. This British mathematician was one of the founders of recursion theory, invented the Turing machine (an abstract model of a computer), did important work in cryptography, and invented the computer. *Alan Turing. The Enigma by Andrew Hodges, 1983.

**1941 Ivor Owen Grattan-Guinness**(23 June 1941 – 12 December 2014) was a historian of mathematics and logic. Grattan-Guinness was born in Bakewell, England; his father was a mathematics teacher and educational administrator. He gained his bachelor degree as a Mathematics Scholar at Wadham College, Oxford, and an MSc (Econ) in Mathematical Logic and the Philosophy of Science at the London School of Economics in 1966. He gained both the doctorate (PhD) in 1969, and higher doctorate (D.Sc.) in 1978, in the History of Science at the University of London. He was Emeritus Professor of the History of Mathematics and Logic at Middlesex University, and a Visiting Research Associate at the London School of Economics.

He was awarded the Kenneth O. May Medal for services to the History of Mathematics by the International Commission for the History of Mathematics (ICHM) on 31 July 2009, at Budapest, on the occasion of the 23rd International Congress for the History of Science. In 2010, he was elected an Honorary Member of the Bertrand Russell Society.

He spent much of his career at Middlesex University. He was a fellow at the Institute for Advanced Study in Princeton, New Jersey, and is a member of the Académie Internationale d'Histoire des Sciences. *Wik

**1944 Richard Peter Stanley**(June 23, 1944; New York City, New York - ) is the Norman Levinson Professor of Applied Mathematics at the Massachusetts Institute of Technology, in Cambridge, Massachusetts. He received his Ph.D. at Harvard University in 1971 under the supervision of Gian-Carlo Rota. He is an expert in the field of combinatorics and its applications to other mathematical disciplines.

Stanley is known for his two-volume book Enumerative Combinatorics (1986–1999). He is also the author of Combinatorics and Commutative Algebra (1983) and well over 100 research articles in mathematics. He has served as thesis advisor to more than 45 doctoral students, many of whom have had distinguished careers in combinatorial research.

Stanley's distinctions include membership in the National Academy of Sciences (elected in 1995), the 2001 Leroy P. Steele Prize for mathematical exposition, the 2003 Schock Prize, a plenary lecture at the 2006 meeting of the ICM (in Madrid, Spain), and election in 2012 as a fellow of the American Mathematical Society

Stanley created the symbol \( (\binom{n}{k}) = \binom {n+k-1}{k} \) for binomial selection with replacement. *John D Cook, Wik

**1891 Wilhelm Eduard Weber**(24 October 1804 – 23 June 1891) German physicist who investigated terrestrial magnetism. For six years, from 1831, Weber worked in close collaboration with Gauss. Weber developed sensitive magnetometers, an electromagnetic telegraph (1833) and other magnetic instruments during this time. His later work (1855) on the ratio between the electrodynamic and electrostatic units of charge proved extremely important and was crucial to Maxwell in his electromagnetic theory of light. (Weber found the ratio was 3.1074 x 108 m/sec but failed to take any notice of the fact that this was close to the speed of light.) Weber's later years were devoted to work in electrodynamics and the electrical structure of matter. The magnetic unit, weber, is named after him.*TIS

**1891 Norman Robert Pogson**(23 Mar 1829; 23 Jun 1891 at age 62) English astronomer who devised the magnitude scale of the brightness of stars (1850) now in use. He divided the classical scale in which a first magnitude star is one hundred times brighter than a sixth magnitude star using five integer steps. Each step represents a fifth-root of 100 (about 2.512) increase in brightness. The Sun's magnitude on this scale is -26.91, whereby negative numbers denote objects brighter than first magnitude. Sirius is magnitude -1.58, Aldebaran is 1 and the faintest star detected is 30. His interest in astronomy began in his youth; by age 18 he had calculated orbits for two comets. He discovered 8 asteroids, 21 new variable stars and compiled a massive star catalogue. In 1860 he moved to India for the remainder of his life's work.*TIS

**1892 Pierre Ossian Bonnet**(22 December 1819, Montpellier – 23 June 1892, Paris)died. He worked on minimal surfaces, geodesics, and integral geometry. *VFR Bonnet made major contributions introducing the notion of geodesic curvature. A formula for the line integral of the geodesic curvature along a closed curve is known as the Gauss-Bonnet theorem. Gauss published a special case.

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