Gaspard Monge

Gaspard Monge, Comte de Péluse was a French mathematician, the inventor of descriptive geometry, the father of differential geometry. During the French Revolution he served as the Minister of the Marine, was involved in the reform of the French educational system, helping to found the École Polytechnique. Monge was born at the son of a merchant, he was educated at the college of the Oratorians at Beaune. In 1762 he went to the Collège de la Trinité at Lyon, one year after he had begun studying, he was made a teacher of physics at the age of just seventeen. After finishing his education in 1764 he returned to Beaune, where he made a large-scale plan of the town, inventing the methods of observation and constructing the necessary instruments. An officer of engineers who saw it wrote to the commandant of the École Royale du Génie at Mézières, recommending Monge to him and he was given a job as a draftsman. L. T. C. Rolt, an engineer and historian of technology, credited Monge with the birth of engineering drawing.

Those studying at the school were drawn from the aristocracy, so he was not allowed admission to the institution itself. His manual skill was regarded, but his mathematical skills were not made use of, he worked on the development of his ideas in his spare time. At this time he came to contact with Charles Bossut, the professor of mathematics at the École Royale. "I was a thousand times tempted," he said long afterwards, "to tear up my drawings in disgust at the esteem in which they were held, as if I had been good for nothing better."After a year at the École Royale, Monge was asked to produce a plan for a fortification in such a way as to optimise its defensive arrangement. There was an established method for doing this which involved lengthy calculations but Monge devised a way of solving the problems by using drawings. At first his solution was not accepted, since it had not taken the time judged to be necessary, but upon examination the value of the work was recognized, Monge's exceptional abilities were recognized.

After Bossut left the École Royale du Génie, Monge took his place in January 1769, in 1770 he was appointed instructor in experimental physics. In 1777, Monge married Cathérine Huart; this led Monge to develop an interest in metallurgy. In 1780 he became a member of the French Academy of Sciences. In 1783, after leaving Mézières, he was, on the death of É. Bézout, appointed examiner of naval candidates. Although pressed by the minister to prepare a complete course of mathematics, he declined to do so on the grounds that this would deprive Mme Bézout of her only income, that from the sale of the textbooks written by her late husband. In 1786 he published his Traité élémentaire de la statique; the French Revolution changed the course of Monge's career. He was a strong supporter of the Revolution, in 1792, on the creation by the Legislative Assembly of an executive council, Monge accepted the office of Minister of the Marine, held this office from 10 August 1792 to 10 April 1793, when he resigned.

When the Committee of Public Safety made an appeal to the academics to assist in the defence of the republic, he applied himself wholly to these operations, distinguished himself by his energy, writing the Description Le l'art de Fabriquer Les canons and Avis aux ouvriers en fer sur la fabrication de l'acier. He took a active part in the measures for the establishment of the Ecole Normale, of the school for public works, afterwards the École Polytechnique, was at each of them professor for descriptive geometry. Géométrie descriptive. Leçons données aux écoles normales was published in 1799 from transcriptions of his lectures given in 1795, he published Application de l'analyse à la géométrie, which enlarged on the Lectures. From May 1796 to October 1797 Monge was in Italy with C. L. Berthollet and some artists to select the sculptures being levied from the Italians. While there he became friendly with Napoleon Bonaparte. Upon his return to France, he was appointed as the Director of the École Polytechnique, but early in 1798 he was sent to Italy on a mission that ended in the establishment of the short-lived Roman Republic.

From there Monge joined Napoleon's expedition to Egypt, taking part with Berthollet in the scientific work of the Institut d'Égypte and the Egyptian Institute of Sciences and Arts. They accompanied Bonaparte to Syria, returned with him in 1798 to France. Monge was appointed president of the Egyptian commission, he resumed his connection with the École Polytechnique, his mathematical papers are published in the Journal and the Correspondence of the École Polytechnique. On the formation of the Sénat conservateur he was appointed a member of that body, with an ample provision and the title of count of Pelusium, he became the Senate conservateur's president during 1806–7. On the fall of Napoleon he had all of his honours taken away, he was excluded from the list of members of the reconstituted Institute. Monge was an atheist. Monge died in Paris on 28 July 1818, his funeral was held 30 July 1818 at St. Thomas Aquinas Church in Paris, his remains were first interred in a mausoleum in Le Père Lachaise Cemetery in Paris and transferred to the Panthéon in Paris.

A statue portraying him was erected in Beaune in 1849. Monge's name is one of the 72 names inscribed on the base of the Eiffel Tower. Since 4 November 1992 the Marine Nationale operate the MRIS FS Monge, named after him. B

Euler diagram

An Euler diagram is a diagrammatic means of representing sets and their relationships. They are useful for explaining complex hierarchies and overlapping definitions, they are similar to another set diagramming technique, Venn diagrams. Unlike Venn diagrams, which show all possible relations between different sets, the Euler diagram shows only relevant relationships; the first use of "Eulerian circles" is attributed to Swiss mathematician Leonhard Euler. In the United States, both Venn and Euler diagrams were incorporated as part of instruction in set theory as part of the new math movement of the 1960s. Since they have been adopted by other curriculum fields such as reading as well as organizations and businesses. Euler diagrams consist of simple closed shapes in a two dimensional plane that each depict a set or category. How or if these shapes overlap demonstrates the relationships between the sets; each curve divides the plane into two regions or "zones": the interior, which symbolically represents the elements of the set, the exterior, which represents all elements that are not members of the set.

Curves which do not overlap represent disjoint sets. Two curves which overlap represent sets that intersect. A curve, contained within the interior of another represents a subset of it. Venn diagrams are a more restrictive form of Euler diagrams. A Venn diagram must contain all 2n logically possible zones of overlap between its n curves, representing all combinations of inclusion/exclusion of its constituent sets. Regions not part of the set are indicated by coloring them black, in contrast to Euler diagrams, where membership in the set is indicated by overlap as well as color; as shown in the illustration to the right, Sir William Hamilton in his posthumously published Lectures on Metaphysics and Logic erroneously asserts that the original use of circles to "sensualize... the abstractions of Logic" was not Leonhard Paul Euler but rather Christian Weise in his Nucleus Logicae Weisianae that appeared in 1712 posthumously, the latter book was written by Johann Christian Lange rather than Weise.

He references Euler's Letters to a German Princess In Hamilton's illustration the four categorical propositions that can occur in a syllogism as symbolized by the drawings A, E, I and O are: A: The Universal Affirmative, Example: "All metals are elements". E: The Universal Negative, Example: "No metals are compound substances". I: The Particular Affirmative, Example: "Some metals are brittle". O: The Particular Negative, Example: "Some metals are not brittle". In his 1881 Symbolic Logic Chapter V "Diagrammatic Representation", John Venn comments on the remarkable prevalence of the Euler diagram: "...of the first sixty logical treatises, published during the last century or so, which were consulted for this purpose:-somewhat at random, as they happened to be most accessible:-it appeared that thirty four appealed to the aid of diagrams, nearly all of these making use of the Eulerian Scheme." But he contended, "the inapplicability of this scheme for the purposes of a general Logic" and on page 101 observed that, "It fits in but badly with the four propositions of the common Logic to which it is applied."

Venn ends his chapter with the observation illustrated in the examples below—that their use is based on practice and intuition, not on a strict algorithmic practice: “In fact... those diagrams not only do not fit in with the ordinary scheme of propositions which they are employed to illustrate, but do not seem to have any recognized scheme of propositions to which they could be affiliated.” In his Chapter XX HISTORIC NOTES Venn gets to a crucial criticism. The weak point about these consists in the fact that they only illustrate in strictness the actual relations of classes to one another, rather than the imperfect knowledge of these relations which we may possess, or wish to convey, by means of the proposition. Accordingly they will not fit in with the propositions of common logic, but demand the constitution of a new group of appropriate elementary propositions.... This defect must have been noticed from the first in the case of the particular affirmative and negative, for the same diagram is employed to stand for them both, which it does indifferently well"..

Whatever the case, armed with these observations and criticisms, Venn demonstrates how he derived what has become known as his Venn diagrams from the "...old-fashioned Euler diagrams." In particular he gives an example, shown on the left. By 1914, Louis Couturat had labeled the terms. Moreover, he had labeled the exterior region as well, he succinctly explains how to use the diagram – one must strike out the regions that are to vanish: "VENN'S method is translated in geometrical diagrams which represent all the constituents, so that, in order to obtain the result, we need only s


Imagotaria is an extinct monotypic genus of walrus with the sole species Imagotaria downsi. Fossils of Imagotaria are known from the early late Miocene of California; the 1.8 metres long pinniped more resembled in its overall shape a sea lion rather than a walrus. Unlike the extant walrus, Imagotaria did not possess elongate, ever-growing tusks, but instead bore enlarged canines. Imagotaria is an example of a primitive walrus that does not grossly appear similar to a modern walrus. However, the walrus family is a more inclusive group, that includes walruses without tusks, walruses with upper and lower tusks, walruses with upper tusks like the extant walrus, it is possible to classify these pinnipeds as walruses because they share many other skull features as well as many skeletal features, all of which indicate common ancestry. The teeth of Imagotaria indicate that its feeding ecology was markedly different from that of modern walrus, more similar to that of less specialized pinnipeds like seals, fur seals, sea lions.

Conical, unworn teeth and the lack of a vaulted palate indicate that Imagotaria did not feed on molluscs like modern walrus. Modern walruses do not use their teeth to chew molluscs. Instead, they hold a clam in their lips, the vaulted palate allows them to use their tongue as a powerful piston to suck the soft parts right out of the clam shell; the shell is dropped to the seafloor, never entering the oral cavity. Additionally, fossils of Imagotaria demonstrate that early walruses had, by the middle and late Miocene developed extreme sexual dimorphism, it is unclear whether extreme sexual dimorphism is ancestral to all pinnipeds, or if it has been independently acquired in multiple pinniped lineages