# Search Results for parallel

## Biographies

1. Cavalieri biography
• Cavalieri's treatise on the method of indivisibles is voluble and not clearly written, and it is not easy to learn from it precisely what Cavalieri meant by an "indivisible." It seems that an indivisible of a given planar piece is a chord of the piece, and a planar piece can be considered as made up of an infinite parallel set of such indivisibles.
• Similarly, it seems that an indivisible of a given solid is a planar section of that solid, and a solid can be considered as made up of an infinite parallel set of this kind of indivisible.
• Now, Cavalieri argued, if we slide each member of a parallel set of indivisibles of some planar piece along its own axis, so that the endpoints of the indivisibles still trace a continuous boundary, then the area of the new planar piece so formed is the same as that of the original planar piece, inasmuch as the two pieces are made up of the same indivisibles.
• A similar sliding of the members of a parallel set of indivisibles of a given solid will yield another solid having the same volume as the original one.
• If two planar pieces are included between a pair of parallel lines, and if the lengths of the two segments cut by them on any line parallel to the including lines are always equal, then the areas of the two planar pieces are also equal.
• If two solids are included between a pair of parallel planes, and if the areas of the two sections cut by them on any plane parallel to the including planes are always equal, then the volumes of the two solids are also equal.

2. Saccheri biography
• In Euclides ab Omni Naevo Vindicatus Ⓣ, published in 1733, he did important early work on non-euclidean geometry, although he did not see it as such, rather an attempt to prove the parallel postulate of Euclid.
• He was certainly aware of the work of John Wallis and Nasir al-Din al-Tusi on the Parallel Postulate since he criticises both in his book.
• Let us look at how he approached the question of the Parallel Postulate.
• Saccheri knew that if he could prove that the angles at C and D were right angles without using the Parallel Postulate, then he could deduce the Parallel Postulate from the other axioms.
• His aim now was, working with all of Euclid's axioms except the Parallel Postulate, to obtain a contradiction.

3. Playfair biography
• Firstly, there was the contentious "parallel" postulate.
• Robert Simson of Glasgow University had, in his 1756 edition of the Elements, given a proof of the parallel axiom based on another assumption.
• Playfair solved this difficulty in 1795 with a retatement of Euclid's Euclid's parallel axiom:- .
• Two straight lines cannot be drawn through the same point, parallel to the same straight line, without coinciding with one another.
• Given a line and a point not on the line, it is possible to draw exactly one line through the given point parallel to the line.
• He also included a section of notes in the form of an appendix, which gave his reasons for the alterations made throughout the volumes, and an illuminating discussion on the difficult topic of parallel lines.

4. Taylor biography
• The phrase "linear perspective" was invented by Taylor in this work and he defined the vanishing point of a line, not parallel to the plane of the picture, as the point where a line through the eye parallel to the given line intersects the plane of the picture.
• He also defined the vanishing line to a given plane, not parallel to the plane of the picture, as the intersection of the plane through the eye parallel to the given plane.
• The main theorem in Taylor's theory of linear perspective is that the projection of a straight line not parallel to the plane of the picture passes through its intersection and its vanishing point.

5. Geminus biography
• Proclus quotes from Geminus (see for example [',' T L Heath, A History of Greek Mathematics (2 Vols.) (Oxford, 1921).','3]), saying that in the case of the parallel postulate:- .
• Geminus tried the following approach giving a definition of parallel lines:- .
• Parallel straight lines are straight lines situated in the same plane and such that the distance between them, if they are produced without limit in both directions at the same time is everywhere the same.
• The 'proof' which Geminus then gave of the parallel postulate is ingenious but it is false.
• It is interesting, however, that Geminus attempts to prove the parallel postulate and, although it is unlikely to be the first such attempt, at least it is the earliest one for which details have survived.

6. Valiant biography
• Parallel and distributed computing.
• In addition to computational learning theory and computational complexity, a third broad area in which Valiant has made important contributions is the theory of parallel and distributed computing.
• An example of a simple insight is his parallel routing scheme, described in the paper "A scheme for fast parallel communication" (1982).

7. Hendricks biography
• Hendricks was appointed as Deputy Surveyor and given the 'First Standard Parallel South' survey, the 'Second Standard Parallel South' survey and the 'Third Standard Parallel South' survey in the state of Colorado which he completed in September 1861.

8. Kaestner biography
• However he did influence Gauss, in particular with his interest in Euclid's parallel postulate.
• Perhaps the most important feature of Kastner's contributions was his interest in the parallel postulate which indirectly influenced Bolyai and Lobachevsky too.
• Kastner, in spite of his rather great inclination for Euclid's Elements, based his version of the axiomatics of geometry in his Kompendium on other principles (e.g., on motions) and attempted both to seize on other fundamental properties (continuity, ordering) and to determine the selection of the parallel axiom as a foundation.

9. Martin Emilie biography
• in Euclidean space one line and only one can be drawn through a point so as to be parallel to a given line.
• The two non-Euclidean geometries that are to be discussed here arose when geometers first followed to their logical conclusions the hypotheses that through a point more than one line could be drawn parallel to a given line, and that through a point no line could be drawn parallel to a given line.

10. Levi biography
• It is interesting to note that Levi was interested in Euclid's parallel postulate and appears to have been part of a lively debate about whether it could be deduced from the other axioms.
• He proved the parallel postulate with an argument based on an assumption on the convergence or divergence of straight lines that is (as of course it must be) equivalent to the parallel postulate; see [',' T Levy, Gersonide, le Pseudo-Tusi et le Postulat des Paralleles, Arabic Science and Philosophy 2 (1992), 39-82.','43] for further details.

11. Bennett biography
• The object of Part II is to make, for complex numbers, an investigation which shall be as nearly as possible parallel to that of Part I for real numbers.
• He published his historical findings in The parallel motion of Sarrut and some allied mechanisms in 1905.
• a game for two players using as apparatus 48 pebbles and a board hollowed out into two parallel rows of six cups.

12. Monte biography
• In fact he attacked them for their claims that bodies would descend along parallel paths if dropped, saying that all bodies would move along paths which converged to the centre of the Earth.
• The most important result in Guidobaldo's treatise was that any set of parallel lines, not parallel to the plane of the picture, will converge to a vanishing point.

13. Byrne biography
• Byrne also wrote, under the pseudonym E B Revilo (this is just Oliver Byrne reversed!), the strange book The creed of St Athanasius proved by a mathematical parallel.
• Daniel J Cohen writes in [',' D J Cohen, The creed of St Athanasius proved by a mathematical parallel, in Equations from God: Pure Mathematics and Victorian Faith (John Hopkins University Press, Baltimore, 2007), 73.','8]:- .
• Byrne then erected two vertical columns: the left containing the English Book of Common Prayer translation of the Quicunque Vult (the traditional description of the Athanasian Creed), the right containing parallel mathematical equations involving infinity that purported to establish the truth of the statements on the left.

14. Beltrami biography
• Beltrami in this 1868 paper did not set out to prove the consistency of non-Euclidean geometry or the independence of the Euclidean parallel postulate.
• Houel translated both Lobachevsky's and Beltrami's work into French in 1870 and he noted how Beltrami's paper proved the independence of the Euclid's parallel postulate.
• This appears in a 1889 publication in which Beltrami brought to the attention of the mathematical world Saccheri's 1733 study of the parallel postulate.

15. Rapcsak biography
• But this was not the only problem he faced for, by this time, his parents were in financial difficulties and so, in parallel with his studies, he worked as a tutor in a student hostel in Hodmezovasarhely from 1937 to 1942 to help support his parents.
• Hyperplanes are defined by the condition that the unit transversal vectors should be parallel (with respect to the metric of the imbedding space).

16. Samarskii biography
• One of the major advances made by Samarskii in the atomic weapons programme was to discover a method to allow parallel calculations by the 30 girl computers.
• By doing parallel calculations, the problem was solved about 15 times quicker than it would have been had this method not been used and the solution was completed in two months.

17. Lobachevsky biography
• Since Euclid's Elements and his theory of parallel lines are discussed in detail in Montucla's book, it seems likely that Lobachevsky's interest in the Fifth Postulate was stimulated by these lectures.
• The fifth postulate states that given a line and a point not on the line, a unique line can be drawn through the point parallel to the given line.

18. Lambert biography
• In 1766 Lambert wrote Theorie der Parallellinien Ⓣ which was a study of the parallel postulate.
• By assuming that the parallel postulate was false, he managed to deduce a large number of non-euclidean results.

19. Schwartz Jacob biography
• A brief list of some of the areas to which Schwartz has made major contributions gives some notion of his breadth: spectral theory of linear operators, von Neumann algebras, macro economics, the mathematics of quantum field theory, parallel computation, computer time-sharing, high-level programming languages, compiler optimization, transformational programming, computational logic, motion planning in robotics, and, most recently, multimedia.
• He also became interested in parallel computing, robotics, computer vision, and computer design.

20. Craig Thomas biography
• In August Crelle's Journal fur die reine und angewandte Mathematik he published On the parallel surface to an ellipsoid (1882) and Note on parallel surfaces (1883).

21. Butzer biography
• This parallel treatment easily lends itself to an understanding of abstract harmonic analysis; the underlying classical theory is therefore presented in a form that is directed towards the case of arbitrary locally compact abelian groups.
• Paul L Butzer presents a broad but un-specialized survey of scholarship in mathematics and astronomy during Carolingian times, in the explicit context of the older sources available to early medieval writers and in comparison with parallel developments in the Byzantine and Islamic worlds.

22. Borelli biography
• He also took the opportunity to examine the parallel postulate of Euclid [',' G B Halsted, Non-Euclidean Geometry: Historical and Expository, Amer.
• This most learned author [Borelli] blames Euclid, because he defines parallel straight lines to be those, which lying in the same plane do not meet on either side, even if produced into the infinite.

23. Ree biography
• Soon there were dramatic changes in his homeland with Korea being divided by the 38th parallel into a Soviet administered north and a United States administered south.
• Both armies had provoked the other in the preceding months with raids across the 38th parallel.

24. Euclid biography
• The famous fifth, or parallel, postulate states that one and only one line can be drawn through a point parallel to a given line.

25. Al-Jawhari biography
• This work contained nearly fifty propositions additional to those given by Euclid and included an attempt by al-Jawhari to prove the parallel postulate.
• Al-Tusi quotes six of the nearly fifty propositions which together form what al-Jawhari believed was a proof of the parallel postulate.

26. Orszag biography
• Massively parallel supercomputers seem the best hope for achieving progress on ’grand challenge’ problems such as understanding high-Reynolds-number turbulent flows, Physics Today (March 1993), 34-42.','4]:- .
• Among topics covered are fundamentals of LES; LES of incompressible, compressible, and reacting flows; LES of atmospheric, oceanic, and environmental flows; and LES and massively parallel computing.

27. Snyder biography
• This situation also occurs with the Plucker line-coordinates so that the parallel between line geometry in three-space and Lie's "Kugelgeometrie" was apparent.
• Of twenty-one papers he published in the next ten years, twelve were concerned with the metric side of this parallel and dealt with annular, tubular, and developable surfaces, their asymptotic lines, and lines of curvature, or with the development of collateral algebra.

28. Taurinus biography
• the two corresponded on mathematical topics and, largely due to Schweikart's influence, he began to investigate the problem of parallel lines and Euclid's fifth postulate:- .
• Taurinus's works on the problem of parallel lines.

29. Reynolds biography
• The 1883 paper is called An experimental investigation of the circumstances which determine whether the motion of water in parallel channels shall be direct or sinuous and of the law of resistance in parallel channels.

30. Davies biography
• Later, in Parallel distributions and contact transformations (1966) he re-examines the invariant theory of contact transformations from the point of view of parallel distributions.

31. Hipparchus biography
• occupies three degrees of Leo along its parallel circle..
• He has therefore divided each small circle parallel to the equator into 12 portions of 30° each and this means that the right ascension of the star referred to in the quotation is 123°.

32. Bolyai biography
• Around 1820, when he was still studying in Vienna, Bolyai began to follow the same path that his father had taken in trying to replace Euclid's parallel axiom with another axiom which could be deduced from the others.
• What was contained in this mathematical masterpiece? After setting up his own definitions of 'parallel' and showing that if the Fifth Postulate held in one region of space it held throughout, and vice versa, he then stated clearly the different systems he would consider:- .

33. Hippias biography
• As the radius AB rotates about A to move to the position AD then the line BC moves at the same rate parallel to itself to end at AD.
• Draw a line through P parallel to AD to meet the quadratrix at Q.

34. Schreier biography
• During the beginning of the 1928/29 session Schreier lectured on function theory giving parallel courses in Hamburg and Rostock.
• The treatment of projectivities is particularly interesting and, not-withstanding an unfortunate impression, given in the opening paragraph, that the main purpose of projective geometry is to eliminate the distinction between intersecting and parallel lines, the book can be recommended as a rigorous introduction to n-dimensional geometry for second-year university students.

35. Democritus biography
• If a cone were cut by a plane parallel to the base [by which he means a plane indefinitely close to the base], what must we think of the surfaces forming the sections? Are they equal or unequal? For, if they are unequal, they will make the cone irregular as having many indentations, like steps, and unevennesses; but, if they are equal, the sections will be equal, and the cone will appear to have the property of the cylinder and to be made up of equal, not unequal, circles, which is very absurd.
• Firstly notice, as Heath points out in [',' T L Heath, A History of Greek Mathematics I (Oxford, 1921).','7], that Democritus has the idea of a solid being the sum of infinitely many parallel planes and he may have used this idea to find the volumes of the cone and pyramid as reported by Archimedes.

36. Descartes biography
• Not only can questions of solvability and geometrical possibility be decided elegantly, quickly and fully from the parallel algebra, without it they cannot be decided at all.
• And the Moon itself, neither in the plane of the Earth's equator nor in a plane parallel to this? .

37. Tarina biography
• Paper: On the mobility of a space An without torsion which admits fields of parallel vectors (1965).
• Comment: Starting from the fact that there are some general relations between conditions under which a space An with an affine connection without torsion possesses a group of motions or fields of parallel vectors, Țarină considers this problem for different kinds of spaces, known to have maximal groups of motions.

38. Wallis biography
• to regard the parabola as a section of a cone by a plane parallel to a generator than to regard a circle as a section of a cone by a plane parallel to the base, or even a triangle as a plane through the vertex.

39. Apery biography
• It was at this school that, in 1928, he first met Euclid's axioms and became fascinated by the parallel postulate.
• For many years he continued to have parallel political and mathematical careers.

40. Simplicius biography
• 32 (1969), 1-24.','6] where the author discusses the fact that the commentary does not contain an attempt at a proof of the parallel postulate by Simplicius himself, despite the evidence that indeed Simplicius did attempt such a proof.
• 32 (1969), 1-24.','6] there is a discussion of how Simplicius's attempted proof of the parallel postulate entered Arabic mathematics and was first criticised, then incorporated into a new 'proof' designed to take the criticism into account.

41. Wolf biography
• On a plane surface draw a sequence of parallel, equally spaced straight lines; take an absolutely cylindrical needle of length a, less than the constant interval d which separates the parallels, and drop it randomly a great number of times on the surface covered by the lines.
• In the first 50 trials I dropped the needle parallel to the parallels on the plate and in the second 50 ones perpendicular to them, whereas in the third 50 trials I sought to induce all kinds of positions by constantly rotating the plate.

42. Bolyai Farkas biography
• All his life Bolyai was interested in the foundations of geometry and the parallel axiom.
• His attempts to stop his son studying the parallel axiom fortunately failed! Farkas Bolyai wrote to his son:- .

43. Hopkinson biography
• In one of his papers he proved mathematically that alternating current dynamos could be connected to work in parallel.
• He used his mathematical expertise to give a general theory of alternating currents and he applied this theory to the operation of alternating current generators in parallel.

44. Al-Nayrizi biography
• In his work on proofs of the parallel postulate, al-Nayrizi quotes work by a mathematician named Aghanis.

45. Hoehnke biography
• The natural starting point for the structure theory of semigroups is the theory of transformation semigroups, in parallel to Jacobson's structure theory of rings which uses linear transformations of vector spaces.

46. Santalo biography
• Santalo completed his studies in Madrid in 1934, having undertaken in parallel his part-time military service, and was awarded a Degree in Mathematics.

47. Suschkevich biography
• However, he had not given up on his aim of becoming a university professor and so, in parallel with his work as a mathematics teacher, he studied for his Master's Degree (equivalent to a Ph.D.).

48. Chebyshev biography
• The Chebyshev parallel motion is three linked bars approximating rectilinear motion.

49. Frechet biography
• This parallel is drawn by Frechet himself who requires sufficient structure on his abstract systems so that limits and continuity can be studied.

50. Pascal Ernesto biography
• The input unit for x (a two-wheeled carriage which rolls parallel to the x-axis and which carries the entire instrument), the input and output units for the dependent variables (small carriages carrying pointers and pencils, respectively, which roll on tracks attached to the main carriage), and the integrator (a friction wheel which constrains the motion of the output pencil) are retained in substantially the form used in the original integraph.

51. Schrodinger biography
• The paradox was that both universes, one with a dead cat and one with a live one, seemed to exist in parallel until an observer opened the box.

52. Speiser biography
• Although much work was done with the Z4 at ETH, in parallel Speiser and Rutishauser were working on the construction of ETH's own computer ERMETH (Elektronische Rechenmaschine an der ETH).

53. McMullen biography
• In this monograph, the author presents a comprehensive study of a theory which brings into parallel two recent and very deep theorems, involving geometry and dynamics.

54. Salem biography
• He even began working for a doctorate in law but quickly decided that he had to change direction to science, which he had been studying for years in parallel to his work in law.

55. Mansur biography
• The work is in three books: the first book studies properties of spherical triangles, the second book investigates properties of systems of parallel circles on a sphere as they intersect great circles, while the third book gives a proof of Menelaus's theorem.

56. Balogh biography
• Show that we can always construct line segments parallel to the sides of the square of total length 25 or less, so that each Pi is linked by the segments to both of the sides AB and CD.

57. Boone biography
• One book he much admired was Thomas Wolfe's 'Of Time and the River' whose hero, afflicted with a Faustian thirst for knowledge, moves from the rural South of the United States to Boston and then to England and France, and there is a superficial parallel to this in Bill's all-inquisitive progress from the Mid-West to sophisticated Princeton and subsequent regular visits to Europe.

58. Bisacre biography
• In this method the diamond is given a uniform chordal displacement, from line to line, as in the present method of ruling, but during its displacement from line to line it is constrained to rotate about an axis parallel to the ruled lines and passing through the centre of curvature of the face of the grating.

59. Dodgson biography
• Although Dodgson argued here for retaining Euclid's way of treating parallels, in Curiosa Mathematica, Part I: A New Theory of Parallels published nine years later, he presented his own ideas on dealing with the parallel axiom.

60. Proclus biography
• Proclus on the Parallel Postulate .

61. Auzout biography
• In a letter sent on 28 December 1666 to Henry Oldenburg, the first secretary of the Royal Society of London, Auzout explained how his new micrometer, with two parallel wires either of silk of silver, one of which could be moved by a screw, could be used to calculate the diameters of the planets and the parallax of the moon.

62. Bartels biography
• We should note that Lobachevsky took Bartels' course on the History of Mathematics which, following Montucla, considered in detail Euclid's Elements and his theory of parallel lines.

63. Fresnel biography
• Let parallel light impinge on an opaque disk, the surrounding being perfectly transparent.

64. Heaton biography
• Henry Heaton began his two parallel careers at the age of eighteen.

65. Roberts biography
• His writings on geometry included several important papers on parallel curves and surfaces.

66. Rocard biography
• After the award of his two doctorates, Rocard spent ten years working in industry with a parallel academic career.

67. Vekua biography
• During that period he wrote his major works devoted to the theory of distribution of elastic waves in an infinite layer with parallel plane boundaries.

68. Egervary biography
• Two such parallel events took place.

69. Goldstein biography
• He held these two appointments in parallel, dividing his time between the two.

• He had drawn a parallel between the axiomatic method, where theorems are deduced from axioms, and deducibility from primitive terms.

71. Apollonius biography
• In On the Burning Mirror Apollonius showed that parallel rays of light are not brought to a focus by a spherical mirror (as had been previously thought) and discussed the focal properties of a parabolic mirror.

72. Lonie biography
• However, this is explained by the fact that Lonie was training to be a teacher and began working in schools in 1838 in parallel with his university studies.

73. Chernikov biography
• The reader will have noticed the parallel in Chernikov moving from finite to infinite systems of linear inequalities in a similar spirit to moving from finite to infinite groups.

74. Barocius biography
• Among his many books is one on 13 ways to draw two parallel lines Admirandum illud geometricum problema tredecim modis demonstratum quod docet duas lineas in eodem plano designare, quae nunquam invicem coincidant, etiam si in infinitum protrahantur: et quanto longius producuntur, tanto sibiinuicem propiores euadant (1586) written in Latin.

75. Antonelli biography
• Each of these had to be routed to the proper bank of electronics and performed in sequence - not simply a linear progression but a parallel one, for the ENIAC, amazingly, could conduct many operations simultaneously.

• We mentioned above the 20 joint papers by Faddeeva and her husband, noting that some of the last few of these were: Natural norms in algebraic processes (1970), On the question of the solution of linear algebraic systems (1974), Parallel calculations in linear algebra (Part 1 in 1977, Part 2 in 1982), and A view of the development of numerical methods of linear algebra (1977).

77. Childs biography
• Parallel to this interest, with a touch of science intruding, was his interest in the cinema: he held successively the offices of Vice-Chairman (1948 - 51), Chairman (1951 - 54) and Secretary (1954 - 56) of the British Universities' Film Council.

78. Lemoine biography
• He also proved that if parallels are drawn through the Lemoine point parallel to the three sides of the triangle then the six points lie on a circle, now called the Lemoine circle.

79. Cesaro biography
• Cesaro later pointed out that in fact his geometry did not use the parallel axiom so constituted a study of non-euclidean geometry.

80. Zeno of Elea biography
• The resolutions will be parallel.

81. Lewis John biography
• The idea of the project is to use mathematics in the way we did in the earlier project - in telecommunications, as before, but also in multi-media computer operating systems and in parallel processing computer systems.

82. Bhaskara I biography
• He was the first to open discussion on quadrilaterals with all the four sides unequal and none of the opposite sides parallel.

83. Basso biography
• In a theoretical-experimental study, 'Fenomeni di polarizzazione cromatica in aggregati di corpi birifrangenti' Basso was the first to consider, from the point of view of optical behaviour, regular agglomerations of small double refraction elements each of which acts as a small isolated crystal, while forming a considerably continuous system on the whole, that he calls a 'radiated' system; he studied theoretically the phenomena of thin chromatic polarisation that show a thin sheet of radiated structure placed between two Nichol prisms with parallel light and convergent light and then experimentally verified the most salient characteristics predicted by the theory in the numerous special cases considered.

84. Auslander Louis biography
• At Bell Labs I was assigned the task of designing a parallel algorithm for adaptive beamforming.

85. Fricke biography
• In this talk he "outlined what Professor Perry is endeavoring to do in England and compared the Perry movement in England with a parallel movement in the German universities.

86. More Henry biography
• And the Moon itself, neither in the plane of the Earth's equator nor in a plane parallel to this? .

87. Grave biography
• Now Grave was a very talented as a musician and he would have liked to have studied music at the conservatory in parallel with his mathematical studies at the university.

88. Menelaus biography
• He used arcs of great circles instead of arcs of parallel circles on the sphere.

89. Rolle biography
• When one sees that if the desired curve has some maximum or minimum that meets a tangent parallel to the [x-axis], this can only be at the extremity of x = a; and that if it has one that meets ..

90. Beyel biography
• The latter is a concise collection of the basic principles of orthogonal parallel projection intended for students; it is based on Beyel's lectures [',' E Lemoine, review of C Beyel: Darstellende Geometrie, L’Enseignement Mathematique 4, 1902, 456-457 ','2].

91. Poncelet biography
• Since the "points of convergence" of parallel lines on the "mapped plane" do not correspond to real points of the projective plane, Poncelet added "ideal" or "infinitely distant" points to all planes, points that project to "points of convergence." Poncelet introduced infinitely distant points using Carnot's principle of correlation, which he called "the principle of continuity." Developing an idea of Carnot on "complex correlation," Poncelet introduced imaginary points of the plane, and, in particular, imaginary infinitely distant points, such as, for example, "cyclic points" - points belonging to all circles in the plane.

92. Heisenberg biography
• It had formerly been determined already that certain kinds of motions within the atom must be viewed as independent from one another to a certain degree, in the same way that a specific difference is made in classical mechanics between parallel motion and rotational motion.

93. Castelnuovo Emma biography
• The 1950s are the years of important movements of reforms, as the well-known New Math movement in the United States and the parallel Modern Mathematics in Europe, see [',' F Furinghetti, J M Matos and M Menghini, M..

94. Segal biography
• conviction that quantum field theory is on the verge of becoming mathematically firmly established, an will in fact in a few years be recognised as closely parallel to the analytical theory of functionals over infinite dimensional non-linear manifolds admitting group-invariant differential-geometric structures.

95. Vernier biography
• It has two graduated scales, a main scale like a ruler and a second scale, the vernier, that slides parallel to the main scale and enables readings to be made to a fraction of a division on the main scale.

96. Le Tenneur biography
• Le Tenneur essentially refutes this by showing that in a right-angled triangle every point on one of the shorter sides corresponds to a point on the hypotenuse by drawing a line parallel to the other short side, and visa-versa.

97. Collingwood biography
• he had great intellectual powers which enabled him to achieve excellence in diverse activities conducted in parallel and not in series.

98. Perseus biography
• A spiric section is then the curve produced when a plane parallel to the axis of revolution cuts the spiric surface.

99. Lehto biography
• In Chapter 4 the author fills this gap in one important instance by giving an existence proof for the parallel slit mappings (in the case of simply-connected domains this is identical with the Riemann mapping theorem) within the framework of the orthonormal function theory.

100. Cartier biography
• Let us note, however, that in parallel with these topics, he has continued to publish on algebraic geometry.

101. Fasenmyer biography
• I remember feeling that I was about to connect to a parallel universe that had always existed but which until then had remained very well hidden, and I was about to find out what sort of creatures lived there.

102. Quine biography
• And corresponds to terminals in series, or to those in parallel, so that if you simplify mathematical logical steps, you have simplified your wiring.

103. Hunt biography
• During these years, Hunt had continued his tennis career in parallel with his university studies.

104. Darmois biography
• In parallel with the descriptive order, the explanatory or logical orders, scientific theories and laws develop.

105. Recorde biography
• He justifies using two parallel line segments:- .

106. Theodosius biography
• Theodosius considered that it was 'day' if the sun was less than 15° below the horizon for then no stars were visible and he seemed to fail to understand that in the polar regions the sun can move almost parallel to the horizon.

107. Klugel biography
• Again following Kastner's advice he wrote a thesis on the parallel postulate entitled Conatuum praecipuorum theoriam parallelarum demonstrandi recensio Ⓣ.

108. Schramm biography
• In Illuminating sets of constant width (1988) he looked at the minimum number of directions required to illuminate the entire boundary of an n-dimensional body by sets of light rays parallel to these directions.

109. Menaechmus biography
• Menaechmus is famed for his discovery of the conic sections and he was the first to show that ellipses, parabolas, and hyperbolas are obtained by cutting a cone in a plane not parallel to the base.

110. Fredholm biography
• It is tempting to think that with two mathematical careers running in parallel, namely applications to physical applied mathematics and applications to actuarial science, Fredholm would have had little time for other interests.

111. Eratosthenes biography
• He assumed that the sun was so far away that its rays were essentially parallel, and then with a knowledge of the distance between Syene and Alexandria, he gave the length of the circumference of the Earth as 250,000 stadia.

112. Pick biography
• The plane becomes a lattice on setting up two systems of parallel equally spaced straight lines in the plane.

• We list a few of the papers in wrote, in collaboration with his colleagues, while in Malvern: A Practical Comparison of the Systolic and Wavefront Array Processing Architectures (1985); Phase Spaces from Experimental Time (1989); A Parallel Architecture for Nonlinear Adaptive Filtering and Pattern Recognition (1989); A Systolic Array for Nonlinear Adaptive Filtering and Pattern Recognition (1990), and Signal Processing for Nonlinear Systems (1991).

114. Lions biography
• In a long series of notes published in the Comptes Rendus until 2001, Lions returned to numerical analysis, and in particular to parallel computation and domain decomposition methods.

115. Valdivia biography
• He now undertook two studies in parallel.

116. Hertz Heinrich biography
• Hertz's work on the electromagnetic fields associated with a circular disk turning about its axis of symmetry in a magnetic field parallel to the axis of the disk is considered in [',' J Z Buchwald, A potential disagreement between Helmholtz and Hertz, Arch.

117. Mocnik biography
• He did not seek ordination when he completed the course at Gorizia but now took on two parallel careers.

118. Spencer biography
• Spencer's work with Kodaira was one of the most remarkable mathematical collaborations of the twentieth century: its only parallel is the famous Hardy-Littlewood work.

119. Ollerenshaw biography
• There is a strong parallel between mountain climbing and mathematics research.

120. Stephansen biography
• As a curiosity we can mention that this year [the school] has hired a very attractive young lady as an assistant in second year physics and mathematics since the class this spring was so large it had to be divided into two parallel sessions.

121. Wilf biography
• I remember feeling that I was about to connect to a parallel universe that had always existed but which had until then remained well hidden, and I was about to find out what sorts of creatures lived there.

122. Golub biography
• The same two authors published Scientific computing: An introduction with parallel computing in the following year.

123. Freedman biography
• He taught aeronautical engineering at Curtiss-Wright during the 1940s but had a parallel career as a scriptwriter of radio shows, drama critic and newspaper editor.

124. Haughton biography
• He was talented in applying mathematics to a wide range of different topics and the two young men were close friends and became collaborators [',' M DeArce, The parallel lives of Joseph Allen Galbraith (1818-90) and Samuel Haughton (1821-97): religion, friendship, scholarship and politics in Victorian Ireland, Proceedings of the Royal Irish Academy 112C (2011), 1-27.','5]:- .

125. Legendre biography
• Legendre's attempt to prove the parallel postulate extended over 30 years.

126. Chung biography
• Her interests are wide and among her nearly 200 publications there are contributions to spectral graph theory, extremal graphs, graph labelling, graph decompositions, random graphs, graph algorithms, parallel structures and various applications of graph theory in Internet computing, communication networks, software reliability, and discrete geometry.

127. Mathisson biography
• There was also some works to show that the condition of integrability of the magnitude of the angular momentum requires the electric momentum to vanish (and not merely to be parallel to the magnetic moment), but I was not able to follow the argument and have omitted this part.

128. Mellin biography
• The use of the inverse form of the transform, expressed as an integral parallel to the imaginary axis of the variable of integration, was developed by Mellin as a powerful tool for the generation of asymptotic expansions.

129. Anthemius biography
• parallel rays can be reflected to one single point from a parabolic mirror of which the point is the focus.

130. Lewis biography
• After a good resume of the classical theory of equations and inequations, he proceeds to a parallel development of the foundations of the logic of propositions, propositional functions, and classes on the Boole-Peirce-Schroder basis and on that of the 'Principia', exhibiting both the formal identity of the two systems and the inadequacy of Peirce's enumerative method of defining universal and particular propositions in terms respectively of iterated logical multiplication and iterated logical addition.

131. Flett biography
• Again this has a parallel with William Braid's arithmetic exercise book.

132. Schmidt Harry biography
• Parallel curves and normals of a plane arc as coordinate lines.

133. Blum biography
• We've developed a parallel theory ..

134. Chrysippus biography
• Plutarch in Common notions against the Stoics reports on a dilemma proposed by Democritus as reported by Chrysippus about a cone cut by a plane parallel to its base.

135. Al-Kindi biography
• He gave a lemma investigating the possibility of exhibiting pairs of lines in the plane which are simultaneously non-parallel and non-intersecting.

136. Xu Guangqi biography
• However the new Chinese terminology which Xu Guang-qi had to invent for point, curve, parallel line, acute angle, obtuse angle etc.

137. Vacca biography
• Despite his change of topic in mid career, Vacca continued his Chinese and mathematical studies in parallel.

138. Solitar biography
• in Mathematics from Adelphi when he was awarded the degree in 1961 for his thesis Non-Desarguian Planes and Parallel Geometries.

139. Weatherburn biography
• An elementary account of Levi-Civita's theory of parallel displacements is given.

140. Dehn biography
• By absolute geometry, we mean geometry satisfying the axioms of Euclidean geometry except for the parallel postulate.

141. Shannon biography
• Working with John Riordan, Shannon published a paper in 1942 on the number of two-terminal series-parallel networks.

142. Bernoulli Daniel biography
• Bernoulli determined the shape that a perfectly flexible thread assumes when acted upon by forces of which one component is vertical to the curve and the other is parallel to a given direction.

143. Orr biography
• Orr is best remembered today by applied mathematicians through the Orr-Sommerfeld equation that is an eigenvalue problem which models 2-dimensional modes of disturbance in a parallel shear flow.

144. Baker biography
• Its contents are as follows: Euclid's theory of parallel lines; Propositions of incidence; The symbolic representation and Pappus's theorem; Theorems proved from the propositions of incidence; The fundamental hypothesis; The symbols of the real points of a line; Involution and harmonic ranges; Related ranges and pencils; Conics; Assignment of two absolute points, properties of circles; The parabola; The rectangular hyperbola; Theorems on conics; Length and distance; Equation of conic and line.

145. Saint-Vincent biography
• He considers solid figures generated by two plane parallel surfaces on a common base where the solid is bounded by equidistant parallels.

146. Neuberg biography
• In parallel with this position, he was also a lecturer at the Ecole des Mines from October 1878 to October 1880, then he was promoted to professor at this college holding this position from October 1880 to October 1884.

147. Smith Karen biography
• In high school, I essentially discovered for myself the projective plane, a two-dimensional geometry not unlike the geometry one studies in high school but in which parallel lines meet "at infinity".

148. Adams Edwin biography
• The present paper extends this method to the case where the bars are mid-way between parallel planes.

149. Bosworth biography
• Professor Anne Bosworth, of Rhode Island, has followed this up by actually constructing in her doctor's dissertation at Gottingen (1900), under Hilbert, a sect-calculus independent of the parallel-axiom.

150. Bellavitis biography
• Given the plane, he called two line segments equipollent if they are parallel, of equal lengths, and equally directed.

151. Doob biography
• This is the long-awaited book by the author, developing in parallel potential theory and part of the theory of stochastic processes.

152. Brunelleschi biography
• He understood that there should be a single vanishing point to which all parallel lines in a plane, other than the plane of the canvas, converge.

153. Fiedler Wilhelm biography
• It was certainly appropriate to have Fiedler's book alongside Cremona's 'Projective Geometry' in the parallel course of descriptive geometry at the Technical Institutes.

154. Los biography
• In the same year, 1949, Łoś joined the Real Functions group in the Mathematical Institute of the Polish Academy of Sciences and continued to hold various positions in the Institute, in parallel to his university posts, until he retired in 1991.

155. Cooper biography
• In this paper results for Fourier integrals parallel to those of Bosanquet for series are obtained.

156. Robinson Raphael biography
• Imagine the plane cut with two sets of parallel lines into an infinite grid of unit squares called cells.

157. Thue biography
• an original statement that has no parallel in the literature.

158. Evelyn biography
• In the first theorem each circle touches a seventh, in the second the circles alternately touch a pair of parallel lines, in the third each circle touches two of the sides of a triangle and in the fourth each circle touches two out of three fixed circles making a configuration of nine circles in all.

159. Winkler biography
• Both Winkler's careers progressed in parallel.

• In a previous work [his thesis], I have looked at a system of n contiguous pendulums rotating about parallel axes and the existence of n sets of periodic motions around the stable equilibrium position.

161. Linfoot biography
• The "see-saw diagram" method of C R Burch (1942) [Burch worked at the H H Wills Laboratory] is applied to obtain general formulas for the Seidel errors, excluding distortion, of optical systems consisting of two mirrors and a nearly plane parallel plate.

162. Rohrbach biography
• At this time "working students" who financed their studies by taking on paid work in parallel to their studies were becoming increasingly common.

163. Dahlin biography
• He would work at the navigation School in parallel to his work at the prison service until 31 August 1903 when he retired from that position.

164. Cassini Jacques biography
• Cassini had some legal training but not so much that one would expect him to play a substantial role in this field but, in addition to his scientific work, he did undertake a second parallel career.

165. Bukreev biography
• Formula for the parallel angle; III.

166. Varga Otto biography
• A few years later Varga resumed the theme, and gave an elegant, geometrical construction for the metrical parallel translation, and thus for the metrical linear connection in Finsler spaces.

167. Wenninger biography
• However, in parallel to this he had entered Saint John's Abbey and professed Benedictine monastic vows on 11 July 1940.

168. Pogorelov biography
• Parallel with the method of approximation of the surfaces by polyhedra, Pogorelov significantly develops the regular theory of surfaces in the Euclidean and Riemannian spaces.

169. Chaplygin biography
• In the same year he published his results in the paper On the pressure exerted by a plane-parallel flow on an obstructing body, which gave the lift on a wing section [',' A T Grigorian, Biography in Dictionary of Scientific Biography (New York 1970-1990).','1]:- .

170. Henrici Peter biography
• He keeps reminding us to ask what Gauss would have done with a parallel computer - or with a pocket calculator.

171. Motwani biography
• But whereas there have been several books on parallel computation (though fewer good ones), Motwani and Raghavan's is the first textbook, to the best of the reviewer's knowledge, devoted to randomised algorithms.

## History Topics

1. Mathematics and Art
• He understood that there should be a single vanishing point to which all parallel lines in a plane, other than the plane of the canvas, converge.
• The square tiles are assumed to have one edge parallel to the bottom of the picture.
• The diagonals of the squares will all converge to a point D on a line through the centric point parallel to the bottom of the picture.
• The most important result in del Monte's treatise is that any set of parallel lines, not parallel to the plane of the picture, will converge to a vanishing point.
• He considers the representation in the picture plane of lines which meet at a point and also of lines which are parallel to each another.
• The phrase "linear perspective" was invented by Taylor in this work and he defined the vanishing point of a line, not parallel to the plane of the picture, as the point where a line through the eye parallel to the given line intersects the plane of the picture.
• He also defined the vanishing line to a given plane, not parallel to the plane of the picture, as the intersection of the plane through the eye parallel to the given plane.
• The main theorem in Taylor's theory of linear perspective is that the projection of a straight line not parallel to the plane of the picture passes through its intersection and its vanishing point.

2. Fair book
• What is the content of a field in the form of a trapezoid whose parallel sides are 1260 and 984 links and their perpendicular distance 567 links.
• What is the area of a field in the form of a trapezoid, its parallel sides being 1051 and 850 links, and their perpendicular distance 436 links.
• If 2 ac 20 p were to be cut off from the triangle ABC, which contains 4 ac, parallel to AC, the length of AB being 1125 links, in what point of AB must the line of division begin.
• If the 2 parallel ends of a zone of a parabola be 10 and 6, and the part of the absciss perpendicular to and connecting the middle of those ends be 4; what will be the area of the zone.
• If the two parallel ends of a zone of a parabola are 200 and 180 and the part of the absciss connecting the middle of those ends are 120 and makes an angle of 50°; required area of zone.
• Find the area of a trapezoid, of which the parallel sides are 143 and 121 inches, and their perpendicular distance 10 inches, or its content at 1 inch deep, in imperial gallons and bushels.
• Find the content of a vessel in the form of a frustum of an elliptical cone, the diameter of the bottom being 40 and 35 in, those at the top respectively parallel to these 20 and 17 in, and the depth 36 in, in imperial gallons.
• Find the content of a vessel in the form of a frustum of an elliptical cone, the diameter of the bottom being 50 and 40 in, those at the top respectively parallel to these 30 and 27 in, and the depth 50 in, in imperial gallons.
• If from a right-angled triangle, of which the base is 18 feet, and the perpendicular 24 feet, a triangle of which the area is 54 square feet be cut off by a line parallel to the perpendicular, what will be the sides of the latter.
• If from a triangle, of which the three sides are 13, 14, 15, a triangular area of 24 was cut off by a line parallel to the longest side, what will be the length of the sides of the triangle containing that area.

3. Non-Euclidean geometry
• 15">Playfair's Axiom:- Given a line and a point not on the line, it is possible to draw exactly one line through the given point parallel to the line.
• Legendre spent 40 years of his life working on the parallel postulate and the work appears in appendices to various editions of his highly successful geometry book Elements de Geometrie.
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• Elementary geometry was by this time engulfed in the problems of the parallel postulate.
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• He began to work out the consequences of a geometry in which more than one line can be drawn through a given point parallel to a given line.
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• Gauss discussed the theory of parallels with his friend, the mathematician Farkas Bolyai who made several false proofs of the parallel postulate.
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• The boundary lines of the one and the other class of those lines will be called parallel to the given line.
• Lobachevsky's Parallel Postulate.
• There exist two lines parallel to a given line through a given point not on the line.

4. Sundials
• The spaces marked off by the parallel lines running from top to bottom of the face showed where the shadow was to be read during the different months of the year, starting with the summer solstice at one edge and turning back again with the winter solstice at the other.
• 8-12.','7] This is as opposed to modern designs that have their gnomon slanted parallel to the earth's axis.
• The basic construction involved hollowing out a hemisphere (or smaller wedge of a sphere) with its top parallel to the horizon.
• These daily arcs were all parallel, and the arc of the equinox was half of a circle with the same centre as the hemisphere (a great circle).[',' S L Gibbs, Greek and Roman sundials / Sharon L Gibbs.
• Two or three parallel circular arcs were all that were needed for ease of reading (being the corresponding lines of "latitude").

5. Weather forecasting
• Here, u is the zonal wind, parallel to the circles of latitude; v is the meridional wind, parallel to the circles of longitude; and w is the vertical wind component [','U Langematz, Vorlesung 6: Dynamik I (FU Berlin, 2009) ','16].

6. Fair book insert
• What is the area of a trapezoid, the parallel sides of which are 20 and 30 yds and the perpendicular distance between them 18 yds.
• The first solution given to this problem is incorrect but then the correct solution is given using the rule area = (a + b)h/2 where a, b are the lengths of the parallel sides and h is the perpendicular distance between them.

7. Classical time
• In fact two calendars ran in parallel, the one which was used for practical purposes such as the sowing of crops, harvesting crops etc.
• There was a parallel, said Boyle, between the creator of the Strasbourg clock who built a mechanism which ran on its own without the intervention of the builder and the universe made by God which operated according to his laws but without his intervention.

8. Abstract linear spaces
• The parallel development in analysis was to move from spaces of concrete objects such as sequence spaces towards abstract linear spaces.
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• He defines two line segments as 'equipollent' if they are equal and parallel, so, in modern notation, two line segments are equipollent if they represent the same vector.
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9. Cartography
• He chose a defining line for the north-south lines of his grid through Rhodes and drew seven parallel lines to each of his defining lines to form a rectangular grid.
• He then marked off where the lines of longitude crossed the parallel of Rhodes, taking 400 stadia per degree.

10. Classical light
• Let parallel light impinge on an opaque disk, the surrounding being perfectly transparent.
• He discovered what is now called the Faraday effect, namely that if a beam of light is passed through a substance which polarises it, then the plane of polarisation is rotated by a magnetic field parallel to the ray of light.

11. Alcuin's book
• Let us assume that the top half of the field is a trapezium with the parallel sides of length 50 and 60 yards and the other two sides of length 50 yards.
• We can use Pythagoras to work out the distance between the parallel lines in the trapezium as 49.75 metres (approximately).

12. Indian Sulbasutras
• Now draw RE parallel to YP and complete the square QEFG.

13. Calculus history
• Fermat also investigated maxima and minima by considering when the tangent to the curve was parallel to the x-axis.
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14. Brachistochrone problem
• From the given point A let there be drawn an unlimited straight line APCZ parallel to the horizontal, and on it let there be described an arbitrary cycloid AQP meeting the straight line AB (assumed drawn and produced if necessary) in the point Q, and further a second cycloid ADC whose base and height are to the base and height of the former as AB is to AQ respectively.

15. Doubling the cube
• So what was the machine which Eratosthenes invented to solve the problem? It consists to two parallel lines with triangles between them as shown in the top diagram.

16. Pi history
• If we have a uniform grid of parallel lines, unit distance apart and if we drop a needle of length k < 1 on the grid, the probability that the needle falls across a line is 2k/π.
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17. Egyptian mathematics
• In fact two calendars ran in parallel, the one which was used for practical purposes of sowing of crops, harvesting crops etc.

18. Planetary motion
• The focus A is constructed geometrically by drawing FM parallel to CD to cut the circle at M, and dropping a perpendicular from M to cut CD at A (thus making AM = BF).

19. Trisecting an angle
• Finally draw from A the radius AX of the circle with AX parallel to EC.

20. Bourbaki 2
• Some wanted to see two parallel approaches, that of Grothendieck in the spirit of the founders, and also advanced texts in areas they wanted to cover.

21. Brachistochrone problem references
• G J Tee, Isochrones and brachistochrones, Neural Parallel Sci.

22. Nine chapters references
• R B Wang, A preliminary exploration on the logical order of the Shang gong chapter of Jiu zhang suanshu - a parallel discussion on the understanding of the operational rule ab+ac=a(b+c) in 'Jiu zhang' (Chinese), in Collected research papers on the history of mathematics Vol.

23. Brachistochrone problem references
• G J Tee, Isochrones and brachistochrones, Neural Parallel Sci.

24. Nine chapters references
• R B Wang, A preliminary exploration on the logical order of the Shang gong chapter of Jiu zhang suanshu - a parallel discussion on the understanding of the operational rule ab+ac=a(b+c) in 'Jiu zhang' (Chinese), in Collected research papers on the history of mathematics Vol.

25. Modern light
• The clearest example of how this works is to look again at Thomas Young's experiment of passing rays of light through two parallel slits and observing the interference patterns on a screen behind (see the article Light through the ages: Ancient Greece to Maxwell).

26. Tait's scrapbook
• This seems to be made when the motion towards the audience is combined with that in a direction parallel to it; but the full resolution into its components still remains for some bright young student.

27. Infinity
• If a line is moved parallel to itself across two areas and if the ratio of the lengths of the line within each area is always a : b then the ratio of the areas is a : b.

28. Measurement
• The French proposed 45° which conveniently fell in France, the British proposed London, and the United States proposed the 38th parallel which was conveniently close to Thomas Jefferson's estate.

29. Egyptian numerals
• The two systems ran in parallel for around 2000 years with the hieratic symbols being used in writing on papyrus, as for example in the Rhind papyrus and the Moscow papyrus, while the hieroglyphs continued to be used when carved on stone.

30. Real numbers 1
• Magnitudes, being distinct entities from numbers, had to have a separate definition and indeed Nicomachus makes such a parallel definition for magnitudes.

31. Hirst's diary
• (23 Dec 1859) What a wonderful head he has, not merely round but spheroidal with the largest diameter parallel to his eyes, or rather to the line joining his ears.

32. Jaina mathematics
• A circle is divided by parallel lines into regions of prescribed widths.

## Societies etc

1. Academy of Sciences of Belarus
• It has Divisions of: algebra; number theory; control processes theory; differential equations; mathematical cybernetics; mathematical theory of systems; non-linear analysis; numerical methods of mathematical physics; numerical modelling; parallel computational processes; and stochastic analysis.

## Honours

1. Galway Group Theory.html
• M Batty (Dublin) Parallel algorithms in hyperbolic groups .

## References

1. References for Orszag
• Massively parallel supercomputers seem the best hope for achieving progress on 'grand challenge' problems such as understanding high-Reynolds-number turbulent flows, Physics Today (March 1993), 34-42.

2. References for Beltrami
• M J Scanlan, Beltrami's model and the independence of the parallel postulate, Hist.

3. References for Al-Tusi Nasir
• G D Mamedbeii, Muhammed Nasir al-Din al-Tusi on the theory of parallel lines and the theory of ratios (Azerbaijani), Izdat.

4. References for Schwartz Jacob
• Parallel Computing Pioneers : Jacob T Schwartz .

5. References for Al-Samarqandi
• A E-A Hatipov, The theory of parallel lines in the medieval East (Russian), Trudy Samarkand.

6. References for Al-Jawhari
• A E-A Hatipov, The theory of parallel lines in the medieval East (Russian), Trudy Samarkand.

7. References for Byrne
• D J Cohen, The creed of St Athanasius proved by a mathematical parallel, in Equations from God: Pure Mathematics and Victorian Faith (John Hopkins University Press, Baltimore, 2007), 73.

8. References for Haughton
• M DeArce, The parallel lives of Joseph Allen Galbraith (1818-90) and Samuel Haughton (1821-97): religion, friendship, scholarship and politics in Victorian Ireland, Proceedings of the Royal Irish Academy 112C (2011), 1-27.

9. References for Kaluza
• M Kaku, Hyperspace : A scientific odessey through parallel universes, time warps, and the 10th dimension (Oxford, 1994), 99-107.

1. Proclus on the Parallel Postulate
• Proclus on the Parallel Postulate .
• We give below an extract from Proclus concerning the parallel postulate, but first let us quote the Postulates from Book I of Euclid's Elements:- .
• Proclus Diadochus, in his Commentary on Euclid's Elements, discusses the Parallel Postulate.
• The Parallel Postulate .
• http://www-history.mcs.st-andrews.ac.uk/Extras/Proclus_parallel_postulate.html .

2. Valiant Turing Award
• Parallel and distributed computing.
• In addition to computational learning theory and computational complexity, a third broad area in which Valiant has made important contributions is the theory of parallel and distributed computing.
• An example of a simple insight is his parallel routing scheme, described in the paper "A scheme for fast parallel communication" (SIAM J.
• Valiant's main contribution to parallel computing is the introduction of the "bulk synchronous parallel" (BSP) model of computing.
• One of the main articles introducing this model is his paper "A bridging model for parallel computation" (CACM, 1990).
• This paper is a must read for both technical and pedagogical reasons It lays out the case that a model of parallel computing should attempt to bridge software and hardware.
• Specifically the model should capture parallel computing hardware by a small collection of numerical parameters.
• Similarly, parallel compilers should only need to know these few parameters from the model when compiling programs.
• Such a model should be judged by how well it can predict the actual running times of parallel algorithms.
• In contrast to the case of sequential computing, where von Neumann's model easily satisfied these requirements (at least to a first approximation), coming up with a similar model in parallel computing has been hard.
• A parallel computer is further specified by parameters for the number of processors as well as for the latency and throughput of the interconnect.
• The debate on the right model for parallel computing remains unresolved to this day, with several competing suggestions.

3. Edinburgh Mathematics Examinations
• A straight line which divides the sides of a triangle proportionally is parallel to the base of the triangle.
• Through the point of intersection of the diagonals of a trapezium a line is drawn parallel to the parallel sides; prove that the parallel sides have the same ratio as the parts into which the line cuts the non-parallel sides.
• The difference of longitude between two places is 5°, and the latitude of both is 45°; find the distance between them along the parallel of latitude.
• Prove geometrically that the locus of the middle points of a series of parallel chords of a conic section is a straight line.
• Shew that the locus of the middle points of a series of parallel chords of a conic section is a straight line.
• Two circular plates of equal diameters, and placed parallel to and over one another, are kept charged with constant quantities of electricity.

4. University of Glasgow Examinations
• One of the parallel sides of a trapezoid is double the other.
• One of the two parallel sides of a trapezoid exceeds the other in length by 4 feet, and if the former were made equal to the latter, the area would be increased by 8 square feet and in the ratio of 6 : 7.
• Find the lengths of the parallel sides.
• Two parallel forces, each of 8 poundals, act in opposite directions, and in a line midway between their lines of action, a third force of 16 poundals acts.
• From an equilateral triangle whose side is a inches long a triangular piece is cut off by a line drawn from the middle point of one of the sides parallel to another side.
• Enunciate and prove the theorem regarding the difference between the moments of inertia with respect to an axis through the centre of inertia and an axis parallel to this.

5. Valdivia aspects of maths
• nnnnnnnna single parallel to the given line.
• In the background, Saccheri is using a method that in mathematics is called 'reduction to the absurd', starting from the hypothesis that is equivalent to that in the plane, through a point not on a given straight line more than one straight line parallel to the given straight line can be drawn and, reasoning from here, to try to come to a contradiction.
• This geometry is constructed from the postulates that appear explicitly or implicitly in the 'Elements' of Euclid, suppressing the postulate of the parallels and putting in its place the following: "In a plane, through any point not on a given straight line, can be drawn more than one parallel to the given line." Many of the propositions of this geometry contradict other results of Euclidean geometry.
• There is in the plane a line and a point not on it through which it is possible to draw more than one parallel to that line.
• In a plane, through a point not on a straight line, you can draw a single straight line parallel to it.

6. Euclid on elementary astronomy
• And, since this star appears to be equidistant in all directions from the circumferences of the circles in which the rest of the stars move, we must assume that the circles are all parallel, so that all the fixed stars move in parallel circles having for one pole the aforesaid star.
• Further, the circle of the Milky Way and the zodiac circle, which are both obliquely inclined to the parallel circles and cut one another, appear in their revolution always to show semicircles above the earth.
• For, if a cone or a cylinder be cut by a plane not parallel to the base, the section arising is a section of an acute-angled cone, which is like a shield (an ellipse).
• But if a sphere rotates about its own axis all the points on the surface of the sphere describe, in the same time, similar arcs of the parallel circles on which they are carried; therefore the points in question traverse similar arcs of the equinoctial circle, on one side the arc above the earth, on the other the arc under the earth; therefore the arcs are equal; therefore both are semicircles, for the distance from rising to rising, or from setting to setting, is the whole circle; therefore the zodiac circle and the equinoctial circle bisect one another.

7. St Andrews Mathematics Examinations
• Hence prove that if a straight line falls on two parallel straight lines, it makes the two interior angles on the same side equal to two right angles.
• Planes to which the same straight line is perpendicular are parallel.
• What does the equation to any locus give? What do m and c stand for in y = mx + c? Find the equation to the straight line through (5, -7), (i) parallel to 3x + 5y = 8 ; and (ii) perpendicular to the same line.
• A floor is ruled with equidistant parallel lines; a rod shorter than the distance between each pair is thrown at random on the floor: find the chance of its falling on one of the lines.

8. Santalo honorary doctorate
• As an example, he presents the case of a needle that is thrown randomly onto a plane in which equidistant parallel lines have been drawn, the distance between them being greater than the length of the needle, with the agreement that the player wins if the needle does not cut any parallel and lose in the opposite case.
• In order to calculate the prize that the player will receive in the case of winning, it is necessary to calculate the probability that the needle does not cut any parallel.
• Laplace (1749-1827) in his Analytical Theory of Probabilities (1812), considers the plane divided into congruent rectangles by two series of parallel lines and calculates the probability that a needle thrown at random on the plane does not cut any of those straight lines (problem of the Laplace needle).

9. Heaton problems
• Problem: A certain solid has a square, side = a, for its base, and all parallel sections are squares, the two sections through the middle points of the opposite side of the square are semi-circles, however.
• Prove that the other intersections of the third circle with the first two are in a line parallel to the common tangent of the first two.
• What is the average length of all straight lines that can be drawn within a given square parallel to one of the diagonals? .
• What is the average length of all straight lines that can be drawn within a given square parallel to one of the diagonals? .

10. Rouche and de Comberousse
• 4 - Parallel lines.
• 2 - Straight lines and parallel planes.
• - Left quadrilateral cut by any plane and, in particular, by a plane parallel to two opposite sides.
• A condition for straight lines to have their perspectives parallel.

11. St Andrews Physics Examinations
• Find the magnitude and line of action of the resultant of two parallel forces acting on a body in the same direction.
• A circular wire is hung up, and another circular wire carrying an electric current is gradually approached to it in a parallel plane: describe what happens.
• Find the resultant of two unequal parallel forces acting in opposite directions.
• Prove that if two parallel conducting plates distant t from one another be respectively at potentials V1 and V2 , and if S be the amount of the surface of each, and F the total mechanical force urging the conductors, then .

12. Carl Runge: 'Graphical Methods
• If the area of any closed curve is to be found, the way to proceed is to choose two parallel lines that cut off two segments on either side (see Fig.
• As a rule it is convenient to draw a and b at right angles and the similar triangle either with its hypotenuse parallel (Fig.
• Then the drawing of a parallel to the hypotenuse of the rectangular triangle a, b through the end of the line corresponding to c will always lead to the number .

13. James Jeans addresses the British Association in 1934, Part 2
• A shower of parallel-moving electrons forms in effect an electric current.
• The newer and more accurate wave-picture, which transcends the framework of space and time, recombines the photons into a single beam of light, and the shower of parallel-moving electrons into a continuous electric current.
• The same is true, mutatis mutandis, of the electrons of a parallel-moving shower.

14. Ahrens book of quotes
• In the theory of parallel lines we have come no further than Euclid did.
• Few topics in Mathematics will be as prominently covered in literature as the hole at the foundation of geometry in the justification of the theory of parallel lines.
• I am pleased that you have the courage to express yourself in such a way as if to accredit the possibility that our theory of parallel lines, and thus our whole geometry, is wrong.

15. Eulogy to Euler by Fuss
• Euler had imagined that the magnetic body possessed pores which formed continuous piping, parallel and bristling, similar to veins or valves and so narrow so as to only allow passage for the most subtle parts of the aether, of which the elasticity pushes the relaxed parts into the magnet's pores.
• Euler had seized upon this similarity and followed its parallel by allowing us to see that light is born from of vibratory movement in the aether and sound is produced by a similar movement in the air.

16. W H Young addresses ICM 1928 Part 2
• A modern, still somewhat embryonic parallel is the idea of Ophelimity in Social Economics, due, I believe, chiefly to Pareto.
• The two forms of limitation are indeed the precise analogue of those found in the most current arguments from Analogy; whenever a scheme A is compared to a scheme B, the comparison unfailingly breaks down at some point; entities exist in A which have no correlatives in B, and vice versa; properties of A may be adduced which find no parallel in the scheme B, and vice versa.

17. Mathematics in Glasgow
• Three parallel subdivisions of the class meet daily (except on Saturdays), one from 9 to 10, one from 10 to 11, one from 12 to 1.
• Two parallel subdivisions of the Class meet daily (except on Saturdays), one from 9 to 10, one from 12 to 1.

18. Arthur Eddington's 1927 Gifford Lectures
• There is a familiar table parallel to the scientific table, but there is no familiar electron, quantum or potential parallel to the scientific electron, quantum or potential.

19. The Dundee Numerical Analysis Conferences
• There was a Conference on the Applications of Numerical Analysis from 23 - 26 March, 1971, with 170 participants, 18 one hour lectures by invited speakers and 17 submitted talks given in parallel sessions.
• There were 234 participants, with 20 invited speakers, and 43 submitted papers presented in parallel sessions.

20. Nevil Maskelyne measures the Earth's density
• It had also the advantage, by its steepness, of having but a small base from north to south; which circumstance, at the same time that it increases the effect of attraction, brings the two stations on the north and south sides of the hill, at which the sum of the two contrary attractions is to be found by the, experiment, nearer together; so that the necessary allowance of the number of seconds, for the difference of latitude due to the measured horizontal distance of the two stations, in the direction of the meridian, would be very small, and consequently not subject to sensible error from any probable uncertainty of the length of a degree of latitude in this parallel.
• Thus the less latitude appearing too small by the attraction on the south side, and the greater latitude appearing too great by the attraction on the north side, the difference of the latitudes will appear too great by the sum of the two contrary attractions; if, therefore, there is an attraction of the hill, the difference of latitude by the celestial observations ought to come out greater than what answers to the distance of the two stations measured trigonometrically, according to the length of a degree of latitude in that parallel, and the observed difference of latitude subtracted from the difference of latitude inferred from the terrestrial operations, will give the sum of the two contrary attractions of the hill.

21. Caius Iacob: 'Applied mathematics and mechanics
• Parallel forces.
• Addition of parallel gravity forces.

22. Poincaré on non-Euclidean geometry
• (3) Through one point only one parallel can be drawn to a given straight line.
• The number of parallel lines that can be drawn through a given point to a given line is one in Euclid's geometry, none in Riemann's, and an infinite number in the geometry of Lobachevsky.

23. Dahlin Extracts
• The triangle is split into required parts by drawing lines that are parallel to one of the other sides.
• This is followed by the "praxis", as the author puts it; "the square root of the ratio between the square of the basis and the number of parts, determines the distance between the point of intersection on the base and the angle opposite the side with which the division line is parallel".

24. Kurosh: 'Lectures on general algebra' Introduction
• Topological algebra sprang up and soon occupied a very prominent position, and a parallel development took place in the theory of ordered algebraic structures.
• The theory of lattices made its appearance and developed rapidly; and the last few years have seen the rise of the parallel theory of categories, which undoubtedly has a most important future.

25. Turnbull lectures on Colin Maclaurin, Part 2
• He drew an axis PE, and allowed the point P to move along PE, while each of the lines PA and Pa moved parallel to itself.
• (This formula is nothing else than that of , for the subtangent, when E is the origin, and EP = x, PA = y are the coordinates of A referred to oblique axes, namely EP and a line through E parallel to PA.) Hence , and on taking the sum of n such results .

26. Prufer's reviews
• Synthetic projective geometry is presented based upon an axiomatic structure comprised of postulates of connection (including the parallel postulate, in Playfair's form) for three dimensions, postulates of order and continuity.
• The Playfair statement is the form adopted for the parallel axiom.

27. H M Macdonald addresses the British Association in 1934
• The difference between Cauchy's hypothesis as to the nature of the mutual actions of the medium and Green's hypothesis has been referred to above; another important difference in their treatments is that Cauchy assumes that the direction of the disturbance in the medium is parallel to the plane of polarisation, while Green, in accordance with Fresnel's view, assumes that this direction is perpendicular to the plane of polarisation.
• With these limitations he proves that, if the direction of a disturbance is parallel to the plane of polarisation and the medium is free from the action of any external forces, the directions of polarisation and the velocities of propagation are the same as in Fresnel's theory.

28. Horace Lamb addresses the British Association in 1904
• It is noteworthy, however, that the development of the modern German school of mathematical physics, represented by Helmholtz and Kirchhoff, in linear succession to Franz Neumann, ran in many respects closely parallel to the work of Stokes and his followers.
• It would, however, be going too far to claim this tendency as the exclusive characteristic of English physicists; for example, the elastic investigations of Green and Stokes have their parallel in the independent though later work of Kirchhoff; and the beautiful theory of dynamical systems with latent motion which we owe to Lord Kelvin stands in a very similar relation to the work of Helmholtz and Hertz.

29. Klein Elementary Mathematics
• Finally, with regard to the method of presentation in what follows, it will suffice if I say that I have endeavored here, as always, to combine geometric intuition with the precision of arithmetic formulas, and that it has given me especial pleasure to follow the historical development of the various theories in order to understand the striking differences in methods of presentation which parallel each other in the instruction of today.

30. Henry Baker addresses the British Association in 1913
• You are familiar with the axiom that, given a straight line and a point, one and only one straight line can be drawn through the point parallel to the given straight line.

31. ELOGIUM OF EULER
• This observation gave reason to believe that two prisms both unequal and comprised of different substances combined together might divert a ray from its course without decomposing it or rather by replacing, by a triple refraction, the elementary rays in a parallel direction.

32. Who was who 1852
• Lobachevsky had been pondering over the parallel postulate since 1815; around 1826 he arrived at the alternate postulate of two parallels and his investigations were published in seven memoirs 1829-1856.

33. The Tercentenary of the birth of James Gregory
• Since the days of Ancient Greece no parallel to the brilliancy of that half century of scientific thought can be found.

34. Halmos popular papers
• What does mathematics really consist of? Axioms (such as the parallel postulate)? Theorems (such as the fundamental theorem of algebra)? Proofs (such as Godel's proof of undecidability)? Concepts (such as sets and classes)? Definitions (such as the Menger definition of dimension)? Theories (such as category theory)? Formulas (such as Cauchy's integral formula)? Methods (such as the method of successive approximations)? Mathematics could surely not exist without these ingredients; they are all essential.

35. Airy on Thales' eclipse
• There is one road from Kaisarieh falling on a branch of the Euphrates, which flows by Malatieh (Melitene); a rugged road parallel to it from Guroun; and finally, the road which is the best of all, descending from the southern mountains into the plain of Tarsus and Adana, then skirting the sea by Issus to Antioch.

36. Knorr's papers
• Moreover, we know that about the same period or earlier the geometers Archytas and Eudoxus had initiated studies in solid geometry to which such a study by Menaechmus could easily be related: Indeed, the natural philosopher Democritus had posed a problem in connection with the parallel sectioning of the cone.

37. Flett's books
• I have assumed that any student beginning this book will be familiar with the techniques of elementary calculus, and that while reading the book he will be following a parallel course on abstract algebra.

38. Charles Bossut on Leibniz and Newton Part 2
• They required the curves, which being constructed in two contrary directions on one axis of a given position, and then moving parallel to themselves with unequal velocities, should constantly intersect each other at a given angle.

39. Napier Tercentenary
• Professor David Eugene Smith, New York, read a paper on the "Law of Exponents in the Works of the Sixteenth Century." The nature of the geometrical progression and its correspondence with a parallel arithmetical progression were traced from the works of Chuquet (1484) and Boethius (1499) through the writings of Rudolff (1525) and Stifel (1544) to those of later date.

40. Smith's History Papers
• This is a long street running parallel to the Boulevard de Strasbourg from near the Conservatoire des Arts et Metiers to the Seine.

41. Gibson History 8 - James Stirling
• He showed that a curve of the nth degree is determined in general by n(n + 3)/2 points, and that parallel lines meet any algebraic curve in the same number of real or imaginary points.

42. Byrne: Doctrine of Proportion
• Nor do we defend the system of showing how inadequate numbers or letters are to express magnitudes, or their relations to one another; for it is admirable to see how the parallel propositions agree, although managed by means essentially different.

43. Groups St Andrews proceedings
• An invitation to Professor J Neubuser (Aachen) to arrange a workshop on Computational Group Theory and the use of GAP was taken up so enthusiastically by him that the workshop became effectively a fully-fledged parallel meeting throughout the second week, with over thirty hours of lectures by experts and with practical sessions organised by M Schonert (Aachen).

44. McBride equal bisectors
• (Transversal across two parallel lines makes interior angle-sum on one side equal to two right angles.) We have already rejected, as indirect, proofs of the Contrapositive, like Casey's, Steiner's, etc.

45. Simplicius on astronomy and physics
• But he must go to the physicist for his first principles, namely, that the movements of the stars are simple, uniform, and ordered, and by means of these principles he will then prove that the rhythmic motion of all alike is in circles, some being turned in parallel circles, others in oblique circles.

46. Atiyah reviews
• However, it appears that parallel problems were being investigated in the past but a common language and framework were missing.

47. Teixeira on da Silva
• It is well known that Louis Poinsot, in his beautiful treatise on Statics, replaced the moments of forces, employed before him by geometers as subsidiary means for deducing the equilibrium conditions of bodies, by pairs of equal forces, in parallel and opposite directions (couples), and in this way managed to simplify and illuminate most of the theories of Mechanics.

48. Howie Thanksgiving Service
• In parallel with his academic pursuits, John kept a continuous involvement with Scottish education matters.

49. Mathematicians and Music 3
• He, too, occupied himself with the problem of the vibrating string and constructed a model of a surface, certain parallel sections of which give the form of the curve of the vibrating string at any time under conditions which Monge states.

50. Heath: Everyman's Library 'Euclid' Introduction
• Thus "Axiom 11," that all right angles are equal to one another, was Euclid's "Postulate 4," while "Axiom 12" (the well-known parallel-axiom) was "Postulate 5." Further, of the first ten "Axioms" only five can, with any probability, be attributed to Euclid himself (1, 2, 3, 8, and 9 in Todhunter's edition).

51. Hardy on the Tripos
• I was asked whatever could you do, if you could not tell the quality of a man by looking at his examination record? I wonder whether my questioner realised that these elaborate honours examinations, so far from being one of the fundamental necessities of modern civilisation, are a phenomenon almost entirely individual to Oxford and Cambridge, copied in a half-hearted fashion by other English universities, and, beyond that, having hardly a parallel in the world? Does Germany suffer from intellectual stagnation, because there are no honours examinations in her universities? Germany does not think in terms of firsts and seconds; we think in terms of them, so far as we do so think, and perhaps the practice is to some extent abating, merely because we have heard so much about them that they have become to us like bitter ale or eggs and bacon, and we have forgotten that we could get on quite happily without them.

52. Gibson History 11 - John Playfair, Sir John Leslie
• His Elements of Euclid was long in use in, Scottish Schools; in it he uses the Parallel Axiom now known by his name, though he expressly states that it had been "assumed by others, particularly by Ludlam in his very useful little tract entitled Rudiments of Mathematics" (p.

53. Harold Jeffreys on Logic and Scientific Inference
• At present we are faced with the inaccuracy of Euclid's parallel axiom, which for millennia was considered intuitively obvious; with the inaccuracy of Newton's law of gravitation, which had been well established by experience and had been believed for centuries to be exact; with the failure in stars of the law of the indestructibility of matter; and with the discordance of the classical undulatory theory of light with the group of facts known as quantum phenomena.

54. Proclus on pure and applied mathematics
• The divisions of optics are: (a) the study which is properly called optics and accounts for illusions in the perception of objects at a distance, for example, the apparent convergence of parallel lines or the appearance of square objects at a distance as circular; (b) catoptrics, a subject which deals, in its entirety, with every kind of reflection of light and embraces the theory of images; (c) scenography (scene-painting), as it is called, which shows how objects at various distances and of various heights may so be represented in drawings that they will not appear out of proportion and distorted in shape.

55. Durell and Robson: 'Advanced Trigonometry
• The authors are planning text-books parallel to the present volume on Advanced Algebra and Calculus, written from a similar point of view.

56. Noneuclidean tesselations
• The fact that all axioms of Hilbert other than the parallel postulate are satisfied is verified explicitly.

57. Finlay Freundlich's Inaugural Address, Part 2
• parallel to the surface of the earth, from which we try to derive the maximum of riches to embellish our lives.

58. Adam Ries: 'Coss
• The selection provides glimpses into the hand-written original of the Coss, accompanied by parallel transliterated passages.

59. Ptolemy's hypotheses of astronomy
• They saw the sun and the moon and the other stars moving from east to west in circles always parallel to each other; they saw the bodies begin to rise from below, as if from the earth itself, and gradually to rise to their highest point, and then, with a correspondingly gradual decline, to trace a downward course until they finally disappeared, apparently sinking into the earth.

60. R L Wilder: 'Cultural Basis of Mathematics III
• The restricted mathematics known as Intuitionism has won only a small following, although some of its methods, such as those of a finite constructive character, seem to parallel the methods underlying the treatment of formal systems in symbolic logic, and some of its tenets, especially regarding constructive existence proofs, have found considerable favour.

61. L E Dickson: 'Linear algebras
• Running parallel with the general theory is an illustrative example treated independently but in the spirit of the theory.

62. Twenty-Five Years of Groups St Andrews Conferences
• The workshop became effectively a fully-fledged parallel meeting throughout the second week, with over thirty hours of lectures by experts together with practical sessions.

63. Poincaré on intuition in mathematics
• (4) through a given point there is not more than one parallel to a given straight.

64. The St Andrews Schmidt-Cassegrain Telescope
• Gregory showed that spherical aberration can be avoided by the use of a parabolic instead of a spherical mirror and that light from a distant object falling on a parabolic mirror, parallel to its axis, produces a clearly defined image of the object.

65. A D Aleksandrov's view of Mathematics
• Another example, equally impressive, is provided by non-Euclidean geometry, which arose from the efforts, extending for 2000 years from the time of Euclid, to prove the parallel axiom, a problem of purely mathematical interest.

66. Madras College exams
• It is a fact to which even in Scotland there is no parallel, that in a town containing not much more than four thousand inhabitants, there should exist a school attended by between eight and nine hundred pupils - a proportion which, according to the common statistical estimate applicable to ordinary circumstances, would more than account for all the children of the community between six and twelve years of age.

67. Big Game Hunting
• The lions are then oriented parallel to the earth's magnetic field, and the resulting beam of lions is focussed by the catnip upon the cage.

68. Big Game Hunting
• The lions are then oriented parallel to the earth's magnetic field, and the resulting beam of lions is focussed by the catnip upon the cage.

69. Ball books
• There is a new chapter of 20 pages on "The Parallel Postulate," and one of 6 pages on the "Insolubility of the Algebraic Quintic." Those who amused themselves in their youth by making figures known as 'Cat's Cradles' by twisting on the hands a loop of string will be interested in the new chapter on "String Figures." The subject is more extensive than most people think.

70. Lehrer Songs
• He had parallel careers as a musician and song-writer, and as an academic teaching mathematics as part of liberal arts courses first at the Massachusetts Institute of Technology and then, from 1972, at the University of California at Santa Cruz.

71. Pack wartime papers
• Abstract: When a jet of gas issues from an orifice as a parallel stream with a given supersonic velocity and flows in a steady state through an outer medium at rest, its behaviour is governed by the ratio between the exit pressure of the jet and the pressure of the outer medium.

72. Smith's Teaching Books
• This book is written for first year students at colleges and technical schools in America, and is in many respects parallel with the recent book on Analytical Geometry by the same authors.

73. Aitken: 'Statistical Mathematics
• One has no difficulty for example in conceiving a die which might be an irregular hexahedron, heterogeneous in density and with non-parallel and unequal opposite edges and faces.

74. Philip Jourdain and Georg Cantor
• A great part of Dedekind's work has developed along a direction parallel to the work of Cantor, and it is instructive to compare with Cantor's work Dedekind's Stetigkeit und irrationale Zahlen and Was sind und was sollen die Zahlen?, of which excellent English translations have been issued by the publishers of the present book.

75. G C McVittie papers
• J L Synge writes: The author's method of transforming the equations of hydrodynamics [see the preceding review] is applied to the case of motion parallel to a surface; two coordinates are orthogonal coordinates in this surface and the third coordinate is orthogonal to it.

76. Turnbull and Aitken: 'Canonical Matrices
• We have, in fact, allowed ourselves a free hand in dealing with the results of earlier writers, in the belief that the outcome would prove to be an easier approach to a subject that has often failed to win affection; and the methods of H J S Smith, Sylvester, Frobenius, and Dickson proved in themselves quite adequate without the inclusion of other parallel theories.

77. Landau and Lifshitz Prefaces
• The reader is not assumed to have any previous knowledge of tensor analysis, which is presented in parallel with the development of the theory.

78. history of reliability
• The importance of the reliability of subsea systems is in many respects parallel to the reliability of spacecrafts.

79. Analysis of Variance
• For each treatment in each locality there is a mixing tank from which the fluid is pumped to all the tanks on this treatment, connected "in parallel:" We do not want a "series" connection, where the outflow from one tank is the inflow to another, because this would confound the effects of the varieties in these two tanks with the effects (if any) of order in the "series" connection.

80. Sommerville: 'Geometry of n dimensions
• In fact, in Euclidean geometry this is not true since parallel lines have no point in common.

81. Phillip S Jones on Brook Taylor
• Taylor gave "perspective" proofs, terming lines "parallel" if they met on the vanishing line of their common plane.

82. Élie Cartan reviews
• He develops the theory of spinors (he discovered the general mathematical form of spinors in 1913) systematically by giving a purely geometrical definition of these mathematical entities; this geometrical origin makes it very easy to introduce spinors into Riemannian geometry, and particularly to apply the idea of parallel transport to these geometrical entities.

83. O'Brien Physics
• U representing any directed magnitude and u any distance, the translation of U to any parallel position in space, in such wise that every point or element of U is caused to describe the distance u, is termed the translation of U along u.

84. Percy MacMahon addresses the British Association in 1901
• From a map of London of 1746 it appears to have run parallel to the present Brick Lane and to have corresponded to the present Wilks Street.] The members of the Society at the beginning were for the most part silk-weavers of French extraction; it was little more than a working man's club at which questions of mathematics and natural philosophy were discussed every Saturday evening.

## Quotations

1. Quotations by Lambert
• This hypothesis (Parallel hypothesis) would not destroy itself at all easily.
• Proofs of the Euclidean [parallel] postulate can be developed to such an extent that apparently a mere trifle remains.

2. Quotations by Littlewood
• To someone who wants them he would say that the ideal system runs parallel to the usual theory: "If this is what you want, try it: it is not my business to justify application of the system; that can only be done by philosophizing; I am a mathematician".

3. Quotations by Lagrange
• Lagrange, in one of the later years of his life, imagined that he had overcome the difficulty (of the parallel axiom).

4. Quotations by Manin
• One could try to find a parallel in the humanities by comparing this to hermeneutics: the art of finding hidden meanings of sacred texts.

5. Quotations by Wiener Norbert
• Bound up with it is a judgement of values, quite parallel to the judgement of values that belongs to the painter or the musician.

6. A quotation by Ollerenshaw
• There is a strong parallel between mountain climbing and mathematics research.

## Famous Curves

1. Curve definitions
• Parallel curves .
• Any parallel curve to C is also an involute of C'.
• Parallel curves : Two curves are parallel if every normal to one curve is a normal to the other curve and the distance between where the normals cut the two curves is a constant.
• Although parallel curves are at a fixed distance apart they can look rather different.
• For example Cayley's sextic and the nephroid are parallel.
• Leibniz was the first to consider parallel curves.
• Let P1 be the point with P1O a line segment parallel and of equal length to PQ.

2. Cycloid
• The caustic of the cycloid, where the rays are parallel to the y-axis is a cycloid with twice as many arches.

3. Cochleoid
• The points of contact of parallel tangents to the cochleoid lie on a strophoid.

4. Circle
• The caustic of a circle with radiant point on the circumference is a cardioid, while if the rays are parallel then the caustic is a nephroid.

5. Spiric
• After Menaechmus constructed conic sections by cutting a cone by a plane, around 150 BC which was 200 years later, the Greek mathematician Perseus investigated the curves obtained by cutting a torus by a plane which is parallel to the line through the centre of the hole of the torus.

6. Nephroid
• The involute of the nephroid is Cayley's sextic or another nephroid since they are parallel curves.

7. Curve definitions
• Any parallel curve to C is also an involute of C'.

8. Parabola
• Gregory and Newton considered the properties of a parabola which bring parallel rays of light to a focus.

9. Tricuspoid
• The caustic of the tricuspoid, where the rays are parallel and in any direction, is an astroid.

10. Freeths
• If the line through P parallel to the y-axis cuts the nephroid at A then angle AOP is 3π/7.

11. Cardioid
• There are exactly three parallel tangents to the cardioid with any given gradient.

## Chronology

1. Mathematical Chronology
• He tries to prove the parallel postulate.
• Desargues begins the study of projective geometry, which considers what happens to shapes when they are projected on to a non-parallel plane.
• In Euclides ab Omni Naevo Vindicatus Saccheri does important early work on non-euclidean geometry, although he considers it an attempt to prove the parallel postulate of Euclid.
• Lambert writes Theorie der Parallellinien which is a study of the parallel postulate.
• By assuming that the parallel postulate is false, he manages to deduce a large number of results about non-euclidean geometry.
• D'Alembert calls the problems to elementary geometry caused by failure to prove the parallel postulate "the scandal of elementary geometry".
• Legendre points out the flaws in 12 "proofs" of the parallel postulate.
• Reynolds publishes An experimental investigation of the circumstances which determine whether the motion of water in parallel channels shall be direct or sinuous and of the law of resistance in parallel channels.

2. Chronology for 1760 to 1780
• Lambert writes Theorie der Parallellinien which is a study of the parallel postulate.
• By assuming that the parallel postulate is false, he manages to deduce a large number of results about non-euclidean geometry.
• D'Alembert calls the problems to elementary geometry caused by failure to prove the parallel postulate "the scandal of elementary geometry".

3. Chronology for 1880 to 1890
• Reynolds publishes An experimental investigation of the circumstances which determine whether the motion of water in parallel channels shall be direct or sinuous and of the law of resistance in parallel channels.

4. Chronology for 1625 to 1650
• Desargues begins the study of projective geometry, which considers what happens to shapes when they are projected on to a non-parallel plane.

5. Chronology for 1AD to 500
• He tries to prove the parallel postulate.

6. Chronology for 1830 to 1840
• Legendre points out the flaws in 12 "proofs" of the parallel postulate.

7. Chronology for 1720 to 1740
• In Euclides ab Omni Naevo Vindicatus Saccheri does important early work on non-euclidean geometry, although he considers it an attempt to prove the parallel postulate of Euclid.

## EMS Archive

1. EMS 1913 Colloquium
• After explaining how non-Euclidean Geometry arose from attempts to prove the axiom about parallel lines, the lecturer proceeded to give an exposition of the system of geometry which was discovered by Lobachevsky, in which Playfair's axiom was directly contradicted and the sum of the angles of a triangle was always less than two right angles.
• Dr Sommerville's second lecture on Non-Euclidean Geometry was devoted to the geometry of Riemann, in which parallel lines do not exist, and the sum of the angles of a triangle is always greater than two right angles.
• While there are no parallel lines in this geometry, lines in space may be equidistant, and a remarkable surface is obtained by revolving one line about another to which it is equidistant.
• When the method of denial was applied to these as to the parallel-postulate, new forms of non-Euclidean geometry emerged.

2. Edinburgh Mathematical Society Lecturers 1883-2016
• (Ladies' College, Edinburgh) Theorems in connection with lines drawn through a pair of points parallel and antiparallel to the sides of a triangle .
• (Wilson College, Bombay) A proof that the middle points of parallel chords of a conic lie on a fixed straight line .
• (StnAndrews) Note on Legendre's and Bertrand's proof of the parallel postulate by infinite areas .

3. EMS 2003 Colloquium
• The meeting was of a somewhat different format from past Colloquia in that it was held in parallel with an LMS-EPSRC Instructional Conference.
• The Course was held in parallel with the 'St Andrews Colloquium' attended by about 15 self-supporting participants which allowed further academic and social interaction.

4. EMS Proceedings papers
• Note on Legendre's and Bertrand's proof of the parallel postulate by infinite areas .

5. EMS Proceedings papers
• Theorems in connection with lines drawn through a pair of points parallel and antiparallel to the sides of a traingle .

6. EMS 1913 Colloquium
• After we had been taught that velocities did not compound according to the parallelogram law, it was a positive delight to find that the Fourier series remained ordinarily additive; and with this in possession we had no great difficulty in apprehending the possibility of a space devoid of parallel lines.

7. 1911-12 Dec meeting
• Sommerville, Duncan M Y: "Note on Legendre's and Bertrand's proof of the parallel postulate by infinite areas" .

8. 1905-06 May meeting
• Jack, John: "A proof that the middle points of parallel chords of a conic lie on a fixed straight line" .

9. Napier Tercentenary
• Professor David Eugene Smith, New York, read a paper on the "Law of Exponents in the Works of the Sixteenth Century." The nature of the geometrical progression and its correspondence with a parallel arithmetical progression were traced from the works of Chuquet (1484) and Boethius (1499) through the writings of Rudolff (1525) and Stifel (1544) to those of later date.

10. 1901-02 Nov meeting
• Burgess, Alexander Gordon: "Theorems in connection with lines drawn through a pair of points parallel and antiparallel to the sides of a triangle" .

## BMC Archive

1. Scientific Committee minutes 2004
• Thus BAMC parallel sessions run alongside BMC splinter groups and Special Sessions, and there will be 'BMC plenary talks', 'BAMC plenary talks' and 'Plenary plenary talks'.
• It was pointed out by Niels Jacob that in Germany conferences like the BMC sometimes have public lectures for schoolchildren given by world class figures, in parallel with the scientific programme.

2. Mathematics 2005
• In addition there were 12 Morning Speakers in the BMC style, whose talks were in pairs, taking place at the same time as BAMC parallel sessions.
• Because of the large number of parallel sessions, it was necessary to use lecture rooms in three adjacent buildings of the University.

3. Scientific Committee minutes 2004
• Thus BAMC parallel sessions run alongside BMC splinter groups and Special Sessions, and there will be 'BMC plenary talks', 'BAMC plenary talks' and 'Plenary plenary talks'.
• It was pointed out by Niels Jacob that in Germany conferences like the BMC sometimes have public lectures for schoolchildren given by world class figures, in parallel with the scientific programme.

4. Report2013.html
• Morning lectures were given, in parallel pairs, by Stuart White (Glasgow), Gavin Brown (Loughborough), Zinaida Lykova (Newcastle), Tim Dokchitser (Bristol), June Barrow-Green (Open), Bruno Vallette (Nice, Isaac Newton Institute), Lasse Rempe-Gillen (Liverpool), Konstantin Ardakov (Queen Mary, London), Tom Leinster (Edinburgh), Anthony Dooley (Bath), Tom Bridgeland (Oxford) and Gesine Reinert (Oxford).
• For one pair, the audience sizes were very unbalanced and, in retrospect, it might have been better not to have conformed to the traditional parallel format throughout the morning sessions.

5. Minutes for 1998
• Dr Garling suggested that two parallel conferences might be held at the same time and place, with free transfer between the two.
• There could be a common theme for such parallel meetings, with perhaps a special session or some plenary lectures in common.

6. Scientific Committee 2002
• The meeting would be in parallel rather than in series (Warwick was in series).
• BAMC has 6 parallel morning sessions and over 100 contributed talks.

7. Minutes for 2003
• The meeting will run from Monday to Thursday (4-7 April) with Pure and Applied Mathematics sessions running in parallel throughout.

8. Minutes for 1993
• The specialist sessions should run in parallel with the "normal" programme (and perhaps with each other) from coffee time until just before the evening lecture.

• Usually 12/14; in 1 hour parallel slots.

10. BMC Report
• These were complemented by a full BAMC programme of principal speakers, contributed talks and minisymposia, in parallel with a full BMC programme of principal speakers, morning speakers, splinter groups, and special sessions in .

11. Minutes for 1976
• (i) The advantages and disadvantages of the present system of parallel morning lectures were discussed.

12. Minutes for 1975
• (a) It was decided that the morning sessions should continue to be two parallel streams of three lectures each.

13. Minutes for 2002
• The Warwick programme was different in many ways from the standard BMC format: morning speakers and special sessions were run together, in three parallel streams rather than two.

14. Minutes for 1974
• The organising committee were asked to look into the possibility of providing short abstracts of the morning talks to help members in choosing between the parallel sessions.

15. LMS report
• The meeting at Liverpool will also be a joint one (with a somewhat different format to the Warwick one, parallel rather than series).

16. Minutes for 2003
• The meeting will run from Monday to Thursday (4-7 April) with Pure and Applied Mathematics sessions running in parallel throughout.

17. Minutes for 2010
• Richard Pinch commented that the scientific committee had done a good job of cross over encouragement and that the joint meeting did not have the feel of two parallel meetings.

18. Minutes for 2003
• The meeting will run from Monday to Thursday (4-7 April) with Pure and Applied Mathematics sessions running in parallel throughout.

19. Minutes for 1977
• 3) It was decided to continue the present pattern of lectures with 9 morning sessions with two speakers in parallel, and 3 afternoon lectures.

## Gazetteer of the British Isles

1. London Museums
• The mathematically and mechanically minded should look closely to see Watt's parallel motion-the first linkage, of which he was most proud - and the 'sun and planet' gearing, developed because someone had been allowed to patent the crank! .

2. Oxford professorships
• Savile is said to have been distressed by the status of the parallel postulate and unable to resolve it, so he founded the Savilian Chair of Geometry in the hope that one of the holders would be able to succeed [Math.

## Astronomy section

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JOC/BS August 2001