I have taken the liberty of renumbering the 40 footnotes so that they run consecutively throughout the paper, rather than beginning at the number "one" on each page. Footnotes are shown enclosed in asterisks, e.g. **5** denotes the fifth footnote.
11. Euler Integrals and Euler's Spiral--Sometimes called Fresnel Integrals and the Clothoide or Cornu's Spiral. **1**
The integrals in question are
and the equations of the spiral are
These were considered by Euler at least as early as 1743 in a problem of his celebrated work on
the calculus of variations: Methodus inveniendi lineas curvas maxlimeminimire proprietate
gaudentes **2**. . . The discussion of the problem was somewhat as follows.**3** Consider an
elastic spring freely coiled up in the form of a spiral. Let us suppose that the interior extremity is
fixed and that the spring can be developed into a horizontal position by a weight p suspended at
the other extremity. Under these conditions the action of the weight on an element of the spring
placed at a distance s from the extremity is ps; and the elasticity of the element preserves it in
equilibrium. This elasticity is "the reciprocal of the osculating radius of the spring in its
unrestricted state." Setting r equal to this radius with respect to the part s of the spring, taken
from its exterior extremity, we have ps = Ek^2/r, Ek^2 being a constant depending upon
the elasticity of the spring. Let Ek^2/p = a^2. For the spring in its natural position we
will thus have (1) rs = a^2, "quae est aequatio naturam curvae. . . complectens.." Whence,
introducing rectangular coordinates,
Euler then remarks: "From the fact that the osculating radius steadily decreases the longer the arc taken it is evident that the curve is not produced to infinity. The curve therefore will be in the nature of a spiral so that when the spiral is completed it is rolled up, as it were, in a certain point which may be called the center. The point seems to be very difficult to discover by this construction. Therefore we must admit that analysis will make no small gain should anyone find a method whereby, approximately at least, the value of this integral would be determined in the case that s is taken to be infinite. This problem does not seem to be unworthy the best strength of geometers."
Euler then expresses x and y as converging series in s for approximating to these values but adds
that if s be made infinite the values of x and y cannot be determined in this way. He sets, finally,
s^2/2a^2 = v, obtaining
and shows that approximate values of x and y could be found by considering the intervals v = 0 to v = pi, v = pi to v = 2pi to v = 3pi, . . ., and certain converging series "requiring long operations and very tedious calculations evaluate them."
Thirty-eight years later, however, Euler had solved the problem completely. This solution is to be
found in one of the last papers which he wrote (he died in 1785). It is entitled "De valoribus
integralium a terminius varliailis x = 0 usque ad x = infinity extensorum" and was presented to the
Academy at Petrograd on April 30, l78l. **4** Again he considers the curve the radius of
curvature at each point of which is inversely proportional to the arc of the curve and is led, as
before, to the equation rs = a^2 from which, Euler says, it would not be difficult to discuss
the form of such a curve. He refers to the infinite number of whorls
(infinitas spiras) about a fixed point "which may be called the pole
of this curve." He then proceeds to determine the coordinates of this pole.
Introducing the angle of contingence (delta) he is led to the form (3) **5** s^2 = 2a^2v,
from which he finds readily the equations (2). Concerning the evaluation of these
integrals for the coordinates of the pole he remarks that he had "recently found by a happy chance
and in an exceedingly peculiar manner" that
Euler's method of evaluation is based upon that of
which in turn, leads to
where p = r cos alpha, q = r sin alpha. Euler then sets q = 1, p = 0, n = 1/2 and finds the required result. He notes also:
and the well known result
the evaluation of which "up to the present has defeated all known artifices of calculation."
While Euler was not the first to discuss some of the problems mentioned above, he was the first to publish any results of importance in connection with them. In 1694 (t the close of his memoir "Curvata Laminae Elasticase" **9**) James Bernoulli (1654 - 1705) mentions among problems which might be worked out: To find the curvature a lamina should have in order to be straightened out horizontally by a weight suspended at one end. In the posthumous edition of his works, in 1744, there is a reference from this passage to a fragment entitled "Invenire Curvarum, cujus curvedo in singulis punctis est proportionalis longitudini arcus; id est, quae ab appenso pondere flectitur in rectam." **10** Here there is the equation rs = a^2 and a construction for points on the curve; but there is not the slightest indication that Bernoulli had any conception, such as Euler had, of the real form of the curve.
In the nineteenth century the Euler integrals and spiral became of special
interest through discoveries of Fresnel (1788 - 1827) in connection with the
diffraction of light. By making certain assumptions and approximations
Fresnel deduced (in 1814) **11** for the intensity of the illumination at
any point of a diffraction pattern
For this reason the integrals which here occur are often called Fresnel's integrals. From what has been indicated above the value of each, for the limits v = 0 to v = infinity, is 1/2.
In his Note of 1818, Fresnel gave a table of the values of
for values of v (differing by 0.1) from 0.1 to 5.1 (later extended to 5.5), to 4 places of decimals. This table is reproduced by E. Verdet in his Lecons d'Optique physique (1869). **12** More detailed tables (to the nearest hundredth) were calculated by Abria. **13** Modifications of Fresnel's method of evaluating A and B, and criticisms and corrections of his results were given by Knochenbauer, **14**, Cauchy, **15**, Gilbert. **16** Knochenbauer's method was good for small values and Cauchy's for large values of v. Peters gives (l. c., page 48) a table similar to that of Fresnel but slightly more extensive, in that intervals of 0.01 are considered from v = 0.01 to v = 0.10, and intervals of 0.05 from v = 0.10 to v = 1.00. This is the table most frequently quoted; **17** but it should be remembered that, except for some corrections and slight additions, it is identical with Fresnel's given some forty years earlier.
The tables of Ignatowsky **18** give (among other things), from
v = 0.0 to v = 8.5 (for intervals of 0.1), the values of A and B to four
places of decimals, and of log A and log B to six places. Lommel
published **19** a table for
from z = 0 to z = 50 at unit intervals. From z = 0.0 to z = 50.0 at intervals fo 0.1 and to four places, it is printed in Jahnke and Emde's tables. **17**
In 1874 Cornu plotted Euler's spiral accurately **20** by means of Peter's table. (Euler had already given half the spiral.) In a sketch of Cornu, Poincare has written as follows: **21** "Aussi, quand il aborda l'etude de la diffraction, il eut bientot fait de remplacer cette multitude rebarbative de formule herissees d'integrales par une figure unique et harmonieuse, que l'oeil suit avec plaisir et ou l'esprit se dirige sans effort. Tout le monde aujourd'hui pour prevoir l'effet d'un ecrau quelconque sur un faisceau lumineux, se sert de la spirale de Cornu." The expression "Cornu's Spiral" was used by Preston, **22**, Wood, **23**, and others before Poincare **24** employed it in the sketch quoted; but the term is evidently highly inappropriate in the light of Euler's discoveries set forth above.
Besides the works on physics to which reference has been made already in connection with our topic we may note those by Drude, **25** Pockels, **26** and Chwolson. **27**
Lommel seems to have been the first **28** to observe the connection
between A', B' and Bessel functions:
Amongst the many methods for evaluating A and B references may be given:
(1) to the method of Godefroy **29** who, by a slight modification of
Laurent's discussion, **30** starts with
and avoids all use of imaginaries;
(2) to a paper by Cayler **31** discussing several interesting points which later occupied the attention of Glaisher **32**, Jamet **33**, and Humbert **34**;
(3) to other methods illustrated by Pierpont **35** and d'Adhemar **36**; and
(4) to Noumoff's "Interpretation geometriques des integrales de Fresnel" **37** in which the projection of helics generated by a certain parabola rolling on a right circular cylinder are curves the sum of the areas under which give the required values. The latter part of the article contains a description of a mechanical integrator for calculating the integrals A and B.
In recent times Cesaro has given to Euler's Spiral the name Clothoide **38** and exhibited a number of remarkable properties of the curve. **39** Among these the following may be mentioned:
(a) the clothoide is the only curve enjoying the property that the center of gravity of any arc is a center of similitude of the circles osculating the extremities of the arc;
(b) when a clothoide rolls on a straight line, the locus of the center of curvature corresponding to the point of contact is an equilateral hyperbola asymptotic to the line considered.
Wieleitner discussed "Die Parallelkurve der Klothoide." **40** For different values of m the intrinsic equation rs^m = a^2 represents a clothoide, a logarithmic spiral, a circle, the involute of a circle, and a straight line.
As noted above, the 40 footnotes have been renumbered to be consecutive throughout this web page. In the original article, they began anew at one on each page.
**1** For historical sketches and properties of this curve see F. Gomes Teixeira, Traite des courbes speciales remarquables planes et gauches, tome 2, Coimbre, 1909, pp. 102-107; G. Loria, Spezielle algebraische und transzendente ebene Kurven ... 2. Auflage. Band 2, Leipzig, Teubner, 1911, pp. 70-73.
**2** Lausannae & Genevae, MDCCXLIV, pp. 276-7. Cf. Verzeichais der Schriften Leonhard Eulers bearbeitet von G. Enestrom. Erste Lieferung, Leipzig, Teubner, 1910, p. 16. See also P. S. Laplace, "Sur la reduction des fonctions en tables," Journal de l'Ecole Polytechnique, tome 8, cahier 15, 1809, pp. 250-251.
**3** As Euler's writings to which reference must be made are very scarce, it would seem best to give more details than otherwise would be necessary.
**4** Leonhardi Euleri Institutionum calculi integralis, Vol. 4, Petropoli, 1794, pp. 337-345; editio tertia, Petropoli, 1845, pp. 337-345. German edition: Vollstandige Anleitung zur Integralrechnung ... ubersetzt von J. Salomon, Wien, 1830, pp. 321-328.
**5** A curve with an equation of this same form was named by K. C. F. Krause "parabola originaris longitudinaris" (Nova theoria linearum curvarum, Monachii, anno MDCCCXXXV, p. 79). The reason for the name is clear. None of Krause's discussion of the curve is worth referring to; Loria's mention of it seems misleading in part.
**6** These were the integrals investigated by Poisson in his "Memoire sur les integrales definies," Journal de l'Ecole Polytechnique, Paris, tome 9, cahier 16, pp. 215-246, 1813, especially pp. 215-219. See also Poisson, Nouveau Bulletin des Sciences par la societe philomathique de Paris, 3 annee, 1811, tome 2, p. 251; Lacroix, Traite du calc. diff. et integ., tome 3, 2e ed., Paris, 1819, pp. 486-490; Grunert, Crelle's Journal, Band 8, 1832, pp. 146 - 151; J. Plans, "Sur trois integrales definies," Acad. Sci. Mem., Bruxelles, Vol. 10, 1837; A. De Morgan, Differential and Integral Calculus, London, 1842, p. 630; Schlomilch, "Ueber einige Integrale welchegoniometrische Funktionen involvieren," Arch. d. Math. u. Phys., 1845, Band 6, pp. 200 ff.; E. F. A. Minding, "Ueber [Integral from 0 to infinity sin x^m * x^-n dx] wo m >= n > 0," Arch. d. Math. u. Phys., 1858, pp. 171-183; W. Walton, "On a Pair of Definite Integrals," Quarterly Journal of Mathematics, 1871, Vol. 11, pp. 373-375; J. W. L Glaisher, "On certain Definite Integrals," Reporrt of the British Association for the Advancement of Science, 1871, London, 1872, Report, pp. 10 - 12.
**7** For p = 1 and q = 0 we find, on substituting t^2 for x, that [Integral from 0 to infinity e^(-t^2) dt = sqrt(pi) / 2, an integral (of great importance in many parts of applied mathematics) definitely evaluated by Laplace in a memoir published in 1781 (Mem. Acad. Paris, 1778; (Euvres, Paris, Vol. 9, "Memoire sur les probabilites"). The Euler integrals, and spiral in connection with the elastic spring, of these notes were also discussed by Laplace in "Sur la reduction des fonctions en tables," Journal de l'Ecole Polytechnique, tome 8, cahier 15, pp. 229-265, 1809. A slip made by Mascheroni is here corrected; in his Adnotationes ad calculum integralen Euleri ec. (Ticini, MDCCXC; [also L. Euler, Opera Omnia, series 1, VOl. 12, Leipzig, Teubner, 1914]) Mascheroni has a note on Cap. V, Sect I, Vol. 1, entitled "De integratione Formularum x^n dx sin x, x^n dx cos x," pp. 38-57 [pp. 454-471]. In the special case of n = -1/2 he gives (p. 53) sqrt(2*pi) instead of sqrt(pi/2).
**8** See G. H. Hardy's discussion of eleven proofs of this in Mathematical Gazette, July 1916, Vol. 8, pp. 301-303; see also Vol. 5, pp. 98-103, 1909, and Vol. 6, pp. 223-4, 1912.
**9** Acta Eruditorum, 1694, p. 276; Opera, Genevae, MDCCXLIV, tome 1, p. 600.
**10** J. Bernoulli, Opera, tome 2, pp. 1084-1086.
**11** OEurres completes d'Augustin Fresnel, tome 1, Paris, MDCCCLXVI, pp. 176-181; see also pp. 198-9, 315-352.
**12** Publiees par A. Levistal, tome 1, Paris, 1869; cf. pp. 343 ff. German edition by K. Exner, Band 1, Braunschewig, Vieweg, 1881, pp. 236-309; the table is given on p. 240 and on p. 241 A and B are graphed as oscillating curves.
**13** "Sur la diffraction de la lumiere," Journal de mathematique pures et appliquees, 1839, tome 4, pp. 248-260.
**14** K. W. Knochenhauer, (1) "Ueber die Oerter der Maxima und Minima des gebeugten Lichtes nach den Fresnel'schen Beobachtungen," Annalen der Physik und Chemie, Leipzig, 1837, Band 41, pp. 103-110; (2) "Ueber eine besondere Klasse von Beugungserscheinungen," idem, 1838, Band 43, pp. 286-292; (3) Die Undulationstheorie des Lichtes, Berlin, 1839, p. 36f.
**15** A. Cauchy, Comptes Rendus, Paris, 1842, tome 15, "Note sur la diffraction de la lumiere," pp. 554-6; "Addition a la Note sur la Note sur la diffraction de la lumiere," pp. 573-578.
**16** Ph. Gilbert, "Recherches analytiques sur la diffraction de la diffraction de la lumiere" (memoire presente le 3 aout, 1861). Memoires couronnes ... acad. roy. d. sc. ... de Belgique, 1863, tome 31, pp. 1-52.
**17** For example: E. Jahnke und F. Emde, Funktionentafeln mit Formeln und Kurven, Leipzig, 1909, pp. 23-26; R. W. Wood, Physical Optics, New York, Macmillan, 1905, p. 198.
**18** W. v. Ignatowsky, Annalen der Physik, 1907, Band 328, pp. 895-898.
**19** E. Lommel, Abh. Munch. Ak., Band 15, 2. Abtheilung, 1880, p. 230.
**20** A. Cornu, (1) "Methode nouvelle pour la discussion des problems de diffraction dans le cas d'une onde cylindrique," Journal de physique theorique et appliquee, Paris, 1874, pp. 5-15; (2) "Etudes sur la diffraction; methode geometrique pour la discussion des problemes de diffraction," Comptes Rendus, tome 78, 1874, pp. 113-117.
**21** H. Poincare, Savants et ecrivains, Paris, Flammarion, , p. 106.
**22** T. Preston, The Theory of Light, 2d edition, London, Maccmillan, 1895, p. 274, [4th ed., 1912, p. 291].
**23** R. W. Wood, Physical Optics, New York, Macmillan, 1905, p. 158 and on the plate at the end of the volume.
**24** Loria seems to be in error here (l. c., p. 71).
**25** P. Drude, Lehrbuch der Optik, Leipzig, Hirzel, 1900, pp. 174-187; English translation by C. R. Mann and R. A. Millikan, London, Longmans, 1902, pp. 188-202.
**26** F. Pockels, pp. 1051-1064 of Handbuch der Physik, Zweite Auflage herausgegeben von A. Winkelmann, Band 6: Optik, Leipzig, Barth, 1906.
**27** O. D. Chowlson, Traite de physique, ouvrage traduit sur les editions russe et allemande, tome 2, Paris, Hermann, 1909, pp. 652-656.
**28** E. Lommel, "Ueber die Anwendung der Bessel'schen Funktionen in der Theorie der Beugung," Zeitschrift fur Mathematik und Physik, Leipzig, 1870, pp. 141-169. Cf. A. Gray and G. B. Mathews, Treatise on Bessel Functions, London, 1895, p. 41.
**29** A. Godefroy, "Sur les integrales de Fresnel," Nouvelles annales de mathematiques, 1898, (3), Vol. 17, pp. 205-206.
**30** H. Laurent, Traite d'analyse, tome 3, Paris, 1888, p. 137.
**31** A. Cayley, "Note on the Integrals [Integral from 0 to infinity cos x^2 dx] and [Integral from 0 to infinity siin x^2 dx]," Quarterly Journal of Mathematics, Vol. 12, 1873, pp. 118-126; also Collected Mathematical Papers, Cambridge, Vol. 9, 1896, pp. 56-63.
**32** J. W. L. Glaisher, "On the Integrals [Integral from 0 to infinity sin x^2 dx] and [Integral from 0 to infinity cos x^2 dx]," Quarterly Journal, 1875, Vol. 13, pp. 343-349.
**33** V. Jamet, "Sur les Integrales de Fresnel," Nouvelles annales de mathematiques, 1896 (3), tome 15, pp. 372-376.
**34** G. Humbert, Cours d'analyse, Paris, Gauthier-Villars, tome 1, 1903, pp. 307-08.
**35** J. Pierpont, Lectures on the Theory of Functions of Real Variables, Boston, Ginn, Vol. 1, 1905, pp. 499-500.
**36** R. d'Adhemar, Exercises et Lecons d'Analyse, Paris, Gauthier-Villars, 1908, pp. 23-25.
**37** Journal de physique theorique et appliquee, Paris, 1847 (3), tome 6, pp. 281-289.
**38** From the Greek word meaning to twist by spinning -- since the curve spins or turns about its asymptotic points.
**39** E. Cesaro, (1) "Les lignes barycentriques," Nouvelles annales de mathematiques, 1886 (3), tome 5, pp. 511-520; (2) "Sur la courbe representative des phenomenes de diffraction," Comptes Rendus, 1890, tome 110, pp. 1119-1122; (3) Nouvelles annales de mathematiques, 1905, 4e serie, tome 5, pp. 570-573. See also L'Intermediaire des mathematiciens, 1916, tome 23, pp. 187-189.
**40** Archiv der Mathematik und Physik, 1907 (3) Band 11, pp. 373-375.