Search Results for length*

Biographies

1. Eudoxus biography
   
   • Eudoxus made important contributions to the theory of proportion, where he made a definition allowing possibly irrational lengths to be compared in a similar way to the method of cross multiplying used today.
     
   • A major difficulty had arisen in mathematics by the time of Eudoxus, namely the fact that certain lengths were not comparable.
     
   • The method of comparing two lengths x and y by finding a length t so that x = m × t and y = n × t for whole numbers m and n failed to work for lines of lengths 1 and √2 as the Pythagoreans had shown.
     
   • By this Eudoxus meant that a length and an area do not have a capable ratio.
     
   • But a line of length √2 and one of length 1 do have a capable ratio since 1 × √2 > 1 and 2 × 1 > √2.
     
   • Hence the problem of irrational lengths was solved in the sense that one could compare lines of any lengths, either rational or irrational.
     
   • In the seventh book Eudoxus wrote at length on the Pythagorean Society in Italy again about which he was clearly extremely knowledgeable.
     

2. Guo Shoujing biography
   
   • The simplest astronomical instruments was the gnomon, nothing other than a stick which was erected and the length of its shadow measured.
     
   • The minimum length of shadow during a day is less in summer than in winter and at the solstices it changes from lengthening to shortening or visa versa.
     
   • The work was completed by 1280, Guo having calculated the length of the year correct to within 26 seconds, and in the following year Kublai Khan introduced the use of this extremely accurate calendar.
     
   • In the diagram d is the diameter of the circle, a is the length of the arc AB and x is the length of NB which Guo wanted to calculate.
     
   • The equation has two real roots, the smaller being the solution to the problem while the other, being numerically larger than the length of the arc, was rightly discarded by Guo.
     
   • Two of the coefficients of the equation, namely the constant term and the coefficient of x2, involve the length a of the arc, so require a value to be chosen for π.
     

3. Zu Chongzhi biography
   
   • He was able to make a calendar with this degree of accuracy since he had calculated the length of the tropical year (time between two successive occurrences of the vernal equinox) as 365.24281481 days (an error of only 50 seconds from its true value of 365 days 5 hours 48 minutes 46 seconds), and a nodal month for the moon of 27.21233 days (compare the modern value of 27.21222 days).
     
   • Having accurate knowledge of the lengths of the year and the month were necessary, but it is still not clear how Zu translated this into a cycle of 391 years.
     
   • Zu's calculations of the length of the year were well within the range that allowed him to differentiate between the tropical and sidereal year.
     
   • To compute this accuracy for π, Zu must have used an inscribed regular 24,576-gon and undertaken the extremely lengthy calculations, involving hundereds of square roots, all to 9 decimal place accuracy.
     

4. Ptolemy biography
   
   • For the sake of completeness in our treatment we shall set out everything useful for the theory of the heavens in the proper order, but to avoid undue length we shall merely recount what has been adequately established by the ancients.
     
   • However, those topics which have not been dealt with by our predecessors at all, or not as usefully as they might have been, will be discussed at length to the best of our ability.
     
   • He confirmed the length of the tropical year as 1/300 of a day less than 365 1/4 days, the precise value obtained by Hipparchus.
     
   • Based on his observations of solstices and equinoxes, Ptolemy found the lengths of the seasons and, based on these, he proposed a simple model for the sun which was a circular motion of uniform angular velocity, but the earth was not at the centre of the circle but at a distance called the eccentricity from this centre.
     
   • In these two book Ptolemy also discusses precession, the discovery of which he attributes to Hipparchus, but his figure is somewhat in error mainly because of the error in the length of the tropical year which he used.
     

5. Hemchandra biography
   
   • A line of length n contains n units where each short syllable is one unit and each long syllable is two units.
     
   • Clearly a line of length n units takes the same time to articulate regardless of how it is composed.
     
   • Hemchandra asks: How many different combinations of short and long syllables are possible in a line of length n? .
     
   • Suppose that there are f (n) possibilities for a line of length n.
     
   • The line of length n either ends in a short syllable or in a long syllable.
     
   • If it is the former than there remains a line of length n-1 which can be composed in f (n-1) ways and if the line of length n ends in a long syllable then there is a line of length n-2 remaining which can be composed in f (n-2) ways.
     

6. Callippus biography
   
   • Callippus made accurate determinations of the lengths of the seasons and constructed a 76 year cycle comprising 940 months to harmonise the solar and lunar years which was adopted in 330 BC and used by all later astronomers.
     
   • Meton's observations were made in Athens in 432 BC but he gave a length for the year which was 1/76 of a day too long.
     
   • Instead of having totals of 440 hollow and 500 full months, Callippus adopted 441 hollow and 499 full, thus reducing the length of four Metonic cycles by one day.
     
   • Other contributions of Callippus to mathematical astronomy included his observation of the inequality in the lengths of the seasons.
     

7. Hipparchus biography
   
   • Hipparchus calculated the length of the year to within 6.5 minutes and discovered the precession of the equinoxes.
     
   • This work came from Hipparchus's attempts to calculate the length of the year with a high degree of accuracy.
     
   • There are two different definitions of a 'year' for one might take the time that the sun takes to return to the same place amongst the fixed stars or one could take the length of time before the seasons repeated which is a length of time defined by considering the equinoxes.
     
   • Of course the data needed by Hipparchus to calculate the length of these two different years was not something that he could find over a few years of observations.
     
   • 21 (4) (1979/80), 291-309.',20)">20] suggests that Hipparchus calculated the length of the tropical year using Babylonian data to arrive at the value of 1/300 of a day less than 3651/4 days.
     
   • Hipparchus also calculated the length of the sidereal year, again using older Babylonian data, and arrived at the highly accurate figure of 1/144 days longer than 3651/4 days.
     

8. Theodorus biography
   
   • root) of a square of three square units and of five square units, that these roots are not commensurable in length with the unit length, and he went on in this way, taking all the separate cases up to the root of seventeen square units, at which point, for some reason, he stopped.
     
   • There is no doubt that Theodorus would have constructed lines of length √3, √5 etc.
     
   • This proof generalises easily (for a modern mathematicians thinking in terms of numbers rather than lengths) to show √n is irrational for any non-square n.
     

9. Brahmagupta biography
   
   • In the Brahmasphutasiddhanta Brahmagupta gave remarkable formulae for the area of a cyclic quadrilateral and for the lengths of the diagonals in terms of the sides.
     
   • Brahmagupta believed in a static Earth and he gave the length of the year as 365 days 6 hours 5 minutes 19 seconds in the first work, changing the value to 365 days 6 hours 12 minutes 36 seconds in the second book the Khandakhadyaka.
     
   • This second values is not, of course, an improvement on the first since the true length of the years if less than 365 days 6 hours.
     
   • One has to wonder whether Brahmagupta's second value for the length of the year is taken from Aryabhata I since the two agree to within 6 seconds, yet are about 24 minutes out.
     

10. Liu Hui biography
    
    • He found a recurrence relation to express the length of the side of a regular polygon with 3 × 2n sides in terms of the length of the side of a regular polygon with 3 × 2n-1 sides.
      
    • We know AB, it is pn-1 , the length of the side of a regular polygon with 3×2n-1 sides, so AY has length pn-1/2.
      
    • Thus OY has length .
      
    • Then YX has length r - √[r2 - (pn-1/2)2].
      
    • Then pn= AX is the length of a side of a regular polygon with N = 3 × 2n sides.
      

11. Theodosius biography
    
    • Theodosius considers the length of the night and day at various points on the earth and claims that the day lasts for seven months at the north pole and the night for five months.
      
    • The other work On days and nights is in two books, the first of which has 13 propositions, the second 19 propositions, which give conditions on the lengths of the night and day depending on the location of the observer.
      
    • Theodosius also considers the two possibilities, that the length of the year is a rational multiple of the length of the day and that it is an irrational multiple.
      

12. Khayyam biography
    
    • Khayyam measured the length of the year as 365.24219858156 days.
      
    • We know now that the length of the year is changing in the sixth decimal place over a person's lifetime.
      
    • For comparison the length of the year at the end of the 19th century was 365.242196 days, while today it is 365.242190 days.
      
    • Find a point on a quadrant of a circle in such manner that when a normal is dropped from the point to one of the bounding radii, the ratio of the normal's length to that of the radius equals the ratio of the segments determined by the foot of the normal.
      
    • to any length, which has not been made before now.
      

13. Bisacre biography
    
    • The critical (optimum) length of the grating for automatic focusing is determined by the condition that the quadratic term in the expansion for the optical path in powers of distance measured along the grating face from its centre must be three-eighths of a wave-length.
      
    • Third-order effects have been considered and it is shown that in the conditions contemplated in the use of these optimum gratings, the third-order term affects the length of the optical path by something like one 650th part of a wave-length, and is consequently negligible.
      
    • This second approximation becomes important if either the radius of curvature is comparable to the wave-length of light or the angle of incidence is very nearly 90° , as it may be in soft X-ray experiments.
      

14. Wolf biography
    
    • From this data Wolf was the first to calculate an accurate length of the cycle, obtaining a value of 11.1 years.
      
    • 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.
      
    • The error will be the smallest for a given number of trials if the length a of the needle is equal to one-fourth of the product of the interval length d and the ratio π.
      
    • On a plate of about one square foot I drew a series of parallels at a distance of 45 mm, and from a knitting needle I broke a piece of 36 mm length - thus getting as close as 1/100 to the ideal ratio according to the instruction above.
      

15. Brunelleschi biography
    
    • Also important was his understanding of scale, and he correctly computed the relation between the actual length of an object and its length in the picture depending on its distance behind the plane of the canvas.
      
    • It was still a lengthy construction process, but by the time Brunelleschi died in 1446 the dome was almost completely finished.
      

16. Arago biography
    
    • Using their data the length of the metre was set but Mechain had remained keen to obtain more data.
      
    • Working with Biot, Arago made measurements of arc length on the Earth which led to the standardisation of the metric system of lengths.
      

17. Zhu Shijie biography
    
    • Hence x = 12 or 5 but the base having the shorter length gives x = 5 bu, y = 12 bu, so z = √(52 + 122) = 13.
      
    • Let x, y be the lengths of the two legs and z the length of the hypotenuse of the triangle.
      

18. Roberval biography
    
    • He compared the lengths of curves, a topic not considered since the times of the ancient Greeks, equating the spiral and parabola in their ordinary forms.
      
    • Before August 1648 he had discovered the equality of the length of the generalised cycloid and the ellipse.
      
    • He also computed the arc length of a spiral.
      

19. Li Zhi biography
    
    • Chapter 1 contains three sections, the first giving the names of the constituents, the second section lists all the values of the lengths of the segments, so in essence contains all the answers to the problems, while the third section comprises of 692 formulae for areas of triangles and lengths of segments.
      
    • Find the length of the side of the farm and the diameter of the pond.
      

20. Heuraet biography
    
    • Sluze, Huygens, van Schooten, Hudde and van Heuraet corresponded regarding the properties of curves, in particular van Heuraet was interested in methods of rectification, that is methods to determine the length of a curve.
      
    • This was particularly important since at this time mathematicians believed that it was not possible to compare the length of a curved arc with a straight line segment.
      
    • We should note that William Neile, independently of van Heuraet, found the arc length of an algebraic curve in 1657 when he rectified the cubical parabola.
      
    • When van Heuraet learned that I had measured the surface of the parabolic conoid and had determined the length of the parabola equal to a given quadrature of the hyperbola (concerning both of which I wrote you previously), he found not only both of them by his own technique but, in addition, he rectified completely all other curves of those genera that we allow in geometry.
      

21. Thales biography
    
    • The claims that Thales used the Babylonian saros, a cycle of length 18 years 10 days 8 hours, to predict the eclipse has been shown by Neugebauer to be highly unlikely since Neugebauer shows in [The exact sciences in antiquity (Providence, R.I., 1957).',11)">11] that the saros was an invention of Halley.
      
    • Hieronymus says that [Thales] even succeeded in measuring the pyramids by observation of the length of their shadow at the moment when our shadows are equal to our own height.
      
    • This appears to contain no subtle geometrical knowledge, merely an empirical observation that at the instant when the length of the shadow of one object coincides with its height, then the same will be true for all other objects.
      
    • Thales discovered how to obtain the height of pyramids and all other similar objects, namely, by measuring the shadow of the object at the time when a body and its shadow are equal in length.
      

22. Mouton biography
    
    • He also suggested a standard linear measurement, which he called the mille, based on the length of the arc of one degree of longitude on the Earth's surface and divided decimally.
      
    • Mouton wanted a practical means to determine the length of a virgula.
      
    • Certainly one could not measure the circumference of the earth, so he proposed a standard based on the length of a pendulum.
      
    • He conducted experiments which led him to the conclusion that a simple pendulum of length one virgula would oscillate 3959.2 times in 30 minutes.
      

23. Napier biography
    
    • And having thought upon many things to this purpose, I found at length some excellent brief rules to be treated of (perhaps) hereafter.
      
    • Consider two lines AB of fixed length and A'X of infinite length.
      
    • Napier chose the length AB to be 107, based on the fact that the best tables of sines available to him were given to seven decimal places and he thought of the argument x as being of the form 102.sin X.
      

24. Landen biography
    
    • is a special relation between the length of an ellipse, the length of a hyperbolic segment, and the length of a circle.
      

25. Pythagoras biography
    
    • Pythagoras noticed that vibrating strings produce harmonious tones when the ratios of the lengths of the strings are whole numbers, and that these ratios could be extended to other instruments.
      
    • However, because of his belief that all things are numbers it would be a natural task to try to prove that the hypotenuse of an isosceles right angled triangle had a length corresponding to a number.
      

26. Koch biography
    
    • His results were all fairly accessible, although many of the calculations are lengthy.
      
    • It gives a continuous curve which is of infinite length and nowhere differentiable.
      

27. Geminus biography
    
    • It describes the main constellations, the variation of the length of night and day at different latitudes, the rising of the signs of the zodiac, and the length of the lunar month.
      
    • But Geminus proves an interesting classification theorem, namely that the helix, the circle and the straight line are the only curves with the property that any part of the curve will coincide with any other part of the same length.
      

28. Karp biography
    
    • The thesis, Languages with expressions of infinite length, was supervised by L Henkin and submitted to the University of Southern California in 1959.
      
    • In 1964 she published a book on her research Languages with expressions of infinite length but she had hoped to write another work which would take her ideas considerably further.
      
    • Carol R Karp's Languages with expressions of infinite length .
      

29. De L'Hopital biography
    
    • Florimond de Beaune had asked for a curve for which the subtangent had a fixed length and Bernoulli had included the solution in the course he had given l'Hopital.
      
    • Grant that a curved line may be considered as the assemblage of an infinite number of infinitely small straight lines; or (what is the same thing) as a polygon with an infinite number of sides, each of infinitely small length such that the angle between adjacent lines determines the curvature of the curve.
      
    • Given his definition of a curve as a polygon with an infinite number of sides each of infinitely small length, he can define the tangent at a point on the curve as being the straight line produced from the infinitely small straight line at that point.
      

30. Zhang Qiujian biography
    
    • Then divide the length of the road by the greatest common divisor to get 325/30= 10 5/6days.
      
    • The chord of the segment is given, as is its area, and the student is asked to compute its height (the length of the perpendicular bisector of the chord to the circle).
      
    • Martzloff [A history of Chinese mathematics (Berlin-Heidelberg, 1997).',2)">2] points out that the length as calculated by Zhang in this problem is in error by about 14%.
      

31. Theaetetus biography
    
    • This means that it was Theaetetus's work on irrational lengths which is described in the Book X, thought by many to be the finest work of the Elements.
      
    • For Theaetetus had distinguished square roots commensurable in length from those which are incommensurable, and who divided the more generally known irrational lines according to the different means, assigning the medial line to geometry, the binomial to arithmetic and the apotome to harmony, as stated by Eudemus..
      

32. Dinostratus biography
    
    • so the length of the circumference of the circle is expressed in terms of the lengths of straight lines.
      

33. Bouguer biography
    
    • In April 1735 Bouguer set out on an expedition, organised by the Academie Royale des Sciences, to Peru to measure the length of a degree of meridian at the equator.
      
    • In 1741 Bouguer discovered a small error in the joint measurements made with La Condamine to determine the length of a degree of meridian.
      
    • In a medium of uniform transparency the light remaining in a collimated beam is an exponential function of the length of the path in the medium.
      

34. Al-Quhi biography
    
    • To construct a sphere segment equal in volume to a given sphere segment and equal in area to a second sphere segment - a problem similar to but more difficult than related problems solved by Archimedes - Al-Quhi constructed the two unknown lengths by intersecting an equilateral hyperbola with a parabola and rigorously discussed the conditions under which the problem is soluble.
      
    • Al-Quhi also described a conic compass, a compass with one leg of variable length, for drawing conic sections in the treatise On the perfect compass [Dictionary of Scientific Biography (New York 1970-1990).',1)">1]:- .
      

35. Picard Jean biography
    
    • He measured the length of the arc of the meridian; the measurements appear in Mesure de la Terre (1671).
      
    • Picard was also involved with the measurement of the length of the second pendulum.
      
    • Finding that the length was not constant led to the first proof that the Earth was not a perfect sphere but was flattened at the poles.
      

36. Al-Umawi biography
    
    • Before describing the Marasim we should make some brief comments about al-Umawi's work calculating lengths and areas.
      
    • In it al-Umawi gives rules for calculating: lengths of chords and lengths of arcs of circles (using Pythagoras's theorem); areas of circles, areas of segments of circles, areas of triangles and quadrilaterals; volumes of spheres, volumes of cones and volumes of prisms.
      

37. Clavius biography
    
    • One of these I observed about midday at Coimbra in Lusitania [Portugal] in the year 1559 [sic], in which the Moon was placed between my sight and the Sun with the result that it covered the whole Sun for a considerable length of time.
      
    • Of course, the study of theology was a lengthy process at this time and, despite becoming a teacher in 1564, he continued with his studies and did not become a full member of the Jesuit Order until 1575.
      

38. Klein Oskar biography
    
    • Klein's adaptation of Kaluza's work had a major difference from the original in that the extra or fifth dimension was curled up into a ball that was on the order of the Planck length, 10-33 cm.
      
    • The dimension was on the order of the Planck length.
      

39. Lorentz biography
    
    • Lorentz is also famed for his work on the FitzGerald-Lorentz contraction, which is a contraction in the length of an object at relativistic speeds.
      
    • They describe the increase of mass, the shortening of length, and the time dilation of a body moving at speeds close to the velocity of light.
      

40. Mersenne biography
    
    • In this work he was the first to publish the laws relating to the vibrating string: its frequency is proportional to the square root of the tension, and inversely proportional to the length, to the diameter and to the square root of the specific weight of the string, provided all other conditions remain the same when one of these quantities is altered.
      
    • He stated the obvious properties including the length of the base line equals the circumference of the rolling circle.
      

41. Delambre biography
    
    • The report was approved by the National Assembly one week later and it remained to calculate a more accurate value of the length of the meridian.
      
    • By June of that year, after Mechain had also reported, a definitive platinum bar of length one metre was made to become the basis of the metric system.
      

42. Laplace biography
    
    • Laplace knew well that the proposed scheme did not really work because the length of the proposed year did not fit with the astronomical data.
      
    • Applications to mortality, life expectancy and the length of marriages are given and finally Laplace looks at moral expectation and probability in legal matters.
      

43. Mathieu Claude biography
    
    • In the same year, together with Biot, he embarked on a series of measurements of the length of the seconds pendulum at different points on the meridian, in particular at Bordeaux and at Dunkirk.
      
    • The fact that the project had been so major and unrepeatable was its great virtue, argued Mathieu, for the metre was now fixed for all time and even if the Archive Metre were damaged, its length was known in relation to the pendulum so could be reconstructed.
      

44. Pascal biography
    
    • Wren had been working on Pascal's challenge and he in turn challenged Pascal, Fermat and Roberval to find the arc length, the length of the arch, of the cycloid.
      

45. Bell John biography
    
    • [Contains lengthy accounts of discussions with John Bell.]',1)">1], Bell reported being perplexed by the usual statement of the Heisenberg uncertainty or indeterminacy principle (Δx Δp ≥ planck , where Δx and Δp are the uncertainties or indeterminacies, depending on one's philosophical position, in position and momentum respectively, and planck is the (reduced) Planck's constant).
      
    • [Contains lengthy accounts of discussions with John Bell.]',1)">1]: .
      

46. Wren biography
    
    • He found the length of an arc of the cycloid using an exhaustion proof based on dissections to reduce the problem to summing segments of chords of a circle which are in geometric progression.
      
    • It is impossible in an article of this length to give even an indication of the range of architectural commissions which Wren carried out.
      

47. Cantor biography
    
    • The next question he asked himself, in January 1874, was whether the unit square could be mapped into a line of unit length with a 1-1 correspondence of points on each.
      
    • During the visit he apparently began to behave eccentrically, talking at great length on the Bacon-Shakespeare question; then he travelled down to London for a few days.
      

48. Buffon biography
    
    • He emphasised the importance of natural history and the great length of geological time.
      
    • to the area of part of the cycloid whose generating circle has diameter equal to the length of the needle.
      

49. Goldbach biography
    
    • In 1710 he set off on a lengthy journey around Europe, meeting many of the leading scientists on his travels.
      
    • Goldbach continued his lengthy tour and was in Venice in 1721.
      

50. Richard Jules biography
    
    • (It is easy to do this - just order the descriptions in terms of the length of the sentence describing the real number and within sentences of the same length use lexicographic order.) Using Cantor's diagonal argument, he then constructed a real number which could not be described in English.
      

51. Al-Khwarizmi biography
    
    • To the square we must add 10x and this is done by adding four rectangles each of breadth 10/4 and length x to the square (Figure 2).
      
    • But the side is of length 5/2 + x + 5/2 so x + 5 = 8, giving x = 3.
      

52. Hertz Heinrich biography
    
    • In mechanics Hertz followed Kirchhoff and considered only length, time and mass as the fundamental entities, force being a derived concept.
      
    • Hertz explained in the Introduction to the 'Principles' that to construct a mechanics capable of accounting for the lawful interaction of perceptible bodies it was necessary to add a hypothesis to the three concepts [length, time and mass].
      

53. Rheticus biography
    
    • In August 1541 Rheticus presented a copy of his work on a map of Prussia to Duke Albert of Prussia and the following day he sent him an instrument he had made to determine the length of the day.
      
    • He designed many instruments such as sea compasses and the instrument to show the length of the day throughout the year which he gave to Duke Albert as we mentioned above.
      

54. Camus biography
    
    • Jean Picard had measured the length of the arc of the meridian, the measurements appear in Mesure de la Terre (1671).
      
    • Camus, Maupertuis, Clairaut and Lemonnier, after their joint work on the Lapland expedition, also worked on measuring the length of the arc of the meridian and examining Jean Picard's work.
      

55. 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.
      
    • Of course how accurate this value is depends on the length of the stadium and scholars have argued over this for a long time.
      

56. Lyndon biography
    
    • These include the development of 'small cancellation theory', work on Fuchsian groups and the Riemann-Hurwitz formula, his introduction of 'aspherical' presentations of groups and his work on length functions in free products of groups.
      
    • They include: Groups rings and dimension subgroups; Two investigations on the borderline of logic and algebra; Decision problems of finite automata design and related arithmetic; On Dehn's algorithm and the conjugacy problem; Projectivities of free products; Continuous model theory and set theory; Real length functions in groups; Automorphisms of the fundamental group of an orientable 2-manifold; Some algorithmic problems for semigroups; and Groups acting on trees.
      

57. Borda biography
    
    • It considered a proposal which had already been made to the French government to base the metre on the length of a pendulum which beat at the rate of one second.
      
    • Borda died shortly before the project to determine the length of the metre was completed.
      

58. Nicomedes biography
    
    • ABC is drawn perpendicular to XY cutting it at B and having the length BC some fixed value, say b.
      
    • prided himself inordinately on his discovery of this curve, contrasting it with Eratosthenes's mechanism for finding any number of mean proportionals, to which he objected formally and at length on the ground that it was impracticable and entirely outside the spirit of geometry.
      

59. Koksma biography
    
    • The approximation theorem of Kronecker is discussed at length.
      
    • On 19 September 1945 Koksma spoke at length in his opening address for the session about the war and what had happened during the war years to the people and the buildings of the Free University.
      

60. Tait biography
    
    • and then the knot would be described by the sequence of crossings of length 2n where each of A, B, C, ..
      
    • It would be quite impossible in an article of this length to cover all the topics which Tait worked on.
      

61. Cassini de Thury biography
    
    • The results of the survey seemed to support the views of Jacques Cassini but, after a while, the opponents to his theory in the Academy planned expeditions to Peru, led by Bouguer and La Condamine, in 1735, and Lapland, under Maupertuis in 1736, to measure the length of a meridian degree and to settle the argument.
      
    • It relied on the fact that if one built up a system of triangles, each successive one standing on one side of the previous one, then measuring the angles and finding an accurate distance for the length of just one side of one triangle was sufficient to give accurate measurements of the sides of all the triangles.
      

62. Wang Xiaotong biography
    
    • Previously a man was able to cover a horizontal road of length 192 bu 62 times per day carrying 2 dou 4 sheng and 8 he of earth on his back.
      
    • The length, upper width, lower widths of the east and west ends and the height of the west end are all known as functions of the height x of the east end.
      

63. Oenopides biography
    
    • Toomer believed that in fact despite Oenopides' Great Year of 59 years, he did not have this accurate value for the length of the month, and later calculations were made using better data than would have been available to Oenopides to give this very accurate value for the length of the month, more accurate than Oenopides could ever have known.
      

64. Bachelier biography
    
    • Bachelier in his Thesis, in progressing from a 'drunkards' random walk with n (discrete) steps in time t, each step being of length d, to a (continuous) distribution for where the drunkard might be at time t, realised that there had to be a relationship between n and d - d equal to (t/n)(1/2) for the limit process to 'work'.
      
    • Appell.',38)">38] he showed, effectively, that if a random walk on the y-axis is represented as a graph in time with the 'drunkard' making n steps in time t, each step of length d, the path was such that the tangent of the path angle {i.e.
      

65. Ampere biography
    
    • This work attempted to solve the problem of constructing a line of the same length as an arc of a circle.
      
    • But more than his creativity, it was Savary's discipline and ability to concentrate at length on specific problems that proved especially valuable to Ampere.
      

66. Archimedes biography
    
    • A ship was frequently lifted up to a great height in the air (a dreadful thing to behold), and was rolled to and fro, and kept swinging, until the mariners were all thrown out, when at length it was dashed against the rocks, or let fall.
      
    • In On spirals Archimedes defines a spiral, he gives fundamental properties connecting the length of the radius vector with the angles through which it has revolved.
      

67. Menelaus biography
    
    • Paul Tannery in [Bulletin des sciences mathematique 7 (1883), 289-292.',8)">8] argues that this make it likely that a curve which it is claimed by Pappus that Menelaus discussed at length was the Viviani's curve of double curvature.
      

68. Ulugh Beg biography
    
    • Data from his Observatory allowed Ulugh Beg to calculate the length of the year as 365 days 5 hours 49 minutes 15 seconds, a fairly accurate value.
      

69. Tunstall biography
    
    • Although the length of Cuthbert's time in Oxford is in some doubt, there is no doubt in the fact that he left both Oxford and Cambridge without taking a degree.
      

70. Lovelace biography
    
    • The notes of the Countess of Lovelace extend to about three times the length of the original memoir.
      

71. Wiener Norbert biography
    
    • Sometimes difficult results appeared with hardly a proof as if they were obvious to Wiener, while at other times he would give a lengthy proof of a triviality.
      

72. Al-Battani biography
    
    • He refined the existing values for the length of the year, which he gave as 365 days 5 hours 46 minutes 24 seconds, and of the seasons.
      

73. Castelnuovo biography
    
    • After lengthy discussions it was decided not to make an award because Castelnuovo and Enriques had made a joint submission.
      

74. Neile biography
    
    • In 1657 he became the first to find the arc length of an algebraic curve when he rectified the cubical parabola.
      

75. Fuller biography
    
    • With a manoeuvrability which allowed it to turn within its own length this was a remarkable conception and development continued until 1943.
      

76. Nash-Williams biography
    
    • Not only was it a remarkable piece of mathematical work but the thesis was also remarkable for its length being over 500 pages.
      

77. Tisserand biography
    
    • In order to keep the Toulouse Observatory operating during this lengthy absence Jules Gruey, a teacher at the Toulouse Faculty of Science was appointed temporary director.
      

78. Bowditch biography
    
    • Bowditch was helped by Benjamin Peirce in this project and his commentaries doubled the length of the book.
      

79. Bolzano biography
    
    • But he was reading and recording his ideas on a host of other subjects as well, including the problem of how best to approach the proper mathematical understanding of zero; Legendre's work on surfaces, convexity, concavity, and conditions for congruity; analysis of other geometric concepts, including lengths, areas, volumes, and spheres; trigonometric formulas and spherical trigonometry; imaginary and exponential numbers; definition of the differential and discussion of the infinite and various opinions about it, as well as aspects of maxima and minima.
      

80. Molyneux William biography
    
    • The second part of the book contains miscellaneous material such as refraction and light, grinding lens for telescopes, how to find foci of lenses, testing a telescope, an the relationship between the focal lengths of the objective and the eyepiece.
      

81. Boscovich biography
    
    • Earlier we analysed his treatise from this point of view in an attempt to cast light upon Boscovich's ideas concerning the question, much debated at the time, of what one might call 'the nature and constitution of the geometric continuum', a problem associated with the question of geometric indivisibles, which originated with the works of Bonaventura Cavalieri and was debated at length.
      

82. Barkla biography
    
    • Professor Barkla! Before it was known that the nature of X-rays is the same as that of light, with a difference only in wave-length, you had found a form of polarization of those rays, and by your investigation of their absorption you had developed a form of spectroscopy, before it was known that there is a spectrum in the real sense of the word.
      

83. Peirce Benjamin biography
    
    • Peirce expands on this at length in [The American Association (1853).',17)">17].
      

84. Ehrenfest-Afanassjewa biography
    
    • The Dutch papers only reported his sudden death and gave lengthy accounts of his achievements.
      

85. Chowla biography
    
    • Among a long list of other results we mention just a very few such as his generalisation of Wolstenholme's theorem; his work on classes of quintics not soluble by radicals; his closed form for the Bernoulli numbers; and his work on the length of the period of the continued fraction expansion of √N.
      

86. Ruan Yuan biography
    
    • Again it is not surprising, if the length of the entry in any way indicates importance, that certain biographies look strange.
      

87. Adams Frank biography
    
    • Over the past few years, various topologists have been heard to complain about the lengthy and technical nature of infinite loop space theory.
      

88. Jordan biography
    
    • He also originated the concept of functions of bounded variation and is known especially for his definition of the length of a curve.
      

89. Poncelet biography
    
    • The distance of two points is not projectively invariant, but in looking for projectively invariant configurations he finds the harmonic one, and this he develops at length.
      

90. Lidstone biography
    
    • He further considers at some length Aitken's method of inverse interpolation by quadratic crossmeans.
      

91. Posidonius biography
    
    • Cleomedes explains in his work the method used by Posidonius to calculate the length of the circumference of the earth.
      

92. Whiteside biography
    
    • To prove this point, the author dwells at some length on two important topics treated by Newton for the first time: the determining of the general orbit traversed in an arbitrary central force field (Book 1, Props.
      

93. Fowler David biography
    
    • difficulty in reading it, a combination of its length, its meticulousness, its minutely referenced notes, and the novelty, boldness, and revisionism of Knorr's point of view, as long-held and cherished opinions were subject to new investigation ..
      

94. Weyl biography
    
    • He explained to me one day that it was for him an absolute necessity to review, by lecturing, his subject of concern in all its length and breadth.
      

95. Regiomontanus biography
    
    • [for] it added later observations, revised computations, and critical reflections - one of which revealed that Ptolemy's lunar theory required the apparent diameter of the moon to vary in length much more than it really does.
      

96. Eckert John biography
    
    • It contained roughly 18000 vacuum tubes and measured about 2.5 metres in height and 24 metres in length.
      

97. Sylvester biography
    
    • On 7 July 1831 Sylvester matriculated as a student at St John's College, Cambridge, although his studies were interrupted when he was forced to take most of the two years 1833-34 and 1834-35 out due to a lengthy illness.
      

98. Nicolson biography
    
    • The instability was not recognised until lengthy numerical computations were carried out by Crank, Nicolson and others.
      

99. Wiener Christian biography
    
    • Wiener extended work on descriptive geometry to physics and calculated the amount of solar radiation received at different latitudes during the varying lengths of days in the course of the year.
      

100. Gregory biography
     
     • The tube of the Gregorian telescope is thus shorter than the sum of the focal lengths of the two mirrors.
       

101. Sierpinski biography
     
     • The length of the curve is infinity, while the area enclosed by it is 5/12 that of the square.
       

102. Kaprekar biography
     
     • In fact applying Kaprekar's process to almost any four-digit number will result in 6174 after at most 7 steps (so our last example was one where the process has maximal length).
       

103. Bessel biography
     
     • a correction in 1826 to the seconds pendulum, the length of which is precisely calculated so that it requires exactly one second for a swing.
       

104. Pacioli biography
     
     • Mathematics and art were topics which they discussed at length, both gaining greatly from the other.
       

105. Pappus biography
     
     • When Ptolemy in the chapter on the apparent diameter of the sun, moon and shadow simply remarks that the tangential cones in question contact the spheres within a negligible error in great circles, then Pappus refers to Euclid's "Optics" to show that the circle of contact has a smaller diameter than the sphere, only to add a lengthy argument to demonstrate that the error committed in Ptolemy's construction is nevertheless negligible.
       

106. Sporus biography
     
     • Hence we cannot regard as appropriate the censure of Sporus of Nicaea, who seems to charge Archimedes with having failed to determine with accuracy the length of the straight line which is equal to the circumference of the circle., to judge by his passage in his Keria where Sporus observes that his own teacher, meaning Philon of Gadara, reduced the matter to more accurate numerical expression than Archimedes did..
       

107. Veblen biography
     
     • He then worked on the problem of transferring raw data into tables of results and for this he had a staff of men to undertake the lengthy hand calculations.
       

108. Gnedenko biography
     
     • Kolmogorov was a connoisseur of art, and [Gnedenko and Kolmogorov] talked at length about ancient Russian icons and architecture, poetry and history.
       

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

110. Alison biography
     
     • After holding the post of second Mathematical master in Edinburgh Academy for the same length of time, he entered the service of the Edinburgh Merchant Company as second mathematical and science master in George Watson's College, and for the last ten years he has been head mathematical master in the same institution.
       

111. Frohlich biography
     
     • In the present paper the fields of at most class two over the rational field are studied, the class of a field being defined as the length of the central series of the Galois group.
       

112. Abraham Max biography
     
     • He lectured at Gottingen as a Privatdozent until 1909 which is an unusual length of time for anyone to hold such an unpaid lecturing position.
       

113. Dionysodorus biography
     
     • In this work Dionysodorus calculates the volume of a torus and shows that it is equal to the product of the area of the generating circle with the length of the circle traced by its centre rotating about the axis of revolution.
       

114. Bassi biography
     
     • She did not achieve this easily but only after a lengthy debate did the University agree to appoint her to a professorship.
       

115. Book biography
     
     • Every quasi-realtime language can be accepted in real time by a nondeterministic one stack, one pushdown store machine, and can be expressed as the length-preserving homomorphic image of the intersection of three context-free languages.
       

116. Tarski biography
     
     • We have looked briefly at some of Tarski's work and we shall examine a little more of his work but it is impossible in a biography of this length to give a proper view of the range of his contributions.
       

117. Besicovitch biography
     
     • The problem had been posed in 1917 by a Japanese mathematician S Kakeya and asked what was the smallest area in which a line segment of unit length could be rotated through 2p.
       

118. Stormer biography
     
     • But many of the papers are very substantial in length and content.
       

119. Qin Jiushao biography
     
     • The novelty here is that the coefficients are not numbers but are functions of lengths in the figure which are left as unspecified.
       

120. Liu Hong biography
     
     • His measurements of the length of the shadow of a pole at the summer and at the winter solstices give results which are accurate to within 1% of their true value.
       

121. D'Ocagne biography
     
     • Edgar Odell Lovett begins a lengthy review of the book as follows:- .
       

122. Proclus biography
     
     • for the qualities he possessed that are exceedingly rare in any age and were almost unique in his: the logical clarity and firmness of his thought, the acuteness of his analyses, his eagerness to understand and readiness to present the views of his predecessors on controversial issues, the sustained coherence of his lengthy expositions, and the large horizon, as broad as the whole of being, within which his thinking moved.
       

123. Al-Mahani biography
     
     • However, he was led to an equation involving cubes, squares and numbers which he failed to solve after giving it lengthy meditation.
       

124. Eckert Wallace biography
     
     • In order to bring the Tables within even their present length, various parts of the basic equations were curtailed whenever permissible in the light of observational requirements (as then visualised).
       

125. Deligne biography
     
     • Alone or in collaboration, Pierre Deligne has written about a hundred papers, most of them of sizeable length.
       

126. Godel biography
     
     • He was not prepared to risk this, and after lengthy negotiation to obtain a U.S.
       

127. Kostrikin biography
     
     • As it gives a good indication of Kostrikin's thoughts we give a fairly lengthy quote:- .
       

128. Lissajous biography
     
     • By turning the mirror in the hand, the image upon the screen was resolved into a bright sinuous track many feet in length.
       

129. Neugebauer biography
     
     • always insisted that the length of the review was not intended to be directly proportional to the importance of the paper; indeed, a bad paper needed to have a review sufficiently detailed so that nobody needed to look at the paper itself, whereas a really important paper needed only to be called to the world's attention.
       

130. Newton biography
     
     • Taking differentiation as the basic operation, Newton produced simple analytical methods that unified many separate techniques previously developed to solve apparently unrelated problems such as finding areas, tangents, the lengths of curves and the maxima and minima of functions.
       

131. Simplicius biography
     
     • In his commentary on Aristotle's Physics Simplicius quotes at length from Eudemus's History of Geometry which is now lost.
       

132. Machin biography
     
     • There are various other ways of finding the lengths or areas of particular curve lines, or planes, which may very much facilitate the practice; as for instance, in the circle, the diameter is to the circumference as 1 to (16/5- 4/239) - 1/3(16/53- 4/2393) &c.
       

133. Vernier biography
     
     • He also describes his most famous invention, that of the vernier caliper, an instrument for accurately measuring length.
       

134. Rahn biography
     
     • Quite independently, both the English translation and Latin translation had doubled in length.
       

135. Cheng Dawei biography
     
     • In the right-angled triangle with sides of length a, b and c with a > b > c, we know that a + b = 81 ken and a + c = 72 ken.
       

136. Thompson John biography
     
     • Despite the importance of the paper several journals declined to publish it because of its length.
       

137. Hooke biography
     
     • However I am well pleased to find that the truth will at length prevail when men have laid aside their prepossessions and prejudices.
       

138. Peano biography
     
     • We could continue at length enumerating the absurdities that the author has piled up.
       

139. Hankel biography
     
     • constitutes a lengthy presentation of much of what was then known of the real, complex, and hypercomplex number systems.
       

140. Yavanesvara biography
     
     • We know of Rudradaman because information is recorded in a lengthy Sanskrit inscription at Junagadh written around 150 AD.
       

141. Higman biography
     
     • After working on finitely generated nilpotent groups and infinite simple permutation groups, Higman, together with Philip Hall, produced another of his landmark papers in 1956 On the p-length of p-soluble groups and reduction theorems for Burnside's problem.
       

142. Lighthill biography
     
     • all the most generally useful fundamental ideas of the science of waves in fluids can be developed at length, one after another.
       

143. Rado biography
     
     • The lectures he gave at this time formed the basis of his major text Length and Area which was published by the American Mathematical Society in 1948.
       

144. Redei biography
     
     • It is based on the notion of a basis (of minimal length) of an arbitrary p-group.
       

145. Fagnano Giulio biography
     
     • In 1751 Euler was asked to examine Produzioni matematiche and he found in this treatise relations between special types of elliptic integrals, that express the length of an arc of a lemniscate, which were quite unexpected to him.
       

146. Crofton biography
     
     • Crofton's lengthy article Probability which appeared in 1885 is still worth reading and is one of many outstanding articles in what many consider to be the greatest encyclopaedia ever produced.
       

147. Al-Tusi Sharaf biography
     
     • In this work I wanted to summarise the art of algebra and al-muqabala, adapt what has survived from the great philosopher Sharaf al-Din al-Muzaffar ibn al-Muzaffar ibn Muhammad al-Tusi, and reduce his over lengthy exposition to a moderate size; I eliminated the tables he drew up to make his computations and solve his problems.
       

148. Gorenstein biography
     
     • Simultaneously with this burgeoning research effort, finite simple group theory was establishing a well-deserved reputation for inaccessibility because of the inordinate lengths of the papers pouring out.
       

149. Adler biography
     
     • On the contrary, his new methods were not as elegant, either in simplicity or length, as the original proof by Mascheroni.
       

150. Baker biography
     
     • Its contents are as follows: Euclid's theory of parallel lines; Propositions of incidence; The symbolic representation and Pappus' 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.
       

151. Paramesvara biography
     
     • Aryabhata gave a rule for determining the height of a pole from the lengths of its shadows in the Aryabhatiya.
       

152. Fine biography
     
     • It begins with setting geometry up in a similar axiomatic way to Euclid's Elements, but then it goes on to more practical considerations of measuring length, height, surface area, and volumes.
       

153. Kolmogorov biography
     
     • In 1953 and 1954 two papers by Kolmogorov, each of four pages in length, appeared.
       

154. Gellibrand biography
     
     • Epitome of Navigation first appeared 62 years after his death; a rather remarkable length of time.
       

155. Petersson biography
     
     • On 1 May 1937 Petersson joined the Nazi party, after a lengthy explanation of why his name was not given a German spelling, still trying to gain favour.
       

156. Wittgenstein biography
     
     • we first get a part of 693 distinct, numbered remarks, varying in length from one line to several paragraphs, and a second part of fourteen sections, half a page to thirty-six pages long ..
       

157. Geocze biography
     
     • In this paper he constructed a function which was continuous everywhere but in every interval, no matter how small, it had infinite length.
       

158. Russell Scott biography
     
     • He became embroiled in a lengthy financial dispute about an armaments contract; the Great Eastern suffered a serious breakdown; and he was controversially expelled from the Institute of Civil Engineers.
       

159. Plato biography
     
     • For example a line is an object having length but no breadth.
       

160. La Condamine biography
     
     • In April 1735 La Condamine set out on the expedition to Peru to measure the length of a degree of meridian at the equator.
       

161. Arnauld biography
     
     • In particular early in his career he corresponded at great length with Descartes and then much later with Leibniz.
       

162. Valyi biography
     
     • One of his favourite number theory problems was to find all triangles with sides of integer length whose area and perimeter are given by the same number.
       

163. Ruffini biography
     
     • The order of a permutation is the least common multiple of the lengths in the decomposition into disjoint cycles.
       

164. Gateaux biography
     
     • Since 23 March 1905 ([44]), a new law replacing the law of 16 July 1889 for the organization of the army had been voted by the Parliament, where the length of the active military service had been reduced to 2 years, but there were many more candidates.
       

165. Ramanathan biography
     
     • Of course, in a paper of only 18 pages in length, the author can only discuss a small portion of Ramanujan's modular equations and he concentrates therefore on equations of composite degree.
       

166. Kober biography
     
     • Kober's wife provided him with the sort of back-up which allowed him to make lengthy visits to Cambridge in England, for she simply took over teaching his classes in Breslau while he spent time at Cambridge doing research.
       

167. Birkhoff biography
     
     • Because Birkhoff worked on so many different mathematical topics it is difficult to do justice to the range of his contributions in a biography of this length.
       

168. Oughtred biography
     
     • He added and subtracted lengths by using a pair of dividers, operations that were equivalent to multiplying and dividing.
       

169. Van Kampen biography
     
     • Five papers appeared in 1940, one of them a major article over 30 pages in length in the American Journal of Mathematics with the title Infinite product measures and infinite convolutions.
       

170. Kurschak biography
     
     • He proves that the dodecagon can be dissected into a set of triangles which can be rearranged so as to fill three squares with sides having length 1.
       

171. Al-Tusi Nasir biography
     
     • In the latter work al-Tusi discussed objections raised by earlier mathematicians to comparing lengths of straight lines and of curved lines.
       

172. Taylor biography
     
     • He tried to find the shape of the vibrating string and the length of the isochronous pendulum rather than to find its equations of motion.
       

173. Milnor biography
     
     • Among other results discussed are Milnor's result showing that we cannot necessarily "hear the shape" of a 16-dimensional torus, and another result giving upper and lower bounds on the number of distinct words of a given length in a finitely generated subgroup of the fundamental group.
       

174. Shewhart biography
     
     • "Dr Shewhart prepared a little memorandum only about a page in length.
       

175. Hobbes biography
     
     • If the magnitude of a body which is moved (although it must always have some) is considered to be none, the path by which it travels is called a line, and the space it travels along a length, and the body itself is called a point.
       

176. Young Thomas biography
     
     • The credit for deciphering hieroglyphic eventually went to the French linguist Champollion, but he benefited greatly from Young's efforts and the two corresponded at great length.
       

177. Jonquieres biography
     
     • During 1860-61 he made lengthy sea voyages and this provided him with much time free from distractions during which he could work out many of the mathematical ideas which he was developing.
       

178. Zenodorus biography
     
     • To do this Zenodorus makes use of Archimedes result that the area of a circle is equal to that of a right-angled triangle of perpendicular side equal to the radius of the circle and base equal to the length of the circumference of the circle.
       

179. Crank biography
     
     • The instability was not recognised until lengthy numerical computations were carried out by Crank, Nicolson and others.
       

180. Levi biography
     
     • of a staff of 41/2 feet long and about one inch wide, with six or seven perforated tablets which could slide along the staff, each tablet being an integral fraction of the staff length to facilitate calculation, used to measure the distance between stars or planets, and the altitudes and diameters of the Sun, Moon and stars.
       
     • The topic of the third book, God's omniscience, was a subject debated at length by philosophers and Levi gives his own views [',56)">56]:- .
       

181. Gauss biography
     
     • He discussed this topic at length with Farkas Bolyai and in his correspondence with Gerling and Schumacher.
       

182. Al-Baghdadi biography
     
     • It is concerned with the measurement of lengths, areas and volumes.
       

183. Knuth biography
     
     • For his quite remarkable contributions Knuth has received many honours - far too many to be mentioned in an article of this length.
       

184. Mackenzie biography
     
     • (ii) wave-length determinations; .
       

185. Ceva Giovanni biography
     
     • Although he wrongly concluded that the periods of oscillation of two pendulums were in the same ratio as their lengths, he later corrected the error.
       

186. Euwe biography
     
     • In 1929 he published a mathematics paper in which he constructed an infinite sequence of 0's and 1's with no three identical consecutive subsequences of any length.
       

187. Gemma Frisius biography
     
     • For example in De Radio Astronomico (1545) he described his work constructing a cross-staff about 1.5 metres long with one cross piece about 3/4 of a metre in length.
       

188. Aryabhata I biography
     
     • His value for the length of the year at 365 days 6 hours 12 minutes 30 seconds is an overestimate since the true value is less than 365 days 6 hours.
       

189. Leibniz biography
     
     • He made a lengthy trip to search archives for material on which to base this history, visiting Bavaria, Austria and Italy between November 1687 and June 1690.
       

190. McAfee biography
     
     • He said, "If you were being drafted to fight the war or if for some particular need, you had some particular talent and they were drafting you to do a job for a certain length of time, we would give you the leave of absence.
       

191. Hippocrates biography
     
     • The quadratures of lunes, which were considered to belong to an uncommon class of propositions on account of the close relation of lunes to the circle, were first investigated by Hippocrates, and his exposition was thought to be correct; we will therefore deal with them at length and describe them.
       

192. Bocher biography
     
     • Perhaps 'extended version' doesn't do it justice since this book was now four times the length of his doctoral thesis.
       

193. Nicomachus biography
     
     • But, unlike Euclid, who attempts to prove musical propositions through mathematical theorems, Nicomachus seeks to show their validity by measurement of the lengths of strings.
       

194. Greenhill biography
     
     • An important contribution Greenhill made to the theory of elasticity was his study of the greatest length that a cylinder can have before it bends under its own weight.
       

195. Goldstine biography
     
     • We envisage that a properly organized automatic, high speed establishment will include an extensive collection of such subroutines, of lengths ranging from about 15 - 20 words upwards.
       

196. Boersma biography
     
     • A lengthy report is given of the work of other authors on this subject, and the paper itself contains still a different version.
       

197. Wexler-Kreindler biography
     
     • In the case of a lattice S of finite length L and L' coincide.
       

198. Salmon biography
     
     • A characteristic of Salmon's work was his love of carrying out lengthy calculations.
       

199. Cleomedes biography
     
     • The only certainty here is that On the Circular Motions of the Celestial Bodies discusses the work of Posidonius at length and so is clearly written after the middle of the first century BC.
       

200. Mylon biography
     
     • He also attempted to prove the result found by Wren concerning the length of the arc of the cycloid.
       

201. West biography
     
     • Many students estimate the difficulty of their task by its length; they wish to continue to lay the burden on their memory, and they imagine that to repeat is the same thing as to comprehend.
       

202. Ohm biography
     
     • The result was not contained in Ohm's firsts paper published in 1825, however, for this paper examines the decrease in the electromagnetic force produced by a wire as the length of the wire increased.
       

203. Hazlett biography
     
     • Her description of the lengths she had to go to in order to keep her work secret makes interesting reading [University of Illinois Archives (23 February, 1964; 4 March, 1964).',5)">5]:- .
       

204. Narayana biography
     
     • In particular he gave a rule of finding integral triangles whose sides differ by one unit of length and which contain a pair of right-angled triangles having integral sides with a common integral height.
       

205. Doppelmayr biography
     
     • This had the lengthy title Historische Nachricht von den Nurnbergischen Mathematicis und Kunstlern, welche fast von dreyen Seculis her durch ihre Schriften und Kunst-Bemuhungen die Mathematic und mehrere Kunste in Nurnberg vor andern trefflich befordert und sich um solche sehr wohl verdient gemacht zu einem guten Exempel, und zur weitern ruhmlichen Nachahmung and was published in 1730.
       

206. Cassini Jacques biography
     
     • By 1738 the geodesic measurements carried out in Peru by Bouguer and La Condamine in 1735 and Lapland by Maupertuis in 1736 to measure the length of a meridian degree had produced very strong evidence for the flattening at the poles.
       

207. Fizeau biography
     
     • This was an important first step in a rather lengthy process which eventually led to the discarding of the ether hypothesis in the early years of the 20th century.
       

208. Mackenzie Gladys biography
     
     • (ii) wave-length determinations; .
       

209. Fredholm biography
     
     • In his Stockholm lectures Fredholm loved to talk at length about the great problems and methods of classical mathematical physics which had been the main theme of his scientific work.
       

210. Hadamard biography
     
     • There is no way that an article of this length can even indicate the range of Hadamard's mathematical contributions.
       

211. Maupertuis biography
     
     • A second expedition was sent to Lapland headed by Maupertuis, also to measure the length of a degree along the meridian.
       

212. Bennett biography
     
     • Then it follows as a consequence that the sines of the twists are proportional to the lengths of the bars.
       

213. Vijayanandi biography
     
     • It deals with the topics of: units of time measurement; mean and true longitudes of the sun and moon; the length of daylight; mean longitudes of the five planets; true longitudes of the five planets; the three problems of diurnal rotation; lunar eclipses, solar eclipses; the projection of eclipses; first visibility of the planets; conjunctions of the planets with each other and with fixed stars; the moon's crescent; and the patas of the moon and sun.
       

214. Kochina biography
     
     • This book, devoted to a nineteenth-century mathematician who has long deserved a good full-length biography, is very welcome indeed.
       

215. Lacroix biography
     
     • When he was only 14 years old he spent many happy hours engrossed in making lengthy calculations of the motion of the planets.
       

216. Levinson biography
     
     • To gain a full appreciation of Levinson's mathematical contribution we quote at length from the Preface to [Norman Levinson : Selected papers of Norman Levinson (2 Vols.) (Boston, MA, 1998).',2)">2]:- .
       

217. Hunayn biography
     
     • As an example of the lengths that Hunayn went in order to find a particular manuscript we quote his description of a search for a medical manuscript (see for example [Dictionary of Scientific Biography (New York 1970-1990).',1)">1]):- .
       

218. Cayley biography
     
     • He united projective geometry and metrical geometry which is dependent on sizes of angles and lengths of lines.
       

219. De Forest biography
     
     • He returned to Europe in 1863 for a lengthy trip which lasted until 1865.
       

220. Schutzenberger biography
     
     • Later he published a series of results on variable-length codes all of them reported in our book with Jean Berstel (Theory of Codes, Academic Press, 1984).
       

221. Cavalieri biography
     
     • He developed a general rule for the focal length of lenses and described a reflecting telescope.
       

222. Herschel Caroline biography
     
     • She carried out the lengthy calculations necessary to reduce William's data with remarkable accuracy.
       

223. Feller biography
     
     • Thus if it is supposed that the probability that each individual in a time interval of length dt has probability l dt of producing a second individual, the exact value of the probability of having n individuals at time t is found.
       

224. Brouncker biography
     
     • In 1659 Brouncker's improvement of Neile's computation of the arc length of the semicubical parabola ay2 = x3 appeared in Wallis's work De Cycloide et de Corporibus inde Genitis.
       

225. Jacobson biography
     
     • This attack Jacobson answered at length with a reply which appears in the June 1980 Notices of the American Mathematical Society.
       

226. Libri biography
     
     • Indeed many of the precious documents were returned to France after lengthy negotiations with the English authorities.
       

227. Frank biography
     
     • They were joint authors of the lengthy two volume book Differentiagleichungen und Integralgleichungen der Mechanik und Physik which was published in 1925.
       

228. Riemann biography
     
     • However, the brilliant ideas which his works contain are so much clearer because his work is not overly filled with lengthy computations.
       

229. Dirac biography
     
     • Dirac unified the theories of quantum mechanics and relativity theory, but he also is remembered for his outstanding work on the magnetic monopole, fundamental length, antimatter, the d-function, bra-kets, etc.
       

230. McCowan biography
     
     • We quote the letter at length since it gives a feel for the times in the town when the subject of this biography was growing up [1]:- .
       

231. Finsler biography
     
     • A Finsler space is a generalisation of a Riemannian space where the length function is defined differently and Minkowski's geometry holds locally.
       

232. Hamilton biography
     
     • The book ended up double its intended length and took seven years to write.
       

233. Hammersley biography
     
     • Hammersley published a variety of papers in 1951 including A theorem on multiple integrals, On a certain type of integral associated with circular cylinders, The sums of products of the natural numbers, and The total length of the edges of the polyhedron.
       

234. Ehrenfest biography
     
     • The Dutch papers only reported his sudden death and gave lengthy accounts of his achievements.
       

235. Banu Musa biography
     
     • Muhammad and Ahmad measured the length of the year, obtaining the value of 365 days and 6 hours.
       

236. Kuczma biography
     
     • Many of the theorems are of general interest; the occasional theorem requiring lengthy and tedious proof should not discourage the general reader.
       

237. Escher biography
     
     • He made a number of attempts at using this style of artwork over the next couple of years but was unhappy about both the length of time this passion was taking (due to its trial and error nature) and the poor quality of his final work, and he left aside regular division for a number of years.
       

238. Wright Sewall biography
     
     • But, paradoxically, when he did start to talk about something of interest - his childhood, his experience on the railroad surveying team, his ancestors, guinea pigs, evolution, genetics, politics - he could, and would, talk at length.
       

239. Hall biography
     
     • In 1956 Hall published, jointly with Graham Higman, On the p-length of p-soluble groups and reduction theorems for Burnside's problem.
       

240. Askey biography
     
     • Published by Cambridge University Press, this new work is six times the length of the earlier one.
       

241. Mauchly biography
     
     • It contained roughly 18000 vacuum tubes and measured about 2.5 metres in height and 24 metres in length.
       

242. Kutta biography
     
     • He wrote a paper on Wallis's 1659 work on integration and the length of an ellipse.
       

243. Shen Kua biography
     
     • In fact Shen demonstrated a remarkable ability to view spatial arrangements and he gave an approximate formula for the length of a circular arc in terms of the cord subtending the arc.
       

244. Kolosov biography
     
     • It showed that the concentration of stress could become far greater, as the radius of curvature at an end of the hole becomes small compared with the overall length of the hole.
       

245. La Hire biography
     
     • In 1708 he calculated the length of the cardioid.
       

246. Mayer Tobias biography
     
     • The Difficulty and Length of the necessary Calculations seemed the only Obstacles to hinder them from becoming of general Use.
       

247. Legendre biography
     
     • The committee worked on the metric system and undertook the necessary astronomical observations and triangulations necessary to compute the length of the metre.
       

248. Boruvka biography
     
     • the total length of the net will be minimal.
       

249. Lax Peter biography
     
     • We should remark that the difficulty in giving a description of Lax's contributions is that they are so numerous and important that in an article of this length it is impossible to do them justice.
       

250. Heisenberg biography
     
     • Heisenberg hoped this mathematical property would lead to a fundamental property of nature with a 'fundamental length' as one of the constants of nature.
       

251. Mandelbrot biography
     
     • It was one of the shortest lengths of time that anyone would study there, for he left after just one day.
       

252. Yule biography
     
     • The correlation of lengths or measurements on portions of the body form examples of the first kind; of numbers of children in families, petals or other parts of flowers, are examples of the second.
       

253. Ajima biography
     
     • In this figure we have a segment of a circle on the chord AB of length a.
       
     • It has length m.
       

History Topics

1. Fair book
   
   • If the length of the keel of tonnage be 100 feet and the extreme breadth of the ship 35 feet.
     
   • If the length of the keel of tonnage be 80 feet and the extreme breadth of the ship 27 feet.
     
   • If the length of the keel of tonnage be 96 feet and the extreme breadth of the ship 33 feet.
     
   • What is the value of an ox measuring 7 ft 3 in in girth and 5 ft 4 in in length at 5 / 10 a stone, reckoning the offal 1/3 the value of the four quarters.
     
   • He uses a rule squaring the girth in feet, multiplying by the length in feet, then multiplying the answer by 5/21 to obtain the weight in stones.
     
   • What is the dead weight of the four quarters of an ox measuring 5 ft 7 1/2 in in length and 9 ft 3 1/2 in in girth, adding 1/12 part to the weight found by the rule as an allowance for the beasts extraordinary fatness, also living weight.
     
   • The answer is correct but an extremely lengthy and difficult method to obtain the solution.
     
   • What is the area of a rectangular field, its length being 1156 links and its breadth 948 links.
     
   • Takes the length times breadth and converts to acres, roods, poles, square yards.
     
   • What is the area and rent of a ridge of grass, the length being 965 links, the breadth 23 links at one end and 21 links at the other, at 10 guineas per acre.
     
   • There follow a number of problems which involve fields which are drawn with given measurements (all lengths of lines).
     
   • After these rather lengthy sums, we go back to simpler examples.
     
   • What length of a rectangular field, of which the breadth is 500 links, will make 1 acre, 2 roods, 30 poles.
     
   • What length of a ridge 21 links broad will make 8 poles.
     
   • 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.
     
   • Now he uses the fact that the ratio of the squares of the lengths of the sides of the triangles will be the ratio of the areas to compute the length as 770.23 links.
     
   • The length of a base line within a field curvilinear on the other side is 315 links and 11 equidistant ordinates erected thereon measure 70, 86, 96, 104, 109, 110, 108, 105, 99, 90 and 85 links, respectively, what is the area of the space between the base line and the curvilinear side of the field.
     
   • For the circumference he uses the approximate expression π/2 × √(a2 + b2) where a, b are the lengths of the major and minor axes.
     
   • The length of a cask composed of two equal frustums of a paraboloid is 45 inches: what is its content in imperial gallons, the bung diameter being 40 and the head diameter 20 inches.
     
   • What is the solidity of a parabolic spindle of which the length is 30, and the greatest diameter 12.
     
   • What is the solidity of a parabolic spindle of which the length is 20, and the greatest diameter 8.
     
   • What is the solidity of a parabolic spindle of which the length is 50, and the greatest diameter 20.
     
   • What is the solidity of the middle frustum of parabolic spindle its length being 25 in, greater diameter 20, and less diameter 15.
     
   • The length of a cask in the form of the middle frustum of a parabolic spindle is 46 inches, its bung diameter 31, and head diameter 24 in.
     
   • The length of a cask in the form of the middle frustum of a parabolic spindle is 38 inches, its bung diameter 35.5, and head diameter 32 in.
     
   • Find the content of a block of freestone 15 ft in length; the lower part being a parallelepiped of which the end is 8 feet in breadth and 6 ft in depth, and the upper part a triangular prism, of which the height is 3 ft.
     
   • Find the content of a block of freestone of which the dimensions taken in different places are as follows; the lengths 13 ft 5 in and 12 ft 7 in, breadths 5 ft 10 in, 5 ft 7 in and 5 ft 1 in, and depths 4 ft 9 in, 4 ft 7 in and 4 ft 2 in.
     
   • Walker computes the average length, breadth and depth which are 13..
     
   • Find the content of a block of freestone of which the dimensions taken in different places are as follows: the lengths 16 feet 5 inches and 14 feet 7 inches; breadths 7 feet 10 inches and 7 feet 7 inches and 7 feet 1 inch, and depths 6 feet 9 inches, 6 feet 7 inches and 6 feet 2 inches.
     
   • The average length, breadth and depth is calculated.
     
   • Find the area of a rhombus, of which the length is 60 in and the perp.
     
   • Using six figure logs, Walker squares the length of the side and multiplies by 3.6339124.
     
   • Using six figure logs, Walker squares the length of the side and multiplies by 4.8284271.
     
   • Find the content of a vessel in the form of a parallelepiped, of which the length is 80 inches, the breadth 24 inches, and the depth 25 inches in imp gallons and bushels.
     
   • How many imperial gallons of wort will a back contain, of which the length is 110 in, the breadth 90 in, and the depth 10 in.
     
   • Find the content of a vessel in the form of a pyramid, of which the depth is 48 in, the length and breadth at the top 64 and 36 in, and the length and breadth of the bottom 48 and 27 in, in imp gall and bushels.
     
   • Find the content of a vessel in the form of a pyramid, of which the depth is 68 in, the length and breadth at the top 84 and 56 in, and the length and breadth of the bottom 68 and 57 in respectively, in imp gall and bushels.
     
   • Find the content of a vessel in the form of a pyramid, of which the depth is 68 in, the length and breadth at the top 84 and 56 in, and the length and breadth of the bottom 68 and 57 in respectively, in imp gall and bushels.
     
   • Find the content of a cask, of which the bung diameter is 31 inches, the head diameter 24 inches, and the length 32 1/2 inches and the perpendicular distance mn being 8 inches.
     
   • Find the content of a rum cask, of which the bung diameter is 31.7 inches, the head diameter 26.8 inches, and the length 32.7 inches and the perpendicular distance mn being 58 inches.
     
   • Find the content of a pipe of Madeira wine, of which the bung diameter is 29 inches, the head diameter 21.2 inches, and the length 47.4 inches and the perpendicular mn 9 3/4 inches.
     
   • Find the content of a pipe of Spanish wine, of which the bung diameter is 31 inches, the head diameter 24 inches, and the length 46 inches and the perpendicular mn 8 3/4 inches.
     
   • Find the content of a cask, of which the bung diameter is 29 inches, the head diameter 23 inches, and the length 36 inches and the perpendicular mn 9 inches.
     
   • Find the content of a cask, of which the bung diameter is 31 inches, the head diameter 23 inches, and the length 40 inches and the perpendicular mn 1.4 inches.
     
   • Find the mean diameter, and thence the content in imperial gallons of a cask, of which the bung diameter is 31 inches, the head diameter 23 inches, and the length 50 inches and the perpendicular mn 1.4 inches.
     
   • Find the mean diameter, and thence the content in imperial gallons of a spherical cask, of which the bung diameter is 31 inches, the head diameter 27 inches, and the length 301/2 inches.
     
   • Find the mean diameter, and thence the content in imperial gallons of a spherical hogshead, of which the bung diameter is 29 inches, the head diameter 23 inches, and the length 27 inches.
     
   • Find the mean diameter, and thence the content in imperial gallons of a pipe of wine, of the variety of which the bung diameter is 28.8 inches, the head diameter 22.3 inches, and the length 47 inches.
     
   • Find the content of a cask, of which the bung diameter is 21 inches the head diameter 18 inches, and the length 30 inches.
     
   • Find the content of a cask, of which the bung diameter is 32 inches the head diameter 25.3 inches, and the length 47 inches.
     
   • Find the ullage of a lying aulm, of which the length is 24 inches the bung diameter 22 inches, the head diameter 19 inches, and the depth of liquor 12 inches.
     
   • What is the ullage of a lying hogshead, of which the length is 27 inches, the bung diameter 29 inches, the head diameter 23 inches, and the depth of liquor 10 inches.
     
   • What is the ullage of a standing hogshead, of which the length is 29.1 inches, the bung diameter 28.5 inches, the head diameter 24 inches, and the depth of liquor 18 inches.
     
   • What is the ullage of a standing butt, of which the length is 39.5 inches, the bung diameter 33.8 inches, the head diameter 29.8 inches, and the depth of liquor 30 inches.
     
   • Find the number of balls in a rectangular pile, the length and breadth of the base row being 50 and 30 respectively.
     
   • 13/4, the length of the three sides was 80 feet, find the sum of the other two sides.
     
   • 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.
     
   • Walker actually missed out the length of the third side! .
     
   • A perpendicular drawn from one of the angles of an equilateral triangle to the opposite side measures 12 feet, find the length of a side of the triangle.
     
   • A field in the form of an equilateral triangle contains half an acre, what must be the length of the tether fixed to a horses nose to allow him to graze exactly half of it.
     
   • A piece of cable 3 feet in length and 9 inches in girt weighs 22 lbs, what will the cable weigh per fathom of which the girt is 12 inches.
     
   • The length of the keel of a ship of 250 tons is 72 feet, find the tonnage of another ship of the same form, of which the keel is 81 feet long.
     
   • length 10.09, breadth 6.49.
     
   • If a cubic foot of brass were drawn out into wire 1/40 of an inch in diameter, what would be the length of the wire, supposing no loss in the metal.
     
   • A gentleman has a bowling green 300 feet in length, and 200 feet in breadth which he wishes to be raised one foot higher by means of the earth to be dug out of a ditch by which he intends to surround it; to what depth must be ditch be dug, if the breadth be everywhere 8 feet.
     
   • length breadth .
     
   • Find the depth of a tub in the form of a conic frustum, of which the greater diameter is 60 inches, the diagonal 66 inches, and the length of the staves 30 in.
     
   • The base of a plane triangle is 384 feet, and the other sides 288 and 192, find the length of the perpendicular upon the base, and the length of the segments of the base made by the line bisecting the vertical angle.
     

2. Measurement
   
   • Ancient measurement of length was based on the human body, for example the length of a foot, the length of a stride, the span of a hand, and the breadth of a thumb.
     
   • Based on the human body, it was taken to be the length of an arm from the elbow to the extended fingertips.
     
   • Since different people have different lengths of arm, the Egyptians developed a standard royal cubit which was preserved in the form of a black granite rod against which everyone could standardise their own measuring rods.
     
   • To measure smaller lengths required subdivisions of the royal cubit.
     
   • Although we might think there is an inescapable logic in dividing it in a systematic manner, this ignores the way that measuring grew up with people measuring shorter lengths using other parts of the human body.
     
   • Their basic unit of length was, like the Egyptians, the cubit.
     
   • Several scales for the measurement of length were also discovered during excavations.
     
   • Of course ten units is then 13.2 inches (33.5 centimetres) which is quite believable as the measure of a "foot", although this suggests the Harappans had rather large feet! Another scale was discovered when a bronze rod was found to have marks in lengths of 0.367 inches.
     
   • Now 100 units of this measure is 36.7 inches (93 centimetres) which is about the length of a stride.
     
   • Measurements of the ruins of the buildings which have been excavated show that these units of length were accurately used by the Harappans in their construction.
     
   • The Greeks used as their basic measure of length the breadth of a finger (about 19.3 mm), with 16 fingers in a foot, and 24 fingers in a Greek cubit.
     
   • These units of length, as were the Greek units of weight and volume, were derived from the Egyptian and Babylonian units.
     
   • Scientists had long seen the benefits of rationalising measures and those such as Wren had proposed a new system based on the yard defined as the length of a pendulum beating at the rate of one second in the Tower of London.
     
   • Gabriel Mouton, in 1670, had suggested that the world should adopt a uniform scale of measurement based on the mille, which he defined as the length of one minute of the Earth's arc.
     
   • He proposed that decimal subdivisions should be used to determine the lengths of shorter units of length.
     
   • Talleyrand put to the National Assembly a proposal due to Condorcet, namely that a new measurement system be adopted based on a length from nature.
     
   • The system should have decimal subdivisions, all measures of area, volume, weight etc should be linked to the fundamental unit of length.
     
   • The basic length should be that of a pendulum which beat at the rate of one second.
     
   • An immediate problem was that the pendulum length depended on the latitude at which the experiment was performed so a latitude had to be chosen.
     
   • Diplomatic wording allowed an international agreement to be reached, but in March 1791 Borda, as chairman of the Commission of Weights and Measures, proposed using instead of the length of a pendulum, the length of 1/10,000,000 of the distance from the pole to the equator of the Earth.
     
   • The Royal Society in London declared this was based on a measurement of France, the Americans were not prepared to accept the word of the French mathematicians for its length and even in France it was claimed that the whole project was really proposed in order to gain information on the shape of the Earth.
     
   • Indeed, probably Laplace and others were more interested in finding the shape of the Earth rather than the length of the metre.
     
   • Note that in all these redefinitions, the length of the metre was always taken as close as possible to the value fixed in 1799 by data from the Delambre-Mechain survey.
     
   • Now Borda had argued against using the length of a pendulum which beats at the rate of one second to define the metre in 1791 on the reasonable grounds that the second was not a fixed unit but could change with time.
     
   • Indeed the second, then defined as 1/86,400 of the mean solar day, does change but a fixed definition was introduced in 1956 by the International Bureau of Weights and Measures, as 1/31,556,925.9747 of the length of the tropical year 1900.
     
   • Although this fixed the value, it was seen as an unsatisfactory definition since the length of the year 1900 could never be measured after 1900.
     

3. Real numbers 1
   
   • And the strength of the thread does not reside in the fact that some one fibre runs through its whole length, but in the overlapping of many fibres.
     
   • The first of these might refer to the length of a geometrical line while the second concept, namely number, was thought of as composed of units.
     
   • (the natural numbers in the terminology of today) in a geometrical way, not as lengths of a line as we do, but rather in the form of discrete points.
     
   • All numbers, essentially by definition, were, as we have seen, (positive integer) multiples of a base unit but ratios of lengths were shown not to have the property of being ratios of numbers (integers).
     
   • The usual example given of this comes from a right angled triangle whose shorter sides are both of unit length.
     
   • Such a triangle has as hypotenuse a line of length √2 times the lengths of the shorter sides.
     
   • There is no length x such that 1 and √2 are both multiples (remember integer multiples) of x.
     
   • Euclid goes on to prove, among many other results, those of Theodorus, namely that segments of length √3, √5, ..
     
   • , √17 are incommensurable with a segment of unit length.
     
   • Also magnitudes were considered and these were essentially lengths constructible by ruler and compass from a line of unit length.
     
   • Hence mathematicians studied magnitudes which had lengths which, in modern terms, could be formed from positive integers by addition, subtraction, multiplication, division and taking square roots.
     
   • The Arabic mathematicians went further with constructible magnitudes for they used geometric methods to solve cubic equations which meant that they could construct magnitudes whose ratio to a unit length involved cube roots.
     
   • Well the obvious question the reader might feel they want to ask Stifel is: what about the length of the circumference of a circle with radius of unit length? In fact Stifel gives an answer to this in an appendix to the book.
     
   • Not too good an argument, but nevertheless a remarkable insight that there were lengths which did not correspond to radical expressions but which could be approximated as closely as one wished.
     

4. Golden ratio
   
   • Of course if AB has length 1 and AC = x where C divides AB in the golden ratio, then we can use simple algebra to find x.
     
   • Al-Khwarizmi does indeed give several problems on dividing a line of length 10 into two parts and one of these does find a quadratic equation for the length of the smaller part of the line of length 10 divided in the golden ratio.
     
   • Abu Kamil gives similar equations which arise from dividing a line of length 10 in various ways.
     
   • In Liber Abaci he gives the lengths of the segments of a line of length 10 divided in the golden ratio as √125 -5 and 15 - √125.
     
   • He also states the result given in Liber Abaci on the lengths of the segments of a line of length 10 divided in the golden ratio.
     
   • He gives "about 0.6180340" for the length of the longer segment of a line of length 1 divided in the golden ratio.
     

5. Mathematics and Architecture
   
   • But the first gardener in history to lay out a perfect ellipse with three stakes and a length of string certainly held no degree in the theory of cones! Nor did Egyptian architects have anything more than simple devices -- "tricks", "knacks" and methods of an entirely empirical kind, no doubt discovered by trial and error -- for laying out their ground plans.
     
   • Pythagoras saw the connection between music and numbers and clearly understood how the note produced by a string related to its length.
     
   • This led to the use of a module, a basic unit of length for the building, where the dimensions were now small integer multiples of the basic length.
     
   • Let us look briefly at the dimensions of the Parthenon to see how the lengths conform to the mathematical principles of proportion of the Pythagoreans.
     
   • The length of the Temple is 69.5 m, its width is 30.88 m and the height at the cornice is 13.72 m.
     
   • To a fairly high degree of accuracy this means that the ratio width : length = 4 : 9 while also the ratio height : width = 4 : 9.
     
   • height : width : length = 16 : 36 : 81 .
     
   • which gives a basic module of length 0.858 m.
     
   • Then the length of the Temple is 92 modules, its width is 62 modules and its height is 42 modules.
     
   • The module length is used throughout, for example the overall height of the Temple is 21 modules, and the columns are 12 modules high.
     

6. Elliptic functions
   
   • For example the period of a simple pendulum was found to be related to an integral which expressed arc length but no form could be found in terms of 'simple' functions.
     
   • The study of elliptical integrals can be said to start in 1655 when Wallis began to study the arc length of an ellipse.
     Go directly to this paragraph
     
   • In fact he considered the arc lengths of various cycloids and related these arc lengths to that of the ellipse.
     Go directly to this paragraph
     
   • Both Wallis and Newton published an infinite series expansion for the arc length of the ellipse.
     Go directly to this paragraph
     
   • In 1679 Jacob Bernoulli attempted to find the arc length of a spiral and encountered an example of an elliptic integral.
     Go directly to this paragraph
     
   • whose arc length is given by the integral from 0 to x of .
     
   • The other good features of the lemniscate integral are the fact that it is general enough for many of its properties to be generalised to more general elliptic functions, yet the geometric intuition from the arc length of the lemniscate curve aids understanding.
     

7. Sundials
   
   • Depending on the design of the dial, either the side of the shadow's length or the position of the tip of the shadow was used to determine the time.) Unfortunately, Vitruvius ends his discussion of sundials with the list given above and writes of water clocks for the rest of Book 9.
     
   • With the head to the east 4 hours are marked off by decreasing shadow lengths after which the instrument is reversed with head to the west to mark 4 afternoon hours.
     
   • (The shadow at sunrise would be infinite in length, and so useless for marking the hour.) Two hours similarly passed in the evening.[Ancient Egyptian Astronomy.
     
   • At six in the morning the shadow would strike the top of the dial; as the sun rose higher the shadow would decrease in length until at noon it touched the lowest line; it reached the top of the dial again at six in the evening.
     
   • Both Vitruvius and Ptolemy describe analemmas which for given solar positions serve to determine length and direction of the shadow cast by a gnomon on the face of a planar sundial.
     
   • Though he does not say so explicitly, each pair of spherical coordinates is singularly suited for finding the length and direction of a gnomon's shadow for a type of plane sundial.
     

8. Trigonometric functions
   
   • Given a circle of fixed radius, 60 units were often used in early calculations, then the problem was to find the length of the chord subtended by a given angle.
     Go directly to this paragraph
     
   • For a circle of unit radius the length of the chord subtended by the angle x was 2sin (x/2).
     Go directly to this paragraph
     
   • They became important for calculating heights from the length of the shadow that the object cast.
     Go directly to this paragraph
     
   • The length of shadows was also of importance in the sundial.
     Go directly to this paragraph
     
   • Thales used the lengths of shadows to calculate the heights of pyramids.
     Go directly to this paragraph
     

9. Classical time
   
   • The beginnings of civilisation on Earth required a knowledge of the seasons, and the mysteries surrounding the length of the year, the length of the day and the length of the month began to be studied.
     
   • The fact that knowing the length of a year was vitally important, yet much less visible from the timekeepers in the sky, led to calculation.
     
   • Later a more accurate value of 3651/4 days was worked out for the length of the year but the civil calendar was never changed to take this into account.
     
   • We should also note that many early units of time varied throughout the year as the length of the day and night varied with the seasons.
     
   • Sand was also used in the still familiar hour glass where sand trickles from a container, taking a set length of time to run out.
     

10. Indian mathematics
    
    • Several scales for the measurement of length were also discovered during excavations.
      
    • A similar measure based on the length of a foot is present in other parts of Asia and beyond.
      
    • Another scale was discovered when a bronze rod was found which was marked in lengths of 0.367 inches.
      
    • Measurements of the ruins of the buildings which have been excavated show that these units of length were accurately used by the Harappans in construction.
      

11. Chinese problems
    
    • Find the length of the side of the farm and the diameter of the pond.
      
    • In the right-angled triangle with sides of length a, b and c with a > b > c, we know that a + b = 81 ken and a + c = 72 ken.
      
    • Let x, y be the lengths of the two legs and z the length of the hypotenuse of the triangle.
      

12. Trisecting an angle
    
    • Given the angle CAB then mark off equal lengths AB and AC.
      
    • Just mark off a length of 2 cross AC at the right hand end of the ruler and then slide the ruler with one mark on CD, the other on FC extended until the ruler defines a line passing through A.
      
    • Again this can be done in a mechanical way by marking a length equal to the radius of the circle on the ruler and moving it keeping one mark on BA produced and having the second mark on the circle.
      
    • Pappus tells us that in practice the conchoid was not always actually drawn but that some, for greater convenience, moved a ruler about the fixed point until by trial the intercept was found to be equal to the given length.
      

13. Nine chapters
    
    • This chapter contains 24 problems and takes its name from the first eleven problems which ask what the length of a field will be if the width is increased but the area kept constant.
      
    • What must its length be if its area is 1? .
      
    • What methods are used to try to date the material? Perhaps the most important is to examine the units of length, volume and weight which appear in the various problems.
      
    • Standard decimal units of length were established in China around 200 BC and later further subdivisions occurred.
      
    • Of course, the dating using units of length is not conclusive.
      
    • The Nine Chapters on the Mathematical Art was certainly an important text, so may have had its units of length brought up to date as it evolved.
      

14. Infinity
    
    • Although the circumference of A is twice the length of the circumference of B they have the same number of points.
      
    • Galileo proposed adding an infinite number of infinitely small gaps to the smaller length to make it equal to the larger yet allow them to have the same number of points.
      
    • 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.
      

15. Calculus history
    
    • They got round this difficulty by using lengths, areas and volumes in addition to numbers for, to the Greeks, not all lengths were numbers.
      
    • Leibniz was to have a lengthy correspondence with Barrow.
      Go directly to this paragraph
      

16. Water-clocks
    
    • The shortening and lengthening of the days must be corrected day by day and month by month through the addition or removal of wedges.
      
    • Other devices were employed to vary the hour lengths by regulating the flow of water from the reservoir.
      
    • The hour markings around the column would reflect the shortening and lengthening of the hours throughout the year.
      

17. 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.
      Go directly to this paragraph
      
    • Still on the theme of phoney experiments, Gridgeman, in a paper which pours scorn on Lazzerini and others, created some amusement by using a needle of carefully chosen length k = 0.7857, throwing it twice, and hitting a line once.
      
    • Is π normal to base 10? That is does every block of digits of a given length appear equally often in its decimal expansion in an asymptotic sense? .
      
    • Is π normal ? That is does every block of digits of a given length appear equally often in the expansion in every base in an asymptotic sense? The concept was introduced by Borel in 1909.
      

18. Mayan mathematics
    
    • With such crude instruments the Maya were able to calculate the length of the year to be 365.242 days (the modern value is 365.242198 days).
      
    • Two further remarkable calculations are of the length of the lunar month.
      
    • This gives 29.5302 days as the length of the lunar month.
      
    • This gives 29.5308 days as the length of the lunar month.
      

19. Doubling the cube
    
    • Draw the circle OBA having OA as diameter where OA is the greater [OA = a b]; and inscribe OB, of length b and produce it to meet at C the tangent to the circle at A.
      
    • Here AE and DH are the two lengths for which it is required to find two mean proportionals.
      

20. Indian Sulbasutras
    
    • The rope which is stretched along the length of the diagonal of a rectangle produces an area which the vertical and horizontal sides make together.
      
    • Increase a unit length by its third and this third by its own fourth less the thirty-fourth part of that fourth.
      
    • We now have a square the length of whose sides is .
      

21. Jaina mathematics
    
    • The lengths of the boundary chords and the areas of the regions are given, based on stated rules.
      
    • For example in the Surya Prajnapti data exists which implies a ratio of 3:2 for the maximum to the minimum length of daylight.
      

22. Ten classics
    
    • Its side therefore has length √( a2 + b2).
      
    • Therefore the hypotenuse of the right angled triangle with sides of length a and b has length √( a2 + b2).
      

23. Prime numbers
    
    • Is there an arithmetic progression of consecutive primes for any given (finite) length? e.g.
      
    • 251, 257, 263, 269 has length 4.
      
    • The largest example known has length 10.
      

24. Squaring the circle
    
    • Then Archimedes proves in Proposition 19 of On spirals that OT is the length of the circumference of the circle with radius OP.
      
    • Note.- If the area of the circle be 140,000 square miles, then [the side of the square] is greater than the true length by about an inch.
      
    • For a circle of diameter 8000 miles, the error in the length of the side of the square constructed was only a fraction of an inch.
      

25. Mathematics and Art
    
    • Also important was his understanding of scale, and he correctly computed the relation between the actual length of an object and its length in the picture depending on its distance behind the plane of the canvas.
      
    • The length of CD determines the correct viewing distance, that is the distance the observer has to be from the picture to obtain the correct perspective effect.
      

26. Burnside problem
    
    • P Hall and G Higman [p-length of p-soluble groups and reduction theorems for Burnside\'s Problem, Proc.
      
    • (3) 6 (1956), 1-42',11)">11] showed that B0(m, 6) exists and has order 2a3b where a = 1 + (m - 1)3c , b = 1 + (m - 1)2m , c = m + mC2 + mC3 and is hence soluble of derived length 3.
      
    • Theorem (Hall-Higman, 1956 [p-length of p-soluble groups and reduction theorems for Burnside\'s Problem, Proc.
      

27. Pell's equation
    
    • which recurs with length 6.
      
    • A polynomial time method in the length of the input n would be an algorithm which took time bounded by a fixed power of log n (the length of the input).
      

28. Mathematical classics
    
    • Its side therefore has length sqrt ( a2 + b2).
      
    • Therefore the hypotenuse of the right angled triangle with sides of length a and b has length sqrt ( a2 + b2).
      

29. History overview
    
    • Earlier a place value notation number system had evolved over a lengthy period with a number base of 60.
      
    • A more precise formulation of concepts led to the realisation that the rational numbers did not suffice to measure all lengths.
      Go directly to this paragraph
      

30. Fair book insert
    
    • 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.
      
    • He first computes the area of the base by squaring the length of the side of the regular pentagon and multiplying by 1.7204774.
      

31. Tartaglia versus Cardan
    
    • There is a tree, 12 braccia high, which was broken into parts at such a point that the height of the part which was left standing was the cube root of the length of the part that was cut away.
      
    • What is the length of one of the sides? .
      
    • and you ask the length of one of the sides.
      

32. Knots and physics
    
    • would then be described by the sequence of crossings of length 2n where each of A, B, C, ..
      
    • However there were some other problems, for example although a sequence of length 10, say, might represent a knot it might be one with less than 5 crossings.
      

33. Longitude1
    
    • 24 hours a day, the other sidereal time of 23 hours, 56 minutes and 4 seconds to the day (the length of time until the stars reach the same position as the previous day).
      
    • Varin and des Hayes found that, like those of Picard, their clocks did not run correctly and they had to shorten the length of the pendulums.
      Go directly to this paragraph
      

34. Babylonian mathematics
    
    • For example if the area is given and the amount by which the length exceeds the breadth is given, then the breadth satisfies a quadratic equation and then they would apply the first version of the formula above.
      
    • A problem on a tablet from Old Babylonian times states that the area of a rectangle is 1, 0 and its length exceeds its breadth by 7.
      

35. Planetary motion
    
    • This by definition involves the dimensions of length and time alone, while excluding altogether the dimension of mass.
      
    • This length AF is subsequently recognized as 'the mean distance', which is of great significance in Part III below.
      

36. Modern light
    
    • It gets back to the detector in the same length of time irrespective of whether the detector is moving or not.
      
    • Using a light path of length 35 km from the Mount Wilson observatory to the telescope on Mount San Antonio, he found the value of 299,796 km per sec.
      

37. test.html
    
    • Degrees of longitude and latitude are marked in the margins, and the maps themselves are traversed by parallels and meridians; but while the meridians are fixed by the numbers in the margins, just as on modern maps, with one for every five degrees, the parallels are fixed according to the length of the longest day, at intervals of a quarter, half, or whole hour of difference.
      
    • This anomaly is a vestige of the fact that latitude was early associated with the length of the day.
      

38. Christianity and Mathematics
    
    • He was an extremely religious man and discussed God at length in his works.
      
    • We have discussed Greek philosophers at some length in an article on Christianity and the mathematical sciences.
      

39. Ptolemy mss.html
    
    • Degrees of longitude and latitude are marked in the margins, and the maps themselves are traversed by parallels and meridians; but while the meridians are fixed by the numbers in the margins, just as on modern maps, with one for every five degrees, the parallels are fixed according to the length of the longest day, at intervals of a quarter, half, or whole hour of difference.
      
    • This anomaly is a vestige of the fact that latitude was early associated with the length of the day.
      

40. Egyptian mathematics
    
    • Not that anyone believes that the Egyptians knew of the secant function, but it is of course just the ratio of the height of the sloping face to half the length of the side of the square base.
      
    • Later a more accurate value of 365 1/4 days was worked out for the length of the year but the civil calendar was never changed to take this into account.
      

41. Classical light
    
    • In about 60 AD Heron made the interesting observation that when light is reflected by a mirror it travels along the path of least length.
      
    • In 1647 Cavalieri published an important contribution to optics when he gave the relationship between the curvature of a thin lens and its focal length.
      

42. Quadratic etc equations
    
    • However all Babylonian problems had answers which were positive (more accurately unsigned) quantities since the usual answer was a length.
      
    • In about 300 BC Euclid developed a geometrical approach which, although later mathematicians used it to solve quadratic equations, amounted to finding a length which in our notation was the root of a quadratic equation.
      Go directly to this paragraph
      

43. Greek astronomy
    
    • Yet a knowledge of the approximate length of the year was vital for food production and so schemes had to be devised.
      
    • Meton worked in Athens with another astronomer Euctemon, and they made a series of observations of the solstices (the points at which the sun is at greatest distance from the equator) in order to determine the length of the tropical year.
      

44. Babylonian numerals
    
    • On the other hand many measures do involve 12, for example it occurs frequently in weights, money and length subdivisions.
      

45. test2.html
    
    • } of elements generating K, then there exists a function L(n) such that all elements of K of length not exceeding n are contained in the subgroup KL generated by the subset { a1 , a2 , ..
      

46. Kepler's Laws
    
    • Tycho developed, refined and cross-checked his instruments and sometimes attained an accuracy of 2' (which is approximately the breadth of a hair held at arm's length).
      

47. Cubic surfaces
    
    • In his lengthy Memoir on Cubic Surfaces Cayley presented Schlafli's complete classification of cubic surfaces into 23 distinct species and he also added further investigations of his own.
      

48. Poincaré - Inspector of mines
    
    • Poincare then gave a precise description of the lengths of the three galleries and the accurate measurements of the underground passages.
      

49. Set theory
    
    • For example Albert of Saxony, in Questiones subtilissime in libros de celo et mundi, proves that a beam of infinite length has the same volume as 3-space.
      Go directly to this paragraph
      

50. Non-Euclidean geometry
    
    • Riemann, who wrote his doctoral dissertation under Gauss's supervision, gave an inaugural lecture on 10 June 1854 in which he reformulated the whole concept of geometry which he saw as a space with enough extra structure to be able to measure things like length.
      Go directly to this paragraph
      

51. Real numbers 3
    
    • We can represent c as a point on a line segment of length 1.
      

52. Jaina mathematics references
    
    • S D Sharma, and S S Lishk, Length of the day in Jaina astronomy, Centaurus 22 (3) (1978/79), 165-176.
      

53. Bolzano publications.html
    
    • Contains reprints of the following papers by Bolzano: Considerations on some points in elementary geometry (1804), Contributions to a better founded exposition of mathematics (1810), The binomial theorem (1816), Pure analytical proof of the intermediate value theorem (1817), and The three problems of curve length, surface area and volume (1817).
      

54. Debating topics
    
    • What is wrong with having the hypotenuse of a right angled triangle, whose shorter sides are each of one unit, not corresponding to a number? Why should there be a number corresponding to the length of every line we draw? .
      

55. Greek astronomy references
    
    • Y Maeyama, The length of the synodic months : The main historical problem of the lunar motion, Arch.
      

56. Jaina mathematics references
    
    • S D Sharma, and S S Lishk, Length of the day in Jaina astronomy, Centaurus 22 (3) (1978/79), 165-176.
      

57. Tait's scrapbook
    
    • He writes a permutation as the product of disjoint cycles, then notes that a cycle of odd length can be written as the product of an even number of transpositions.
      

58. Special relativity
    
    • the length of material bodies changes, according as they are moving through the ether or across it, by an amount depending on the square of the ratio of their velocities to that of light.
      

59. Cosmology
    
    • For many years it seemed a purely academic point, whether the universe was eternal and unchanging, or had only existed for a finite length of time.
      

60. Greek astronomy references
    
    • Y Maeyama, The length of the synodic months : The main historical problem of the lunar motion, Arch.
      

61. Real numbers 2
    
    • He also defined the notion of quantity as that which can be continuously increased or diminished and thought of length, area, volume, mass, velocity, time, etc.
      

62. Burnside problem references
    
    • P Hall, P and G Higman, On the p-length of p-soluble groups and reduction theorems for Burnside's Problem, Proc.
      

63. Word problems
    
    • } of elements generating K, then there exists a function L(n) such that all elements of K of length not exceeding n are contained in the subgroup KL generated by the subset { a1 , a2 , ..
      

64. Cartography
    
    • He suggested that a grid should be chosen with astronomical significance so that, for example, points on the same line would all have the same length of longest day.
      

65. Babylonian Pythagoras
    
    • 4 is the length and 5 the diagonal.
      

66. function concept
    
    • Although the circumference of A is twice the length of the circumference of B they have the same number of points.
      

67. Ledermann interview
    
    • Turnbull had to go to the lengths of saying that he would resign if Walter's post was not confirmed before the protest at the appointment of a foreigner was ended.
      

68. Gravitation
    
    • Although the time taken for the bob to rise and fall depended on the length of the pendulum, it did not depend on the weight of the bob.
      

69. Brachistochrone problem
    
    • Here the problem was to find curves of minimum length where the curves were constrained to lie on a given surface.
      

70. Burnside problem references
    
    • P Hall, P and G Higman, On the p-length of p-soluble groups and reduction theorems for Burnside's Problem, Proc.
      

71. Zero
    
    • In other words Greek mathematicians did not need to name their numbers since they worked with numbers as lengths of lines.
      

72. Chinese overview
    
    • It is impossible in an article of this length to mention many of the numerous contributions from this period on.
      

73. Euclid's definitions
    
    • A line is a breadthless length.
      

Famous Curves

1. Neiles
   
   • It was the first algebraic curve to have its arc length computed.
     
   • Neile's parabola was the first algebraic curve to have its arc length calculated; only the arc lengths of transcendental curves such as the cycloid and the logarithmic spiral had been calculated before this.
     

2. Cardioid
   
   • Its length had been found by La Hire in 1708, and he therefore has some claim to be the discoverer of the curve.
     
   • In the notation given above the length is 16a.
     
   • The length of any chord through the cusp point is 4a and the area of the cardioid is 6πa2.
     

3. Cycloid
   
   • Mersenne gave the first proper definition of the cycloid and stated the obvious properties such as the length of the base equals the circumference of the rolling circle.
     
   • Wren's contribution was the most remarkable as he found the arc length, the length of the arch being 8a.
     

4. Equiangular
   
   • Torricelli worked on it independently and found the length of the curve.
     
   • If P is any point on the spiral then the length of the spiral from P to the origin is finite.
     
   • Therefore the length of the curve from a point at distance d from the origin along a radius vector is about 5.126 d.
     

5. Cissoid
   
   • Newton gave a method of drawing the Cissoid of Diocles using two line segments of equal length at right angles.
     
   • He studied the cissoid in his attempt to solve the problem of finding the length of the side of a cube having volume twice that of a given cube.
     

6. Tractrix
   
   • What is the path of an object dragged along a horizontal plane by a string of constant length when the end of the string not joined to the object moves along a straight line in the plane? .
     
   • Among the properties of the tractrix are the fact that the length of a tangent from its point of contact to an asymptote is constant.
     

7. Astroid
   
   • The length of the astroid is 6a and its area is 3πa2/8.
     
   • Then the length XY is a constant and is equal to a.
     

8. Tricuspoid
   
   • The length of the tangent to the tricuspoid, measured between the two points P, Q in which it cuts the curve again is constant and equal to 4a.
     
   • The length of the curve is 16a and the area it encloses is 2πa2.
     

9. Ellipse
   
   • There is no exact formula for the length of an ellipse in elementary functions and this led to the study of elliptic functions.
     
   • Ramanujan, in 1914, gave the approximate length .
     

10. Durers
    
    • He drew lines QRP and P'QR of length 16 units through Q (q, 0) and R (0, r) where q + r = 13.
      

11. Curve definitions
    
    • Let P1 be the point with P1O a line segment parallel and of equal length to PQ.
      

12. Hypocycloid
    
    • If a = (n + 1)b where n is an integer, then the length of the epicycloid is 8nb and its area is πb2(n2 - n).
      

13. Lemniscate
    
    • Euler's investigations of the length of arc of the curve (1751) led to later work on elliptic functions.
      

14. Epicycloid
    
    • If a = (m - 1)b where m is an integer, then the length of the epicycloid is 8mb and its area is πb2(m2 + m).
      

15. Spiral
    
    • Archimedes was able to work out the lengths of various tangents to the spiral.
      

16. Nephroid
    
    • The nephroid has length 24a and area 12π2.
      

17. Watts
    
    • Suppose that a rod of length 2c is fixed at each end to the circumference of the two wheels.
      

18. Circle
    
    • For the circle with formula given above the area is πa2 and the length of the curve is 2πa.
      

Societies etc

1. Memorandum 1976
   
   • The Bangor committee discussed this at length and decided to retain the £10 fee.
     

2. BMC 1977
   
   • Davies, R OSemi-continuity of length .
     

3. Minutes for 2004
   
   • It is difficult to persuade very famous foreign speakers to speak at a BMC, and so the list of possibilities should be at least twice the length of the number of slots; the best chance of securing an acceptance is to use as an intermediary a UK colleague who knows the person personally.
     

4. New York Academy of Sciences
   
   • book-length reviews of a subject, combining both original research and review articles from researchers working in a variety of disciplines.
     

5. Minutes for 1985
   
   • Professor Rogers asked that, in view of the length of time being taken over meals and the subsequent journey to the Sidgwick site, that Thursday's splinter groups begin at 2.15 p.m., and Professor Hoare's talk begin at 8.45 p.m.
     

6. Paris Academy of Sciences
   
   • The idea for the Academie des Sciences arose from a number of sources, but one was certainly Mersenne's group which met regularly and corresponded at length with eminent figures, including Descartes, Desargues, Fermat, Etienne Pascal, Blaise Pascal, Gassendi, Roberval, and Galileo.
     

7. BMC 2004
   
   • Carne, T K How an analytic functio changes the length of a radius .
     

References

1. References for Bell John
   
   • [Contains lengthy accounts of discussions with John Bell.] .
     

2. References for Bhaskara II
   
   • S A Naimpally, Approximate Formula For The Length of a Chord, Ganita Bharati 9 (1987) 57-58.
     

3. References for Lambert
   
   • L Giacardi, On the approximate calculation of the length of the circumference in Huygens and in Lambert : Conclusion of the Archimedean procedures and introduction of infinity (Italian), Rend.
     

4. References for Yunus
   
   • W Hartner, An unusual value for the length of the meridian degree : 66 1/2 miles, in Ibn Yunus' 'Hakimitic Zij', Centaurus 24 (1980), 148-152.
     

5. References for Huygens
   
   • L Giacardi, On the approximate calculation of the length of the circumference in Huygens and in Lambert, Rend.
     

6. References for Archimedes
   
   • S E Brodie, Archimedes' axioms for arc-length and area, Math.
     

7. References for Infeld
   
   • L Stern, Shadowing Infeld : Secret documents show the lengths to which Canadian spies went to try to prove exiled physicist was a communist, The Ottawa Citizen (Sunday, January 24, 1999).
     

8. References for Hipparchus
   
   • N M Swerdlow, Hipparchus's determination of the length of the tropical year and the rate of precession, Arch.
     

Additional material

1. Carl Runge: 'Graphical Methods
   
   • - Any quantity susceptible of mensuration can be graphically represented by a straight line, the length of the line corresponding to the value of the quantity.
     
   • A quantity might also be and is sometimes graphically represented by an angle or by the length of a curved line or by the area of a square or triangle or any other figure or by the anharmonic ratio of four points in a straight line or in a variety of other ways.
     
   • The addition of two positive quantities represented by straight lines of given length is effected by laying them off in the same direction, one behind the other.
     
   • Lay the edge in succession over the different lines and run a pointer along it through an amount equal in each case to the length of the line and in the positive or negative direction according to the sign of the quantity.
     
   • wide are substituted for the area so that, measured in square centimetres, it is equal to half the sum of the lengths of the strips measured in centimetres.
     
   • The drawing of the strips may be dispensed with, their lengths being estimated, only their width must be shown.
     
   • If the scale should be too short for the whole length, the only thing we have to do is to break any of the lengths that range over the end of the scale and to count how many times we have gone over the whole scale.
     
   • If the curves of the segments may with sufficient accuracy be regarded as arcs of parabolas the area would be two thirds the product of length and width.
     
   • need only represent the numbers by the ratios of the lengths of straight lines to a certain fixed line.
     
   • The ratio of the length of the sum of the lines to the length of the fixed lines is equal to the sum of the numbers.
     
   • The construction also applies to positive and negative numbers, if we represent them by the ratio of the length of straight lines of opposite directions to the length of a fixed line.
     
   • In order to multiply a given quantity c by a given number, let the number be given as the ratio of the lengths of two straight lines a/b.
     
   • If the quantity c is also represented by a straight line, all we have to do is to find a straight line x whose length is to the length of c as a to b.
     

2. University of Glasgow Examinations
   
   • If the length of the larger segment be 10 in., what is the length of the smaller? .
     
   • 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.
     
   • If the unit of length were m feet, and the unit of time n seconds, what number would represent an acceleration of a feet per second per second.
     
   • Investigate the relation between the length of a simple pendulum and the time of oscillation.
     
   • A "second's" pendulum is lengthened 1 per cent.
     
   • According to experiments of Sabine, the length of the seconds pendulum at London is 39.139 inches.
     
   • Find also the wave-length for the note C (256 vibrations per second), and compare it with the wavelength for light of any particular colour.
     

3. Max Planck: 'The Nature of Light
   
   • The wave-length or frequency determines the colour in the same manner as it determines the pitch in sound.
     
   • What Maxwell could only prophecy, Heinrich Hertz was able to verify a generation later, when he showed how to produce the electro-magnetic waves calculated by Maxwell, and thereby ensured the final acceptance of the electro-magnetic theory of light, according to which electric waves only differ from heat and light rays in that they have very much greater wave-length.
     
   • These rays, too, have the character of light waves, and are electro-magnetic oscillations, but have a very much shorter wave-length.
     
   • Instead of the eye, special pieces of apparatus have been devised for receiving and measuring the different wave-lengths of the remainder of the spectrum.
     
   • The velocity increases as the wave-length diminishes.
     
   • Like ultra-violet rays, Rontgen rays and Gamma rays give us the same effect, though, owing to the very much shorter wave-lengths of these rays, the velocities of the liberated electrons are much greater.
     
   • Imagine a tall apple tree, all its branches weighed down with ripe fruit, all of the same size, but with stalks of different lengths; the apples are arranged so that those with short stalks are higher than those with long stalks.
     
   • by the lengths of their stalks; all the other apples remain on the tree.
     
   • It follows that a short wave-length corresponds to a large amount of energy, considered as a light quantum.
     

4. Heinrich Tietze on Numbers, Part 2
   
   • But the fraction 5/3 can also be interpreted as the measurement of a length (by dividing the length into three equal parts and then considering five such parts side by side).
     
   • Negative and positive numbers can also be used to measure lengths by taking a fixed point on a line and specifying that measurements to the right of the point are positive and to the left negative.
     
   • 7, in which AB is the unit of length and the length of AC is 5/3.
     
   • Therefore, if AB is the unit of length, the length of the segment AC cannot be represented by any fraction m/n.
     
   • Greek mathematics could compare two segments as equal (or larger or smaller); what it lacked was the concept of assigning numbers to lengths of segments incommensurable with a prescribed unit segment.
     
   • While now we can simply say: If the side AB is the unit of length, the length of the diagonal AC is equal to √2, this would have been impossible for the Greeks because √2 does not exist in the domain of rational numbers.
     

5. Mathematicians and Music 2.1
   
   • He proclaimed the remarkable fact, of which the proof existed in his famous experiments with stretched strings of different lengths, that the ratios of the intervals perceived as consonant could all be expressed by the numbers 1, 2, 3, 4.
     
   • If a string be divided into two parts by a bridge, in such a manner as to give two consonant sounds when struck, the lengths of those parts will be in the ratio of two of the first four positive integers.
     
   • If the bridge be so placed that two thirds of the string lie to the right and one third to the left, so that the two lengths are in the ratio of 1 : 2, they produce the interval of the octave, the greater length being given to the deeper note.
     
   • If the bridge be so placed that three fifths of the string lie to the right and two fifths to the left, the ratio of the two lengths is 2 : 3 and the interval produced is the fifth.
     
   • Thus corresponding to the successively higher notes c, f, g and c we have the numbers 1 , 3/4 , 2/3 , and 1/2 for the relative lengths of the strings corresponding to the different notes.
     
   • Nearly two thousand years passed before Galileo went one step further, and proved that the lengths of strings of the same size and tension were in the inverse ratios of the numbers of the vibrations of the tones they produced.
     
   • The next experiment was to divide the length that produced the fourth of the prime into two equal parts, when the sound, the octave of the fourth, was established.
     

6. St Andrews Physics Examinations
   
   • How can the focal length of a convex lens be determined by experiment? How can the focal length of a concave lens be determined by experiment? .
     
   • What is meant by Ohm's law? If a given battery of internal resistance one ohm sends a certain current through a wire of eight ohms' resistance, what current will the same battery send through a wire of the same material of one half the diameter and three times the length? .
     
   • How may the length of a simple pendulum and its time of vibration be determined by observations made with a compound pendulum? (Kater's method.) .
     
   • The length of a seconds pendulum at a certain place is 39.15 inches: find the local value of g.
     
   • An open and a stopped organ-pipe are of equal lengths what is the difference in pitch between the notes they produce? .
     
   • A tuning-fork makes 256 vibrations per second, and the velocity of sound is 340 metres per second: what is the wave length of the note produced? .
     
   • Compare the currents which the same electromotive force is capable of producing in two wires of the same material whose lengths are as 5 to 1, and cross sections as 3 to 2.
     

7. Carol R Karp: 'Languages with expressions of infinite length
   
   • Carol R Karp: Languages with expressions of infinite length .
     
   • Carol Ruth Karp published Languages with expressions of infinite length in 1964.
     
   • Techniques for proving completeness theorems in logic and representation theorems for Boolean algebras combined to yield a completeness theorem: Valid formulas of denumerable length in which only finitely many variables can be quantified at a time are provable in a system very much like the ordinary first-order predicate calculus.
     
   • The central problem could now be formulated as, "For which cardinals a, b do there exist definable complete formal systems for formulas of length less than a in which fewer than b variables can be quantified at a time?", a question that is almost completely answered in this monograph.
     

8. Eddington on the Expanding Universe
   
   • Length is necessarily relative.
     
   • I am going to refer to another much more elementary relativity of length, viz.
     
   • that length always implies comparison with a standard of length.
     
   • It is only the ratio of lengths that enters into our experience.
     
   • Suppose that every length and every distance in the universe were suddenly to be doubled; nothing would seem altered.
     
   • Intrinsically Brobdingnag and Lilliput are precisely the same; it needs an intruding Gulliver - an extraneous standard of length - to make them appear different.
     
   • It was Professor Weyl who first called attention to the very big hiatus involved, when we speak of a length such as the radius of a hydrogen atom being a certain fraction of the standard metre.
     

9. Eddington on the Expanding Universe
   

10. James Jeans: 'Physics and Philosophy' I
    
    • A physicist may announce, for instance, that the density of gold is 19.32, by which he means that the ratio of the weight of any piece of gold to that of a volume of water of equal size is 19.32; or that the wave-length of the line Ha in the spectrum of atomic hydrogen is 0.000065628 centimetre, by which he means that the ratio of the length of a wave of Ha light to that of a centimetre is 0.000065628, a centimetre being defined as a certain fraction of the diameter of the earth, or of the length of a specified bar of platinum, or as a certain multiple of the wave-length of a line in the spectrum of cadmium.
      
    • Our minds can never step out of their prison-houses to investigate the real nature of the things - gold, water, atomic hydrogen, centimetres or wave-lengths - which inhabit that mysterious world out beyond our sense-organs.
      
    • The wave-lengths of these lines can be measured, and are found to be related with one another in a very simple way which can be expressed by a quite simple mathematical formula.
      

11. Muir on research in Scotland
    
    • So long as there are different nationalities, and especially nationalities with a lengthened past history, so long must their institutions, including their Universities, be cast in different moulds.
      
    • The policy of having such a lengthened holiday has been more than once called in question.
      
    • The masters are men with a lengthened educational training, much of it in some cases devoted to a particular department of knowledge, and each of them is engaged day by day teaching his own special subject, and that subject alone.
      
    • The bulk of the University students came for a lengthened period from the parochial schools, and they it was who practically fixed the University standard.
      
    • School work and University lecturing, apart from inequality in the length of time consumed by them, are two very different things.
      
    • It is unnecessary, however, to speak at any length of what societies can accomplish in this way, or of the means they employ in attaining their ends; - these are things of everyday knowledge.
      

12. Muir on research in Scotland
    
    • So long as there are different nationalities, and especially nationalities with a lengthened past history, so long must their institutions, including their Universities, be cast in different moulds.
      
    • The policy of having such a lengthened holiday has been more than once called in question.
      
    • The masters are men with a lengthened educational training, much of it in some cases devoted to a particular department of knowledge, and each of them is engaged day by day teaching his own special subject, and that subject alone.
      
    • The bulk of the University students came for a lengthened period from the parochial schools, and they it was who practically fixed the University standard.
      
    • School work and University lecturing, apart from inequality in the length of time consumed by them, are two very different things.
      
    • It is unnecessary, however, to speak at any length of what societies can accomplish in this way, or of the means they employ in attaining their ends; - these are things of everyday knowledge.
      

13. Eulogy to Euler by Fuss
    
    • Euler proceeded from the principle that the perception of any perfection brings forth the feeling of pleasure, and since order is one of the perfections which provides the soul with pleasant feelings, then all the pleasure that a beautiful music allows us to enjoy consists of the perception of the relationships that sounds have between themselves and relative to the length of time between the notes as in relation to the frequency of the vibrations in the air which produces them.
      
    • Naval architecture which by the omission of informed principles had been obliged and at the mercy to routine developments and even a lengthy experience could not prevent the many mistakes in construction of ship and in their masting, saw themselves suddenly transformed by a complete theory, which other arts did not have the advantage of receiving only after successive false starts and by minor almost negligible adjustments.
      
    • The excessive length that was necessary to provide for glasses prior to the discovery of composed objective and the confusion of images had obliged the astronomers to abandon these entirely telescopes and to limit themselves to the use of reflexive telescopes.
      
    • Euler has known to utilize this analysis to reconcile all of the possible advantages for all types of instruments; those concerning image clarity, the greatest field of vision, the shortest possible length concerning all of the magnifications and for the number of eyecups that are to be used.
      
    • Instead of being stopped in his tracks as in the past due to the inability to integrate the three differential equations of the second degree that the mechanical principles furnish, he regrouped them at first into the three coordinates which determine the location of the moon, he then distributed all the inequalities of the moon into separate classes based on whether they depended either on the mean length of the sun from the moon or concerning the eccentricity or the parallax or the inclination of the lunar orbit.
      
    • Within this work is found the expression of the ways in which to make these instruments of shorter length with a greater field of vision, advantages which were impossible to provide for in the glasses prior to the last adjustments to the requisite calculations.
      

14. Eulogy to Euler by Fuss
    
    • Euler proceeded from the principle that the perception of any perfection brings forth the feeling of pleasure, and since order is one of the perfections which provides the soul with pleasant feelings, then all the pleasure that a beautiful music allows us to enjoy consists of the perception of the relationships that sounds have between themselves and relative to the length of time between the notes as in relation to the frequency of the vibrations in the air which produces them.
      
    • Naval architecture which by the omission of informed principles had been obliged and at the mercy to routine developments and even a lengthy experience could not prevent the many mistakes in construction of ship and in their masting, saw themselves suddenly transformed by a complete theory, which other arts did not have the advantage of receiving only after successive false starts and by minor almost negligible adjustments.
      
    • The excessive length that was necessary to provide for glasses prior to the discovery of composed objective and the confusion of images had obliged the astronomers to abandon these entirely telescopes and to limit themselves to the use of reflexive telescopes.
      
    • Euler has known to utilize this analysis to reconcile all of the possible advantages for all types of instruments; those concerning image clarity, the greatest field of vision, the shortest possible length concerning all of the magnifications and for the number of eyecups that are to be used.
      
    • Instead of being stopped in his tracks as in the past due to the inability to integrate the three differential equations of the second degree that the mechanical principles furnish, he regrouped them at first into the three coordinates which determine the location of the moon, he then distributed all the inequalities of the moon into separate classes based on whether they depended either on the mean length of the sun from the moon or concerning the eccentricity or the parallax or the inclination of the lunar orbit.
      
    • Within this work is found the expression of the ways in which to make these instruments of shorter length with a greater field of vision, advantages which were impossible to provide for in the glasses prior to the last adjustments to the requisite calculations.
      

15. Eddington: 'Mathematical Theory of Relativity' Introduction
    
    • The vocabulary of the physicist comprises a number of words such as length, angle, velocity, force, work, potential, current, etc., which we shall call briefly "physical quantities." Some of these terms occur in pure mathematics also; in that subject they may have a generalised meaning which does not concern us here.
      
    • Consider, for example, a length or distance between two points.
      
    • Or again, instead of cutting short the astronomical calculations when we reach the parallax, we might go on to take the cube of the result, and so obtain another manufactured quantity, a "cubic parallax." For some obscure reason we expect to see distance appearing plainly as a gulf in the true world-picture; parallax does not appear directly, though it can be exhibited as an angle by a comparatively simple construction; and cubic parallax is not in the picture at all The physicist would say that he finds a length, and manufactures a cubic parallax; but it is only because he has inherited a preconceived theory of the world that he makes the distinction.
      
    • If the length AB is double the length CD, the parallax of B from A is half the parallax of D from C; there is undoubtedly some world-relation which is different for AB and CD, but there is no reason to regard the world-relation of AB as being better represented by double than by half the world-relation of CD.
      
    • We do not need to ask the physicist what conception he attaches to "length"; we watch him measuring length, and frame our definition according to the operations he performs.
      
    • But to catalogue all the precautions and provisos in the operation of determining even so simple a thing as length, is a task which we shirk.
      
    • I should be puzzled to say off-hand what is the series of operations and calculations involved in measuring a length of 10-15 cm; nevertheless I shall refer to such a length when necessary as though it were a quantity of which the definition is obvious.
      
    • Indeed it has been suspected that the perplexities of quantum phenomena may arise from the tacit assumption that the notions of length and duration acquired primarily from experiences in which the average effects of large numbers of quanta are involved, are applicable in the study of individual quanta.
      

16. Menger on the Calculus of Variations
    
    • History does not describe the form of the territory she chose, but if she was a good mathematician she covered the territory in the form of a circle, for today we know: Of all surfaces bounded by curves of a given length, the circle is the one of largest area.
      
    • In the first example (that of Queen Dido) the family consists of all closed curves with a given length, and the associated number is the area of the enclosed surface; in the second example (that of Newton) the number is the resistance which a body somehow associated with the curve meets in the air; in the third example (that of the brothers Bernoulli) the family of curves consists of all curves joining two given points, and the number associated with each curve is the time it takes a body to fall along this curve.
      
    • For example, we consider the two following extremely simple problems: two given points may be joined by all possible curves; which of them has the shortest length, and which of them has the greatest length? The first problem is soluble: The straight line segment joining the two points is the shortest line joining them.
      
    • The length is a number associated with each curve which for no curve assumes a finite maximum.
      
    • The Italian mathematician Tonelli found out twenty years ago that the deeper reason for the solubility of the minimum problem concerning the length, that is, for the existence of a shortest line between every two points, is the following property of the length: A curve between two fixed points being given, there are always other curves as near as you please to it, and yet much longer than the given curve (e.g., some zigzag lines near the given curve).
      
    • This property of the length is called the semi-continuity of the length.
      

17. Mathematicians and Music 2.2
    
    • This work begins by laying down at length the general rules and principles of the art, and then goes on to treat of ancient music in all its forms; of music as Cardano knew and enjoyed it; of the system of counterpoint and composition, and of the construction of musical instruments.
      
    • In the early part of the third period in the development of music, namely, the period of Harmonic or Modern Music, we have the first opera and the first oratorio, and, as I have already said, the discovery by Galileo that the simple ratios of the lengths of strings existed also for the pitch numbers of the tones they produced, an observation later generalized by Newton.
      
    • when a string is plucked or struck, or, as we may add 'bowed' at any point in its length which is the node of any of its so-called harmonics those simple vibrational forms of the string which have a node in that point are not contained in the compound vibrational form.
      
    • Because of this law piano makers eliminate certain undesirable upper partials by striking the middle strings of their instruments at a point 1/7 to 1/9 of their lengths from their extremities.
      

18. Poincaré on the future of mathematics
    
    • It is for the same reason that, when a somewhat lengthy calculation has conducted us to some simple and striking result, we are not satisfied until we have shown that we might have foreseen, if not the whole result, at least its most characteristic features.
      
    • Why is this? What is it that prevents our being contented with a calculation which has taught us apparently all that we wished to know? The reason is that, in analogous cases, the lengthy calculation might not be able to be used again, while this is not true of the reasoning, often semi-intuitive, which might have enabled us to foresee the result.
      
    • Only, is it always necessary to state it so many times? Those who were the first to pay special attention to exactness have given us reasonings that we may attempt to imitate; but if the demonstrations of the future are to be constructed on this model, mathematical works will become exceedingly long, and if I dread length, it is not only because I am afraid of the congestion of our libraries, but because I fear that as they grow in length our demonstrations will lose that appearance of harmony which plays such a useful part, as I have just explained.
      

19. H L F Helmholtz: 'Theory of Music' Introduction
    
    • 540-510) knew that when strings of different lengths but of the same make, and subjected to the same tension, were used to give the perfect consonances of the Octave, Fifth, or Fourth, their lengths must be in the ratios of 1 to 2, 2 to 3, or 3 to 4 respectively, and if, as is probable, his knowledge was partly derived from the Egyptian priests, it is impossible to conjecture in what remote antiquity this law was first known.
      
    • Later physics has extended the law of Pythagoras by passing from the lengths of strings to the number of vibrations, and thus making it applicable to the tones of all musical instruments, and the numerical relations 4 to 5 and 5 to 6 have been added to the above for the less perfect consonances of the major and minor Thirds, but I am not aware that any real step was ever made towards answering the question: What have musical consonances to do with the ratios of the first six numbers? Musicians, as well as philosophers and physicists, have generally contented themselves with saying in effect that human minds were in some unknown manner so constituted as to discover the numerical relations of musical vibrations, and to have a peculiar pleasure in contemplating, simple ratios which are readily comprehensible.
      
    • Hitherto it is the physical part of the theory of sound that has been almost exclusively treated at length, that is, the investigations refer exclusively to the motions produced by solid, liquid, or gaseous bodies when they occasion the sounds which the ear appreciates.
      

20. Born Inaugural
    
    • Now it is evident and trivial that not every grammatically correct question is reasonable; take, for instance, the well-known conundrum: Given the length, beam, and horse-power of a steamer, how old is the captain? - or the remark of a listener to a popular astronomical lecture: "I think I grasp everything, how to measure the distances of the stars and so on, but how did they find out that the name of this star is Sirius?" Primitive people are convinced that knowing the "correct" name of a thing is real knowledge, giving mystical power over it, and there are many instances of the survival of such word-fetishism in our modern world.
      
    • Exactly the same kind of pattern can be observed when two beams of light cross one another, the only difference being that you need a magnifying-lens to see them; the inference is that a beam of light is a train of waves of short wave-length.
      
    • Experiment then shows that the corresponding train of waves has the simplest form possible, which is called harmonic, and is characterised by a definite sharp frequency and wave-length.
      
    • A train of waves is by definition harmonic only if it fills the whole of space and lasts from eternity to eternity! [The latter point may not appear so evident; but a mathematical analysis made by Fourier more than a hundred years ago has clearly shown that every train of waves finite in space and time has to be considered as a superposition of many infinite harmonic waves of different frequencies and wave-lengths which are arranged in such a way that the outer parts destroy one another by interference; and it can be shown that every finite wave can be decomposed into its harmonic components.] Bohr has emphasised this point by saying that Planck's principle introduces an irrational feature into the description of nature.
      

21. Edinburgh Mathematics Examinations
    
    • The two sides of a right angled triangle being taken as axes, find the equations to the sides of the square described on the hypotenuse in terms of the lengths a and b of the two sides.
      
    • Hence calculate the length of the sub-normal in terms of the abscissa from the centre, the latus rectum, and the eccentricity.
      
    • An elliptic plot is described in a garden by means of a string 20 feet in length and passing round two pegs distant by 5 feet.
      
    • A smooth cylindric glass rod, the radius of which is small compared to the length, is placed in a smooth hemispherical mortar, one end projecting; find the position of rest.
      

22. Charles Bossut on Leibniz and Newton
    
    • Sluze and Gregory had each separately found a method for tangents; and Newton, in a letter to Collins dated December the 10th, 1672, proves that he had likewise found one; he applies it to an example without adding the demonstration; and he afterwards says that it is only a corollary of another general method which he has for drawing tangents, squaring curves, finding their lengths and centres of gravity, etc., without being stopped by the radical quantities, as Hudde was in his method for maxima and minima.
      
    • Mr de Fontenelle, who however meant me well, was wrong when he contented himself with saying at the beginning of his Geometrie de l'Infini, that, after having at first admitted infinitely small quantities, I had at length receded so far as to reduce the infinities of different orders to mere incomparables in the sense in which a grain of sand would be incomparable to the globe of the Earth.
      
    • He has the advantage over Newton of having invented and carried to a great length the integral calculus of differential equations.
      
    • To conclude, whatever length of time the completion of the Principia may have required, we ought not to forget that this work did not appear till two or three years after Leibniz had published his differential calculus, and some sketches of the integral.
      

23. W H Young addresses ICM 1928 Part 2
    
    • In so far as it appeals to Mathematics, this theory borrows a number of its mathematical tools from Physical Optics, and, in particular, seems to have adopted the mode of characterising a monochromatic light-radiation by its Wave-length.
      
    • The assumption of the wave- length as the characteristic of a monochromatic light, does not seem to have availed much towards the solution of the problem.
      
    • Thus, for instance, in the natural enquiry into the mutual relation of Complementary Colours, the attempt to discover a relation between their Wave Lengths might with fair probability have been condemned on a priori grounds: the unconcerned retention of the wave length as a working variable, after it had served only as a characterising Order-Number, and the ignoration of the second variable in the case, seem sufficiently crave errors to warrant any failure.
      

24. University of Edinburgh Examinations
    
    • The two sides of a right angled triangle being taken as axes, find the equations to the sides of the square described on the hypotenuse in terms of the lengths a and b of the two sides.
      
    • Hence calculate the length of the sub-normal in terms of the abscissa from the centre, the latus rectum, and the eccentricity.
      
    • An elliptic plot is described in a garden by means of a string 20 feet in length and passing round two pegs distant by 5 feet.
      
    • A smooth cylindric glass rod, the radius of which is small compared to the length, is placed in a smooth hemispherical mortar, one end projecting; find the position of rest.
      

25. Sommerfeld: 'Atomic Structure
    
    • The electric waves produced by Hertz had a wave-length of several metres.
      
    • From them an almost unbroken chain of phenomena leads by way of heat rays and infra-red rays to the true light rays, whose wave-lengths amount to only fractions of ??.
      
    • The greatest link in this chain came later as a direct result of Hertz's experiments, namely, the waves of wireless telegraphy, whose wave-length have to be reckoned in kilometres.
      
    • (Nauen {A German radio transmitter] sends out waves having a wave-length of 12 kilometres, or 71/2 miles); the smallest and most delicate link is added at the other end of the chain, as we shall see, in the form of Rontgen rays, and the still shorter ??-rays which are of a similar nature; likewise the ultra- ?? - or cosmic radiation.
      

26. Studies presented to Richard von Mises' Introduction
    
    • In drawing such a picture, the central task is to understand the relation between the direct sense observation of the experimental physicist and the conceptual system of science, which consists of expressions such as "increase of entropy" or "principle of relativity." Most physicists are inclined to say that the picture drawn and the principles devised by our inductive ability are eventually checked by actual measurement of physical quantities like length, weight, electric charge, etc., but they use the expression "measurement of a length" in a perfunctory way, forgetting that no numerical value can ever be assigned to a length by a single measurement.
      
    • In fact, a long series of measurements is needed from which eventually "the value of the length" can be computed.
      
    • In contrast to the procedure of the physicist, applied mathematics concentrates its efforts on the problem: how can "values of length" be computed from sets of different readings? And, in a general way, it has become the business of applied mathematics to investigate the connection between "direct pointer readings" and the abstract conceptions (as length, or electromagnetic field) that occur in all laws of science - in Newton's mechanics as well as in Maxwell's theory of the electromagnetic field.
      

27. Aitken: 'Statistical Mathematics
    
    • The technique of collecting data and the principles to be heeded in order to avoid bias in the interpretation are described at length and exemplified in chapters of more extensive treatises which the reader may consult.
      
    • Now the question of assigning a measure to such aggregates has been deeply studied in modern pure mathematics, the guiding idea being that of extending as widely as possible the scope of a concept familiar in simple cases, namely the cardinal number of a finite set of objects, the length of a line, the area of a surface, the volume of a solid.
      
    • For example, if the aggregate were of points on a continuous line segment, and the measure were ordinary length, then we have implied in this description that all points in the segment are equally likely.
      
    • For example, given a circle, let a chord be drawn across it at random: what is the probability that the length of the chord exceeds half the diameter? It depends entirely on the manner in which the chord is drawn.
      
    • This prologue, though it has omitted many subtler points which could be amplified at very great length, must now be cut short.
      

28. Mathematicians and Music 3
    
    • To find the number of vibrations that a string will make in a certain time having given its length, its weight, and the weight that stretches it.
      
    • where the origin of coordinates was at the end of the chord whose length is l, the axis of x in the direction of the chord, and y the displacement at any time t.
      
    • From these laws we learn the nature of consonance and dissonance, knowledge so necessary for building up a system of harmony; we learn the principles which determined those degrees of musical sound selected by various nations at various times; we understand the reasons for the simple ratios of the lengths of strings producing consonant tones and the limitation of the numbers of these ratios; and we appreciate the value of temperaments for different instruments.
      

29. Al-Biruni: 'Coordinates of Cities
    
    • However, he found that its actual length [i.e.
      
    • the stade's] was not sufficiently known to the translators to enable them to identify it with local standards of length.
      
    • While on their way back, they verified, by a second survey, their former estimates of the lengths of the courses they had followed, until both parties met at the place whence they had departed.
      

30. Poincaré on non-Euclidean geometry
    
    • Imagine this figure to be traced on a flexible and inextensible canvas applied to the surface, in such a way that when the canvas is displaced and deformed the different lines of the figure change their form without changing their length.
      
    • If we resume the comparison that we made just now, and imagine beings without thickness living on one of these surfaces, they will regard as possible the motion of a figure all the lines of which remain of a constant length.
      
    • All depends, he says, on the manner in which the length of a curve is defined.
      
    • Now, there is an infinite number of ways of defining this length, and each of them may be the starting-point of a new geometry.
      

31. Euler Elogium.html.html
    
    • By reading this last work, one is no less astonished to see the lengths to which a great man of genius, animated exclusively by the desire to leave nothing to chance concerning important issues, can push the limits of patience and the obstinacy to work.
      
    • Magnetic theory, the propagation of fire, the laws of body adhesion and that of friction provided the opportunity for lengthy calculations applied to hypothesis which unfortunately should have been based on experimentation.
      

32. Gibson History 4 - John Napier
    
    • Suppose one point P to set out from the point A and to move along the line AX (of unlimited length) with a uniform velocity V; then suppose a second point Q to set out from B on the line BY, of fixed length r, at the same time as P sets out from A, starting with the velocity V and moving not uniformly but so that its velocity at any point, as D, is proportional to the distance DY from D to the end Y of the line BY.
      
    • In Napier's terminology r, the length of BY, is the whole sine; when Q is at B, P is at A so that the logarithm of the whole sine is 0.
      

33. Chrystal.html
    
    • The men who did not know when they were beaten returned to their seats, and doggedly took notes, their faces lengthening daily.
      
    • Some compressed their lips, others were as lively as fireworks dipped in water; there were those who rushed round and round the quadrangle; only one went to the length of saying that he did not want to pass.
      

34. Edinburgh Physics Examinations
    
    • Upon what physical fact does the possibility of constructing a Compensation Pendulum depend? Show the application to a measuring rod, whose length is to be independent of temperature changes.
      
    • If the length of the ladder be twice the height of the rail above the ground, what is the coefficient of friction if the ladder is just about to slide down when inclined 45 degrees to the vertical? .
      
    • Find the length of the sidereal second in terms of the mean solar second.
      

35. The Tercentenary of the birth of James Gregory
    
    • Reflecting telescopes contain both mirrors and lenses, so combined as greatly to reduce the length of the tube, in contrast to the usual refractive pattern which consisted of lenses alone.
      
    • Also, to gain magnifying power it was customary in those days to lengthen the tubes to hundreds of feet.
      

36. Professor Chrystal
    
    • The men who did not know when they were beaten returned to their seats, and doggedly took notes, their faces lengthening daily.
      
    • Some compressed their lips, others were as lively as fireworks dipped in water; there were those who rushed round and round the quadrangle; only one went to the length of saying that he did not want to pass.
      

37. Kelvin on the sun
    
    • The rate of shrinkage corresponding to the present rate of solar radiation has been proved to us, by the consideration of our dynamical model, to be 35 metres on the radius per year, or one ten-thousandth of its own length on the radius per two thousand years.
      
    • Forty years later Langley, in an excellently worked out consideration of the whole question of absorption by our atmosphere, of radiant heat of all wave-lengths, accepts and confirms Forbes's reasoning, and by fresh observations in very favourable circumstances on Mount Whitney, 15,000 feet above the sea-level, finds a number a little greater still than Forbes (1.7, instead of Forbes' 1.6, times Pouillet's number).
      

38. Centenary of John Leslie
    
    • With this seat of honour, forecast of a professorial chair, he was reasonably pleased, but be was at length superseded by an even younger pupil.
      
    • He could indulge in unwarranted applications of mathematics, so as to find an analogy between circulating decimals and the lengthened cycles of the seasons; but we may, with the accumulated knowledge of the present year, smile, though not unkindly, at his speculation that the earth was a thin crust filled with light of an overpowering splendour.
      

39. P G Tait's obituary of Listing
    
    • What a pregnant comment on the conduct of those "British geologists" who, not many years ago, treated with outspoken contempt Thomson's thermodynamic investigations into the admissible lengths of geological periods! .
      
    • From this follow the properties of a large class of knots which form "clear coils." A special example of these, given by Listing for threads, is the well-known juggler's trick of slitting a ring-formed band up the middle, through its whole length, so that instead of separating into two parts, it remains in a continuous ring.
      

40. James Jeans addresses the British Association in 1934
    
    • Typical of its knowledge is the statement that the line Ha in the hydrogen spectrum has a wave-length of so many centimetres.
      
    • The moment we are told that it is a certain fraction of the earth's radius, or of the length of a bar of platinum, or a certain multiple of the wave-length of a line in the cadmium spectrum, our knowledge becomes real, but at that same moment it also becomes purely numerical.
      

41. ELOGIUM OF EULER
    
    • By reading this last work, one is no less astonished to see the lengths to which a great man of genius, animated exclusively by the desire to leave nothing to chance concerning important issues, can push the limits of patience and the obstinacy to work.
      
    • Magnetic theory, the propagation of fire, the laws of body adhesion and that of friction provided the opportunity for lengthy calculations applied to hypothesis which unfortunately should have been based on experimentation.
      

42. Taylor versus Continental mathematicians
    
    • without engaging in the lengthy calculations of my brother or in the obscurity of that of Mr Taylor.
      
    • Mr Taylor - a man of acuteness, and a very skilful geometer, who has successfully penetrated even our most profound discoveries, as it would appear from his book on "Methodus Incrementorum" - well aware of the tedious length of my brother's analysis, and wishing to render it shorter and a little more clear, has himself spread such obscurity over this matter (as well as other in which he has wished to be brief) that he seems to take pleasure in it, and I doubt there be anyone, no matter how penetrating he might be, that would understand all of it, even did he already know that matter in another way.
      

43. Edmund Landau: 'Foundations of Analysis' Prefaces
    
    • I will refrain from speaking at length about the fact that often even Dedekind's fundamental theorem (or the equivalent theorem in the development of the real numbers by means of fundamental sequences) is not included in the basic material; so that such matters as the mean-value theorem of the differential calculus, the corollary of the mean-value theorem to the effect that a function having a zero derivative in some interval is constant in that interval, or, say, the theorem that a monotonically decreasing bounded sequence of numbers converges to a limit, are given without any proof or, worse yet, with a supposed proof which in reality is no proof at all.
      
    • To make it as easy as possible for the reader I have repeated in several chapters, or sometimes in all, certain (not very lengthy) phrases.
      

44. L R Ford: Monthly Editor
    
    • They develop at length results that might be considered as mathematical exercises.
      
    • Many a paper composed in a lengthy, prolix, and awkward prose might have been saved by a sprightly and fluent style.
      

45. Grassmann: 1844 foreword
    
    • In the latter case AB and BC are not interpreted merely as lengths, but rather their directions are simultaneously retained as well, according to which they are precisely oppositely oriented.
      
    • Thus the distinction was drawn between the sum of lengths and the sum of such displacements in which the directions were taken into account.
      
    • While I was pursuing the concept of product in geometry as it had been established by my father, I concluded that not only rectangles but also parallelograms in general may be regarded as products of an adjacent pair of their sides, provided one again interprets the product, not as the product of their lengths, but as that of the two displacements with their directions taken into account.
      

46. Felix Klein on intuition
    
    • In imagining a line, we do not picture a length without breadth, but a strip of a certain width.
      
    • It seems to me, therefore, that Kirchhoff makes a mistake when he says in his Spectral Analyse that absorption takes place only when there is an exact coincidence between the wave-lengths.
      

47. W H Young addresses ICM 1928
    
    • I do not propose to speak at length of the Methods of Mathematics itself, and it is unfortunate that a typographical error in the early edition of the Programme might give this impression.
      
    • What is the nature of the problems confronting the human race for which a method is required? The problems which dire necessity sets before us are not solved by lengthy reflection; for unforeseen accidents we require unforeseen expedients, foresight for apprehended dangers.
      

48. 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.
      

49. De Montmort: 'Essai d'Analyse
    
    • This, the first edition, often passed over because of the greater length of the second edition and the incorporation of the Montmort-Bernoulli correspondence, is worthy of comment.
      
    • The letter, of extreme length, is that of a very angry man.
      

50. Max Planck: 'Quantum Theory
    
    • Later, a universal function was proved to exist, which depended only on temperature and wave-length, and was in no way related to the properties peculiar to any substance.
      
    • This exceedingly simple relation is a complete and adequate expression of Wien's law of distribution of energy; for the dependence upon wave-length is always given immediately as well as the dependence upon energy by Wien's generally accepted law of displacements.
      

51. Mathematics in Edinburgh
    
    • Properties of Matter, Abstract Dynamics (commonly called Mechanics), and Conservation of Energy, together with two or three others of the above divisions of the subject, are treated in detail every Session, the remainder being necessarily discussed in a more superficial manner, as it is impossible to enter at length into all in the course of a single Session.
      
    • A practical class, for the instruction of beginners in the elementary processes of measuring Time, Mass, Length, Angle, Force, etc., will be formed in the Winter and also in the Summer Sessions.
      

52. L'Hôpital: 'Analyse des infiniment petits' Preface
    
    • He found the lengths of some of them, the areas they enclosed, the volumes swept out by these areas, the centres of gravity of these areas and volumes etc etc.
      
    • But M Leibniz wrote to me to say that he himself was engaged upon describing the integral calculus in a treatise he calls De Scientia infiniti, and I did not wish to deprive the public of such a work, which will deal with all the most interesting consequences of this inverse method of tangents, showing how it can be used to find the lengths of curves, to find the area they enclose, to find the volumes and surfaces of their solids of revolution, to find centres of gravity etc.
      

53. Association 1904 Part 2.html
    

54. Henry Baker addresses the British Association in 1913
    
    • Are we sure that human nature is the only continuous variable in the concrete world, assuming it be continuous, which can possess such a vacillating character? Or I may refer to the more elementary fact that all the rational fractions, infinite in number, which lie in any given range, can be enclosed in intervals whose aggregate length is arbitrarily small.
      
    • At length there came a step which to many probably will still seem unintelligible.
      

55. Horace Lamb addresses the British Association in 1904, Part 2
    
    • Everyone will grant, however, that the distance between two clouds, for instance, is not a definable magnitude; and the distance of the earth from the sun, and even the length of a wave of light, are in precisely the same case.
      
    • Most of us have, however, been forced at length to acquiesce in the view that Geometry, like Mechanics, is an applied science that it gives us merely an ingenious and convenient symbolic representation of the relations of actual bodies; and that, whatever may be the a priori forms of intuition, the science as we have it could never have been developed except for the accident (if I may so term it) that we live in a world in which rigid or approximately rigid bodies are conspicuous objects.
      

56. Rota's lecture on 'Mathematical Snapshots
    
    • If P is a parallelotope with orthogonal sides of lengths x1 , x2 , x3 , then v(P) = x1x2x3.
      
    • When multiplied by 4, it equals the perimeter of the parallelotope P, that is, the sum of the lengths of all the edges of the parallelotope P.
      

57. Planetary motion tackled kinematically
    
    • Before that breakthrough, planetary motion involved merely a path (a curve), together with a measure of time, represented geometrically: that is, a strictly kinematical treatment - which by definition involves the dimensions of length and time alone, while excluding altogether the dimension of mass.
      
    • This length AF is subsequently recognized as 'the mean distance', which is of great significance in Part III below.
      

58. 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.
      

59. EMS obituary
    
    • Owing to a lengthy illness which forced him some years ago to retire from his post as Lecturer in the University of Edinburgh, he was not known personally to the students of the last decade.
      
    • There was no alternative to Mathematics then, and what Arts student of those early days does not remember the terrors of the long grim galley slip of examination paper, lengthy as the student's walking stick, then the newly acquired sceptre of academic dignity.
      

60. The St Andrews Schmidt-Cassegrain Telescope
    
    • In order to achieve the first of these aims it was necessary to increase the length of the instrument and this resulted eventually in the construction of telescopes several hundred feet long; the mechanical difficulties associated with the mounting of such instruments need hardly be stressed.
      
    • This arrangement is preferable to the Gregorian system since the combination of a convex and a concave mirror reduces the overall spherical aberration and at the same time reduces the length of the tube.
      

61. Kepler's Planetary Laws
    
    • Tycho developed, refined and cross-checked his instruments and sometimes attained an accuracy of 2' (which is approximately the breadth of a hair held at arm's length).
      
    • (That watershed book also contained a sophisticated treatment of tangential velocity in orbit, as well as formulating a concept of acceleration to accompany the concept of attractive force.) Thus it is clear that Kepler could not have been aware of the modern implication of (the dimension of) mass in the solar system, though his interpretation of an orbit certainly involved the dimensions both of length and of time, as we have demonstrated.
      

62. D'Arcy Thompson on Greek irrationals
    
    • We have set forth this table at greater length than the others because it so happens that Archimedes makes use of certain of its higher convergents.
      
    • Selecting any convenient fraction from the same table, for instance 89/144 we may then define the length of the side of the inscribed pentagon (still in terms of the radius) as .
      

63. George Chrystal's Second Promoter's Address
    
    • Regarding the general principle of that ordinance it would hardly be profitable to speak at length, as it has been tacitly agreed on all hands to give it a trial.
      

64. Sommerville obituary.html
    
    • Approximations to the length of an Arc, 37, 76-79 (1918).
      

65. R L Wilder: 'Cultural Basis of Mathematics I
    
    • In his famous work Der Untergang des Abendlandes [15], 0 Spengler discussed at considerable length the nature of mathematics and its importance in his organic theory of cultures.
      

66. J L Synge: 'Geometrical Optics
    
    • It is possible to justify geometrical optics as a limiting case of physical optics, the wave-length of the light in question tending to zero; [M Born, Optik (Berlin, 1933), 45] but we shall be content with the development of geometrical optics on the basis of its own hypotheses, just as it is customary to develop the dynamics of rigid bodies as a separate theory, and not as a limiting case of the dynamics of elastic bodies whose elastic moduli tend to infinity.
      

67. Three Sadleirian Professors
    
    • We shall deal at some length with this aspect of their work.
      

68. Gregory's Observatory
    
    • The College to which the letter refers is evidently the Library building whose length diverges some 9 or 10 degrees from true east and west.
      

69. Wolfgang Pauli and the Exclusion Principle
    
    • giving the lengths of the periods in the natural system of chemical elements, was zealously discussed in Munich, including the remark of the Swedish physicist, Rydberg, that these numbers are of the simple form 2n2 if n takes on all integer values.
      

70. Elemér Kiss: 'Mathematical Gems from the Bolyai Chests' Preface
    
    • " Although I fully feel the significance of FARKAS BOLYAI's wise words and have considered them at length, I decided, after much pondering and hesitation to try to become a "writer".
      

71. Cassini and the Division in Saturn's Ring
    
    • After the emergence of Saturn from the rays of the Sun as a morning star in the year 1675, the globe of the planet appeared with a dark band, similar to those of Jupiter, extending the length of the ring from East to West, as it is nearly always shown by the 34-foot telescope, and the breadth of the ring was divided by a dark line into two equal parts, of which the interior and nearer one to the globe was very bright, and the exterior part slightly dark.
      

72. Mathematics at Aberdeen 4
    
    • While the new examinations were generally resulting in higher standards, the test for the Gray Bursary was becoming excessively lengthy, eventually extending throughout the night.
      

73. Ford - Mathematics for Field Artillery
    
    • The total length of the French 155 mm.
      

74. Muir obituary.html
    
    • An appreciation of his work would require two separate articles of some length, one to do proper justice to his work for education in Scotland and in the Union of South Africa, the other to give a survey and adequate account of his 307 original papers, his textbooks and his history of determinants.
      

75. Von Neumann: 'The Mathematician' Part 2
    
    • Also, if the deductions are lengthy or complicated, there should be some simple general principle involved, which "'explains" the complications and detours, reduces the apparent arbitrariness to a few simple guiding motivations, etc.
      

76. EMS 1913 Colloquium
    
    • The problem was finally solved by the formula, ln = constant, where l is the name-length and n the hump-frequency.
      

77. Cochran: 'Sampling Techniques' Introduction
    
    • The definition of the population may present no problem, as when sampling a batch of electric light bulbs in order to estimate the average length of life of a bulb.
      

78. Kepler's Planetary Laws
    
    • Tycho developed, refined and cross-checked his instruments and sometimes attained an accuracy of 2' (which is approximately the breadth of a hair held at arm's length).
      

79. Simplicius on astronomy and physics
    
    • However, it consists almost entirely of writings of Geminus on the scope of astronomy contrasted with that of physics which Simplicius quotes at length:- .
      

80. Archimedes on mechanical and geometric methods
    
    • If from [one magnitude another magnitude be subtracted which has not the same centre of gravity, the centre of gravity of the remainder is found by] producing [the straight line joining the centres of gravity of the whole magnitude and of the subtracted part in the direction of the centre of gravity of the whole] and cutting off from it a length which has to the distance between the said centres of gravity the ratio which the weight of the subtracted magnitude has to the weight of the remainder.
      

81. Chrystal: 'Algebra' Preface
    
    • I suppose that the student has gone in this way the length of, say, the solution of problems by means of simple or perhaps even quadratic equations, and that he is more or less familiar with the construction of literal formulae, such, for example, as that for the amount of a sum of money during a given term at simple interest.
      

82. Kaplansky: 'Infinite abelian groups' Introduction
    
    • Furthermore, when this monograph was nearly complete, I had the privilege of reading an unpublished manuscript (of book length) on abelian groups which he prepared several years ago.
      

83. Collected Papers of Paul Ehrenfest' Preface
    
    • He shunned calculations of any length and numerical constants were often considered irrelevant.
      

84. Al-Kashi's letter
    
    • 724]) al-Kashi describes at length the accomplishments of UIugh Beg who, he says:- .
      

85. Christiaan Huygens' article on Saturn's Ring
    
    • But I had only the ordinary form of telescope, which measured five or six feet in length.
      

86. Coolidge: 'Origin of Polar Coordinates
    
    • Pascal used the same transformation to calculate the length of a parabolic are, a problem previously solved by Roberval, but his solution was not universally accepted as valid.
      

87. Fermat's Journal des Sçavans obituary
    
    • He gave a general method for finding the length of curved lines etc.
      

88. Harold Jeffreys on Logic and Scientific Inference
    
    • We do not say that it is the solution of the present difficulty, but a priori knowledge exists, and we shall have occasion later to consider instances of it at length.
      

89. Finlay Freundlich's Inaugural Address, Part 2
    
    • To prove in the first case that A'B'C' is congruent to ABC, one imagines the triangle A'B'C' moved until A' falls upon A; then it is turned until the line A'B' falls in line with AB; since the length A'B' equals AB, B' will fall upon B; and since the angle at A equals that at A', also the side A'C' will fall in line with AC and C' will fall upon C, so that obviously the two triangles can be brought to cover each other completely.
      

90. Christiaan Huygens' article on Saturn's Ring
    
    • But I had only the ordinary form of telescope, which measured five or six feet in length.
      

91. EMS obituary
    
    • On demobilisation in May 1919 he had to his credit a length of service in the first World War equalled by very few, and punctuated by the minimum amount of leave.
      

92. Schrödinger: 'Statistical Thermodynamics
    
    • The treatment of those topics which are to be found in every one of a hundred text-books is severely condensed; on the other hand, vital points which are usually passed over in all but the large monographs (such as Fowler's and Tolman's) are dealt with at greater length.
      

93. G H Hardy addresses the British Association in 1922, Part 2
    
    • There is no difficulty in specifying possible groups of any length we please.
      

94. Mathematics in Aberdeen.html
    

95. Mathematics in Aberdeen
    
    • The examination in the Integral Calculus will be confined to the following subjects:- " Integration, application to lengths and areas of curves, volumes of solids, and to questions of mean value; definition and chief properties of the Gamma functions." .
      

96. Preface of 'ElemŽr Kiss', Mathematical Gems from the Bolyai Chests
    
    • On the pages of his manuscripts, JçNOS BOLYAI quotes his father's opinion about writing as thus: "É As my father has it, there may be nowadays more writers than readers, and perhaps those deserve mentioning who, being able to read and write, do not in want of a considerable and weighty reason become a writer É " Although I fully feel the significance of FARKAS BOLYAI's wise words and have considered them at length, I decided, after much pondering and hesitation to try to become a "writer".
      

97. EMS obituary
    
    • An approximate formula for the length of an are of a suspended rope (ibid.
      

98. John Couch Adams' account of the discovery of Neptune
    
    • Now that the discovery of another planet has confirmed in the most brilliant manner the conclusions of analysis, and enabled us with certainty to refer these irregularities to their true cause, it is unnecessary for me to enter at length upon the reasons which led me to reject the various other hypotheses which had been formed to account for them.
      

99. Whittaker EMS Obituary.html
    
    • I once asked him if he could write for the Edinburgh Mathematical Notes a short obituary notice of a schoolmaster who had been an Edinburgh graduate and received a notice of a page or so in length by return of post.
      

100. Bolzano's publications
     
     • Contains reprints of the following papers by Bolzano: Considerations on some points in elementary geometry (1804), Contributions to a better founded exposition of mathematics (1810), The binomial theorem (1816), Pure analytical proof of the intermediate value theorem (1817), and The three problems of curve length, surface area and volume (1817).
       

101. R A Fisher: 'Statistical Methods' Introduction
     
     • The last possibility may be represented by a frequency curve; the values of the variate are set out along a horizontal axis, the fraction of the total population, within any limits of the variate, being represented by the area of the curve standing on the corresponding length of the axis.
       

102. Laplace: 'Méchanique Céleste
     
     • I shall adopt the decimal division of the right angle, and of the day, and shall refer the linear measures to the length of the metre, determined by the are of the terrestrial meridian comprised between Dunkirk and Barcelona.
       

103. Sommerville: 'Geometry of n dimensions
     
     • The crude ideas of shape, bulk, superficial extent, and length became analysed, refined, and made abstract, and led to the conception of geometrical figures.
       

104. G H Hardy addresses the British Association in 1922
     
     • There is no difficulty in specifying possible groups of any length we please.
       

105. Pólya on Fejér
     
     • "As to earning a living", he said, "a professor's salary is a necessary, but not sufficient, condition." Once he was very angry with a colleague who happened to be a topologist, and explaining the case at length he wound up be declaring "..
       

106. Gauss: 'Disquisitiones Arithmeticae
     
     • And when at length I was ready to present my work to the world, it was YOUR munificence alone which removed all the obstacles that threatened to delay its publication.
       

107. EMS 1914 Colloquium 3.html
     
     • He showed that the length of a straight line and the magnitude of an angle can be expressed in terms of a property which is unaltered by projection - e.g., cross-ratio.
       

108. Cassini and the Division in Saturn's Ring
     
     • After the emergence of Saturn from the rays of the Sun as a morning star in the year 1675, the globe of the planet appeared with a dark band, similar to those of Jupiter, extending the length of the ring from East to West, as it is nearly always shown by the 34-foot telescope, and the breadth of the ring was divided by a dark line into two equal parts, of which the interior and nearer one to the globe was very bright, and the exterior part slightly dark.
       

109. Thomas Bromwich: 'Infinite Series
     
     • The notion of uniform convergence usually presents difficulties to beginners; for this reason it has been explained at some length, and the definition has been illustrated by Osgood's graphical method.
       

110. Gibson History 5 - James Gregory
     
     • If h is taken as unity the integral gives the length of the are; but he extends the proposition to various cases in which h is a function of x and also to cases in which y is taken as the variable of integration.
       

111. Edmund Whittaker: 'Physics and Philosophy
     
     • On the other hand he treats of the fifth way at some length.
       

112. Euclid on elementary astronomy
     
     • Now it is clear that, if such a figure be cut through its centre length-wise and breadth-wise respectively, the segments respectively arising are dissimilar; it is also clear that, even if it be cut in oblique sections through the centre, the segments formed are dissimilar in that case also; but this does not appear to happen in the case of the universe.
       

113. John Couch Adams' account of the discovery of Neptune
     
     • Now that the discovery of another planet has confirmed in the most brilliant manner the conclusions of analysis, and enabled us with certainty to refer these irregularities to their true cause, it is unnecessary for me to enter at length upon the reasons which led me to reject the various other hypotheses which had been formed to account for them.
       

114. Flatland' Second Edition Preface
     
     • When we see a Line, we see something that is long and bright; brightness, as well as length, is necessary to the existence of a Line; if the brightness vanishes, the Line is extinguished.
       

115. Gillespie: 'Integration
     
     • The area of a rectangle is equal to the product of its length and breadth, and from this, by the methods of Euclid, the areas of figures bounded by straight lines can be determined.
       

116. Durell and Robson: 'Advanced Trigonometry
     
     • In all these subjects, it must be admitted, there are certain difficulties which the average student will never face, but which are all-important for the real mathematician; these include, for example, the purely arithmetical treatment of real number, limits, continuity, convergence, mean-value theorems, the analysis of area, length of a curve, etc.
       

117. Airy on Thales' eclipse
     
     • The historical account of the eclipse is that the Medes attacked the Lydians, and that a war continued several years, until at length, when the two armies were preparing for battle, the day suddenly became night (an event which Thales is said to have predicted), and both parties were so much alarmed that they made peace at once.
       

118. Turnbull lectures on Colin Maclaurin, Part 2
     
     • where the dot can be interpreted as indicating the rate of change of the length affected.
       

119. EMS 1913 Colloquium 6.html.html
     
     • The problem was finally solved by the formula, ln = constant, where l is the name-length and n the hump-frequency.
       

120. Mathematicians and Music
     
     • I shall presently consider these at some length.
       

121. A CONTRIBUTION TO THE MATHEMATICAL THEORY OF BIG GAME HUNTING
     
     • lion can move his own length.
       

122. George Gibson: 'Calculus
     
     • The somewhat lengthy discussion of the conceptions of a rate and a limit I have found in practice to be the simplest method of enabling a student to grapple with the special difficulties of the Calculus in its applications to mechanical or physical problems; when these notions have been thoroughly grasped, subsequent progress is more certain and rapid.
       

123. Einar Hille: 'Analytic Function Theory
     
     • Certain topological concepts, such as the notions of neighbourhood, distance, length, and metric space, have also been stressed.
       

124. George William Hill's new theory of Jupiter and Saturn
     
     • At length the whole difficulty with Jupiter and Saturn was removed by Laplace's discovery of the great inequalities in 1786.
       

125. EMS 1914 Colloquium
     
     • He showed that the length of a straight line and the magnitude of an angle can be expressed in terms of a property which is unaltered by projection - e.g., cross-ratio.
       

126. Cochran: 'Sampling Techniques' Preface
     
     • Sampling theory and practice have both grown so much in the past ten years that an adequate coverage of the two aspects of sampling now requires a lengthy volume.
       

127. George William Hill's new theory of Jupiter and Saturn
     
     • At length the whole difficulty with Jupiter and Saturn was removed by Laplace's discovery of the great inequalities in 1786.
       

128. W H and G C Young
     
     • Later, he was put in charge of the Weights and Measures Department and he erected in Trafalgar Square, on the North Side, public standards of length.
       

129. De Thou on François Viète
     
     • Roman followed him there, although it meant a journey of about a hundred leagues, and when he finally had the pleasure of meeting Viete he consulted him at length about all the difficulties he had encountered.
       

130. Heath: 'The thirteen books of Euclid's Elements' Preface
     
     • Every school would have a different standard; matter of assumption in one being matter of demonstration in another; until, at length, GEOMETRY, in the ancient sense of the word, would be altogether frittered away or be only considered as a particular application of Arithmetic and Algebra." It is, perhaps, too early yet to prophesy what will be the ultimate outcome of the new order of things; but it would at least seem possible that history will repeat itself and that, when chaos has come again in geometrical teaching, there will be a return to Euclid more or less complete for the purpose of standardising it once more.
       

131. Bronowski and retrodigitisation
     
     • This puzzle carried the warning that computation might prove lengthy, and, indeed, the answer runs to 16 digits.
       

Quotations

1. Quotations by Russell
   
   • "But," you might say, "none of this shakes my belief that 2 and 2 are 4." You are quite right, except in marginal cases -- and it is only in marginal cases that you are doubtful whether a certain animal is a dog or a certain length is less than a meter.
     

2. Quotations by Recorde
   
   • To avoide the tediouse repetition of these woordes: is equalle to: I will settle as I doe often in woorke use, a paire of paralleles, or gemowe [twin] lines of one lengthe: =, bicause noe .2.
     

3. Quotations by Gauss
   
   • This is chiefly because I am never satisfied until I have said as much as possible in a few words, and writing briefly takes far more time than writing at length.
     

Chronology

1. Mathematical Chronology
   
   • Hipparchus discovers the precession of the equinoxes and calculates the length of the year to within 6.5 minutes of the correct value.
     
   • He measures the length of the year to be 365.24219858156 days, a remarkably accurate result.
     
   • Neile becomes the first to find the arc length of an algebraic curve when he rectified the cubical parabola.
     
   • Wren finds the length of an arc of the cycloid.
     
   • Kochanski gives an approximate method to find the length of the circumference of a circle.
     
   • La Hire calculates the length of the cardioid.
     
   • Gauss publishes a treatise on optics in which he gives a formulae for calculating the position and size of the image formed by a lens with a given focal length.
     
   • It is a continuous curve which is of infinite length and nowhere differentiable.
     

2. Chronology for 1650 to 1675
   
   • Neile becomes the first to find the arc length of an algebraic curve when he rectified the cubical parabola.
     
   • Wren finds the length of an arc of the cycloid.
     

3. Chronology for 1675 to 1700
   
   • Kochanski gives an approximate method to find the length of the circumference of a circle.
     

4. Chronology for 1840 to 1850
   
   • Gauss publishes a treatise on optics in which he gives a formulae for calculating the position and size of the image formed by a lens with a given focal length.
     

5. Chronology for 500BC to 1AD
   
   • Hipparchus discovers the precession of the equinoxes and calculates the length of the year to within 6.5 minutes of the correct value.
     

6. Chronology for 1900 to 1910
   
   • It is a continuous curve which is of infinite length and nowhere differentiable.
     

7. Chronology for 900 to 1100
   
   • He measures the length of the year to be 365.24219858156 days, a remarkably accurate result.
     

8. Chronology for 1700 to 1720
   
   • La Hire calculates the length of the cardioid.