JULY - University of Texas at San Antonio



JULY

1 July, 1646.

What mathematician taught himself to read Greek and Latin while he was a child?

OR

What famous mathematician’s funeral was attended only by his secretary?

Leibniz, the great universal genius of the 17th century and Newton's rival in the invention of the calculus, was born in Leipzig. Having taught himself to read Latin and Greek when he was a mere child, he had, before he was 20, mastered the ordinary textbook knowledge of mathematics, philosophy, theology, and law. At this young age, he began to develop the first ideas of his characteristica generalis, that later blossomed into the symbolic logic of George Boole (1815-1864) and, still later, in 1910, into the great Principia mathematics of Whitehead and Russell. When, ostensibly because of his youth, he was refused the degree of doctor of laws at the University of Leipzig, he moved to Nuremberg. There he wrote a brilliant essay on teaching law by the historical method and dedicated it to the Elector of Mainz. This led to his appointment by the Elector to a commission for the recodification of some statutes. The rest of Leibniz’s life from this point on was spent in diplomatic service, first for the Elector of Mainz and then, from about 1676 until his death, for the estate of the Duke of Brunswick at Hanover.

In 1672, while in Paris on a diplomatic mission, Leibniz met Huygens, who was then residing there, and he prevailed upon the scientist to give him lessons in mathematics. The following year, Leibniz was sent on a political mission to London, where he exhibited a calculating machine to the Royal Society. Before he left Paris to take up his lucrative post as librarian for the Duke of Brunswick, Leibniz had already discovered the fundamental theorem of calculus, developed much of his notation in this subject, and worked out a number of the elementary formulas of differentiation.

Leibniz’ appointment in the Hanoverian service gave him leisure time to pursue his favorite studies, with the result that he left behind him a mountain of papers on all sorts of subjects. He was a particularly gifted linguist, winning some fame as a Sanskrit scholar, and his writings on philosophy have ranked him high in this field. He entertained various grand projects that came to naught, such as reuniting the Protestant and Catholic churches.

In 1682, he and Otto Mencke founded the journal Acta eruditorum, of which he became editor-in-chief. Most of his mathematical papers, which were largely written in the ten-year period from 1682 to 1692, appeared in this journal. In 1700, Leibniz founded the Berlin Academy of Science.

The closing years of Leibniz’ life were embittered by the controversy that others brought upon him and Newton concerning whether he had discovered the calculus independently of Newton. In 1714, his employer became the first German King of England, and Leibniz was left, neglected at Hanover. It is said that when he died two years later, in 1716, his funeral was attended only by his faithful secretary.

Leibniz’ search for his characteristica generalis led to plans for a theory of mathematical logic and a symbolic method with formal rules that would obviate the necessity of thinking. Leibniz invented his calculus sometime between 1673 and 1676. It was on October 29, 1675, that he first used the modern integral sign, as a long letter S derived from the first letter of the Latin word summa (sum). Leibniz had a remarkable feeling for mathematical form and was very sensitive to the potentialities of a well-devised symbolism. His notation in the calculus proved to be very fortunate and is unquestionably more convenient and flexible than the fluxional notation of Newton. We conclude with a closing homage to Leibniz' unique talent. There are two broad and antithetical domains of mathematical thought, the continuous and the discrete; Leibniz is the one man in the history of mathematics who possessed both of these qualities of thought to a superlative degree.

2 July, 1852.

Whose second edition book on group theory is considered a classic?

Birthdate of William Burnside, whose research was in such diverse fields as mathematical physics, complex function theory, geometry, group theory, and the theory of probability. On the basis of the first two fields he was elected a fellow of the Royal Society of London in 1893. It was in the theory of groups, however, that he made his most significant contributions, and he is best known today for his outstanding and comprehensive book: Theory of Groups. The first edition of this work came out in1897, and an improved and augmented second edition in 1911. This second edition is regarded as a classic in its field. Burnside died on August 21, 1927.

3 July, 1822.

What mathematician invented our modern computer/calculator 75 years before it could be produced because of the lack of precision tools used today?

Charles Babbage described his ideas for a “difference engine" to the Royal Society of London. It was about 1812 that the English mathematician Charles Babbage (1792-1871) began to consider the construction of a machine to aid in the calculation of mathematical tables. He resigned the Lucasian professorship at Cambridge in order to devote all his energies to the construction of his machine. In 1623, after investing and losing his own personal fortune in the venture, he presented his ideas to the Royal Society. As a result he secured financial aid from the British government and set to work to make a difference engine capable of employing 26 significant figures and of computing and printing successive differences out to the sixth order.

But Babbage's work did not progress satisfactorily, and 10 years later the governmental aid was withdrawn. Babbage thereupon abandoned his difference engine and commenced work on a more ambitious machine, which he called his analytic engine, which was intended to execute completely automatically a whole series of arithmetic operations assigned to it at the start by the operator. This machine. also, was never completed, largely because the necessary precision tools were not yet made. It wasn't until 73 years after his death the Babbage's dream came true - in the great IBM Automatic Sequence Controlled Calculator (the ASCC), completed at Harvard University in 1944 as a Joint enterprise by the University and the International Business Machines Corporation under contract for the Navy Department. The machine is 51 feet long, 8 feet high, with 2 panels 6 feet long, and weighs about 5 tons.

4 July, 1862.

What author of several children’s books published a number of texts in mathematics?

Charles Lutwidge Dodgson went boating on the Isis River, a tributary of the Thames, with the three daughters of Henry George Liddell, dean of Christ Church, Oxford. It was for the daughter Alice, who at the time was 10, that he later wrote his Alice books, under the pseudonym of Lewis Carroll. Dodgson was an English mathematician and logician holding a mathematical lectureship at Christ Church. Many people, literarily acquainted with him as Lewis Carroll, do not know of Dodgson the mathematician and of the fact that he published a number of texts in the field of mathematics. There is a story that Queen Victoria was so struck with Lewis Carroll's Alice books that she sent out a courtier to bring back a copy of every book that man had written, and the courtier returned with a bundle of mathematics books that the poor Queen could not read. Dodgson carried the art of nonsense-writing to a peak, and there are numerous instances in his literary works of remarkably involved syllogisms of logic. There is some evidence that the changes in Alice's size and proportions in the wonderland adventure form a closed set of projective transformations, and there is no doubt that, Through the Looking Glass is based upon an end-game of chess.

Dodgson was shy and was afflicted with a stammer. It was perhaps at least partly because of his stammer that he was drawn to the society of children, especially little girls, in whose company he felt at ease.

He became an outstanding photographer of young children. He enjoyed children's parties. There once was a children's party held in a house in London, and next door there happened to be at the same time an adults' party. To amuse the children, Dodgson decided on his arrival to walk in on all fours. Unfortunately he crawled into the parlor of the wrong house. Teachers of mathematics will recall the Mock Turtle of Alice in Wonderland, whose "regular course" in school contained, among other things, "the different branches of Arithmetic - Ambition, Distraction, Uglification, and Derision." All problemists have, in their collection, Lewis Carron's Pillow Problems and A Tangled Tale.

5 July, 1687.

Who translated works from Arabic even though he did not know a single word of that language?

Edmund Halley wrote to Isaac Newton that Newton's Principia was finally printed. Halley succeeded John Wallis as Savilian professor of geometry and later became astronomer royal. He restored the lost Book VIII of Apollonius' Conic Sections by inference, and edited various works of the ancient Greeks, translating some of them from the Arabic even though he did not know a single word of that language. He also compiled a set of mortality tables of the sort now basic in the life insurance business. His major original Contributions, however, were chiefly in astronomy and of excellent quality. He was very kind to follow scholars. It was at Halley's urging that Newton completed his great Philosophiae naturalis principle mathematical usually referred to by the briefer title Principia, which was then published in 1687 at Halley's expense.

6 July, 1785.

When did the Continental Congress adopt the decimal system of currency with the dollar as unit?

7 July, 1906.

Who wrote An Introduction to Probability Theory and Its Applications, of which it has been said, “No other work in the subject matches these two volumes, with their combination of purest abstract mathematics and interesting application”?

Birthdate of William Feller in Zagreb, Yugoslavia, where he was educated by a private tutor until he entered the University of Zagreb in 1923, there earning the equivalent of an M.S. degree in 1925. A year later he received a Ph.D. from the University of Gottingen, where he remained until 1928. In 1928 he moved to the University of Liel, to head the applied mathematics laboratory. In 1933, after Hitler came to power, he moved to Copenhagen. The following year he moved to the University of Stockholm, as a research associate in probability. In 1939 he came to the United States as a professor of mathematics and served as the first executive editor of Mathematical Reviews. This international review was founded in 1939 (because the German review had come under Nazi control) and has been of inestimable value to mathematicians. Much of its success is due to Feller's policies. In 1945 Feller accepted a professorship at Cornell University, where he remained until he made his final move - to Princeton University as the Eugene Higgins professor of mathematics.

Feller wrote several papers applying probability theory to genetics, and, with Kolmorgorov (Soviet Union) and Levy (France), transformed mathematical probability into one of the most vigorous branches of present-day mathematics. One of Feller's greatest mathematical bequests is his unique two-volume work: An Introduction to Probability Theory and Its Applications. No other work in the subject matches these two volumes, with their combination of purest abstract mathematics and interesting application -- written in a style reflecting the ebullient enthusiasm of the author.

To listen to Feller lecture was a unique experience.

Shortly before his death Feller was named to receive the 1969 prestigious National Medal of Science, but he died before the awards ceremony took place, and his wife accepted the medal on his behalf. Feller died in Now York City on January 14, 1970.

8 July 1661.

What 18-year-old found Euclid’s Elements obvious?

Isaac Newton matriculated at Trinity College, Cambridge. He was 18 years old. It was not until this stage in his schooling that his attention came to be directed to mathematics, by a book on astrology picked up at the Stourbridge Fair. As a consequence, he first read Euclid's Elements, which he found too obvious, then Descartes’ La geometrie, which he found somewhat difficult. He also read Oughtred's Clavis, works of Kepler and Viete, and the Arithmatica infinitorum by Wallis. From reading mathematics, he turned to creating it.

9 July, 1814.

When did Gauss make his 146th, and final entry in his mathematical diary?

It was in this famous diary that he confided in cryptic fashion many of his greatest mathematical achievements. Because Gauss was both slow and reluctant to publish, this diary, which was not found until 1898, has settled a number of disputes on priority. As an illustration of the cryptic nature of the entries in the diary consider that for July 10, l796, which reads

ERPHKA! ( + ( + (,

and records Gauss' discovery of a proof of the fact that every positive integer is the sum of 3 triangular numbers. (The nth triangular number is Tn = ). All the entries of the diary except 2 have, for the most part, been deciphered. The entry for March 19, 1797, shows that Gauss had already at that time discovered the double periodicity of certain ellipfunctions (he was not yet 20 years old), and a later entry shows that he had recognized the double periodicity for the general case. This discovery alone, had Gauss published it, would have earned him mathematical fame. But Gauss never published it. (See April 30.)

10 July, 1682.

Roger Cotes was born. (See June 5,)

11 July, 1731.

What 11-year-old composed a paper on curves of the third order?

Alexis Claude Clairaut was elected to the French Academy of Sciences. Clairaut was born in Paris in 1713 and died there in 1765. He was a youthful mathematical prodigy, composing in his 11th year a treatise on curves of the third order. This early paper, and a singularly elegant subsequent one on the differential geometry of twisted curves in space, won him his seat in the French Academy at the illegal age of 18.

In 1736, he accompanied Pierre Louis Moreau de Maupertuis (1698-1759) on an expedition to Lapland to measure the length of a degree of one of the earth's meridians. The expedition was undertaken to settle a dispute as to the shape of the earth. Newton and Huygens had concluded, from mathematical theory, that the earth is flattened at the poles. But about 1712, the Italian astronomer and mathematician Giovanni Domenico Casaini (1625-1712), and his French-born son Jacques Cassini (1677-1756), measured an arc of longitude extending from Dunkirk to Perpignan, and obtained a result that seemed to support the Cartesian contention that the earth is elongated at the poles. The measurement made in Lapland unquestionably confirmed the Newton-Huygens belief and earned Maupertuis title of "earth flattener."

In 1743, after his return to France, Clairaut published his definitive work, Theorie de la figure de la Torre. In 1752, he won a prize from the St. Petersburg Academy for his paper Theoris de la Lulle, a mathematical study of lunar motion that cleared up some unanswered questions. Every student of differential equations meets Clairaut's name in connection with the so-called Clairaut's, equation.

Clairaut had a brother, three years his junior and known in the history of mathematics only as "le cadet Clairaut" (1716-1732), who tragically died of smallpox when only 16, but who at 14 read a paper on geometry before the French Academy and at 15 published a work on geometry. The father of the Clairaut children, Jean Baptiste Clairaut (died soon after 1765), was a teacher of mathematics, a correspondent of the Berlin Academy, and a writer on geometry; he had twenty children of whom only one survived him.

12 July

An uneventful day in the history of mathematics.

13 July, 1832.

What award helped Babbage work on his calculating machine?

Charles Babbage received the first Gold Medal of the Royal Astronomical Society. It was awarded to him for his paper "Observations on the Application of Machinery to the Computation of Numerical Tables." It was this presentation that resulted in his securing government aid to work on his difference engine. (See July 3.)

14 July, 1887.

What is Esperonto?

The first textbook on Esperonto was published. The international language Esperonto was invented by Ludwig Zamenhof (1859-1919) of Polanct. Esperonto had a vocabulary based on word roots common to many European languages and a regular system of inflection. The Italian mathematician Giuseppi Peano created his own international language, Latina sina foxione, but it proved to be even less successful than Esperonto. In earlier times, Latin tended to serve as an international languages

15 July, 1662.

Why was The Royal Society of London founded?

The Royal Society of London received its charter. On November 28, 1660, after attending a lecture by Christopher Wren, a group gathered to discuss the founding of "a college for promoting physico-mathematical experiment" learning." The result was the Royal Society of London.

16 July, 1848.

When did Gauss celebrate his golden jubilee? (See May 5.)

17 July, 1879.

Who was first announced as proving that no more than 4 colors are needed to color any map and show all boundaries via color?

It was announced in Nature that A. B. Kempe had proved the four-color conjecture. About 1850 Francis Guthrie, when a graduate student at University College in London, noticed that four colors are sufficient to distinguish the counties on a map of England. From this observation arose the famous four-color conjecture - - that four colors will suffice to color any map on a plane or sphere, where two contrles sharing a common boundary have different colors.

Frederick Guthrie, younger brother of Francis Guthrie, communicated the conjecture to his teacher Augustus De Morgan, who, in turn, communicated it in a letter dated October 25, 1852. to Sir William Rowan Hamilton and stated that he was unable to supply a proof. Hamilton showed no interest in the matter, and so for a time the problem lay dormant. Then, on June 13, 1678, at a meeting of the London Mathematical Society, Arthur Cayley announced that he had been unable to obtain a proof of the conjecture. In the first volume of the Proceedings of the Royal Geographic, Society (1879), Cayley stated the problem again. Shortly after Cayley's announcement, A. B. Kampe, a British barrister-at-law, published, in 1879. a "proof" of the conjecture in the American Journal of Mathematics. A simplified version of the “proof" was published later in the same year in the Transactions of the London Mathematical Society, and again, in the following year, 1880, in Nature. In 1890, in the London Quarterly Journal of Mathematics, P. J. Heawood (1861-1955) pointed out a flaw, which had evaded detection for 11 years, in Kampe's reasoning. For close to 100 years this flaw remained uncircumvented, and for that long period of time the four-color problem stood as one of the most celebrated unverified conjectures in mathematics. For the story of the final solution of the problem, by a computer, see July 26. In addition to his fallacious proof of the four-color conjecture, Tempe published an engaging little book titled How to Draw a Straight Line. When we draw a circle, we do not trace around some circular object (like a 500 piece); we use a compass. But when we draw a straight line, we trace along the edge of, say, a ruler. The matter dealt with in Kempe's little book is how to draw a straight line without tracing along something already straight.

18 July, 1768.

Whose name should be applied to the complex plane?

Birth of Jean Robert Argand, one of the earliest to point out the association of the complex numbers with the real points of a plane. In this he was preceeded by Caspar Wessel (See March 10) and followed by Carl Friedrich Gauss. Argand was a bookkeeper, born in Geneva, Switzerland. His contribution, a short paper entitled "Essai sur uns maniere de reprosenter les quantites imaginaires dans les constructions geometiiques," was published in 1806. The paper was later reported on, in 1814, in Gergonne's Annales de.Mathematiques. Argand's paper lacks the clarity and pointedness of Wessel's paper and fails to make any geometric application of the representation. But Wessel's paper was unknown to the general mathematical world until it was discovered by an antiquary some 98 years after. It was then republished on the hundredth anniversary of its first appearance. This delay in the general recognition of Wessel's accomplishment is why the plane of complex numbers has come to be called by many the Argand PLane, rather than the Wessel plane. Gauss' contribution is found in a memoir presented to the Royal Society of Gottingen in 1831, and later reproduced in his Collected Works. Gauss pointed out that the basic idea of the representation can be found in his doctoral dissertation of 1799. This claim seems to be well taken and explains why the plane of complex numbers is frequently referred to as the Gauss plane.

19 July, 1799.

What 3 languages are on the Rosette Stone?

The Rosette Stone was found in the Nile delta. Ability to read Egyptian hieroglyphic and demotic characters resulted from the successful decipherment by Jean Frangois Champollion (1790-1832) of inscriptions on the Rosetta Stone, a polished basaltic slab that was found in 1799, during Napoleon's fateful Egyptian campaign, by French engineers while they were digging foundations for a fort near the Rosette branch in the delta of the Nile. The stone measures 3 feet and 7 inches by 2 feet and 6 inches, and the inscriptions on it give a common message repeated in Egyptian hieroglyphic, Egyptian demotic, and Greek. Since scholars were able to read the Greek, the stone furnished a clue to the decipherment of ancient Egyptian writing. The stone was engraved in 196 BC, and as part of the treaty of capitulation when the French surrendered to the British, it went to England, where it now rests in the British Museum.

20 July, 1969.

When did Neil Armstrong first walk on the moon?

21 July, 1967.

When did the Mobius strip first appear on a stamp?

Brazil issued a stamp in commemoration of the 6th Brazilian Mathematical Congress. The stamp depicts a Mobius strip, this being the first time that Famous strip has appeared on either a stamp or a coin. A Mobius strip is formed by a band of paper twisted through 180o and then the two ends glued together. A Mobius strip is one-sided and has one unknotted edge.

22 July, 1764.

Who was the first to accurately measure the parallax of a star?

Birthdate of Friedrich Wilhelm Bessel. Bessel was a noted Prussian astronomer, a close friend of Gauss, and the creator of the so-called Bessel functions fundamental in applied mathematics and encountered by all students of that subject. Bessel became director of the new observatory at Konigsberg on May 10, 1810. In 1838 he was the first to measure accurately the parallax of a star. He died in 1846.

23 July, 1754.

How old was Lagrange when he published his first work?

Lagrange published his first work; he was only 18. The work, in the form of a letter, was written in Italian, A month later he realized that he had rediscovered Leibniz’ formula for the nth derivative of a product.

24 July, 1860.

When was the first Ph.D. granted in the United States?

Yale University authorized the granting of Ph.D. degrees. The first such degree in the United States was awarded at Yale in 1861. Today, in the United States, a Ph.D. is almost a prerequisite for obtaining a professorship in mathematics.

25 July, 1783.

When was the Royal Academy of Sciences in Turin founded?

26 July, 1976.

Are there mathematical problems of such complexity that they are beyond an unassisted human mind and therefore must be approached via some computer?

Kenneth Appel and Wolfgang Haken communicated their solution of the four-color conjecture. In the summer of 1976, Appel and Haken, of the University of Illinois, established the conjecture by an immensely intricate computer-based analysis. The proof contains several hundred pages of complex detail and subsumes over a thousand hours of computer calculation. The method of proof involves an examination of 1936 reducible configurations, each requiring a search of up to half a million options to verify reducibility. This last phase of the work occupied six months and was finally completed in June, 1976. Final checking took most of the month of July, and the results were communicated to the, Bulletin of the American Mathematical Society on July 26, 976. The Appel-Haken solution is unquestionably an astonishing accomplishment, but a solution based on computerized analysis of close to 2000 cases with a total of something like a billion logical options seems, to many mathematicians, far indeed from elegant mathematics. Certainly on at least an equal footing with a solution to a problem is the elegance of the solution itself. This is probably why, when the result above was personally presented by Haken to an audience of several hundred mathematicians at the University of Toronto in August 1976, the presentation was rewarded with little more than a mildly polite applause.

Although a second, and considerably less complex, computer proof of the four-color conjecture was given in the following year, 1977, by F. Allaire, the existence and possible necessity of such treatments of mathematical problems have raised philosophical questions as to just what should be allowed to constitute a proof of a mathematical proposition. Of course it is possible that, in time, someone may find a proof of the four-color problem independent of any computer analysis - - a proof elegant and concise enough to be verifiable by an unassisted human mind. Nevertheless, one begins to wonder if mathematics perhaps contains problems beyond this realization, that is, contains problems of such complexity that they are beyond an unassisted human mind and therefore must be approached via some computer. There is every reason to believe that such problems exist. From this point of view, the four-color problem is perhaps more valuable in mathematics than in cartography, in that it may help clarify limitations of purely human solution to problems.

>>>>>> Brumbaugh Side Note >>>>>>

I got the following written note from Professor Eves when I visited him:

“January 13, 2002

Dear Doug:

In the mathematical calendar you mentioned the other day, there should be an entry for October 25, 1994, since it was on this date that Andrew Wiles finally solved Fermat’s Last Theorem.

I don’t recall when I made out the calendar, but it was on July 26, 1976 that the solution of the four-color problem by Kenneth Appel and Wolfgang Haken was published in the Bulletin of the American Mathematical Society. You might check this since I cannot recall when I wrote the calendar.

All best wishes

Howard (Printed, not typed)

H. Eves

787 Field Street

Oviedo, FLORIDA 32765”

Eves has since sold this house and moved permanently to Maine.

>>>>>> End Brumbaugh Side Note >>>>>>

27 July, 1837.

Who proved that an arithmetic sequence with the first term and the common difference being relatively prime contains and infinite number of primes?

Peter Gustav Lejeune Dirichlet presented his first paper in number theory. At a meeting of the Berlin Academy of Sciences, Dirichlet proved the fundamental theorem that today bears his name: Every arithmetic sequence

a, a + d, a + 2d, a + 3d, . . . ,

in which a and d are relatively prime, contains an infinitude of primes. This theorem had long been conjecturer. Adrien-Marie Legendre (1752-1833) tried hard to find a proof but succeeded in establishing it only for special cases.,such as when a = 1, d = 4. The case a = 0, d = 1 was known to Euclid. (See May 5.)

28 July, 1866.

When was it first legal to use the metric system in the United States?

The first act making it legal,(in the United States) to employ the metric system of weights and measures was approved.

29 July, 1958.

When did President Eisenhower sign the NASA into existence?

President Eisenhower signed the National Aeronautics and Space Act. NASA started business on October 1, 1958, and within a few days launched its Project Mercury.

30 July

A blank day.

31 July, 1704.

Why do we say Cramer’s Rule rather than Maclaurin’s Rule?

Birth of Gabriel Cramer. Colin Maclaurin probably knew as early as 1729 the rule for solving systems of simultaneous linear equations by determinants that today is called Cramer's rule. The rule first appeared in print in 1748 in Maclaurin's posthumous Treatise of Algebra. The Swiss mathematician Gabriel Cramer independently published the rule in 1750 in his Introduction 'a l’analyse des lignes courbes algebrigues, and it is probably his superior notation that led the general mathematical world to learn the rule from him rather than from Maclaurin.

Book VII of Pappas' Mathematical Collection contains a solution of the problem: To inscribe in a given circle a triangle whose sides, produced if necessary, shall pass through three given collinear points. This has become known as the Castillon-Cramer problem because the problem was generalized by Cramer to the case where the three points need not be collinear, and a solution of this generalization was published by Castillon in 1776. Solutions were also given by Lagrange, Euler, Lhuilier, Fuss, and Lexell in 1780. A few years later, a gifted Italian lad of sixteen, named Giordano, generalized the problem to that of inscribing in a circle an n-gon whose sides shall pass through n given points, and he furnished an elegant solution. Victor Poncelet extended the problem still further by replacing the circle with an arbitrary conic section. Cramer died in 1752.

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