## Friday, 31 July 2015

### On This Day in Math - July 31

I advise my students to listen carefully the moment
they decide to take no more mathematics courses.
They might be able to hear the sound of closing doors.
~Caballero, James

The 212th day of the year; Besides being the Fahrenheit boiling point of water at sea level, 212 produces a prime of the form k10+k9+...+k2+k+1, when k=212. (Edward Shore@edward_shore sent me a note:" That number would be 184,251,916,841,751,188,170,917.")
(students might explore different values of k, and different maximum exponents to produce primes..ie when k is 2, then 26 +25+...+22+2+1 is prime

EVENTS

1669 Lucasian professor Isaac Barrow sent John Collins a manuscript of Newton’s De analysi and thereby Newton’s anonymity began to dissolve. It was a summary of Newton’s work on the calculus and was written after Newton saw Nicholas Mercator’s Logarithmotechnia (1668). Newton wrote his paper in order that he would not lose credit for his work on inﬁnite series. Collins immediately recognized Newton’s genius. Although not published until 1711, this paper led to Newton’s appointment as Lucasian professor on 29 October 1669.*VFR

1790 The U.S. Patent Office issued its ﬁrst patent to Samuel Hopkins of Vermont for his “process for making pot and pearl ashes,” whatever they are. Since George Washington and Thomas Jefferson signed Hopkins’ patent, more than 4 million have been issued. *VFR In 1790, the first U.S. patent was granted to Samuel Hopkins of Vermont for a process for making potash and pearl ashes. Potash was important as an ingredient in soap and fertilizer. The patent was granted for a term of 14 years and signed by President George Washington, who had the previous month signed the first U.S. patent statute into law on 10 April 1790. Hopkins did not get Patent with a serial No.1 as thousands of patents were issued before the Patent Office began to number them. Congress had passed the Patent Act on 10 Apr 1790. Two other patents were granted that year - one for a new candle-making process and the other the flour-milling machinery of Oliver Evans. The next year, 1791, Samuel Hopkins also was granted the first Canadian patent.*TIS

1851 Gauss witnessed the opening ceremonies when the newly constructed railway from Cassel reached Gottingen. *VFR

1943 Ireland issued—as its ﬁrst stamp with a mathematical theme—two stamps to celebrate the centenary of the discovery of Quaternions by Sir William Rowan Hamilton. [Scott #126-7]. *VFR

2015 The second full moon this month (the other was on the 2nd). This only happens “Once in a blue moon”—and this is the origin of the phrase. Consequently, there were be thirteen full moons this year.  The last "blue moon" was in 1985, and the next is predicted in 2018.

BIRTHS

1704 Gabriel Cramer (31 July 1704 – 4 January 1752). He is best known for “Cramer’s Rule,” a method for solving systems of simultaneous linear equations using determinants. *VFR Gabriel Cramer (31 July 1704 – 4 January 1752) was a Swiss mathematician, born in Geneva. He showed promise in mathematics from an early age. At 18 he received his doctorate and at 20 he was co-chair of mathematics. In 1728 he proposed a solution to the St. Petersburg Paradox that came very close to the concept of expected utility theory given ten years later by Daniel Bernoulli. He published his best known work in his forties. This was his treatise on algebraic curves, "Introduction à l'analyse des lignes courbes algébriques", published in 1750. It contains the earliest demonstration that a curve of the n-th degree is determined by n(n + 3)/2 points on it, in general position. He edited the works of the two elder Bernoullis; and wrote on the physical cause of the spheroidal shape of the planets and the motion of their apsides (1730), and on Newton's treatment of cubic curves (1746). He was professor at Geneva, and died at Bagnols-sur-Cèze.*Wik

1712 Samuel König (July 31, 1712, Büdingen – August 21, 1757, Zuilenstein near Amerongen) was a German mathematician who is best remembered for his part in a dispute with Maupertuis over the Principle of Least Action.*SAU

1718 John Canton (31 July 1718 – 22 March 1772) British physicist and teacher, born Stroud, Gloucestershire. He made a number of minor discoveries in physics and chemistry. As a result of preparing artificial magnets in 1749 he was elected to the Royal Society. In 1762, he demonstrated that water was slightly compressible. He invented a number of devices in connection with electricity. His notable work, between 1756 and 1759, was to record that on days when the aurora borealis was particularly bright, a compass needle behaved with more irregularity than usual. Thus he was the first to record this as an electromagnetic phenomenon for what is now known to be a magnetic storm.*TIS

1826 Daniel Friedrich Ernst Meisse mathematical work covers number theory, work on Möbius inversion and the theory of partitions as well as work on Bessel functions, asymptotic analysis, refraction of light and the three body problem. *SAU

1843 Friedrich Robert Helmert (July 31, 1843 – June 15, 1917) German geodesist and an important writer on the theory of errors.
From 1887 Helmert was professor of advanced geodesy at the University of Berlin and director of the Geodetic Institute.
Helmert received many honours. He was president of the global geodetic association of "Internationale Erdmessung", member of the Prussian Academy of Sciences in Berlin, was elected a member of the Royal Swedish Academy of Sciences in 1905, and recipient of some 25 German and foreign decorations. *TIA

1858 Richard Dixon Oldham (31 July 1858 – 15 July 1936) Irish geologist and seismologist who discovered evidence for the existence of the Earth's liquid core (1906). In studying seismograms of great 1897 Indian Earthquake he identified P (primary) and S (secondary) waves. It is interesting that he did not get a clue to the presence of the core from the S waves, which are actually incapable of being transmitted through the liquid of the outer core. (The liquid core does not transmit the shear wave energy released during an earthquake.) Rather he noted the existence of a shadow zone in which P waves from an earthquake in the opposite hemisphere of the earth failed to appear*TIS

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

1923 Joseph B. Keller (born July 31, 1923, Paterson, New Jersey) is an American mathematician who specializes in applied mathematics. He is best known for his work on the "Geometrical Theory of Diffraction" (GTD).
He worked on the application of mathematics to problems in science and engineering, such as wave propagation. He contributed to the Einstein-Brillouin-Keller method for computing eigenvalues in quantum mechanical systems.
In 1988 he was awarded the U.S. National Medal of Science, and in 1997 he was awarded the Wolf Prize by the Israel-based Wolf Foundation. In 1996, he was awarded the Nemmers Prize in Mathematics.*Wik

1945 John O'Connor (31st July 1945 in Luton, Bedfordshire, England.- )
Lists his Research interests A lapsed topologist, I am interested in Computational Algebra.
I am interested in the History of Mathematics and at present am supervising two research students in this area. * His Personal web page

1927 F. E. Browder born. Worked in Nonlinear monotone operators and convex sets in Banach spaces. and more.

DEATHS

1726 Nikolaus II Bernoulli died (February 6, 1695, Basel, Switzerland – July 31, 1726, St. Petersburg, Russia). *VFR Nicolaus(II) Bernoulli was the favourite of three sons of Johann Bernoulli. He made important mathematical contributions to the problem of trajectories while working on the mathematical arguments behind the dispute between Newton and Leibniz.*SAU

1784 Denis Diderot died. (October 5, 1713 – July 31, 1784) was a French philosopher, art critic, and writer. He was a prominent persona during the Enlightenment and is best-known for serving as co-founder and chief editor of and contributor to the Encyclopédie. *Wik

1896 Ludwig Christian Wiener (7 December 1826 Darmstadt – 31 July 1896 Karlsruhe) was a German mathematician, physicist and philosopher, known for his explanation of Brownian motion , which identified him as a skillful experimenter. He mainly dealt with geometry.*Wik

1913 John Milne (30 December 1850 – 31 July 1913) English seismologist who invented the horizontal pendulum seismograph (1894) and was one of the European scientists that helped organize the seismic survey of Japan in the last half of the 1800's. Milne conducted experiments on the propagation of elastic waves from artificial sources, and building construction. He spent 20 years in Japan, until 1895, when a fire destroyed his property, and he returned home to the Isle of Wight. He set up a new laboratory and persuaded the Royal Society to fund initially 20 earthquake observatories around the world, equipped with his seismographs. By 1900, Milne seismographs were established on all of the inhabited continents and he was recognized as the world's leading seismologist. He died of Bright's disease*TIS

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

## Thursday, 30 July 2015

### On This Day in Math - July 30

I have created a new universe from nothing.
~Janos Bolyai

The 211th day of the year; 211 is a primorial prime,(a prime that is one more, or one less than a primorial  can you find the next larger (or smaller) primorial prime? 211 is also the sum of three consecutive primes (67 + 71 + 73)...

There are also 211 primes on a 24-hour digital clock. (00:00 - 23:59) *Derek Orr@ Derektionary

EVENTS

1738 Euler sends a letter to John Bernoulli with the solution to a question from Danial Bernoulli regarding isoperimetric curves, particularly the  one for which the integral of rm gave a maximum or minimum.

1859 Bernhard Reimann is appointed full professor at Gottingen, succeeding his two former teachers, Gauss and Dirichlet. He also is allowed to occupy Gauss' apartments at the observatory. *John Derbyshire, Prime Obsession, pg 135

In 1898, Corn Flakes were invented by William Kellogg. At Battle Creek Sanitarium, Sanitarium superintendent, Dr. John Harvey Kellogg and Will Keith Kellogg, his younger brother and business manager, invented many grain-based foods, including a coffee substitute, a type of granola, and peanut butter to provide patients a strict nutritious diet. In 1894 they unintentionally invented a flaked cereal process based on wheat. By 1898, W.K. Kellogg had developed the first flaked corn cereal. Patients enjoyed the cereals and wanted more to take home. In 1906, the Battle Creek Toaster Corn Flake Company was founded by W.K. Kellogg.*TIS

1907 The Axiom of Choice is usually given as created by Zermelo in 1908, presumably because that was the year it appeared in Mathematische Annalen, but the date on the actual paper is "Chesières, 30 July 1907.". The paper contains, "AXIOM VI. (Axiom of choice). If T is a set whose elements all are sets that are different from 0 and mutually disjoint, its union "union of T" includes at least one subset S1 having one and only one element in common with each element of T." [The original German read "Axiom der Auswahl".]
Ernst Zermelo used the Axiom of Choice to prove that every set can be well-ordered on a paper of 1904, but did not use the name "Axiom of Choice". *Jeff Miler, Earliest Known Uses of Some of the Words of Mathematics
1918 Richard Courant sat down with Ferdinand Springer and signed a contract for the series of books now famous as the “Yellow Series.” *Constance Reid, Courant in Gottingen and New York, p. 72

1971 Apollo 15 mission became the fourth mission to land on the moon when the Falcon lunar lander touched down. This mission allowed the astronauts to spend more time on the surface of the moon. The lander stayed three days on the surface and the crew conducted over 18 hours of outside work. They also were aided for the first time by a lunar rover vehicle.*Science Today

BIRTHS

1857 Thorstein Bunde Veblen, (July 30, 1857 – August 3, 1929) was an American economist and sociologist, and a leader of the so-called institutional economics movement. Besides his technical work he was a popular and witty critic of capitalism, as shown by his best known book The Theory of the Leisure Class (1899).

1863 Henry Ford (July 30, 1863 – April 7, 1947) American inventor and car manufacturer, born in Dearborn, Mich. Ford first experimented with internal combustion engines while he was an engineer with the Edison Illuminating Company. He completed his first useful gas motor on 24 Dec 1893. The Quadricycle, he designed made its first road test on 4 Jun 1896. In 1903 the Ford Motor Company was incorporated. By 1908, Ford was manufacturing the low cost, reliable Model T, while continuing to revolutionize his industry. Ford introduced precision manufactured parts designed to be standardized and interchangeable parts. In 1913, production was increased using a continuous moving assembly line. By 1918, half of all cars in America were Model T's.*TIS

1878 Joel Stebbins (July 30, 1878 – March 16, 1966) was an American astronomer who pioneered photoelectric photometry in astronomy.
He earned his Ph.D at the University of California. He was director of University of Illinois observatory from 1903 to 1922 and the Washburn Observatory at the University of Wisconsin-Madison from 1922 to 1948. After 1948, Stebbins continued his research at Lick Observatory until his final retirement in 1958.
Stebbins brought photoelectric photometry from its infancy in the early 1900s to a mature technique by the 1950s, when it succeeded photography as the primary method of photometry. Stebbins used the new technique to investigate eclipsing binaries, the reddening of starlight by interstellar dust, colors of galaxies, and variable stars.
Rumford Prize of the American Academy of Arts and Sciences (1913)
Henry Draper Medal of the National Academy of Sciences (1915)
Bruce Medal of the Astronomical Society of the Pacific (1941)
Gold Medal of the Royal Astronomical Society (1950)
Henry Norris Russell Lectureship of the American Astronomical Society (1956)
The Lunar crater Stebbins and the asteroid 2300 Stebbins are named in his honor. *TIA

1887 Felix Andries Vening Meinesz (The Hague July 30, 1887 - Amersfoort August 10, 1966) was a Dutch geophysicist and geodesist who was known for his measurements of gravity at sea for which he devised the Vening Meinesz pendulum apparatus with comparable accuracy as on land. Starting in 1923 he conducted several global gravity surveys on voyages on submarines, particularly to and in the Indonesian Archipelago. He detected strong gravity anomaly belts running parallel to the Indonesian deep sea trenches. He explained these Meinesz belts as sites of downbuckling of the Earth's crust. He introduced the concept of regional isostasy taking flexure of an elastic crust into account. He also contributed to physical geodesy: The Vening Meinesz formula connects the deviation of the vertical from the plumbline to gravity anomalies. *TIS

1888 Vladimir Zworykin (July 29 [O.S. July 17] 1888 – July 29, 1982) was born in Russia. After emigrating to Pittsburgh, Zworykin took a job at Westinghouse Electric Corp., where in 1923 he filed a patent for the iconoscope, the first television transmission tube and a technology that was to become of interest to early computer designers. With a later invention, the kinescope, Zworykin was able to create the first all-electric television system. Zworykin took the technology to RCA in 1929, where he continued his work and earned the title "father of television.*CMH

DEATHS

1762 William Braikenridge (1700; 30 July 1762 in London, England) was an English clergyman who worked on geometry and discovered independently many of the same results as Maclaurin.*SAU

1978 Rufus Bowen (23 February 1947 - 30 July 1978) worked on dynamical systems. Rufus died of a cerebral hemorrhage at the age of 31. *SAU
In 1970, Bowen completed his doctorate in Mathematics at Berkeley under Stephen Smale, and joined the faculty as assistant professor in that year. At this time he began calling himself Rufus, the nickname he had been given because of his red hair and beard.  He was an invited speaker at the 1974 International Mathematical Conference in Vancouver, British Columbia.He was promoted to full professorship in 1977.
Bowen's mature work dealt with dynamical systems theory, a field which Smale, Bowen's dissertation advisor, explored and broadened in the 1960s.

1985 Julia Robinson (December 8, 1919 – July 30, 1985) died of leukemia. After receiving her Ph.D. in 1948 under the direction of Alfred Tarski, she began work on Hilbert’s tenth problem, the problem which occupied most of her professional life.*VFR She also worked on computability, decision problems and non-standard models of arithmetic. *SAU Her sister was Constance Reid who wrote biographies of several mathematicians and several popular math books.

2002 Dr. Lyle B. Borst, (Nov 24, 1912 - July 30, 2002) was a nuclear physicist who helped build Brookhaven National Laboratory's nuclear reactor and was an early member of the Manhattan Project.
In 1950, Dr. Borst led the construction of the Brookhaven Graphite Research Reactor, which was the largest and most powerful reactor in the country and the first to be built solely for research and other peacetime uses of atomic energy.
Within the first nine months of operating the reactor, Dr. Borst announced that it had produced a new type of radioactive iodine, which is used in treating thyroid cancer.
In 1952, based on studies of new types of atomic nuclei created in the reactor, Dr. Borst helped explain the mystery behind giant stars, known as supernovae, that burst with the energy of billions of atomic bombs and flare for several years with the brilliance of several million suns.
Dr. Borst found that beryllium 7, an isotope of beryllium that does not occur naturally on earth, is formed in supernovae by the fusion of two helium nuclei. The fusion takes place after the star has used up its hydrogen supply. This reaction absorbs huge quantities of energy, causing the star to collapse in the greatest cosmic explosion known. *NY Times obit.

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

## Wednesday, 29 July 2015

### Some History Notes and Some Requests for Information on Division Symbols

Recently James Tanton posted a short article about problems that are circulating on the internet such as (and this is the one he used) "What is the value of the following expression: 62 ÷ 2(3)+4, and then asked, "Is the answer 10 or is the answer 58?" (my personal choice for historical reasons explained below is 3.6)

I don't care to argue the possible choices, although Professor Tanton does a good job of that in his blog, but I'm more interested in the history of some symbols for division he mentions there, obelus, vinculum, and one he didn't, the solidus. In particular, I'm interested in how the usage may have changed over time.

The earliest of the three terms to appear was the vinculum, and it came to us from the Hindu or Arabic mathematicians between the seventh and twelfth century. Here is how it is described by Jeff Miller's excellent web page on the first use of math symbols
Ordinary fractions without the horizontal bar. According to Smith (vol. 2, page 215), it is probable that our method of writing common fractions is due essentially to the Hindus, although they did not use the bar. Brahmagupta (c. 628) and Bhaskara (c. 1150) wrote fractions as we do today but without the bar.

The horizontal fraction bar was introduced by the Arabs. "The Arabs at first copied the Hindu notation, but later improved on it by inserting a horizontal bar between the two numbers" (Burton).

Several sources attribute the horizontal fraction bar to al-Hassar around 1200.

Now if you read Prof. Tanton's article, in which he ecstatically plugs the use of the vinculum, this is NOT what he is suggesting. The horizontal fraction bar made its way into western culture mostly on the back of Leonardo Fibonacci, who introduced both Arabic numbers, and some of their symbols. He referred to the fraction bar as "uirgula"; which has become the more modern word virgule, something like a wand or small rod. Unfortunatly, today the virgule is a term interchangeable with the older term solidus, and you recognize it as the slanted fraction bar, as in 3/5 (and occasionally with an s like bend such as the current symbol for integration), but all that would come much later.

The use of the vinculum that has the professor so excited was introduced around 1452 by Nicholas Chuquet The word is from the diminutive of vincere, to tie. Vinculum referred to a small cord for binding the hands or feet often used to keep cattle from wandering too far afield as they grazed in common areas. The meaning in math is mostly unchanged from that original meaning. The vinculum notation was once used in much the same way we now use parenthesis and brackets to "bind together" a group of numbers or symbols. Where today we might write (2x+3)5 the early users of the vinculum would write $\underline {2x+3} \> 5$ . Originally the line was placed under the items to be grouped although a bar over the grouping became the more lasting usage (and still is in the symbols for the radical sign for roots, and the repeat bar for decimal fractions). The bar on top seems to have been first used by Frans van Schooten.

Dr Peterson at the Math Forum disagrees with calling the fraction bar a vinculum and has written, "I find no evidence, by the way, that it has ever properly been called a vinculum, which is a bar OVER an expression and serves to group it as parentheses do today. The fraction bar has something in common with that, but not enough in my opinion to justify the usage. With both vinculum and virgule used for other things, I just call it a fraction bar and am perfectly happy with that term!" (I'm OK with that, too.) Professor Tanton suggest that the vinculum, properly used, would eliminate questions about whether the answer to the question is 58, 10, (or 3.6).

The symbol "÷" which is used to indicate the operation of division is called an obelus. The word comes from the Greek word obelos, for spit or spike, a pointed stick used for cooking.  Perhaps because both are sharp and used for piercing meat, the word is sometimes used for a type of stabbing knife called a dagger and the same name is applied to an editing symbol that looks like a little dagger, . The root also gives rise to the word obelisk for a pointed pillar of stone.
The symbol(s) was used as an editing notation in early manuscripts, sometimes only as a line without the two dots, to indicate material which the editor thought might need to be "cut out". It had also found occasional use as a symbol for subtraction, for instance, by the famed Adam Riese as early as 1525, although he did not use it exclusively, intermixing the standard horizontal subtraction bar. It was first used as a division symbol by the Swiss mathematician Johann H Rahn in his Teutsche Algebra in 1659.
There has long been a controversy about whether the symbol was introduced to him by John Pell. Cajori in his famous book on mathematical notation says there is no evidence for this, but some later historians, Jacqueline A. Stedall for one, now think it quite probably was Pell's creation. Pell had been Rahn's teacher in Zurich and they communicated on the book. Pell was famous for vacillating over whether he would, or would not, let his name be used on information he shared with others.

Let me make it clear I am not an authority on math history and do not read German,  but as I looked at the examples in Teutsche Algebra, I began to think that Pell/Rahn was not introducing this as a mathematical operator as it is now used. I could find no examples where the books used something like the expression in the problem in Prof. Tanton's blog.  Instead it seems to be used exclusively for a shorthand in explaining the operations used.

Here is an image from page 76 of the Algebra, and it is using a method of teaching algebra by use of a 3 column format, which is certainly from the work of Pell. Each line contains a line number in the middle, instructions for what is being done to the equation in the left column, and the result in the right column. Today many solutions would simply show the sequence of equations in the right column.

The first two lines describe the given information. In the third line, the swirl is exponentiation and says that equation 1 has been squared on both sides. It is line 8 that provides the interesting note about the ÷ usage. The left column says equation 7 is divided by GG+1, but if you look at the right side, you will see that 7 ÷ GG+1 treats all the material to the right of the expression as if it were included in a parenthetical enclosure. Don't divide by GG and then add 1, but divide by the total quantity GG+1.

Now the two surprises here, for me, is that a) Rahn/Pell intends that the "÷" breaks the operation into two parts, the left and the right side as if they were enclosed in parentheses or marked with a vinculum. But the second, is that he doesn't use the expression as an operator in his expressions. Instead he uses the common horizontal division bar/vinculum common to others. So when did we begin to use "÷" as an operation with numbers. I do not have access to the great libraries that contain the early English arithmetics and algebras that eagerly adopted the obelus (it was almost never used anywhere except in English speaking countries), so I am hoping some of you who have more experience/access/knowledge can share so the rest of us will know. When did expressions like 62 ÷ 2(3)+4 first appear in arihtmetic/algebra books? (At the moment I suspect they are a 20th century creation.)

So what about the Solidus. The slanted bar, "/", that is used for fractions, and division is often called a solidus. If you think that looks too much like solid to be a coincidence, you are right. The word comes from the same root. From the glory days of Rome to the Fall of the Byzantine Empire, the solidus was a gold coin ("solid" money). The origin of the modern word "soldier" is from the custom of paying them in solidus. According to Steven Schhwartzman's The Words of Mathematics, the coins reverse carried a picture of a spear bearer, with the spear going form lower left to upper right. He suggests that this is the relation to the slanted bar. Cajori seems to indicate (footnote 6, article 275, Vol 1) that the symbol is derived from the old version of the latin letter s. This / symbol is also frequently called a virgule. Prior to the conversion to decimal coinage in the United Kingdom, it was common to use the symbol as a division between shillings and pence; for example 6/3 would indicate six shillings, three pence. Because of this use the symbol is also sometimes referred to as the shilling mark.
The solidus was introduced as a fraction/division symbol first suggested in De Morgan's Calculus of Functions he proposes the use of the slant line or "solidus" for printing fractions in the text, as in 3/4. In 188 G. G. Stokes put this into practice. Cayley would write to Stokes, "I think the solidus' looks very well indeed . . . ; it would give you a strong claim to be President of a Society for the prevention of Cruelty to Printers."
Stokes, in explaining his choice, says that the slanted bar is already in use for fractions, and simply uses it to expand to algebraic division. Then he states an explanation of the operational use, "In the use of the solidus, it seems convenient to enact that it shall as far as possible take the place of the horizontal bar for which it stands, and accordingly that what stands immediately on the two sides of it shall be regarded as welded into one." He then gives examples that make clear that he intends that a / bc means $\frac{a}{bc}$ . He even gives a method for a period stop to indicate that the grouping has ended, so a/b.c would mean $\frac{a}{b} (c)$

So when did this end. When did we make the switch to the confusion of PEMDAS or BEMDAS or whatever it is called in your country. Cajori (1929) suggests that when using division and multiplication, "there is at present no agreement as to which sign shall be used first."  So it seems that the advent of memorized mnemonics independent of the symbol seems to have occurred later than that.  Similarly in 1923 the National Committee on Mathematical Requirements of the MAA recommended that the ÷ and : for division be replaced with the / solidus "(where the meaning is clear}."

So I looked on my bookshelf and found a 1939 copy of The New Curriculum Arithmetics, Grade Seven.  The authors are a professor of elementary education, a dean of a school of education, a superintendent of schools, and an elementary supervisor, surely folks who would be aware of the MAA recommendations, and yet, there was the ÷ all through the problem sets.  What was not there was a section on order of operations, or any problems that went beyond " number ÷ number."  No long strings of numbers and operations strung together.

Certainly the question was in the air, but unsettled in 1938 when Joseph A. Nyberg of Hyde Park HS in Chicago wrote in The Mathematics Teacher

Read the part in Italics again.... multiplication first, then division, without regard to the order.  That is not what you are telling your students today (I hope).  So maybe they were just working it out.... Nope, here is what N. J. Lennes had written in The American Mathematical Monthly in the article Discussions Relating to the Order of Operations in Algebra in February of 1917, 21 years earlier.

Better, right?  then turn the page, and find
So there is our old friend the obelus used exactly as I suspect Pell and Rahn had intended (if they intended it to be used as an operator at all), and lower down the solidus in the manner that Stokes suggested, but apparently used in a way the users thought distinguished it from the use of the obelus.  And you wonder why your students are confused?

I still have yet to resolve when the first use of the obelus appeared for division as an operator in an algebraic or arithmetic problem.  Anyone who has more information, please share.
I will continue my search as time allows and when I find out more I will continue to update this post. Thank you for any information you can share.

### On This Day in Math - July 29

To call in the statistician after the experiment is done may be
no more than asking hm to perform a postmortem examination:
he may be able to say what the experiment died of.
~Ronald Fisher

The 210th day of the year; (21, 20, 29) and (35, 12, 37) are the two least primitive Pythagorean triangles with different hypotenuses and the same area (=210). Students are challenged to find another pair of such PPTs

There are an infinite number of numbers that appear six or more times in Pascal's Arithmetic Triangle, but only three of them; 1, 120, and 210 are year dates.

7! hours is 210 days.

EVENTS
1654 Pascal wrote a letter to Fermat agreeing to a result of Fermat on a probability problem about repeated rolls of a single die for a wager. "Impatience has seized me as well as it has you, and although I am still abed, I cannot refrain from telling you that I received your letter in regard to the problem of the points  yesterday evening from the hands of M. Carcavi, and that I admire it more than I can tell you. I do not have the leisure to write at length, but, in a word, you have found the two divisions of the points and of the dice with perfect justice. I am thoroughly satisfied as I can no longer doubt that I was wrong, seeing the admirable accord in which I find myself with you."  *York Univ Hist of Stats

1698 In a letter to John Bernoulli, Leibniz introduces the dot for multiplication..(cajori 233; vol 1 pg 267) “The dot was introduced as a symbol for multiplication by G. W. Leibniz. On July 29, 1698, he wrote in a letter to John Bernoulli: “I do not like X as a symbol for multiplication, as it is easily confounded with x; … often I simply relate two quantities by an interposed dot and indicate multiplication by ZC · LM. Hence, in designating ratio I use not one point but two points, which I use at the same time for division.”

1739 D’Alembert, age 21, submitted his ﬁrst mathematical paper to the Academy of Sciences. *VFR As his knowledge of mathematics was mainly due to self-study, he often found that others had already established his mathematical discoveries by more elegant and more direct means. In 1739 d’Alembert submitted his first paper to the French Académie Royale des Sciences, in which he described the errors found in the standard textbook, Analyse démontrée, written by Charles Reyneau. *webpage of Robert Nowland

1773 First schoolhouse West of the Alleghenies.*VFR (built in Schoenbrunn, OH.)

1867 Thomas Hill, president of Harvard College, who was also somewhat of a mathematician, wrote Benjamin Peirce, who was a professor there: “I have the honor of informing you that the University, on Commencement Day, conferred on you the Degree of Doctor of Laws in recognition of the transcendent ability with which you have pursued mathematical physical investigations, and in particular for the luster which she has herself for so many years borrowed from your genius.” [P. 10 of Benjamin Peirce, AMM offprint, 1925] *VFR

1878 This was the height of search for the intra-Mercurial planet Vulcan using eclipses to block the Sun. (Vulcan was a small planet proposed to exist in an orbit between Mercury and the Sun. In an attempt to explain peculiarities of Mercury's orbit, in the 19th-century French mathematician Urbain Jean Joseph Le Verrier hypothesized that they were the result of another planet, which he named Vulcan.) Several observers claim sightings, but they are never confirmed. The problem is finally resolved by Albert Einstein (1879-1955) in his general theory of relativity in 1916. *NSEC

1958 President Eisenhower signed the National Aeronautics and Space Act. NASA opened for business on 1 October 1958, and within a week launched Project Mercury—the start of the U.S. manned space program. *VFR

2005, another candidate for tenth planet was announced by Mike Brown of California Institute of Technology. Its diameter is estimated at 2,100 miles - about 1-1/2 times that of Pluto. Its orbit is eccentric and inclined at about 45 degrees to the main plane of the solar system. It was named 2003 UB313 on a photograph made 31 Oct 2003. Later, its motion was recognized, on 8 Jan 2005. With orbits significantly inclined to the others, the status as a planet of either or even Pluto, is a subject for debate. They are in a region of numerous frozen comet-like objects beyond Neptune - the Kuiper Belt. The object Sedna - somewhat smaller than Pluto - was also found there in 2004. NASA also in an official statement referred to 2003 UB313 as a tenth planet*TIS

2015 On July 29, 2015, a 15th type of pentagon that would tile the plane was announced by Casey Mann, Jennifer McLoud, and David Von Derau of the University of Washington Bothell. In 1918, K. Reinhardt discovered five different families of convex pentagons that could tile the plane. This was the complete list until 1968, when Richard Kershner wrote about three more families of tiling pentagons. Martin Gardner wrote about the complete list of eight tiling pentagons in 1975, and then got a message from Richard James III about another type. Martin updated the readers of Mathematical Games, but then got a message from a housewife with no mathematical training, Marjorie Rice, who found four more families of tiling pentagons. In 1985, Rolf Stein found a convex pentagon that can tile the plane. Now, there is one more. *Wolfram
 *guardian.com

BIRTHS

1858 Francesco Gerbaldi (29 July 1858, La Spezia, Italy to 29 June 1934, Pavia, Italy) was an Italian geometer, who proved Gerbaldi's theorem. In geometry, Gerbaldi's theorem, proved by Gerbaldi (1882), states that one can find six pairwise apolar linearly independent nondegenerate ternary quadratic forms. These are permuted by the Valentiner group. (say that three times real fast) *Wik

1862 Eduard Brückner (July 29, 1862–May 20, 1927) pioneer climate researcher. He also studied the glaciers of the Alps and particularly the effect of the ice ages on the Earth's surface features. By analyzing direct and indirect observations of climatic fluctuations, he discovered the 35-year Brückner climatic cycle (1887) of swings between damp-cold and warm-dry conditions. He initiated scientific debate on whether climate change should be interpreted as a natural function of the Earth system, or whether it was influenced by man's activities, such as deforestation. He considered the impact of climate change on the balance of power between nations and its economic significance in agricultural productivity, emigration, river transportation and the spreading of diseases.*TIS

1898 Isidor Isaac Rabi (29 July 1898 – 11 January 1988) was an American physicist who was awarded the Nobel Prize for Physics in 1944 for his invention (in 1937) of the atomic and molecular beam magnetic resonance method of measuring magnetic properties of atoms, molecules, and atomic nuclei. He spent most of his life at Columbia University (1929-67), where he performed most of his pioneering research in radar and the magnetic moment associated with electron spin in the 1930s and 1940s. His Nobel-winning work led to the invention of the laser, the atomic clock, and diagnostic uses of nuclear magnetic resonance. He originated the idea for the CERN nuclear research center in Geneva (founded 1954). *TIS

1912 Noel Bryan Slater, often cited NB Slater, (1912 in Blackburn , January 31 1973) was a British mathematician and physicist who worked on including statistical mechanics and physical chemistry, and probability theory.*Wik

DEATHS

1781 Johann Kies (September 14, 1713—July 29, 1781) a German astronomer and mathematician. Born in Tübingen, Kies worked in Berlin in 1751 alongside Jérôme Lalande in order to make observations on the lunar parallax in concert with those of Nicolas Louis de Lacaille at the Cape of Good Hope.
From 1742 to 1754, at the recommendation of the mathematician Leonhard Euler, he was made professor of mathematics at Berlin's Academy of Sciences and astronomer at its observatory.
He subsequently taught also at the Collegium of Tübingen. From 1754 to 1755, Kies served as director of the Astronomisches Rechen-Institut in Heidelberg.
Kies was one of the first to propagate Newton's discoveries in Germany, and dedicated two of his works to the Englishman: De viribus centralibus (Tübingen, 1758) and De lege gravitatis (Tübingen, 1773). Kies is also the author of a work on lunar influences: De influxu lunae in partes terrae mobiles (Tübingen, 1769). He wrote many other works, both in French and in Latin, on astronomy.
Kies corresponded with Euler from 1747 to 1767. Their correspondence consists of 8 letters, all of which were written by Kies.
The crater Kies on the Moon is named in his honor. *TIA

1839 Gaspard de Prony. (July 22, 1755 - July 29, 1839) Cauchy was elected his successor at the Bureau des Longitudes but was not admitted as he refused to take the oath of allegiance. *VFR
In 1793, de Prony began a major task of producing logarithmic and trigonometric tables for the French Cadastre. The effort was begun at the request of the French National Assembly, which, after the French Revolution wanted to bring uniformity to the multiple measurements and standards used throughout the nation. The tables and their production were vast, with values calculated to between fourteen and twenty-nine decimal places. Inspired by Adam Smith's Wealth of Nations, de Prony divided up the labor, bragging that he "could manufacture logarithms as easily as one manufactures pins." At the top of the organizational hierarchy were scientists and mathematicians who devised the formulas. Next were workers who created the instructions for doing the calculations. At the bottom were about ninety "computers" (as they were called) who were not trained in mathematics, but who followed the instructions."
One of de Prony's important scientific inventions was the 'de Prony brake' which he invented in 1821 to measure the performance of machines and engines. He also was first to propose using a reversible pendulum to measure gravity, which was independently invented in 1817 by Henry Kater and became known as the Kater's pendulum. He also created a method of converting sinusoidal and exponential curves into a systems of linear equations. Prony estimation is used extensively in signal processing and finite element modelling of non linear materials. Prony was a member, and eventually president, of the French Academy of Science. He was also elected a foreign member of the Royal Swedish Academy of Sciences in 1810. His name is one of the 72 names inscribed on the Eiffel Tower. *Wik

1898 John Alexander Reina Newlands, (July 22, 1755 - July 29, 1839) was a British chemist who first established an order of elements by the atomic weights, and observed a periodicity in the properties. Every eighth element has similar properties, hence he named the Law of Octaves (7 Feb 1863). It took another quarter century, and the work of others, such as Mendeleev, for the significance of his discovery to be recognized. He died in London.*TIS

1944 David Eugene Smith (January 21, 1860 in Cortland, New York – July 29, 1944 in New York) died in New York City at the age of eighty-four.*VFR Smith attended Syracuse University, graduating in 1881 (Ph. D., 1887; LL.D., 1905). He studied to be a lawyer concentrating in arts and humanities, but accepted an instructorship in mathematics at the Cortland Normal School in 1884. He also knew Latin, Greek, and Hebrew. He became a professor at the Michigan State Normal College in 1891, the principal at the State Normal School in Brockport, New York (1898), and a professor of mathematics at Teachers College, Columbia University (1901).
Smith became president of the Mathematical Association of America in 1920. He also wrote a large number of publications of various types. He was editor of the Bulletin of the American Mathematical Society; contributed to other mathematical journals; published a series of textbooks; translated Klein's Famous Problems of Geometry, Fink's History of Mathematics, and the Treviso Arithmetic. He edited Augustus De Morgan's Budget of Paradoxes (1915) and wrote many books on Mathematics. *Wik

1962 Ronald Aylmer Fisher FRS (17 February 1890 – 29 July 1962) was an English statistician, evolutionary biologist, eugenicist and geneticist. Among other things, Fisher is well known for his contributions to statistics by creating Fisher's exact test and Fisher's equation. Anders Hald called him "a genius who almost single-handedly created the foundations for modern statistical science" while Richard Dawkins called him "the greatest of Darwin's successors". In 2010 Dawkins named him "the greatest biologist since Darwin". Fisher was opposed to the conclusions of Richard Doll and A.B. Hill that smoking caused lung cancer. He compared the correlations in their papers to a correlation between the import of apples and the rise of divorce in order to show that correlation does not imply causation.
To quote Yates and Mather, "It has been suggested that the fact that Fisher was employed as consultant by the tobacco firms in this controversy casts doubt on the value of his arguments. This is to misjudge the man. He was not above accepting financial reward for his labours, but the reason for his interest was undoubtedly his dislike and mistrust of puritanical tendencies of all kinds; and perhaps also the personal solace he had always found in tobacco."

After retiring from Cambridge University in 1957 he spent some time as a senior research fellow at the CSIRO in Adelaide, Australia. He died of colon cancer there in 1962.
He was awarded the Linnean Society of London's prestigious Darwin–Wallace Medal in 1958.
Fisher's important contributions to both genetics and statistics are emphasized by the remark of L.J. Savage, "I occasionally meet geneticists who ask me whether it is true that the great geneticist R.A. Fisher was also an important statistician"*Wik The stained glass window is from the Greatroom at Caius College.

1994 Dorothy Mary Hodgkin OM FRS (12 May 1910 – 29 July 1994), known professionally as Dorothy Crowfoot Hodgkin or simply Dorothy Hodgkin, was a British biochemist who developed protein crystallography, for which she won the Nobel Prize in Chemistry in 1964.
She advanced the technique of X-ray crystallography, a method used to determine the three-dimensional structures of biomolecules. Among her most influential discoveries are the confirmation of the structure of penicillin that Ernst Boris Chain and Edward Abraham had previously surmised, and then the structure of vitamin B12, for which she became the third woman to win the Nobel Prize in Chemistry.
In 1969, after 35 years of work and five years after winning the Nobel Prize, Hodgkin was able to decipher the structure of insulin. X-ray crystallography became a widely used tool and was critical in later determining the structures of many biological molecules where knowledge of structure is critical to an understanding of function. She is regarded as one of the pioneer scientists in the field of X-ray crystallography studies of biomolecules. *Wik

1996 Marcel-Paul "Marco" Schützenberger (October 24, 1920 – July 29, 1996) was a French mathematician and Doctor of Medicine. His work had impact across the fields of formal language, combinatorics, and information theory.[1] In addition to his formal results in mathematics, he was "deeply involved in [a] struggle against the votaries of Darwinism,"[2] a stance which has resulted in some mixed reactions from his peers and from critics of his stance on evolution. Several notable theorems and objects in mathematics bear his name (for example Schutzenberger group).*Wik

2004 Walter Feit (October 26, 1930 – July 29, 2004)was an Austrian mathematician who (with John Thompson) proved one of the most important theorems about finite simple groups.*SAU

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, 28 July 2015

### On This Day in Math - July 28

It appears to me that if one wishes to make progress in mathematics
one should study the masters and not the pupils.
Quoted in O Ore's, Niels Abel, Mathematician Extraordinary

The 209th day of the year; 209=16+25+34+43+52+61. Also 209 is a "Self number" A self number, Colombian number or Devlali number( (after the town where he lived) is an integer which, in a given base, cannot be generated by any other integer added to the sum of that other integer's digits. For example, 21 is not a self number, since it can be generated by the sum of 15 and the digits comprising 15, that is, 21 = 15 + 1 + 5. No such sum will generate the integer 20, hence it is a self number. These numbers were first described in 1949 by the Indian mathematician D. R. Kaprekar. students might want to explore self numbers for patterns
[The earliest use of Colombian number I can find is by B. Recaman (1974). "Problem E2408". Amer. Math. Monthly 81. Would love to know if there are earlier uses.]

EVENTS
1619 Kepler wrote Napier expressing his enthusiasm for Napier’s invention of logarithms. *VFR

1851  First American eclipse expedition to Europe when George Phillips Bond (1825 - 1865) led a team to Scandinavia. *NSEC   In the transcription of his notes he wrote:

1851 A total solar eclipse was photographed for the ﬁrst time. *VFR The first correctly-exposed photograph of the solar corona was made during the total phase of the solar eclipse of 28 July 1851 at Königsberg (now Kaliningrad) by a local daguerreotypist named Berkowski at the Royal Observatory in Königsberg, Prussia (now Kalinigrad in Russia). Berkowski, whose first name was never published, observed at the Royal Observatory. A small 6-cm refracting telescope was attached to the 15.8-cm Fraunhofer heliometer and a 84-second exposure was taken shortly after the beginning of totality.
United Kingdom astronomers, Robert Grant and William Swan, and Austrian astronomer Karl Ludwig von Littrow observed this eclipse and determined that prominences are part of the Sun because the Moon is seen to cover and uncover them as it moves in front of the Sun.*Wik

In 1858, fingerprints were used as a means of identification for the first time.*TIS The English first began using fingerprints in July of 1858, when Sir William James Herschel, Chief Magistrate of the Hooghly district in Jungipoor, India, first used fingerprints on native contracts. On a whim, and without thought toward personal identification, Herschel had Rajyadhar Konai, a local businessman, impress his hand print on a contract.
The idea was merely "... to frighten [him] out of all thought of repudiating his signature." The native was suitably impressed, and Herschel made a habit of requiring palm prints--and later, simply the prints of the right Index and Middle fingers--on every contract made with the locals. Personal contact with the document, they believed, made the contract more binding than if they simply signed it. Thus, the first wide-scale, modern-day use of fingerprints was predicated, not upon scientific evidence, but upon superstitious beliefs.
As his fingerprint collection grew, however, Herschel began to note that the inked impressions could, indeed, prove or disprove identity. While his experience with fingerprinting was admittedly limited, Sir William Herschel's private conviction that all fingerprints were unique to the individual, as well as permanent throughout that individual's life, inspired him to expand their use. *History of Fingerprints, Onin.com

1866 The ﬁrst act (in the USA) legalizing the employment of the metric system was approved (14 Stat. L. 339). The act provided that it “shall be lawful throughout the United States of America to employ the weights and measures of the metric system.” *VFR

1882 The Institute of Accountants and Bookkeepers was organized in New York City. It was the first accounting society in the United States. *FFF

1899 Cantor asks Dedekind whether the set of all cardinal numbers is itself a set, because if it is it would have a cardinal number larger than any other cardinal. *VFR

1948 Allen Turing writes to Jack Good with an estimate of the number of neurons in the human brain. "I have repeatedly looked in books on neurology ... and never found any numbers offered. My own estimate is 3x108 to 3x109. " *Turing Archives

1984 The town of Eighty-four Pennsylvania celebrated it's centennial on this day.

2061 Halley's comet will next reach perihelion. The comet last reached perihelion on 9 February 1986, and will reach it again on 28 July 2061 *Wik

BIRTHS
1849 Robert Scott studied at Cambridge and was elected to a fellowship. After a short time teaching he studied to be a barrister. He spent most of his career as Bursar and Master of St John's College Cambridge. He published a book on Determinants. *SAU

1867 Charles Dillon Perrine (July 28, 1867; Steubenville, Ohio, – June 21, 1951) U.S. astronomer who discovered the sixth and seventh moons of Jupiter in 1904 and 1905, respectively. In 1904 he published a calculation of the solar parallax (a measure of the Earth-Sun distance) based on observations of the minor planet Eros during one of its close approaches to the Earth. *TIS

 maser components at amhistorymuseum HT to Ben Gross ‏@bhgross144
1915 Charles Hard Townes (July 28, 1915 – January 27, 2015) was an American Nobel Prize-winning physicist and educator. Townes was known for his work on the theory and application of the maser, on which he got the fundamental patent, and other work in quantum electronics connected with both maser and laser devices. He shared the Nobel Prize in Physics in 1964 with Nikolay Basov and Alexander Prokhorov.
In a career that spanned six decades, Dr. Townes developed radar bombing systems and navigation devices during World War II, advised presidents and government commissions on lunar landings and the MX missile system, verified Einstein’s cosmological theories, discovered ammonia molecules at the center of the Milky Way, and created an atomic clock that measured time to within one second in 300 years. He died at the age of 99 in Berkeley, California*Wik *NY Times

1928 John Bell (28 June 1928 – 1 October 1990)   his great achievement was that during the 1960s he was able to breathe new and exciting life into the foundations of quantum theory, a topic seemingly exhausted by the outcome of the Bohr-Einstein debate thirty years earlier, and ignored by virtually all those who used quantum theory in the intervening period. Bell was able to show that discussion of such concepts as 'realism', 'determinism' and 'locality' could be sharpened into a rigorous mathematical statement, 'Bell's inequality', which is capable of experimental test. Such tests, steadily increasing in power and precision, have been carried out over the last thirty years. *SAU

1954 Gerd Faltings (July 28, 1954 - ) was born in Gelsenkirchen-Buer, West Germany. In 1986 he received a Fields Medal for solving Mordell’s Conjecture using arithmetic algebraic geometry. *VFR He has also been closely linked with the work leading to the final proof of Fermat's Last Theorem by Andrew Wiles. In 1983 Faltings proved that for every n greater than 2 there are at most a finite number of coprime integers x, y, z with xn + yn = zn. This was a major step but a proof that the finite number was 0 in all cases did not seem likely to follow by extending Falting's arguments.
However, Faltings was the natural person that Wiles turned to when he wanted an opinion on the correctness of his repair of his proof of Fermat's Last Theorem in 1994.*TIS

DEATHS
1818 Gaspard Monge (9 May 1746 – 28 July 1818) died in disgrace in Bourbon Paris, having been stripped of his place in the reorganized Acad´emie of 1816. Although he contributed to diﬀerential equations and the geom¬etry of surfaces, his special interest was descriptive geometry. Employed as a teacher, he made signiﬁcant contributions to educational reform. [Ivor Grattan-Guiness, Convolutions in French Mathematics, 1800–1840, p. 616]
On the fall of Napoleon he was deprived of all his honors, and even excluded from the list of members of the reconstituted Institute. Monge died at Paris on 28 July 1818 and was interred in Le Père Lachaise Cemetery, in Paris, in a mausoleum. He was later transferred to the Panthéon. The mausoleum and Monge's bust remain in Le Père Lachaise Cemetery.
A statue portraying him was erected in his home town of Beaune, Côte-d'Ors in 1849. His name is one of the 72 names inscribed on the Eiffel Tower.

1944 Sir Ralph Fowler (17 January 1889 – 28 July 1944) a brilliant physicist. But it may be for his influence upon others that he is best known. In fact, no less than fifteen Fellows of the Royal Society and three Nobel Laureates were supervised by Fowler between 1922 and 1939. The total number supervised during this time was a staggering sixty-four giving him an average of eleven research students at any given time. One might be led to believe that this did not allow for any depth of relationship to form between him and his students. However, this was far from the truth of the matter. Those who studied under Fowler had a tremendous admiration for him. In particular, E A Milne [1] was especially taken by the man whom he fondly referred to as "the kind of man you can still remain friendly with, even when he has sold you a motor-bike; it is not possible to say more" and whom he called a "prince amongst men".
Aside from Milne, on whom he had a profound impact, he also had the opportunity of influencing the likes of Sir Arthur Eddington, Subramanian Chandrasekhar, Paul Dirac, Sir William McCrea, Lady Jeffreys and others either directly through supervision or indirectly through collaboration. Even in his personal life he was intimately connected with brilliant people having married Eileen, the only daughter of Lord Rutherford whom he met through Rutherford's Cavendish Laboratory at Cambridge. Sometimes his influence was simply the fact that he was known to so many people. It was Fowler who ultimately introduced Paul Dirac to the burgeoning field of quantum theory in 1923 leading Dirac to the forefront of its ultimate discovery in 1925. Fowler also put Dirac and Werner Heisenberg in touch with each other through Niels Bohr. As Sir William McCrea simply put it: "he was the right man in the right place at the right time." *SAU
1968 Otto Hahn (8 Mar 1879; 28 Jul 1968 at age 89) German physical chemist who, with the radiochemist Fritz Strassmann, is credited with the discovery of nuclear fission. He was awarded the Nobel Prize for Chemistry in 1944 and shared the Enrico Fermi Award in 1966 with Strassmann and Lise Meitner. Element 105 carries the name hahnium in recognition of his work.*TIS

2000 Abraham Pais (May 19, 1918 – July 28, 2000) Dutch-American physicist and science historian whose research became the building blocks of the theory of elemental particles. He wrote Subtle Is the Lord: The Science and Life of Albert Einstein, which is considered the definitive Einstein biography. In Holland, his Ph.D. in physics was awarded on 9 Jul 1941, five days before a Nazi deadline banning Jews from receiving degrees. Later, during WW II, while in hiding to evade the Gestapo, he worked out ideas in quantum electrodynamics that he later shared when working with Niels Bohr (Jan - Aug 1946). In Sep 1946, he went to the U.S. to work with Robert Oppenheimer at Princeton, where Pais contributed to the foundations of the modern theory of particle physics.*TIS

2004 Francis Harry Compton Crick (8 June 1916 – 28 July 2004) was a British biophysicist, who, with James Watson and Maurice Wilkins, received the 1962 Nobel Prize for Physiology or Medicine for their determination of the molecular structure of deoxyribonucleic acid (DNA), the chemical substance ultimately responsible for hereditary control of life functions. Crick and Watson began their collaboration in 1951, and published their paper on the double helix structure on 2 Apr 1953 in Nature. This accomplishment became a cornerstone of genetics and was widely regarded as one of the most important discoveries of 20th-century biology. *TIS

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

## Monday, 27 July 2015

### On This Day in Math - July 27

But just as much as it is easy to find the differential of a given quantity,
so it is difficult to find the integral of a given differential.
Moreover, sometimes we cannot say with certainty
whether the integral of a given quantity can be found or not.

~Bernoulli, Johann

This is the 208th day of the year; 208 is the sum of the squares of the first five primes.
and from Today is 7/27. or 27/7 in some countries. 727 and 277 are both prime.

EVENTS
1630, On July 27 Giovanni Batista Baliani wrote a letter to Galileo Galilei about the explanation of an experiment he had made in which a siphon, led over a hill about twenty-one meters high, failed to work. Galileo responded with an explanation of the phenomena: he proposed that it was the power of a vacuum which held the water up, and at a certain height (in this case, thirty-four feet) the amount of water simply became too much and the force could not hold any more, like a cord that can only withstand so much weight hanging from it.

1794 Jean Baptiste Joseph Fourier (1766?-1830) was a student at the École Normale, c1794. He was sentenced to the guillotine by Robespierre on July 28 of 1794, but Robespierre was overthrown the day before his scheduled execution (27 July, 1794) was due. Fourier went on to both political and scientific success. He was unanimously elected the first Secretary of the Institute of Egypt in 1798. He was Governor of Lower Egypt in 1798‑1801  or Commissioner at the Divan of Cairo .  He led one of the expeditions of exploration which examined ancient monuments and he suggested the publication of the great report on Egypt.  He was was a professor at the École Polytechnique up to 1806.  Napoléon made him a baron and during Napoléon's return from Elba in 1815, he made Fourier a count and Prefect of the Rhone, based at Lyons, from 10 Mar to 1 May.  In 1815, he was penniless in Paris and giving lessons for his living.  The Prefect of Paris found out and made him director of the Bureau de la Statistique of the Préfecture of the Seine.  He was elected to the Académie in 1816, but this was vetoed by the government, so he was elected again in 1817 and this was permitted.    He was Prefect of the Department of Isère, whose capital is Grenoble, from 1802 to 1817 (1815??)  He was Permanent Secretary of the Académie des Sciences in 1822-1830.

1829 By a remarkable coincidence, both Cauchy and Sturm sent papers to the Acad´emie des Sciences dealing with differential equations. Both of them used techniques which we recognize as matrix methods. Thus they are early contributors to linear algebra, a ﬁeld which is usually dated to Cayley’s introduction of matrices in 1858. [Ivor Grattan-Guiness, Convolutions in French Mathematics, 1800–1840, p. 1150]

1861 The Athenaeum magazine carried a review of Charles Dodgson's pamphlet entitled The Formula of Plane Trigonometry in which he suggested new symbols for the six basic trig functions. The reviewer was not convinced.

1837 At a meeting of the Berlin Academy of Sciences, Dirichlet presented his ﬁrst paper on analytic number theory. He proved the fundamental theorem that bears his name: Every arithmetical series an + b, n =0, 1, 2,... of integers where a and b are relatively prime, contains inﬁnitely many primes. The result had long been conjectured. Legendre tried hard for a proof but could only establish special cases such as 4n + 1. *VFR

1866 Cyrus W. Field finally succeeded, after two failures, in laying the first underwater telegraph cable 1,686 miles long across the Atlantic Ocean between North America and Europe. Massachusetts merchant and financier Cyrus W. Field first proposed laying a 2,000-mile copper cable along the ocean bottom from Newfoundland to Ireland in 1854, but the first three attempts ended in broken cables and failure. Field's persistence finally paid off in July 1866, when the Great Eastern, the largest ship then afloat, successfully laid the cable along the level, sandy bottom of the North Atlantic. *TIS

1936 Einstein writes to John Tate, editor of the Physical Review angrily withdrawing a paper that he had submitted for publication but had been rejected after peer review. Einstein and Rosen's paper claimed that gravitational waves did not exist. It was Einstein who introduced gravitational waves in his theory of general relativity in 1916, within a few months of finding the correct form of the field equations for it. However by 1936 he had changed his mind, and wrote to his friend, Max Born, "Together with a young collaborator, I arrived at the interesting result that gravitational waves do not exist,..."
Later he would submit the paper again, but then drastically revise the conclusions before publication. Einstein simply explained why “fundamental” changes in the paper were required because the “consequences” of the equations derived in the paper had previously been incorrectly inferred. The referee of the paper, it is now known, was relativist Howard Percy Robertson. He was on sabbatical at Caltech. When he returned to Princeton he struck up a friendship with Einstein’s then newly arrived assistant Infeld. Robertson then convinced Infield of the problems with the paper he had re-submitted, and after Infield talked to Einstein, the paper was revised. It seems that Einstein had never read the referee's comments.
*physicstoday

1948 Hungary issued a stamp commemorating the centenary of the birth of the physicist Baron Roland E˝otv˝os1 (1848–1919). [Scott #840]. *VFR They issued another in 1991

BIRTHS

1667 Johann Bernoulli (27 July 1667 – 1 January 1748; also known as Jean or John) was a Swiss mathematician who studied reflection and refraction of light, orthogonal trajectories of families of curves, quadrature of areas by series and the brachystochrone.*SAU

1733 Jeremiah Fenwicke Dixon (27 July 1733 – 22 January 1779) was an English surveyor and astronomer who is best known for his work with Charles Mason, from 1763 to 1767, in determining what was later called the Mason-Dixon line.
Dixon was born in Cockfield, near Bishop Auckland, County Durham, the fifth of seven children, to Sir George Fenwick Dixon 5th Bt. and Lady Mary Hunter. His father was a wealthy Quaker coal mine owner and aristocrat. His mother came from Newcastle, and was said to have been "the cleverest woman" to ever marry into the Dixon family. Dixon became interested in astronomy and mathematics during his education at Barnard Castle. Early in life he made acquaintances with the eminent intellectuals of Southern Durham: mathematician William Emerson, and astronomers John Bird and Thomas Wright. In all probability it was John Bird, who was an active Fellow of the Royal Society, who recommended Dixon as a suitable companion to accompany Mason.

Jeremiah Dixon served as assistant to Charles Mason in 1761 when the Royal Society selected Mason to observe the transit of Venus from Sumatra. However, their passage to Sumatra was delayed, and they landed instead at the Cape of Good Hope where the transit was observed on June 6, 1761. Dixon returned to the Cape once again with Nevil Maskelyne's clock to work on experiments with gravity.
Dixon and Mason signed an agreement in 1763 with the proprietors of Pennsylvania and Maryland, Thomas Penn and Frederick Calvert, sixth Baron Baltimore, to assist with resolving a boundary dispute between the two provinces. They arrived in Philadelphia in November 1763 and began work towards the end of the year. The survey was not complete until late 1766, following which they stayed on to measure a degree of Earth's meridian on the Delmarva Peninsula in Maryland, on behalf of the Royal Society. They also made a number of gravity measurements with the same instrument that Dixon had used with Maskelyne in 1761. Before returning to England in 1768, they were both admitted to the American Society for Promoting Useful Knowledge, in Philadelphia.
Dixon sailed to Norway in 1769 with William Bayly to observe another transit of Venus. The two split up, with Dixon at Hammerfest Island and Bayly at North Cape, in order to minimize the possibility of inclement weather obstructing their measurements. Following their return to England in July, Dixon resumed his work as a surveyor in Durham. He died unmarried in Cockfield on 22 January 1779, and was buried in an unmarked grave in the Quaker cemetery in Staindrop.
Although he was recognized as a Quaker, he was not a very good one, dressing in a long red coat and occasionally drinking to excess. *Wik

1801 Sir. George Biddell Airy (27 July 1801 – 2 January 1892) born in Alnwick, England. *VFR English astronomer who became the seventh Astronomer Royal (1836-92). In his life he studied interference fringes in optics, made a mathematical study of the rainbow and computed the density of the Earth by swinging a pendulum at the top and bottom of a deep mine, determined the mass of the planet Jupiter and its period rotation, calculated the orbits of comets and cataloged stars. He designed corrective lenses for astigmatism (1825), the first that worked. His motivation was his own astigmatism. Airy had a long-standing battle with Babbage. In 1854, the conflict continued between the two during the battle of the incompatible railway gauges in England. Airy championed the railway narrow gauge and Babbage for the wide gauge. *TIS

1844 Ágoston Scholtz (27 July 1844 in Kotterbach, Zips district, Austro-Hungary (now Rudnany, Slovakia) - 6 May 1916 in Veszprém,) From 1871 he was a teacher of mathematics and natural philosophy at the Lutheranian Grammar School of Budapest which at that time had been upgraded to become a so called 'chief grammar school', namely one which offered eight years of teaching. This was precisely the school which later was attended by several famous mathematicians such as Johnny von Neumann and Eugene Wigner (or Jenó Pál Wigner as he was called at that time). Scholtz became the school director of the Lutheranian Grammar School in 1875. Unfortunately this excellent school was closed in 1952, and most of its equipment was lost. Due to the initiative and support of its former well-known students, among others Wigner, it was reopened in 1989 after being closed for thirty-seven years. Scholtz's field of research was projective geometry and theory of determinants. His results were recorded by Muir in his famous work The history of determinants *SAU

1848 Roland Baron von Eötvös (27 July 1848 – 8 April 1919) was a Hungarian physicist who studied at Heidelberg where he was taught by Kirchhoff, Helmholtz and Bunsen. Eötvös introduced the concept of molecular surface tension and published on capillarity (1876-86). For the rest of his life he concentrated on study of the Earth's gravitational field. He developed the Eötvös torsion balance, long unsurpassed in precision, which gave experimental proof that inertial mass and gravitational mass, to a high degree of accuracy, are equivalent - which later was a major principle of Albert Einstein.*TIS

1849 John Hopkinson (27 July 1849 – 27 August 1898) British physicist and electrical engineer who worked on the application of electricity and magnetism in devices like the dynamo and electromagnets. Hopkinson's law (the magnetic equivalent of Ohm's law) bears his name. In 1882, he patented his invention of the three-wire system (three phase) for electricity generation and distribution. He presented the principle the synchronous motors (1883), and designed electric generators with better efficiency. He also studied condensers and the phenomena of residual load. In his earlier career, he became (1872) engineering manager of Chance Brothers and Co., a glass manufacturer in Birmingham, where he studied lighthouse illumination, improving efficiency with flashing groups of lights.*TIS

1867 Derrick Norman Lehmer (27 July 1867, Somerset, Indiana, USA — 8 September 1938 in Berkeley, California, USA) was an American mathematician and number theorist.
In 1903, he presented a factorization of Jevons' number (8,616,460,799) at the San Francisco Section of the American Mathematical Society, December 19, 1903.
He published tables of prime numbers and prime factorizations, reaching 10,017,000 by 1909 (In Number Theory and Its History, Ore calls this the "best factor table now (1948) available"). He developed a variety of mechanical and electro-mechanical factoring and computational devices, such as the Lehmer sieve, built with his son Derrick Henry Lehmer.
He is also known for a reversible algorithm that assigns a Lehmer code to every permutation of size n. *SAU

1870 Bertram Borden Boltwood (July 27, 1870 Amherst, Massachusetts - August 15, 1927, Hancock Point, Maine) was an American chemist and physicist whose work on the radioactive decay of uranium and thorium was important in the development of the theory of isotopes. Boltwood studied the "radioactive series" whereby radioactive elements sequentially decay into other isotopes or elements. Since lead was always present in such ores, he concluded (1905) that lead must be the stable end product from their radioactive decay. Each decay proceeds at a characteristic rate. In 1907, he proposed that the ratio of original radioactive material to its decay products measured how long the process had been taking place. Thus the ore in the earth's crust could be dated, and give the age of the earth as 2.2 billion years.*TIS

1871 Ernest Friedrich Ferdinand Zermelo. (27 July 1871; Berlin, German Empire - 21 May 1953 (aged 81) Freiburg im Breisgau, West Germany) In 1904 he formulated the Axiom of Choice in Set Theory. Years later, when he refused to give the Nazi salute, he was threatened with dismissal from his univeristy position. In reply, he resigned. *VFR

2007 Ralph Asher Alpher's belated recognition for his work on the "Big Bang" process. In 2005 Alpher was awarded the National Medal of Science. The citation for the award reads "For his unprecedented work in the areas of nucleosynthesis, for the prediction that universe expansion leaves behind background radiation, and for providing the model for the Big Bang theory." The medal was presented to his son Dr. Victor S. Alpher on July 27, 2007 by President George W. Bush, as his father could not travel to receive the award. *Wik

DEATHS

1759 Pierre-Louis Moreau de Maupertuis (17 July 1698 – 27 July 1759) French mathematician, biologist, and astronomer. In 1732 he introduced Newton's theory of gravitation to France. He was a member of an expedition to Lapland in 1736 which set out to measure the length of a degree along the meridian. Maupertuis' measurements both verified Newton's predictions that the Earth would be an oblate speroid, and they corrected earlier results of Cassini. Maupertuis published on many topics including mathematics, geography, astronomy and cosmology. In 1744 he first enunciated the Principle of Least Action and he published it in Essai de cosmologie in 1850. Maupertuis hoped that the principle might unify the laws of the universe and combined it with an attempted proof of the existence of God.*TIS (he died in the home of Johann II Bernoulli. Johan Bernoulli (above) was born on the day Maupertuis died, but Johann II Bernoulli died on the Calendar date on which Maupertuis was born...)

1844 John Dalton, (6 September 1766 – 27 July 1844) English teacher who, from investigating the physical and chemical properties of matter, deduced an Atomic Theory (1803) whereby atoms of the same element are the same, but different from the atoms of any other element. In 1804, he stated his law of multiple proportions by which he related the ratios of the weights of the reactants to the proportions of elements in compounds. He set the atomic weight of hydrogen to be identically equal to one and developed a table of atomic weights for other elements. He was the first to measure the temperature change of air under compression, and in 1801 suggested that all gases could be liquified by high pressure and low temperature. Dalton recognised that the aurora borealis was an electrical phenomenon.*TIS

1931 Jacques Herbrand (12 February 1908 – 27 July 1931) was a French mathematician who died young but made contributions to mathematical logic.*SAU Although he died at only 23 years of age, he was already considered one of "the greatest mathematicians of the younger generation" by his professors Helmut Hasse, and Richard Courant. *Wik

1999 Aleksandr Danilovic Aleksandrov (4 Aug 1912 in Volyn, Ryazan, Russia
- 27 July 1999) approached the differential geometry of surfaces [by extending the notion of the objects studied], extending the class of regular convex surfaces to the class of all convex surfaces ... . In order to solve concrete problems Aleksandrov had to replace the Gaussian geometry of regular surfaces by a much more general theory. In the first place the intrinsic properties (i.e. those properties that appear as a result of measurements carried out on the surface) of an arbitrary convex surface had to be studied, and methods found for the proof of theorems on the connection between intrinsic and exterior properties of convex surfaces. Aleksandrov constructed a theory of intrinsic geometry of convex surfaces on that basis. Because of the depth of this theory, the importance of its applications and the breadth of its generality, Aleksandrov comes second only to Gauss in the history of the development of the theory of surfaces. *SAU

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

## Sunday, 26 July 2015

### On This Day in Math - July 26

Mathematics is the most exact science, and its conclusions are capable of absolute proof. But this is so only because mathematics does not attempt to draw absolute conclusions. All mathematical truths are relative, conditional
~Steinmetz, Charles P.

This is the 207th day of the year; 207 is the smallest possible sum of primes which are formed using each of the digits 1 through 9 (i.e., 89 + 61 + 43 + 7 + 5 + 2 = 207) *Prime Curios (So how many such sums can there be? And which of such sums are prime?)

There are exactly 207 different matchstick graphs with eight edges ( a matchstick graph is a graph that can be drawn in the plane in such a way that its edges are line segments with length one that do not cross each other)

EVENTS
1609 Thomas Harriot was the first person to make a drawing of the Moon through a telescope, on July 26, 1609, over four months before Galileo. Factoring to solve equations was once frequently called “Harriot’s Method.” *Wik
Thony Christie points out that, "Harriot’s drawings are very primitive, mere sketches, and cannot be compared with the justifiably famous moon drawings published by Galileo Galilei in his Sidereus Nuncius from 1610. Galileo unlike Harriot was a trained artist and realised that what he was seeing through his telescope were three dimensional landscape features, mountains, valleys, etc." Thony has a great blog about the naming of the features on the moon with more great images.

1712 Brooke Taylor describes what we now call a “Taylor series” in a letter to John Machin on July 26, 1712. He would not publish about them until three years later. It would be another fifty years before the power of the method is realized by Lagrange, and another fifty before Cauchy gives a formal proof.

1732 George Berkeley gave his farm near Newport, Rhode Island to the College of New Haven [now Yale University] to endow two graduate Fellows in Greek and Latin. This was the ﬁrst provision for graduate study in America. Berkeley is known in mathematics for his Analyst (1734), which criticized the foundations of the calculus. See G. P. Conroy, “Berkeley and Education in America,” Journal for the History Ideas, 21(1960), pp. 211-221.

1766 “To your care and recommendation am I indebted for having replaced a half-blind mathematician by a mathematician with both eyes, which will especially please the anatomical members of the academy.” So wrote Frederick the Great to d’Alembert, thanking him for his suggestion of hiring Lagrange to succeed Euler at the Berlin Academy. [AMM 34(1927), p 128]

1775 Benjamin Franklin became Postmaster-General of the United States *TIS

1800 Caroline Herschel gets annual salary from George III. "William Herschel was paid £200 in annual salary as King’s Astronomer. His sister Caroline was paid £50 to act as his assistant, making her the first professional female astronomer.
A note from Herschel’s wife Mary says that the handwriting is that of King George III himself. " *sciencemuseum.org.uk

1895 Marie Sklodovska became Marie(CURIE) (1867-1934) entered the Sorbonne in 1891 and came in first in physics in 1893 and second in mathematics in 1894. She first lived with her sister and brother-in-law at 92 Avenue Jean-Jaurès, La Villette, 19e. Married Pierre Curie (1859-1906), a teacher at the École de Physique et Chimie, 42 Rue Lhomond, on 26 Jul 1895
In 1896, Marie Curie decided to investigate Henri Becquerel's discovery of the radiactivity of uranium, as a research topic for her doctoral thesis. Pierre subsequently followed her into research into radioactivity (1898), for which they were later awarded a Nobel Prize. In 1897 she gave birth to a daughter, Irène who later married Frédéric Joliot and became Irène Joliot-Curie (1926). With her husband, she continued the family's work into radioactivity. They, too, received a Nobel Prize *TIS

1976 Kenneth Appel and Wolfgang Haken of the University of Illinois communicated their solution to the Four Color Problem to the Bulletin of the American Mathematical Society. The solution used over 1000 hours of computer calculation. *VFR

1989 A federal grand jury indicts Cornell University student Robert Tappan Morris, Jr. for releasing a computer virus, making him the first person to be prosecuted under the 1986 Computer Fraud and Abuse Act in the United States. *Wik

2009 An event was held at Syon House, West London, to celebrate the 400th anniversary of Thomas Harriot's first observations of the moon. This event, Telescope400, included the unveiling of a plaque to commemorate Harriot by Lord Egremont. The plaque can now be seen by visitors to Syon House, the location of Harriot's historic observations. His drawing made 400 years earlier is believed to be based on the first ever observations of the moon through a telescope. The event (sponsored by the Royal Astronomical Society) was run as part of the International Year of Astronomy (IYA).
The original documents showing Harriot's moon map of c. 1611, observations of Jupiter's satellites, and first observations of sunspots were on display at the Science Museum, London, from 23 July 2009 until the end of IYA. *Wik

BIRTHS
1271 Zhao Youqin's name is sometimes written as Chao Yu-Chhin or Chao Yu-Ch'in. (July 26, 1271, Poyang, China— c. 1335, Longyou Mountains, Zhejiang province) He was born at a time of conflict when the Mongol leader Kublai Khan began attacking the Song Dynasty of China. The Song imperial family surrendered in 1276 and the last of the resistance was crushed in 1296. One source suggests that Zhao was injured in the fighting surrounding these dramatic events. When he was a young man he learnt astronomy and obtained a secret book on alchemy from a Daoist master. He joined the northern branch of the Quanzhen sect of Daoism and became a Daoist hermit, spending ten years writing a commentary on the Book of Changes . No trace of this commentary has survived. He later became the patriarch of the Quanzhen (Complete Perfection) School of Song-Yuan Daoism, ordained by the preceding patriarch, Zhang Mo.
Zhao Youqin was skilled in a large range of topics. He was an expert in astronomy, mathematics and physics, with particular skills in optics. He was also, however, a religious philosopher and a specialist in alchemy. Before he died he gave a copy of the manuscript of his book Ge xiang xin shu, to his disciple Zhu Hui. The manuscript was passed from Zhu Hui to Zhang Jun who published the work. *SAU

1852 Francis Robbins Upton (1852 in Peabody, Massachusetts – March 10, 1921 in Orange, New Jersey) was an American physicist and mathematician.
Upton graduated from Phillips Academy, Andover in 1870. He studied at Bowdoin College in Brunswick, Maine, at Princeton University where he received his M.S., and in Berlin, where he worked together with Hermann von Helmholtz.
In 1878, he joined the laboratory of Thomas Alva Edison in Menlo Park, New Jersey. There he dealt with technical problems in a mathematical way, including electric light, the watt-hour meter, and large dynamos. In October 1879, the first electric light was presented to the public. He was partner and general manager of the Edison Lamp Works, which he founded together with Edison in 1880. Upton published articles in Scribner's Monthly and Scientific American. Since 1958, the Princeton University has had the Francis Upton Graduate Fellowships.
In 1890, Upton patented the first electric fire alarm and detector along with a Mr. Fernando J. Dibble, an accomplishment of his which is often overlooked, stemming most probably from a typographical error that labels the device a "Portable Electric Tire-Alarm." (Google Books; U.S. Congressional Serial Set).*Wik

1863 Paul Walden was a Latvian chemist who, while teaching at Riga, discovered the Walden inversion, a reversal of stereochemical configuration that occurs in many reactions of covalent compounds (1896). Due to this discovery, Walden's name is mentioned almost in all textbooks on organic chemistry published throughout the world. Walden revealed autoracemization and put the foundations to electrochemistry of nonaqueous solutions. Walden is also known for Walden's rule, which relates the conductivity and viscosity of nonaqueous solutions.*TIS

1902 Stanisław Gołąb (July 26, 1902 – April 30, 1980) was a Polish mathematician from Kraków, working in particular on the field of affine geometry.
In 1932, he proved that the perimeter of the unit disc can take any value in between 6 and 8, and that these extremal values are obtained if and only if the unit disc is an affine regular hexagon resp. a parallelogram. *Wik

1903 Kurt Mahler (26 July 1903, Krefeld, Germany – 25 February 1988, Canberra, Australia) was a mathematician and Fellow of the Royal Society. Mahler proved that the Prouhet–Thue–Morse constant and the Champernowne constant 0.1234567891011121314151617181920... are transcendental numbers.
He was a student at the universities in Frankfurt and Göttingen, graduating with a Ph.D. from Johann Wolfgang Goethe University of Frankfurt am Main in 1927. He left Germany with the rise of Hitler and accepted an invitation by Louis Mordell to go to Manchester. He became a British citizen in 1946.
He was elected a member of the Royal Society in 1948 and a member of the Australian Academy of Science in 1965. He was awarded the London Mathematical Society's Senior Berwick Prize in 1950, the De Morgan Medal, 1971, and the Thomas Ranken Lyle Medal, 1977. *Wik

1907 Nachman Aronszajn (26 July 1907, Warsaw, Poland – 5 February 1980 Corvallis, Oregon, U.S) was a Polish American mathematician of Ashkenazi Jewish descent. Aronszajn's main field of study and expertise was mathematical analysis. He also contributed to mathematical logic.
He received his Ph.D. from the University of Warsaw, in 1930, in Poland. Stefan Mazurkiewicz was his thesis advisor. He also received a Ph.D. from Paris University, in 1935; this time Maurice Fréchet was his thesis advisor. He joined the Oklahoma A&M faculty, but moved to the University of Kansas in 1951 with his colleague Ainsley Diamond after Diamond, a quaker, was fired for refusing to sign a newly-instituted loyalty oath. Aronszajn retired in 1977. He was a Summerfield Distinguished Scholar from 1964 to his death.
He introduced, together with Prom Panitchpakdi, the injective metric spaces under the name of "hyperconvex metric spaces". Together with Kennan T. Smith, Aronszajn offered proof of the Aronszajn–Smith theorem. Also, the existence of Aronszajn trees was proven by Aronszajn; Aronszajn lines, also named after him, are the lexicographic orderings of Aronszajn trees.
He also has a fundamental contribution to the theory of reproducing kernel Hilbert space, the Moore–Aronszajn theorem is named after him. *Wik

1926 Joseph F. Engelberger (New York City, July 26, 1925 - ) American engineer who, with George Devol, developed the first industrial robot in the United States, the Unimate, in the 1950's. Engelberger is often referred to as the "Father of Robotics." When he and his partner founded Unimation in 1956, the company was the first major manufacturer of industrial robotic arms in the U.S. By 1962, they had installed their first industrial robots at the auto manufacturer, General Motors. *TIS

DEATHS

1925 Gottlob Frege (8 November 1848 – 26 July 1925) died. He was the greatest logician since Aristotle. *VFR (Friedrich Ludwig) Gottlob Frege was a German mathematician and logician, founder of modern symbolic logic and first to put forward the view that mathematics is reducible to logic. He extended Boole's work by inventing logical symbols (symbols for "or"," if-then", etc.) that improvedon the syllogistic logic it replaced. He also worked on general questions of philosophical logic and semantics. His theory of meaning, based on makig a distinction between what a linguistic term refers to and what it expresses, is still influential. Frege tried to provide a rigorous foundation for mathematics on the basis of purely logical principles, but abandoned the attempt when Bertrand Russell, on whose work he had a profound influence, pointed out a paradox that made the system inconsistent. *TIS

1941 Henri L´eon Lebesgue (June 28, 1875 – July 26, 1941) French mathematician who developed a theory of integration, now known by his name. By extending the work of Camille Jordan and Émile Borel on the Riemann integral, Lebesgue provided a generalization that solved many of the difficulties in using Riemann's theory of integration. Lebesque provided a foundation for subsequent development of integration theory and its further application in calculus, curve rectification and theory of trigonometric theory. He also contributed in several fields of mathematics, including set theory, caluclus of variation and function theory*TIS

1942 Georg Alexander Pick (August 10, 1859 – July 26, 1942) was an Austrian mathematician. He died in the Theresienstadt concentration camp. Today he is best known for Pick's formula for determining the area of lattice polygons. He published it in an article in 1899; it was popularized when Hugo Dyonizy Steinhaus included it in the 1969 edition of Mathematical Snapshots. Pick headed the committee at the (then) German university of Prague which appointed Albert Einstein to a chair of mathematical physics in 1911. Pick introduced Einstein to the work of Italian mathematicians Gregorio Ricci-Curbastro and Tullio Levi-Civita in the field of absolute differential calculus, which later in 1915 helped Einstein to successfully formulate General relativity.*Wik A really nice article about this theorem with references and interactive graphics is Found at Alexander Bogomolny's Cut The Knot web site.

1955 Raymond C Archibald (Colchester County, Nova Scotia, October 7, 1875 - July 26, 1955, in Sackville, New Brunswick) studied in Canada, at Harvard and at Strasbourg. He spent most of his career at Brown University in Rhode Island. His main interests were in the History of Mathematics. *SAU

1977 Oskar Morgenstern (January 24, 1902 – July 26, 1977) German-American economist and mathematician who popularized "game theory" which mathematically analyzes behaviour of man or animals in terms of strategies to maximize gains and minimize losses. He coauthored Theory of Games and Economic Behavior (1944), with John von Neumann, which extended Neumann's 1928 theory of games of strategy to competitive business situations. They suggested that often in a business situation ("game'), the outcome depends on several parties ("players"), each estimating what all of the others will do before determining their own strategy. Morgenstern was a professor at Vienna University, Austria, from 1931 until the Nazi occupation in 1938), when he fled to America and joined the faculty at Princeton University. His later publications included works on economic prediction and aspects of U.S. defence.*TIS

1984 George Horace Gallup (November 18, 1901 – July 26, 1984) was an American pioneer of survey sampling techniques and inventor of the Gallup poll, a successful statistical method of survey sampling for measuring public opinion.
Gallup was born in Jefferson, Iowa, the son of George Henry Gallup, a dairy farmer. His higher education took place at the University of Iowa. He served as a journalism professor at Drake and Northwestern for brief periods. In 1932 he moved to New York City to join the advertising agency of Young and Rubicam as director of research (later as vice president from 1937 to 1947). He was also professor of journalism at Columbia University, but he had to give up this position shortly after he formed his own polling company, the American Institute of Public Opinion (Gallup Poll), in 1935.
In 1936, his new organization achieved national recognition by correctly predicting, from the replies of only 50,000 respondents, that Franklin Roosevelt would defeat Alf Landon in the U.S. Presidential election. This was in direct contradiction to the widely respected Literary Digest magazine whose poll based on over two million returned questionnaires predicted that Landon would be the winner. Not only did Gallup get the election right, he correctly predicted the results of the Literary Digest poll as well using a random sample smaller than theirs but chosen to match it.
Twelve years later, his organization had its moment of greatest ignominy, when it predicted that Thomas Dewey would defeat Harry S. Truman in the 1948 election, by five to fifteen percentage points. Gallup believed the error was mostly due to ending his polling three weeks before Election Day.
Gallup died in 1984 of a heart attack at his summer home in Tschingel, a village in the Bernese Oberland of Switzerland. He was buried in Princeton Cemetery. *Wik

1997 Kunihiko Kodaira (16 March 1915 – 26 July 1997) Japanese mathematician who was awarded the Fields Medal in 1954 for his work in algebraic geometry and complex analysis. Kodaira's work includes applications of Hilbert space methods to differential equations which was an important topic in his early work and was largely the result of influence by Weyl. Through the influence of Hodge, he also worked on harmonic integrals and later he applied this work to problem in algebraic geometry. Another important area of Kodaira's work was to apply sheaves to algebraic geometry. In around 1960 he became involved in the classification of compact, complex analytic spaces. One of the themes running through much of his work is the Riemann-Roch theorem. He won the 1985 Wolf Prize. *TIS

2000 John Wilder Tukey (June 16, 1915 – July 26, 2000) was an American statistician. He was awarded the IEEE Medal of Honor in 1982 "For his contributions to the spectral analysis of random processes and the fast Fourier transform (FFT) algorithm."
Tukey retired in 1985. He died in New Brunswick, New Jersey Tukey coined many statistical terms that have become part of common usage, but the two most famous coinages attributed to him were related to computer science.
While working with John von Neumann on early computer designs, Tukey introduced the word "bit" as a contraction of "binary digit". The term "bit" was first used in an article by Claude Shannon in 1948.
The term "software", which Paul Niquette claims he coined in 1953, was first used in print by Tukey in a 1958 article in American Mathematical Monthly, and thus some attribute the term to him.
In the fall of 2003 a post to the APStats electronic discussion list from Ron Dirkse pointed out that the Japanese word for statistics, toukei, sounds very much like the name of the famous American statistician John Tukey. Ron Dirkse, who taught at the American School in Japan, added that "according to a native speaker the tou means something like 'put together' and the kei is 'measure, calculate or total'. She thought it was interesting that there was a Tukey famous in statistics, but this word pre-dates him by a lot."

Other terms credited to Tukey below are from http://www.stat.berkeley.edu/~brill/Papers/life.pdf That site Also includes a list of his honors, and Ph.D. students
alias (in time series)
ANOVA
bagplot
batch
bispectrum
bit
biweight
bland distribution
borrowing strength
boxplot
cepstrum
coco
complex demodulation
conrmatory data analysis (CDA).

2004 - William A. Mitchell died (October 21, 1911 – July 26, 2004). Mitchell was an American food chemist who was the inventor of Pop Rocks, instant Jell-O, Cool Whip and the orange drink, Tang. While working for the General Foods Corporation, he received over 70 patents.
Pop Rocks were the center of an urban legend where the kid from the Life cereal commercials died when he ate the candy and washed it down with a cola making his stomach explode. General Foods countered the claims with an ad campaign in 45 major publications and 50,000 letters to school principals. Mitchell toured the country to show people that Pop Rocks weren't dangerous. *Science History

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