Young man, if I could remember the names of these particles, I would have been a botanist.
~Enrico Fermi
The 272nd day of the year; 272 = 24·17, and is the sum of four consecutive primes (61 + 67 + 71 + 73).
272 is also a Pronic or Heteromecic number, the product of two consecutive factors, 16x17 (which makes it twice a triangular #).
And 272 is a palindrome, and the sum of its digits, 11, is also a palindrome. (can you find the next?) The product of its digits is a perfect number.
EVENTS
1609 Almost exactly a year after the first application for a patent of the telescope, Giambaptista della Porta, the Neapolitan polymath, whose Magia Naturalis of 1589, well known all over Europe, because of a tantalizing hint at what might be accomplished by a combination of a convex and concave lens: ‘With a concave you shall see small things afar off, very clearly; with a convex, things neerer to be greater, but more obscurely: if you know how to fit them both together, you shall see both things afar off, and things neer hand, both greater and clearly.’sends a letter to the founder of the Accademia dei Lincei, Prince Federico Cesi in Rome, with a sketch of an instrument that had just reached him, and he wrote:" It is a small tube of soldered silver, one palm in length, and three finger breadths in diameter, which has a convex glass in the end. There is another tube of the same material four finger breadths long, which enters into the first one, and in the end. It has a concave [glass], which is secured like the first one. If observed with that first tube, faraway things are seen as if they were near, but because the vision does not occur along the perpendicular, they appear obscure and indistinct. When the other concave tube, which produces the opposite effect, is inserted, things will be seen clear and erect and it goes in an out, as in a trombone, so that it adjusts to the eyesight of [particular] observers, which all differ. *Albert Van Helden, Galileo and the telescope; Origins of the Telescope, Royal Netherlands Academy of Arts andSciences, 2010
(I assume that we can safely date the invention of the trombone prior to 1609 also)
1762 John Winthrop writes Benjamin Franklin about an article Franklin had written in 1754:
"Cambridge, N.E. Sept. 29, 1762.
Sir,
There is an observation relating to electricity in the atmosphere, which seemed new to me, though perhaps it will not to you: However, I will venture to mention it. I have some points on the top of my house, and the wire where it passes within-side the house is furnished with bells, according to your method, to give notice of the passage of the electric fluid. In summer, these bells generally ring at the approach of a thunder cloud; but cease soon after it begins to rain. In winter, they sometimes, though not very often, ring while it is snowing; but never, that I remember, when it rains. But what was unexpected to me was, that, though the bells had not rung while it was snowing, yet the next day, after it had done snowing, and the weather was cleared up; while the snow was driven about by a high wind at W. or N.W. the bells rung for several hours (though with little intermissions) as briskly as ever I knew them, and I drew considerable sparks from the wire. This phaenomenon I never observed but twice; viz. on the 31st of January, 1760, and the 3d of March, 1762.
I am, Sir, &c."
**In September 1753 Franklin wrote Collinson that a year earlier he had “erected an Iron Rod to draw the Lightning down into my House, in order to make some Experiments on it, with two Bells to give Notice when the Rod should be electrified.” That letter was published in Exper. and Obser., 1754 edit., and Winthrop probably read it there. *Founder's Archives
1801 Gauss’s Disquisitiones Arithmeticae published. It is a textbook of number theory written in Latin by Carl Friedrich Gauss in 1798 when Gauss was 21 and first published in 1801 when he was 24. In this book Gauss brings together results in number theory obtained by mathematicians such as Fermat, Euler, Lagrange and Legendre and adds important new results of his own.
The book is divided into seven sections, which are :
Section I. Congruent Numbers in General
Section II. Congruences of the First Degree
Section III. Residues of Powers
Section IV. Congruences of the Second Degree
Section V. Forms and Indeterminate Equations of the Second Degree
Section VI. Various Applications of the Preceding Discussions
Section VII. Equations Defining Sections of a Circle.
Sections I to III are essentially a review of previous results, including Fermat's little theorem, Wilson's theorem and the existence of primitive roots. Although few of the results in these first sections are original, Gauss was the first mathematician to bring this material together and treat it in a systematic way. He was also the first mathematician to realize the importance of the property of unique factorization (sometimes called the fundamental theorem of arithmetic), which he states and proves explicitly.
From Section IV onwards, much of the work is original. Section IV itself develops a proof of quadratic reciprocity; Section V, which takes up over half of the book, is a comprehensive analysis of binary quadratic forms; and Section VI includes two different primality tests. Finally, Section VII is an analysis of cyclotomic polynomials, which concludes by giving the criteria that determine which regular polygons are constructible i.e. can be constructed with a compass and unmarked straight edge alone. *Wik
Gauss was so pleased with his geometric construction of the regular heptadecagon (17-gon) that the design for his memorial in Brunswick have the shape engraved on the pedestal. This proof represented the first progress in regular polygon construction in over 2000 years. During the construction, the heptadecagon looked so much like a circle that a seventeen pointed star was used instead.
1954 The European Organization for Nuclear Research (CERN) was officially established.The organisation runs the world’s largest particle physics laboratory in Geneva, Switzerland. In September 2011 CERN scientists reported that some particles appeared to be travelling faster than light, although it’s now thought that the experiment was flawed. *rsc.org
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| Cern Aerial *wik |
In 1988, the space shuttle Discovery blasted off from Cape Canaveral, Fla., marking America's return to manned space flight following the Challenger disaster. *TIS
1994 HotJava ---- Programmers first demonstrated the HotJava prototype to executives at Sun Microsystems Inc. A browser making use of Java technology, HotJava attempted to transfer Sun's new programming platform for use on the World Wide Web. Java is based on the concept of being truly universal, allowing an application written in the language to be used on a computer with any type of operating system or on the web, televisions or telephones.*CHM
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| HotJava 3.0 under Windows XP. *Wik |
2001 On the 100th anniversary of his birth, the scientist who was known in his life as "The Pope" because he made so few mistakes, was memorialized on the occasion of the centennial of his birth with a stamp...that captured a mental mistake in a photo from 1948. In the upper left of the stamp, the symbols h and e are interchanged. The equation a= h^2/(e*c) should read a= e^2 /(h*c). Whether this was a prop written by someone else, or a mental mistake by his own hand, it remains a tongue in cheek memorial to the great, but fallible, scientist. Something to think about as your tongue comes out to lick a stamp next time, check the address, we all make mistakes.
BIRTHS
1511 (or 1509) Michael Servetus (/sərˈviːtəs/; Spanish: Miguel Serveto as real name, French: Michel Servet), also known as Miguel Servet, Miguel de Villanueva, Michel Servet, Revés, or Michel de Villeneuve (Tudela, Navarre, 29 September 1511 – 27 October 1553), was a Spanish theologian, physician, cartographer, and Renaissance humanist. He was the first European to correctly describe the function of pulmonary circulation, as discussed in Christianismi Restitutio (1553). He was a polymath versed in many sciences: mathematics, astronomy and meteorology, geography, human anatomy, medicine and pharmacology, as well as jurisprudence, translation, poetry and the scholarly study of the Bible in its original languages.
He is renowned in the history of several of these fields, particularly medicine. He participated in the Protestant Reformation, and later rejected the Trinity doctrine and mainstream Catholic Christology. After being condemned by Catholic authorities in France, he fled to Calvinist Geneva where he was burnt at the stake for heresy by order of the city's governing council.
1561 Adriaan van Roomen (29 Sept 1561 , 4 May 1615) is often known by his Latin name Adrianus Romanus. After studying at the Jesuit College in Cologne, Roomen studied medicine at Louvain. He then spent some time in Italy, particularly with Clavius in Rome in 1585.
Roomen was professor of mathematics and medicine at Louvain from 1586 to 1592, he then went to Würzburg where again he was professor of medicine. He was also "Mathematician to the Chapter" in Würzburg. From 1603 to 1610 he lived frequently in both Louvain and Würzburg. He was ordained a priest in 1604. After 1610 he tutored mathematics in Poland.
One of Roomen's most impressive results was finding π to 16 decimal places. He did this in 1593 using 230 sided polygons. Roomen's interest in π was almost certainly as a result of his friendship with Ludolph van Ceulen.
Roomen proposed a problem which involved solving an equation of degree 45. The problem was solved by Viète who realised that there was an underlying trigonometric relation. After this a friendship grew up between the two men. Viète proposed the problem of drawing a circle to touch 3 given circles to Roomen (the Apollonian Problem) and Roomen solved it using hyperbolas, publishing the result in 1596.
Roomen worked on trigonometry and the calculation of chords in a circle. In 1596 Rheticus's trigonometric tables Opus palatinum de triangulis were published, many years after Rheticus died. Roomen was critical of the accuracy of the tables and wrote to Clavius at the Collegio Romano in Rome pointing out that, to calculate tangent and secant tables correctly to ten decimal places, it was necessary to work to 20 decimal places for small values of sine, see [2]. In 1600 Roomen visited Prague where he met Kepler and told him of his worries about the methods employed in Rheticus's trigonometric tables. *SAU
Figure 2: Page in Ideae mathematicae,1593, Adriaan van Roomen.
1803 Jacques Charles-François Sturm (29 Sep 1803; 18 Dec 1855) French mathematician whose work resulted in Sturm's theorem, an important contribution to the theory of equations. .While a tutor of the de Broglie family in Paris (1823-24), Sturm met many of the leading French scientists and mathematicians. In 1826, with Swiss engineer Daniel Colladon, he made the first accurate determination of the velocity of sound in water. A year later wrote a prizewinning essay on compressible fluids. Since the time of René Descartes, a problem had existed of finding the number of solutions of a given second-order differential equation within a given range of the variable. Sturm provided a complete solution to the problem with his theorem which first appeared in Mémoire sur la résolution des équations numériques (1829; “Treatise on Numerical Equations”). Those principles have been applied in the development of quantum mechanics, as in the solution of the Schrödinger equation and its boundary values. *TIS Sturm is also remembered for the Sturm-Liouville problem, an eigenvalue problem in second order differential equations.*SAU
1812 Gustav Adolph Göpel (29 Sept 1812, 7 June 1847) Göpel's doctoral dissertation studied periodic continued fractions of the roots of integers and derived a representation of the numbers by quadratic forms. He wrote on Steiner's synthetic geometry and an important work, Theoriae transcendentium Abelianarum primi ordinis adumbratio levis, published after his death, continued the work of Jacobi on elliptic functions. This work was published in Crelle's Journal in 1847. *SAU
1868 Anne Lucy Bosworth Focke (September 29, 1868 – May 15, 1907) was an American mathematician who became the first mathematics professor at what is now the University of Rhode Island, and later became the first female doctoral student of David Hilbert.
Bosworth attended Woonsocket High School,[2] and graduated from Wellesley College in 1890. At Wellesley, her classmates included mathematicians Grace Andrews and Clara Latimer Bacon.
She worked for two years as a teacher at Amesbury High School in Massachusetts, and was appointed as an instructor of mathematics at the Rhode Island College of Agriculture and Mechanic Arts (later to become the University of Rhode Island) in early 1892, the first year the school became a college. One month later she became its professor of mathematics and physics.
While continuing to work at the college, Bosworth earned a master's degree at the University of Chicago from 1894 through 1896 through summer study with E. H. Moore and Oskar Bolza.
In 1898, taking a leave from her work for the college, Bosworth traveled to the University of Göttingen in Germany, where she worked under the supervision of David Hilbert. She defended her dissertation there in 1899, and was awarded the Ph.D. in 1900. Her dissertation was Begründung einer vom Parallelenaxiome unabhängigen Streckenrechnung, and concerned non-Euclidean geometry. She was David Hilbert's first female doctoral student, part of a group that later included Nadeschda Gernet (1902), Vera Myller (1906), Margarete Kahn (1909), Klara Löbenstein (1910), and Eva Koehler (1912). *Wik
1895 Harold Hotelling, 29 September 1895 - 26 December 1973 He originally studied journalism at the University of Washington, earning a degree in it in 1919, but eventually turned to mathematics, gaining a PhD in Mathematics from Princeton in 1924 for a dissertation dealing with topology. However, he became interested in statistics that used higher-level math, leading him to go to England in 1929 to study with Fisher.
Although Hotelling first went to Stanford University in 1931, he not many years afterwards became a Professor of Economics at Columbia University, where he helped create Columbia's Stat Dept. In 1946, Hotelling was recruited by Gertrude Cox to form a new Stat Dept at the University of North Carolina at Chapel Hill. He became Professor and Chairman of the Dept of Mathematical Statistics, Professor of Economics, and Associate Director of the Institute of Statistics at UNC-CH. (When Hotelling and his wife first arrived in Chapel Hill they instituted the "Hotelling Tea", where they opened their home to students and faculty for tea time once a month.)
Dr. Hotelling's major contributions to statistical theory were in multivariate analysis, with probably his most important paper his famous 1931 paper "The Generalization of Student's Ratio", now known as Hotelling's T^2, which involves a generalization of Student's t-test for multivariate data. In 1953, Hotelling published a 30-plus-page paper on the distribution of the correlation coefficient, following up on the work of Florence Nightingale David in 1938. *David Bee
Probability density function, Hotelling's T square
1901 Enrico Fermi (29 Sep 1901; 28 Nov 1954) Italian-American physicist who was awarded the Nobel Prize for physics in 1938 as one of the chief architects of the nuclear age. He was the last of the double-threat physicists: a genius at creating both esoteric theories and elegant experiments. In 1933, he developed the theory of beta decay, postulating that the newly-discovered neutron decaying to a proton emits an electron and a particle he called a neutrino. Developing theory to explain this decay later resulted in finding the weak interaction force. He developed the mathematical statistics required to clarify a large class of subatomic phenomena, discovered neutron-induced radioactivity, and directed the first controlled chain reaction involving nuclear fission. *TIS
1925 Paul Beattie MacCready (29 Sep 1925; 28 Aug 2007) was an American engineer who invented not only the first human-powered flying machines, but also the first solar-powered aircraft to make sustained flights. On 23 Aug 1977, the pedal-powered aircraft, the Gossamer Condor successfully flew a 1.15 mile figure-8 course to demonstrate sustained, maneuverable manpowered flight, for which he won the £50,000 ($95,000) Kremer Prize. MacCready designed the Condor with Dr. Peter Lissamen. Its frame was made of thin aluminum tubes, covered with mylar plastic supported with stainless steel wire. In 1979, the Gossamer Albatross won the second Kremer Prize for making a flight across the English Channel.*TIS
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| Gossamer Condor |
1928 Ernst Steinitz (13 June 1871 – 29 September 1928) was a German mathematician.
Steinitz was born in Laurahütte (Siemianowice Śląskie), Silesia, Germany (now in Poland), the son of Sigismund Steinitz, a Jewish coal merchant, and his wife Auguste Cohen; he had two brothers. He studied at the University of Breslau and the University of Berlin, receiving his Ph.D. from Breslau in 1894. Subsequently, he took positions at Charlottenburg (now Technische Universität Berlin), Breslau, and the University of Kiel, Germany, where he died in 1928. Steinitz married Martha Steinitz and had one son.
Steinitz's 1894 thesis was on the subject of projective configurations; it contained the result that any abstract description of an incidence structure of three lines per point and three points per line could be realized as a configuration of straight lines in the Euclidean plane with the possible exception of one of the lines. His thesis also contains the proof of Kőnig's theorem for regular bipartite graphs, phrased in the language of configurations.
In 1910 Steinitz published the very influential paper Algebraische Theorie der Körper (German: Algebraic Theory of Fields, Crelle's Journal). In this paper he axiomatically studies the properties of fields and defines important concepts like prime field, perfect field and the transcendence degree of a field extension, and also normal and separable extensions (the latter he called algebraic extensions of the first kind). Besides numerous, today standard, results in field theory, he proved that every field has an (essentially unique) algebraic closure and a theorem, which characterizes the existence of primitive elements of a field extension in terms of its intermediate fields. Bourbaki called this article "a basic paper which may be considered as having given rise to the current conception of Algebra".
Steinitz also made fundamental contributions to the theory of polyhedra: Steinitz's theorem for polyhedra is that the 1-skeletons of convex polyhedra are exactly the 3-connected planar graphs. His work in this area was published posthumously as a 1934 book, Vorlesungen über die Theorie der Polyeder unter Einschluss der Elemente der Topologie, by Hans Rademacher.
1931 James Watson Cronin (September 29, 1931 – August 25, 2016) American particle physicist, who shared (with Val Logsdon Fitch) the 1980 Nobel Prize for Physics for "the discovery of violations of fundamental symmetry principles in the decay of neutral K-mesons." Their experiment proved that a reaction run in reverse does not follow the path of the original reaction, which implied that time has an effect on subatomic-particle interactions. Thus the experiment demonstrated a break in particle-antiparticle symmetry for certain reactions of subatomic particles.*TIS
1935 Hillel (Harry) Fürstenberg (September 29, 1935, ..)) is an American-Israeli mathematician, a member of the Israel Academy of Sciences and Humanities and U.S. National Academy of Sciences and a laureate of the Wolf Prize in Mathematics. He is known for his application of probability theory and ergodic theory methods to other areas of mathematics, including number theory and Lie groups. He gained attention at an early stage in his career for producing an innovative topological proof of the infinitude of prime numbers. He proved unique ergodicity of horocycle flows on a compact hyperbolic Riemann surfaces in the early 1970s. In 1977, he gave an ergodic theory reformulation, and subsequently proof, of Szemerédi's theorem. The Fürstenberg boundary and Fürstenberg compactification of a locally symmetric space are named after him. *Wik
DEATHS
1619 Hans Lipperhey, a German/Dutch lens grinder and eyeglass maker, was buried Sep. 29, 1619, in Middelburg in the Netherlands. We know little about Lipperhey except that, on Oct. 2, 1608, he applied for a patent for an instrument that made far-away things look nearby, or, as we would now call such a device, a telescope. He had discovered that if you take two eyeglass lenses, either two convex lenses or one convex and one concave (we are not sure which he used in his patent application), and space them properly, you can magnify distant objects when you look through them both. HIs patent application was the first written record of a telescope. The application was not successful, because several weeks later, another inventor, Jacob Metius, submitted a similar application, and the States General discovered upon further investigation that many other lens makers, including Zacharias Jensen, claimed to have invented it as well. But Lipperhey was the first to apply for patent recognition.
Lipperhey's claim to have invented the telescope came to public attention when Pierre Borel published a book, De vero telescopii inventore, in 1656, in which he gave credit for the telescope's invention to Lipperhey and Jensen (although Borel opted for Jensen as the earlier inventor and Lipperhey for the “second inventor”).*Linda Hall Org
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| Lipperhey Patent Application |
1939 Samuel Dickstein (May 12, 1851 – September 29, 1939) was a Polish mathematician of Jewish origin. He was one of the founders of the Jewish party "Zjednoczenie" (Unification), which advocated the assimilation of Polish Jews.
He was born in Warsaw and was killed there by a German bomb at the beginning of World War II. All the members of his family were killed during the Holocaust.
Dickstein wrote many mathematical books and founded the journal Wiadomości Mathematyczne (Mathematical News), now published by the Polish Mathematical Society. He was a bridge between the times of Cauchy and Poincaré and those of the Lwów School of Mathematics. He was also thanked by Alexander Macfarlane for contributing to the Bibliography of Quaternions (1904) published by the Quaternion Society.
He was also one of the personalities, who contributed to the foundation of the Warsaw Public Library in 1907.*Wik
1941 Friedrich Engel (26 Dec 1861, 29 Sept 1941)Engel was taught by Klein who recognized that he was the right man to assist Lie. At Klein's suggestion Engel went to work with Lie in Christiania (now Oslo) from 1884 until 1885. In 1885 Engel's Habilitation thesis was accepted by Leipzig and he became a lecturer there. The year after Engel returned to Leipzig from Christiania, Lie was appointed to succeed Klein and the collaboration of Lie and Engel continued.
In 1889 Engel was promoted to assistant professor and, ten years later he was promoted to associate professor. In 1904 he accepted the chair of mathematics at Greifswald when his friend Eduard Study resigned the chair. Engel's final post was the chair of mathematics at Giessen which he accepted in 1913 and he remained there for the rest of his life. In 1931 he retired from the university but continued to work in Giessen.
The collaboration between Engel and Lie led to Theorie der Transformationsgruppen a work on three volumes published between 1888 and 1893. This work was, "... prepared by S Lie with the cooperation of F Engel... "
In many ways it was Engel who put Lie's ideas into a coherent form and made them widely accessible. From 1922 to 1937 Engel published Lie's collected works in six volumes and prepared a seventh (which in fact was not published until 1960). Engel's efforts in producing Lie's collected works are described as, "... an exceptional service to mathematics in particular, and scholarship in general. Lie's peculiar nature made it necessary for his works to be elucidated by one who knew them intimately and thus Engel's 'Annotations' completed in scope with the text itself. "
Engel also edited Hermann Grassmann's complete works and really only after this was published did Grassmann get the fame which his work deserved. Engel collaborated with Stäckel in studying the history of non-euclidean geometry. He also wrote on continuous groups and partial differential equations, translated works of Lobachevsky from Russian to German, wrote on discrete groups, Pfaffian equations and other topics. *SAU
1955 L(ouis) L(eon) Thurstone (29 May 1887, 29 Sep 1955) was an American psychologist who improved psychometrics, the measurement of mental functions, and developed statistical techniques for multiple-factor analysis of performance on psychological tests. In high school, he published a letter in Scientific American on a problem of diversion of water from Niagara Falls; and invented a method of trisecting an angle. At university, Thurstone studied engineering. He designed a patented motion picture projector, later demonstrated in the laboratory of Thomas Edison, with whom Thurstone worked briefly as an assistant. When he began teaching engineering, Thurstone became interested in the learning process and pursued a doctorate in psychology. *TIS
1982 Letitia Chitty (15 July 1897 – 29 September 1982) was an English engineer who became a respected structural analytical engineer, achieving several firsts for women engineers, including becoming the first female fellow of the Royal Aeronautical Society and the second female recipient of the Telford Medal.
She entered Newnham College, Cambridge in 1916, taking the first part of the Tripos. During World War I, as part of a British programme to identify the best female mathematics graduates and current students, she was selected for war work with Alfred Pippard at the Admiralty Air Department at age 19. After the war she returned to her studies, changed subject to engineering, and graduated with a titular degree from Newnham College with first class honours in the Mechanical Sciences Tripos, 1921, the first woman to do so.
Her early career focused on analysing the stresses of airframes, airships and civil engineering structures, initially with the Admiralty Air Department and then, after graduating, at the Air Ministry with Richard Southwell and Alfred Pippard.
Chitty moved to Imperial College in 1934 where she remained for the rest of her career, initially specialising in structural stresses in aircraft. During the 1930s, she was part of a group which analysed the crash of the airship R38, and published various Air Ministry papers on stresses and strains on airship structures. She was an early member of the Women's Engineering Society.
Her World War II work included research into stresses in submarine hulls under shell attack, extensible cables and pulley blocks for barrage balloons, for the Director of Scientific Research of the Admiralty and the Ministry of Supply. Later research interests included arches and arch dams - in particular, the Dukan Dam in Iraq - and she contributed to an international symposium on arched dams in 1968 *Wik
In her will, she left a bequest to Imperial College, which named its Library reading room after her. Imperial College also presents a Letitia Chitty Centenary Memorial Prize, while Newnham College has presented a 'Letitia Chitty Award for Engineering'.
2003 Ovide Arino (24 April 1947 - 29 September 2003) was a mathematician working on delay differential equations. His field of application was population dynamics. He was a quite prolific writer, publishing over 150 articles in his lifetime. He also was very active in terms of student supervision, having supervised about 60 theses in total in about 20 years. Also, he organized or coorganized many scientific events. But, most of all, he was an extremely kind human being, interested in finding the good in everyone he met. *Euromedbiomath
2010 Georges Charpak (1 August 1924 – 29 September 2010) was a French physicist who was awarded the Nobel Prize in Physics in 1992 "for his invention and development of particle detectors, in particular the multiwire proportional chamber". This was the last time a single person was awarded the physics prize. *Wik
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
Since it is often much easier to find Gp and the incenter of inscribable polygons, this provides a clever way to find Gb. Students may be interested to know that Archimedes used the formula Area= 1/2 P r for the area of a circle, where P is the perimeter (circumference). This is, of course, simply and alternate form of the formula they learned in grade school. They may be even more surprised to find that the Archimedean formula gives the area of any inscribable polygon if P is the perimeter of the polygon, and r is the radius of the inscribed circle. The proof is very easy, and is often provided in high school textbooks for the limited case of regular polygons which would always be inscribable. The image below provides non-subtle clues to the proof.
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