You, who wish to study great and wonderful things,

who wonder about the movement of the stars,

must read these theorems about triangles.

Knowing these ideas will open the door to all of astronomy

and to certain geometric problems.

**Johann Regiomontanus**

*De Tringulis Omnimodis*

The 188th day of the year; 188 is the largest known even number that can be expressed as the sum of two (distinct) primes in exactly five ways. *Prime Curios

*Students might seek smaller numbers that can be so expressed.*

**EVENTS**

1656 Huygen, in a letter to Carcavi, gives the solution, without proof, to a dice throwing probability problem posed by Fermat. (If A wins for Throwing a six, and B wins by throwing a seven, and after A throws once, B and A each roll twice in turns. What are the odds of A winning?) *A History of Probability and Statistics and Their Applications Before 1750 By Anders Hald

1708, de Moivre wrote again to Johann Bernoulli about Machin's series, on this occasion giving two different proofs that it converged to π. In 1706 William Jones had published a work Synopsis palmariorum matheseos or, A New Introduction to the Mathematics, Containing the Principles of Arithmetic and Geometry Demonstrated in a Short and Easie Method ... Designed for ... Beginners. (This is the book in which Jones first uses Pi in the mathematical sense it is now used) This contains on page 243 the following passage:-

There are various other ways of finding the lengths or areas of particular curve lines, or planes, which may very much facilitate the practice; as for instance, in the circle, the diameter is to the circumference as 1 to (16/5- 4/239) - 1/3(16/53- 4/2393) &c. = 3.14159 &c. = π. This series (among others for the same purpose, and drawn from the same principle) I received from the excellent analyst, and my much esteemed friend Mr John Machin; and by means thereof, van Ceulen's number, or that in Art. 64.38 may be examined with all desirable ease and dispatch.

Jones also reports that this formula allows π be calculated:-

... to above 100 places; as computed by the accurate and ready pen of the truly ingenious Mr John Machin.

No indication is given in Jones's work, however, as to how Machin discovered his series expansion for π so when de Moivre wrote to Johann Bernoulli on 8 July 1706 telling him about Machin's series for π he suggested that Johann Bernoulli might tell Jakob Hermann about Machin's unproved result. He did so and Hermann quickly discovered a proof that Machin's series converges to π. He produced techniques that show other similar series also converge rapidly to π and he wrote on 21 August 1706 to Leibniz giving details. Two years later, on 6 July 1708, de Moivre wrote again to Johann Bernoulli about Machin's series, on this occasion giving two different proofs that it converged to π.

1785 The Continental Congress of the United States adopted the decimal system of money with the dollar as unit.

1815 Total solar eclipse on the North Pole.

1819 When Marie Madeleine Sopie Blanchard ascended in her hydrogen balloon to watch a firework, she would become the first woman to lose her life while flying. Her craft crashed on the roof of a house and she fell to her death. *yovisto.

1909 Einstein resigns his position at the Bern Patent Office to move to Zurich to take up his first full-time academic position in the newly established chair of theoretical physics at the University of Zurich. His lectures were extremely popular due to his humor, unusual presentations, patience and accommodation to his students to make sure they understood. *Brody & Brody, The Science Class You Wish You Had

In 1920, a radio compass was used for first time for aircraft navigation. In a test of the radio compass as an aid to navigation, an F5L left Hampton Roads and flew directly to the battleship Ohio (BB 12), 94 miles at sea in a position unknown to the pilot. Without landing, the plane made the return trip to Hampton Roads, this time navigating by signals from Norfolk. *TIS

**1849 Alfred Bray Kempe**(6 July 1849, Kensington, London – 21 April 1922, London) published a false "proof" of the four color theorem in 1879 which stood until Heawood showed the mistake 11 years later. The 'proof' is however still the basis for the computer aided proof discovered 100 years later.*SAU

**1883 Ernst Arnold Kohlschütter (July 6, 1883 – May 28, 1969) a German astronomer and astrophysicist from Halle.**

In 1908 he was awarded his Ph.D. from the University of Göttingen.

In 1911 he began working at the Mount Wilson Observatory, studying the spectra of the Sun and stars. In collaboration with Walter Sidney Adams, and in 1914 they discovered that the absolute luminosity of a star was proportional to the relative intensity of the lines in the spectrum. This allowed astronomers to determine the distance of stars, including main sequence and giants, using the spectroscope.

He became the director of the Bonn observatory in 1925. Therein he was dedicated to astrometric studies.

The crater Kohlschütter on the Moon is named in his honor. *Today in Astronomy

In 1908 he was awarded his Ph.D. from the University of Göttingen.

In 1911 he began working at the Mount Wilson Observatory, studying the spectra of the Sun and stars. In collaboration with Walter Sidney Adams, and in 1914 they discovered that the absolute luminosity of a star was proportional to the relative intensity of the lines in the spectrum. This allowed astronomers to determine the distance of stars, including main sequence and giants, using the spectroscope.

He became the director of the Bonn observatory in 1925. Therein he was dedicated to astrometric studies.

The crater Kohlschütter on the Moon is named in his honor. *Today in Astronomy

**1910 Lothar Collatz**(July 6, 1910, Arnsberg, Westphalia – September 26, 1990, Varna, Bulgaria) was a German mathematician. In 1937 he posed the famous Collatz conjecture, which remains unsolved.

The

**Collatz conjecture**is also known as the

**3**, the

*n*+ 1 conjecture**Ulam conjecture**(after Stanisław Ulam),

**Kakutani's problem**(after Shizuo Kakutani), the

**Thwaites conjecture**(after Sir Bryan Thwaites),

**Hasse's algorithm**(after Helmut Hasse), or the

**Syracuse problem**; the sequence of numbers involved is referred to as the

**hailstone sequence**or

**hailstone numbers**, or as

**wondrous numbers**.

Take any natural number

*n*. If

*n*is even, divide it by 2 to get

*n*/ 2, if

*n*is odd multiply it by 3 and add 1 to obtain 3

*n*+ 1. Repeat the process (which has been called "Half Or Triple Plus One", or

**HOTPO**

^{}) indefinitely. The conjecture is that no matter what number you start with, you will always eventually reach 1. The property has also been called

**oneness**.

^{ }

Paul Erdős said about the Collatz conjecture: "Mathematics is not yet ready for such problems." and also offered $500 for its solution.

^{}

In 2006, researchers Kurtz and Simon, building on earlier work by J.H. Conway in the 1970s,

^{}proved that a natural generalization of the Collatz problem is undecidable.

^{}However, as this proof depends upon the generalization, it cannot be applied to the original Collatz problem. *Wik

**DEATHS**

**1476 Regiomontanus, aka**

**Johann Mueller**, (6 Jun 1436, 6 Jul 1476 at age 40)the father of trigonometry as a science independent of astronomy. According to a rumor repeated by Gassendi in his Regiomontanus biography he was assassinated by relatives of George of Trebizond whom he had criticized in his writings. More likely he died in an epidemic raging in Rome at the time.The ideas behind the law of sines, like those of the law of cosines, predate the word sine by over a thousand years. Theorems in Euclid on lengths of chords are essentially the same ideas we now call the law of sines. The law of sines for plane triangles was known to Ptolemy and by the tenth century Abu'l Wefa had clearly expounded the spherical law of sines. It seems that the term "law of sines" was applied sometime near 1850, but I am unsure of the origin of the phrase.

The spherical law of sines was first presented by Johann Muller in his De Triangulis Omnimodis in 1464. This was the first book devoted wholly to trigonometry (a word not then invented). David E. Smith suggests that the theorem was Muller's invention.

The German astronomer and mathematician who was chiefly responsible for the revival and advancement of trigonometry in Europe. His book

In Jan 1472 Muller made observations of a comet which were accurate enough to allow it to be identified with Halley's comet 210 years later (being three returns of the 70 year period comet). He also observed several eclipses of the Moon. His interest in the motion of the Moon led him to make the important observation that the method of lunar distances could be used to determine longitude at sea. However, instruments of the time lacked the necessary accuracy to use the method at sea. *TIS

Thony C who writes the excellent history blog, The Renaissance Mathematicus, noted, "However at least in the case of Regiomontanus appearances are deceptive; what we have here is a date of death that is anything but certain.". See his explanation of the remark here.

The title page of "On Triangles" by Regiomontanus is here. Although the work was written in 1464, it was not published until 1533.

*De triangulis omnimodis*(1464) is a systematic account of methods for solving triangles. Much of the material on spherical trigonometry in Regiomontanus'*On Triangles*was taken directly and without credit from the twelfth-century work of Jabir ibn Aflah otherwise known as Geber, as noted in the sixteenth century by Gerolamo Cardano.In Jan 1472 Muller made observations of a comet which were accurate enough to allow it to be identified with Halley's comet 210 years later (being three returns of the 70 year period comet). He also observed several eclipses of the Moon. His interest in the motion of the Moon led him to make the important observation that the method of lunar distances could be used to determine longitude at sea. However, instruments of the time lacked the necessary accuracy to use the method at sea. *TIS

Thony C who writes the excellent history blog, The Renaissance Mathematicus, noted, "However at least in the case of Regiomontanus appearances are deceptive; what we have here is a date of death that is anything but certain.". See his explanation of the remark here.

The title page of "On Triangles" by Regiomontanus is here. Although the work was written in 1464, it was not published until 1533.

**1854**

**Georg Simon Ohm**(17 March 1789 – 6 July 1854) was a German physicist. As a high school teacher, Ohm began his research with the recently invented electrochemical cell, invented by Italian Count Alessandro Volta. Using equipment of his own creation, Ohm determined that there is a direct proportionality between the potential difference (voltage) applied across a conductor and the resultant electric current. This relationship is now known as Ohm's law.*Wik

1915 Lawrence Hargrave(29 Jan 1850, 6 Jul 1915 at age 65) Australian aeronaut and inventor best known for his invention of the box kite. Hargrave “flew” on 12 Nov 1894, by attaching himself to a huge four kite construction attached to the ground by piano wire. Due to their abilities to carry heavy payloads, steady flight, and capacity for high altitude flight, these kites have had many industrial and military uses in the past. Box kites were used until the 1930's to carry meteorological equipment for high altitude weather studies and by the Royal Air Force as sea rescue equipment to deliver radio aerials. Hargrave also made important studies of wing surfaces and worked with rotary engines and gliders. *TIS

Credits

*CHM=Computer History Museum

*FFF=Kane, Famous First Facts

*NSEC= NASA Solar Eclipse Calendar

*SAU=St Andrews Univ. Math History

*TIS= Today in Science History

*VFR = V Frederick Rickey, USMA

*Wik = Wikipedia

*WM = Women of Mathematics, Grinstein & Campbell