Amsler Polarplanimeter

**Everyone makes for himself a clear idea of the motion of a point, that is to say, of the motion of a corpuscle which one supposes to be infinitely small, and which one reduces by thought in some way to a mathematical point.**

~Louis Poinsot

The 3rd day of the year; 3 is the only prime followed by a square, and every positive integer is the sum of at most 3 triangular numbers.

**EVENTS**

**1657**Fermat made a challenge to the mathematicians of Europe and England. He posed two problems (in words rather than using notation as we shall do) involving

*S*(

*n*), the sum of the proper divisors of

*n*:

1. Find a cube

*n*such that

*n*+

*S*(

*n*) is a square.

2. Find a square

*n*such that

*n*+

*S*(

*n*) is a cube.

We know that Frenicle found four solutions to the first of these problems on the day that he was given the problem, and found another six solutions the next day. He gave solutions to both problems in

*Solutio duorm problematum ...*(1657). In this work he posed some problems of his own, including the following:

Find an integer

*n*such that

*S*(

*n*) = 5

*n*, and

*S*(5

*n*) = 25

*n*.

Find an integer

*n*such that

*S*(

*n*) = 7

*n*, and

*S*(7

*n*) = 49

*n*.

Find

*n*such that

*n*

^{3}- (

*n*-1)

^{3}is a cube.

Frenicle solved other problems posed by Fermat. For example he showed that if a right angled triangle has sides integers

*a*,

*b*,

*c*then its area

*bc*/2 can never be a square. He also showed that the area of a right angled triangle is never twice a square. *SAU

**1825**In 1824, the Renssalaer School was founded in Troy, N.Y., by Stephen van Renssalaer becoming the first engineering college in the U.S. It opened on 3 Jan 1825, with the purpose of instructing persons, who may choose to apply themselves, in the application of science to the common purposes of life." The first class of 10 students graduated on 26 Apr 1826. The first director and senior professor was Amos Eaton who served from Nov 1824 - 10 May 1842. The name of Renssalaer Institute was adopted on 26 Apr 1832, and Renssalaer Polytechnic Institute on 8 Apr 1861.*TIS

**1851**In the basement of his Paris home at the corner of rue de Vaugirard and rud d’Assas Foucault first attempts to observe the turning of the earth on its axis with a pendulum. He mounts a 5 Kg bob on a two meter wire, and only observes the wire snap and the bob fall to the floor of the basement. Three days later he would try again, with much better results. *Amir Aczel, Pendulum, pg 5-7 (It seems that as early as 1661, Vincenzo Viviani, a student of Galileo had written, “we observe that all pendulums hanging on a single thread deviate from their initial vertical plane, and always in the same direction.” This document was discovered after Foucault’s results were released in an 1841 manuscript in the library of the Grand Duke of Tuscany.)

In

**1919**, Professor Ernest Rutherford succeeded in splitting the atom. By bombarding nitrogen atoms with alpha particles emitted by radioactive materials he transmuted the nitrogen atoms into oxygen.*TIS

**1956**Israel issued the world’s ﬁrst postage stamp picturing Albert Einstein, the German born Amer-ican theoretical physicist who invented the theory of Relativity. Naturally his famous equation E = mc

^{2}appears on the stamp. [Scott #117] *VFR

In

**1970**, a fireball was visible over a large area of the U.S. midwest. The meteorite that fell was the first to be detected by the Prairie Network operated by the Smithsonian Institution's Astrophysical Observatory since 1964. Its path was photographed by two of the system's 16 cameras funded by a NASA grant. Using these records, scientists calculated the meteorite's impact point. Gunther Schwartz, field manager of the network found the 21.6-lb meteorite six days later within a half-mile of the predicted site, near the rural hamlet Lost City, about 45 miles east of Tulsa, OK. The fast retrieval enabled examination of radioactivity produced by the meteorite's exposure to cosmic rays, looking for clues to how the universe was created.*TIS

**1977**Apple Computer Corporation is incorporated by Stephen Jobs and Stephen Wozniak. Its IPO, which took place three years later, was the largest one since the Ford Motor Ccompany went public in 1956. The stock rose almost 32% that day giving the company a market valuation of $1.778 billion. Seven years later, on January 24, 1984, the company revealed the Macintosh personal computer in a publicity campaign that compared IBM with Big Brother and Apple as the savior of the masses.*CHM

**1983**TIME magazine alters its annual tradition of naming a "Man of the Year," choosing instead to name the computer its "Machine of the Year." In introducing the theme, Time publisher John A. Meyers wrote: "Several human candidates might have represented 1982, but none symbolized the past year more richly, or will be viewed by history as more significant, than a machine: the computer." *CHM (This is the date of the issue. The decision had been released earlier, it seems)

**BIRTHS**

**1777 Louis Poinsot**(1777–1859) was a French mathematician and physicist. Poinsot was the inventor of geometrical mechanics, showing how a system of forces acting on a rigid body could be resolved into a single force and a couple. The crater Poinsot on the Moon is named after him. A street in Paris is called Rue Poinsot (14th Arrondissement). When Gustave Eiffel built the famous tower, he included the names of 72 prominent French scientists on plaques around the first stage, Poinsot included. *Wik

He discovered four new regular polyhedra, two of which appear in Kepler's work of 1619 but Poinsot was unaware of this. *SAU and is a discoverer of star polyhedra.*VFR

**1906 William Wilson Morgan**(3 Jan 1906, 21 Jun 1994) American astronomer who, in 1951, provided the first evidence that the Milky Way Galaxy has spiral arms. He spent his entire career at the Yerkes Observatory, including three years as director. Eschewing theory, his research was devoted to morphology, the classification of objects by their form and structure. With Keenan and Kellman, he introduced stellar luminosity classes and the two-dimensional classification of stellar spectra strictly based on the spectra themselves. With Osterbrock and Sharpless he demonstrated the existence of spiral arms in the Galaxy using precise distances of O and B stars obtained from spectral classifications. Morgan invented the UBV system of magnitudes and colors.*TIS

**1917 Yurii Alekseevich Mitropolskiy**(3 January 1917 — 14 June 2008) was a renowned Soviet, Ukrainian mathematician known for his contributions to the fields of dynamical systems and nonlinear oscillations.*Wik

**1921 Jean-Louis Koszul**(born January 3, 1921) is a mathematician best known for studying geometry and discovering the Koszul complex. *SAU

**DEATHS**

**1641 Jeremiah Horrocks**(born c. 1617, 3 Jan 1641) English astronomer and clergyman who applied Johannes Kepler's laws of planetary motion to observations of the Moon and Venus. Once Horrocks managed to obtain a small telescope, his observations convinced him that Lansberg's tables were incorrect. He accepted Kepler's elliptical orbits, and in working on the moon he applied an elliptical orbit to it and established that the line of apsides precessed, an effect which he ascribed to the influence of the sun. Horrocks predicted and observed a transit of Venus on 24 Nov 1639, the first one ever observed, and from the observation he corrected the solar parallax, indicating a much greater distance of the sun than anyone before him had admitted. He died at age only 22.*TIS

**1858 Henri-Philibert-Gaspard Darcy**(10 Jun 1803, 3 Jan 1858) French hydraulic engineer who first derived the equation (now known as Darcy's law) that governs the laminar (nonturbulent) flow of fluids in homogeneous, porous media. In 1856, modern studies of groundwater began when Darcy was commissioned to develop a water-purification system for the city of Dijon, France. He constructed the first experimental apparatus to study the flow characteristics of water through the earth. From his experiments, he derived the Darcy's Law equation, describing the flow of water in nature, which is fundamental to understanding groundwater systems. He performed extensive tests on filtration and pipe resistance. He initiated the open-channel studies carried out by Bazin.*TIS

**1891 John Casey**(12 May 1820 in Coolattin, Kilbehenny, Co. Limerick, Ireland - 3 Jan 1891 in Dublin, IrelandCasey wrote over 25 research papers but his mathematical reputation rests on the six textbooks he wrote: A sequel to the first six books of the Elements of Euclid (1881; 8th ed. 1910); A treatise on the analytical geometry of the point, line, circle and conic sections (1885; 2nd ed. 1893); A treatise on elementary trigonometry (1886); A treatise on plane trigonometry (1888); A treatise on spherical trigonometry (1889). He also published his own edition of The first six books of the Elements of Euclid (1882; 17th ed. 1902) which is remarkable for its large store of exercises, collected and devised by himself and Richard Townsend, which occasioned the publication of a separate Key to the exercises of Casey's Elements of Euclid (1885) by Casey's son, Joseph. It was in his Sequel to Euclid that Casey presented for the first time in a textbook those extensions of the theorems of Euclid that became known as the newer geometry of the triangle; indeed, he and the French mathematician Émile Lemoine (1840-1912) are held to be the founders of the so-called Modern Geometry of the circle and triangle. Casey's work was much appreciated in Belgium and France, and Professor Joseph Neuberg of Liège made substantial additions to later editions of Sequel to Euclid.*SAU

**1892 Heinrich Eduard Schroeter**(January 8th 1829 in Königsberg , January 3 1892 in Breslau ) was a German mathematician , who worked in synthetic geometry in the tradition of Jacob Steiner. *Wik

**1908 Charles Augustus Young**(15 Dec 1834, 3 Jan 1908) American astronomer who made the first observations of the flash spectrum of the Sun, proved the gaseous nature of the sun's corona and discovered the reversing layer of the solar atmosphere. He was a pioneer in the study of the spectrum of the sun and experimented in photographing solar prominences in full sunlight. On 22 Dec 1870, at the eclipse in Spain, he saw the lines of the solar spectrum all become bright for perhaps a second and a half (the "flash spectrum") and announced the "reversing layer." By exploring from the high altitude of Sherman, Wy. (1872), he more than doubled the number of bright lines he had observed in the chromosphere, By a comparison of observations, he concluded that magnetic conditions on the earth respond to solar disturbances.*TIS

**1912 Jacob Amsler**(16 Nov 1823 in Stalden bei Brugg, Switzerland - 3 Jan 1912 in Schaffhausen, Switzerland)worked on a problem which had quite a famous history. That was the problem of the attraction of an ellipsoid, which was first studied in depth by Ivory whose solution was later generalised by Poisson. Amsler extended the theorems of both Ivory and Poisson on this topic. It was a promising start to his research career in mathematical physics.

In 1854 Amsler married and this may have been the turning point in his career. His wife, Elsie Laffon, was the daughter of a well known Swiss scientist and, as was the custom in Switzerland at that time, he was known as Amsler-Laffon from this time on. Elsie and Jacob Amsler-Laffon's children were, however, always known by the name Amsler rather than Amsler-Laffon.

Shortly after his marriage Amsler changed his research interests and his career. He began to study the construction of precision mathematical instruments and quite quickly he had an idea for the design of a new type of planimeter. He invented the polar planimeter, a device for measuring areas enclosed by plane curves. It was based on polar coordinates whereas earlier instruments were based on cartesian coordinates. In 1856 Amsler published a paper Über das Planimeter in which he gave details of his idea. As Mahoney writes in , Amsler's planimeter, "... adapted easily to the determination of static and inertial moments and to the coefficients of Fourier series: it proved especially useful to shipbuilders and railway engineers. "

In order to make money from his invention, Amsler set up a workshop in Schaffhausen in 1854 specially designed to produce his polar planimeter. Three years later he had given up al his other interests to concentrate fully on producing instruments in the workshop. His shop produced 50 000 such instruments during his lifetime.

Amsler did not rest his fame on this single inspired idea but continued to invent new precision instruments. None of his other inventions came close to the polar planimeter in importance, but they were of sufficient quality to win him prizes at the world exhibition at Vienna in 1873, at Paris in 1881, and again in Paris in 1889. His brilliance was recognised with election to the Paris Académie des Sciences in 1892. *SAU

**1920 Zygmunt Janiszewski**, the father of Polish mathematics, died. At the end of World War I, Janiszewski was the driving force behind the creation of one of the strongest schools of mathematics in the world. This is all the more remarkable, given Poland's difficult situaltion at war's end.

Janiszewski devoted the family property that he had inherited from his father to charity and education. He also donated all the prize money that he received from mathematical awards and competitions to the education and development of young Polish students.

In mathematics, his main interest was topology.

He was the driving force, together with Wacław Sierpiński and Stefan Mazurkiewicz, behind the founding of the mathematics journal Fundamenta Mathematicae. Janiszewski proposed the name of the journal in 1919, though the first issue was published in 1920, after his death. It was his intent that the first issue comprise solely contributions by Polish mathematicians. It was Janiszewski's vision that Poland become a world leader in the field of mathematics—which she did in the interbellum.

His life was cut short by the influenza pandemic of 1918-19, which took his life at Lwów on 3 January 1920 at the age of 31. He willed his body for medical research, and his cranium for craniological study, desiring to be "useful after his death". *Wik

**1927 Carl David Tolmé Runge**(30 Aug 1856 in Bremen, Germany - 3 Jan 1927 in Göttingen, Germany) worked on a procedure for the numerical solution of algebraic equations and later studied the wavelengths of the spectral lines of elements. *SAU In numerical analysis, the Runge–Kutta methods that are named for him are an important family of implicit and explicit iterative methods for the approximation of solutions of ordinary differential equations. These techniques were developed around 1900 Runge and M.W. Kutta.*Wik When your regular walking partners include Felix Klein, David Hilbert, and Hermann Minkowski, you can't count on easily impressing them with your mental math skills, but it seems that Runge did so frequently. Once on their regular walks Klein brought up some departmental event that required them to know what date Easter would occur the next year. The group immediately turned to the idea of where they might acquire a calendar for the following year along the walk; all that is, except Runge who fell silent for a few yards, and then announced the date.

**1967 Reginald Crundall Punnett**(20 Jun 1875, 3 Jan 1967) English Mendelian geneticist who, with the English biologist William Bateson, were among the first English geneticists. They reported the discovery of two new genetic principles: the first account of genetic linkage in sweet pea; and gene interaction (1905). Punnett devised the "Punnett" square to depict the number and variety of genetic combinations. Punnett had a role in connecting Mendelism with statistics. In 1908, Punnett was asked at a lecture to explain, " if brown eyes were dominant, then why wasn't the whole country becoming brown-eyed?" Punnett in turn asked his friend the mathematician, G. H. Hardy. Out of this conversation came the Hardy-Weinberg Law which calculates how population affects genetic inheritance.*TIS

**1989 Sergei Lvovich Sobolev**(Russian: Серге́й Льво́вич Со́болев; 6 October 1908 – 3 January 1989) was a Soviet mathematician working in mathematical analysis and partial differential equations. He was born in St. Petersburg, and died in Moscow.*Wik

Credits

*VFR = V Frederick Rickey, USMA

*TIS= Today in Science History

*Wik = Wikipedia

*SAU=St Andrews Univ. Math History

*CHM=Computer History Museum

*FFF=Kane, Famous First Facts