The Bulletin became a review journal for topics in vector analysis and abstract algebra such as the theory of equipollence. The mathematical work reviewed pertained largely to matrices and linear algebra as the methods were in rapid development at the time.
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In Euclidean geometry, equipollence is a binary relation between directed line segments. A line segment AB from point A to point B has the opposite direction to line segment BA. Two directed line segments are equipollent when they have the same length and direction.
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Bellavitis anticipated the idea of a Euclidean vector with his notion of equipollence. Two line segments AB and CD are equipollent if they are parallel and have the same length and direction. The relation is denoted AB \bumpeq CD . In modern terminology, this relation between line segments is an example of an equivalence relation.
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Ancient Greek skeptics argued for equipollence, the view that reasons for and against claims are equally balanced. This captures at least one sense of saying that the claims themselves are underdetermined. Underdetermination, again under different labels, arises in the modern period in the work of René Descartes. Among other skeptical arguments, Descartes presents two arguments involving underdetermination.
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The concept of vector, as we know it today, evolved gradually over a period of more than 200 years. About a dozen people made significant contributions to its development.Michael J. Crowe, A History of Vector Analysis; see also his on the subject. In 1835, Giusto Bellavitis abstracted the basic idea when he established the concept of equipollence.
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The concept of equipollent line segments was advanced by Giusto Bellavitis in 1835. Subsequently the term vector was adopted for a class of equipollent line segments. Bellavitis's use of the idea of a relation to compare different but similar objects has become a common mathematical technique, particularly in the use of equivalence relations. Bellavitis used a special notation for the equipollence of segments AB and CD: :AB \bumpeq CD .
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The Social and Political Thought of Ernest Barker (Cambridge University Press, 2006), pp. 64-65. Various committees debated and developed the several proposals, Adams “fighting hard and successfully for the ‘equipollence’ of politics with the other live subjects”.Chester, Economics, Politics and Social Studies in Oxford, 1900-85, pp. 33ff.; The Times, 1 February 1966 (obituary notice, first draft by Thomas Jones). He sat on both the Civil Science and Political Science CommitteesChester, Economics, Politics and Social Studies in Oxford, 1900-85, p.190.
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He published two works in geometric algebra, Introduction à la Méthode des Quaternions (1881)C.-A. Laisant (1881) Introduction a la Méthode des Quaternions, link from Google Books and Théorie et applications des equipollences (1887).C.-A. Laisant (1887) Theorie et Applications des Equipollence, Gauthier-Villars, link from University of Michigan Historical Math Collection He also co-founded a mathematical journal, L'Intermédiaire des Mathématiciens with Émile Lemoine in 1894, and was in 1888 the president of the Société Mathématique de France.Anciens Présidents de la SMF -- 1873-2006 .
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It is asserted by both Priest and Kabay that it is impossible for a trivialist to truly choose and thus act. Priest argues this by the following in Doubt Truth to Be a Liar: "One cannot intend to act in such a way as to bring about some state of affairs, s, if one believes s already to hold. Conversely, if one acts with the purpose of bringing s about, one cannot believe that s already obtains." Ironically, due to their suspension of determination upon striking equipollence between claims, the Pyrrhonist has also remained subject to apraxia charges.
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Geometric equipollence is also used on the sphere: :To appreciate Hamilton's method, let us first recall the much simpler case of the Abelian group of translations in Euclidean three- dimensional space. Each translation is representable as a vector in space, only the direction and magnitude being significant, and the location irrelevant. The composition of two translations is given by the head-to-tail parallelogram rule of vector addition; and taking the inverse amounts to reversing direction. In Hamilton's theory of turns, we have a generalization of such a picture from the Abelian translation group to the non-Abelian SU(2).
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When a line segment is given an orientation (direction) it suggests a translation or perhaps a force tending to make a translation. The magnitude and direction are indicative of a potential change. This suggestion has been absorbed into mathematical physics through the concept of a Euclidean vector.Harry F. Davis & Arthur David Snider (1988) Introduction to Vector Analysis, 5th edition, page 1, Wm. C. Brown Publishers Matiur Rahman & Isaac Mulolani (2001) Applied Vector Analysis, pages 9 & 10, CRC Press The collection of all directed line segments is usually reduced by making "equivalent" any pair having the same length and orientation.Eutiquio C. Young (1978) Vector and Tensor Analysis, pages 2 & 3, Marcel Dekker This application of an equivalence relation dates from Giusto Bellavitis’s introduction of the concept of equipollence of directed line segments in 1835.
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Linear Algebra by Hüseyin Tevfik Pasha In Constantinople in 1882 Tevfik published Linear Algebra with the presses of A. Y. Boyajain. He begins with the concept of equipollence: :By the expression AB = NO, in Linear Algebra and in the science of Quaternions also, it is understood that the length of AB is equal to that of NO, and also that the direction of line AB is the same as that of NO. (page one) The book has five chapters and an appendix "Complex quantities and quaternions" in 68 pages with contents listed on page 69. Tevfik's book refers on page 11 to Introduction to Quaternions by Kelland and Tait which came out with a second edition in 1882.Philip Kelland & P. G. Tait (1882) Introduction to Quaternions with numerous examples But complex numbers and quaternions are missing.
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The following passages, translated by Michael J. Crowe, show the anticipation that Bellavitis had of vector concepts: :Equipollences continue to hold when one substitutes for the lines in them, other lines which are respectively equipollent to them, however they may be situated in space. From this it can be understood how any number and any kind of lines may be summed, and that in whatever order these lines are taken, the same equipollent-sum will be obtained... :In equipollences, just as in equations, a line may be transferred from one side to the other, provided that the sign is changed... Thus oppositely directed segments are negatives of each other: AB + BA \bumpeq 0 . :The equipollence AB \bumpeq n.CD , where n stands for a positive number, indicates that AB is both parallel to and has the same direction as CD, and that their lengths have the relation expressed by AB = n.
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The book has eight chapters: the first on the origins of vector analysis including Ancient Greek and 16th and 17th century influences; the second on the 19th century William Rowan Hamilton and quaternions; the third on other 19th and 18th century vectorial systems including equipollence of Giusto Bellavitis and the exterior algebra of Hermann Grassmann. Chapter four is on the general interest in the 19th century on vectorial systems including analysis of journal publications as well as sections on major figures and their views (e.g., Peter Guthrie Tait as an advocate of Quaternions and James Clerk Maxwell as a critic of Quaternions); the fifth chapter describes the development of the modern system of vector analysis by Josiah Willard Gibbs and Oliver Heaviside. In chapter six, "Struggle for existence", Michael J. Crowe delves into the zeitgeist that pruned down quaternion theory into vector analysis on three-dimensional space.
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In this sense, Kepler already embarked in his MC on a causal investigation by asking for the cause of the number, the sizes and the "motions" (the speeds) of the heavenly spheres. On the other hand, "causality" implies in Kepler, according to the Aristotelian conception of physical science, the concrete "physical cause", the efficient cause which produces a motion or is responsible for keeping the body in motion. Original to Kepler, however, and typical of his approach is the resoluteness with which he was convinced that the problem of equipollence of the astronomical hypotheses can be resolved and the consequent introduction of the concept of causality into astronomy—traditionally a mathematical science. This approach is already present in his MC, where he, for instance, relates for the first time the distances of the planets to a power which emerges from the Sun and decreases in proportion to the distance of each planet, up to the sphere of the fixed stars.
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