DYNAMIC SYMMETRY, Jay Hambidge, Yale University Press, New Haven 1920:16-18

THE ROOT-FIVE RECTANGLE

... A rectangle whose side is divided into five equal parts by a perpendicular has a ratio between its end and its side of one to 2.236, or the square root of five. This area is a root-five rectangle and it possesses properties similar to those of the other rectangles described, except that it divides itself into rectangle similar to the whole with ratios of five and six. A square on the end is to a square on the side as one is to five, that is, the smaller square is exactly one-fifth the area of the larger square. There is an infinite succession of such rectangles, but the Greeks seldom employed a root rectangle higher than the square-root of five. (Figs. 15a, 15b omitted)
    The root-five rectangle, moreover, possesses a curious and interesting property which intimately connects it with another rectangle, perhaps the most extraordinary of all. To understand this strange rectangle, we must consider the phenomena of leaf distribution. This root-five rectangle may be regarded as the base of dynamic symmetry.
    Closely linked with the scheme which nature appears to use in its construction of form in the plant world is a curious system of numbers known as a summation series. It is so called because the succeeding terms of the system arc obtained by the sum of two preceding terms, beginning with the lowest whole number; thus, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, etc. This converging series of numbers is also known as a Fibonacci series, because it was first noted by Leonardo da Pisa, called Fibonacci. Leonardo was distinguished as an arithmetician and also as the man who introduced in Europe the Arabic system of  notation. Gerard, a Flemish mathematician of the 17th century, also drew attention to this strange system of numbers because of its connection with a celebrated problem of antiquity, namely, the eleventh proposition of the second book of Euclid. Its relation to the phenomena of plant growth is admirably brought out by Church, who uses a sunflower head to explain the phenomena:

A fairly large head, 5 to 6 inches in diameter in the fruiting condition, will show exactly 55 long curves crossing 89 shorter ones. A head slightly smaller, 3 to 5 inches across the disk, exactly 34 long and 55 short; very large 11 inch heads give 89 long and 144 short; the smallest tertiary heads reduce to 21 and 34 and ultimately 13 and 21 may be found; but these being developed late in the season are frequently distorted and do not set fruit well. A record head grown at Oxford in 1899 measured 22 inches in diameter, and, though it was not counted, there is every reason to believe that it belonged to a still higher series (144 and 233).... Under normal conditions of growth the ratio of the curves is practically constant. Out of 140 plants counted by Weisse, 6 only were anomalous, the error thus being only 4 per cent. (A. H. Church, On the Relation of Phyllotaxis to Mechanical Law, Williams and Norgate, London, 1904)

    What is called normal phyllotaxis or leaf distribution in plants is represented or expressed by this summation series of numbers. The sunflower is generally accepted as the most convenient illustration of this law of leaf distribution. An average head of this flower possesses a phyllotaxis ratio of 34 x 55.  These numbers are two terms of the converging summation series.
    The present inquiry is concerned with only two aspects of the phyllotaxis phenomena: the character of the curve, and the summation series of numbers which represents the growth fact approximately.The actual ratio can be expressed only by an indeterminate fraction. The plant, in the distribution of its form elements, produces a certain ratio, 1.618, which is obtained by dividing any one term of the summation series by its predecessor. This ratio of 1.618 is used with unity to form a rectangle which is divided by a diagonal and a perpendicular to the diagonal, as in the root rectangles. (Fig. 19.)

Jay Hambidge, 1920, Figures 19 and 20   

  Thus, we may call this "the rectangle of the whirling squares," because its continued reciprocals cut off squares. The line AB in Fig. 19 is a perpendicular cutting the diagonal at a right angle at the point O, and b is the square created. BC is the line which creates a similar figure to the Whole. One or unity should be considered as meaning a square. The number a means two square,,
3, three squares, and so on. In Fig. 19 we have the defined square b, which is unity. The fraction .618 represents a shape similar to the original, or is its reciprocal. Fig. 20 shows the reason for the name "rectangle of the whirling squares."   1, 2, 3, 4, 5, 6, etc., are the squares whirling around the pole O.

Jay Hambidge, 1920, Figure 21    

If the ratio 1.618 is subtracted from 2.236, the square root of 5, the remainder will be the decimal fraction .618. This shows that the area of a root-five rectangle is equal to the area of a whirling square rectangle plus its reciprocal, that is, it equals the area of a whirling square rectangle horizontal plus one perpendicular, as in Fig. 21.
    The writer believes that the rectangles above described form the basis of Egyptian and Greek design. In the succeeding chapters will be explained the technique or method of employment of these rectangles and their application to specific examples of design analysis. (Jay Hambidge, DYNAMIC SYMMETRY, Yale University Press, New Haven 1920:16-18)

 


APPENDIX AND NOTES

NOTE V

NOTES ON THE LOGARITHMIC SPIRAL, GOLDEN SECTION AND THE FIBONACCI SERIES1

by R. C. Archibald

I. The Logarithmic Spiral.2
The first discussions of this spiral occur in letters written by Descartes to Mersenne in 1638, and are based upon the consideration of a curve cutting radii vectores (drawn from a certain fixed point O), under a constant angle, F.3  Descartes made the very remarkable discovery that if B and C are two points on the curve its length from O to B is to the radius vector OB as the length of the curve from O to C is to OC; whence s = ap,5 where s is the length measured along the curve from the pole to the point (p, 01, and a = sec F.6 This leads to the polar equation (I) p = kecF, where k is a constant and c = cot a. The pole O is an asymptotic point. The pole and any two points on the spiral determine the curve; for the bisector of the angle made by the radii vectores is a mean proportional between radii. If c = 1 the ratio of two radii vectores corresponds to a number, and the angle between them to its logarithm; whence the name of the curve.

    The name logarithmic spiral is due to Jacques Bernoulli. The spiral has been called also the geometrical spiral,8  and the proportional spiral; 9 but more commonly, because of the property observed by Descartes, the equiangular spiral.10
    Bernoulli (and Collins at an earlier date) noted the analogous generation of the spiral and loxodrome ("loxodromica") the spherical curve which cuts all meridians under a constant angle. Credit for the first discovery that the loxodrome is the stereographic projection of a logarithmic spiral seems to be due to Collins.11
    As the result of Descartes's Letters distributed by Mersenne, Torricelli also studied the logarithmic spiral. He gave a definition which may be immediately translated into equation (I), and from it he obtained expressions for areas, and lengths of arcs. These results were rediscovered by John Wallis12 and published in 1659.13  Wallis states in this connection that Sir Christopher Wren had written about the logarithmic spiral and arrived at similar results.
    During 1691-93 Jacques Bernoulli gave the following theorems among others: (a) Logarithmic spirals defined by equations (I) for different values of k are equal and have the same asymptotic point; (b) the evolute of a logarithmic spiral is another equal logarithmic spiral having the same asymptotic point;14 (c) the pedal of a logarithmic spiral with respect to its pole is an equal logarithmic spiral;15 (d) the caustics by reflection and refraction of a logarithmic spiral for rays emanating from the pole as a luminous point are equal logarithmic spirals.
    The discovery of such "perpetual renascence" of the spiral delighted Bernoulli. "Warmed with the enthusiasm of genius he desired, in imitation of Archimedes, to have the logarithmic spiral engraved on his tomb, and directed, in allusion to the sublime tenet of the resurrection of the body, this emphatic inscription to be affixed–Eadem mutata resurgo.''16 The engraved spiral (very inaccurately executed) and inscription, in accordance with Bernoulli's desire, may be seen to-day on his tomb in the cloister of the cathedral at Basel.17
    The logarithmic spiral appears in three propositions of Newton's "Principia" 1687).18  From the first there develops that if the force of gravity had been inversely as the cube, instead of the square, of the distance the planets would have all shot off from the sun in "diffusive logarithmic spirals."19  In the second proposition Newton showed that the logarithmic spiral would also be described by a particle attracted to the pole by a force proportional to the square of the density of the medium in which it moves, while this density is at each point inversely proportional to its distance from the pole. In the third proposition the second was generalized by the substitution of "inversely proportional to any power of its distance" for "inversely proportional to its distance"– a result which has been attributed to Jacques Bernoulli (for example, by Comes Teixeira, l. c.).
    There is also considerable discussion of the logarithmic spiral by Guido Grandi in various parts of his Geometria Demonstratio Theorematum Hugenianorum circa Logisticam seu Logarithmicam Lineam . . . , Florentine, 1701.20  A section in the first chapter deals with "spirali logarithmicae per duos motus descriptio," and points are found (page 8) "in Spirali Logistica, alias Spiralis Logarithmicae, quibusdam Spiralis Geometricae riomine appellata' (evidently referring to P. Nicolas, l.c.). In a letter to Ceva, printed at the end of the volume, the gauche spiral cutting the generators of a right circular cone under a constant angle was studied for the first time, and it was shown, by purely geometric methods, that this spiral may be projected into a logarithmic spiral.
    In a memoir read by Pierre Varignon before the French Academy in 1704 21 he discussed a transformation equivalent to x = p, y = Iw,  where p and to are the polar coordinates of the point corresponding to (x, y), and I is a constant. Varignon found, in particular, that from the logarithmic curve x -h  = e y is derived the logarithmic spiral p = e -1/h*w h . So also, if l = 1, the sine curve x = siny becomes a circle. In recent times this latter transformation has been employed in plotting alternating voltage and current curves.22
    In 1892  I. Stringham showed 23 that if the logarithmic spiral is properly defined as a geometric locus it may be used for defining the logarithm and demonstrating its properties, which lead to a classification of logarithmic systems. This classification was somewhat modified by M. W. Haskell and I. Stringham.24
    Cremona's discussion of the logarithmic spiral, and how it may serve, when drawn, for the solution of problems involving extraction of roots25 (higher than the second) should not be forgotten. Then there is A. Steinhauser's Die Elemente des graphischen Rechnens mit besonderer Beriicksichtigung der logarithmischen Spirale. Eine Einleitung zur Construction algebraischer und transcendenter zlusdriicke fur Bau- und Maschinen- Technikers 26--Equiangular spirals appear as "tie-lines" and "strutt-lines" in a problem of W. J. Ibbetson's Elementary Treatise on the Mathematical Theory of Perfect & Elastic Solids27--There is also the little known but notable paper, published by James Clerk Maxwell when only eighteen years of age,28 which contains several properties of logarithmic spirals. Some quotations follow:
    "The involute of the curve traced by the pole of a logarithmic spiral which rolls upon any curve is the curve traced by the pole of the same logarithmic spiral when rolled on the involute of the primary curve." (Page 524 [10].)
    "The method of finding the curve which must be rolled on a circle to trace a given curve is mentioned here because it generally leads to a double result, for the normal to the traced curve cuts the circle in two points, either of which may be a point in the rolled curve.
    "Thus, if the traced curve be the involute of a circle concentric with the given circle, the rolled curve is one of two similar logarithmic spirals." (Page 529 [16].) (Often attributed to Haton de la Goupillière.)
    "If any curve be rolled on itself, and the operation repeated an infinite number of times, the resulting curve is the logarithmic spiral." The curve which being "rolled on itself traces itself is the logarithmic spiral." (Page 532 [19].)
     "When a logarithmic spiral rolls on a straight line the pole traces a straight line which cuts the first line at the same angle as the spiral cuts the radius vector." (Page 535 [23].) (Often attributed to Catalan.)
     Among many other results the following may be noted: Haton de la Goupillière proved 29 that the logarithmic spiral is the only curve whose pedal with respect to a given pole is an equal curve which can be brought into coincidence with the first by a rotation about the pole--Cesàro discussed the tractrix and logarithmic spiral as correlative figures30--From logarithmic spirals H. Dirtrich derived 31 (according to Loria, l. c.) sum and difference spirals which he used for geometrical exposition of hyperbolic functions--If a logarithmic spiral roll on a straight line the locus of its center of curvature at the point of contact is another straight line (A. Mannhelm, 1859)--The involutes of a logarithmic spiral are equal spirals (which is really the same as Bernoulli's result for evolutes)--The inverse of a logarithmic spiral with respect to its pole is an equal spiral with the same pole--Coplanar logarithmic spirals and their orthogonal trajectories, which are again coplanar logarithmic spirals, come up (I) in the discussion of loxodromic substitutions32 and (2) in conformal representations.33 As a consequence of a general theory relative to linear transformations F. Klein and S. Lie obtained the following result:34 The logarithmic spiral is its own polar reciprocal with respect to any equilateral hyperbola which has its center at the pole and is tangent to the spiral.
    In 1833 T. Olivier described to the Sociale Philomathique, Paris, "un compass simple permetrant de traces routes les spirales logarithmiques," 35 and in a letter written by Collins for Tschirnhaus, Sept. 30,1675,36 reference is made to "an instrument invented by M. Tschirnhaus" and its connection with the logarithmic spiral.
    The most practical form of a ship's anchor was discussed in 1796 by F. H. Chapman, vice-admiral in the Swedish Marine.37 He found that the best form for each of the barbed arms would be an arc of a logarithmic spiral cutting the shank of the anchor at an angle of 67o 30'.

    The distinctive properties of the logarithmic spiral which permit it to be used for lines of pitch of cams and non-circular wheels38 are: (a) that the difference of radii vectores of the ends of equal arcs is constant; (b) the curve cuts radii vectores under a constant angle. For these reasons two equal logarithmic spirals may roll together with fixed poles and a fixed distance between the poles. Two arcs (not necessarily equal) of logarithmic are required for the complete line of pitch of a wheel, but any even number of arcs may be used. A wheel with three lobes may act on a wheel with two, which in turn may act on a unilobe wheel. Even with two reacting wheels with the same number of lobes there are varying velocity ratios having maximum and minimum values for the rates of rotation of the shafts.
    The first definite suggestion connecting the logarithmic spiral with organic spirals seems to have been made by Sir John Leslie in his Geometrical Analysis and Geometry of Curve Lines.39 After proving that the involutes of a logarithmic spiral are logarithmic spirals he remarks: "The figure thus produced by a succession of coalescent arcs described from a series of interior centers exactly resembles the general form and the elegant septa of the Nautilus.'' 40 The aptness of this remark has been long since established. One of the earliest mathematical discussions of organic logarithmic spirals was by Canon Moseley, "On the Geometrical Forms of Turbinated and Discoid Shells" 41--a paper written more than eighty years ago which is one of the classics of natural history. In "turbinate" shells we are no longer dealing with a plane spiral as in the nautilus but with a gauche spiral on a right circular cone cutting the generators at a constant angle and such that along a generator the line-segments between successive whorls form a geometric progression.42  For mathematical and other details of Moseley's work as well as of that of many others, on univalve and bivalve shells, Thompson's book, with its many exact references to the literature of the subject, should be consulted. One notable work which Thompson appears to have overlooked is Haton de la Goupillière, "Surfaces Nautiloides.'' 43
    In the field of leaf arrangement or phyllotaxis discussion of the theories of A. H. Church44 and Cook evolved from observations of arrangements in logarithmic spirals of riorots of sunflowers, pine cones, and other growths, should be read in connection with Thompson's criticisms. The fine sunflower photograph by H. Brocard45 ought to be compared with those by Church.
 

II. GOLDEN SECTION.
    In the "Elements" of Euclid (who flourished about 300 B.C.), the following propositions occur: (1) "To cut a given straight line so that the rectangle contained by the whole and one of the segments is equal to the square on the remaining segment" (Book II, proposition 11); (2) "To cut a given finite line in extreme and mean ratio" (Book VI, proposition 30).46  While these propositions are equivalent in statement the methods of construction given by Euclid are quite different. There can be little doubt that the construction in the second is due to Euclid and in the first to the Pythagoreans (fifth century B.C.). The result is used "To construct an isosceles triangle having each of the angles at the base double of the remaining one" (Elements, Book IV, 10) and this leads to the construction of a regular pentagon (Book IV, 11).
    In the Elements, book XIII, the first five propositions, which are preliminary to the construction and comparison of the five regular solids, and deal with properties of a line segment divided in extreme and mean ratio, are usually attributed to Eudoxus, who flourished about 365 B. C. Proclus tells us that Eudoxus "greatly added to the number of the theorems which Plato originated regarding the section"; scholars agree that "the section" refers to the division in extreme and mean ratio.
    The so-called book XIV of Euclid's Elements, written by Hypsicles of Alexandria between 200 and 100 B. C., contains some results concerning "the section."
    In recent times the name golden section has been applied to the division of a line segment as above47 in the ratio (v'5 -- x) : 2. Terquem believed that the expression "extreme and mean ratio" (which is an exact translation of Euclid's Greek phrase) is "une reunion de mots ne presentant aucun sens,''a and following J. F. Lorenz 078 ~) employed the term "continued section." Terquem has also suggested:49 "diviser une droite diagonalement." Leslie introduced the term "medial section.'' 50 "Divine proportion" was used by Fra Luca Pacioli in 1509 51 and possibly earlier by Pier della Francesca; 52 "sectio divina" and "proportio divina" occur in the writings of Kepler.
    Pacioli's work was doubtless influential in inspiring a certain amount of mysticism in the consideration of golden section by later writers. In a work published in 1569, P. Ramus associates the Trinity with the three parts of golden section. A little later Clavius wrote of its "godlike proportions." As noted above Kepler declared himself similarly. He said also: "Geometry has two great treasures, one is the Theorem of Pythagoras, the other the division of a line into extreme and mean ratio; the first we may compare to a measure of gold, the second we may name a precious jewel." 53
    In the Thirteenth Century Campanus proved (in his edition of Euclid's Elements, bk. IX, prop.16) that golden section was irrational. His argument (by mathematical induction) was reproduced in algebraic notation by Genocchi and by Cantor.54
    There is an interesting passage on golden section by Albert Girard in his edition of Stevin's works.55 Girard gives a method of expressing the ratio of the segments of a line (cut in golden section) in rational numbers that converge to the true ratio. For this purpose he takes the sequence

(1)           0, 1, 1, 2, 3, 5, 8, 13, 21, ...,

every term of which (after the second) is equal to the sum of the two terms that precede it, and says, after Kepler, any number in this progression has to the following the same ratios (nearly) that any other has to that which follows it. Thus 5 has to 8 nearly the same ratio that 8 has to 13; consequently any three consecutive numbers such as 8, 13, 21 nearly express the segments of a line cut in golden section. Since the fractions (2)
 

are the various convergents of the continued fraction
 


Maupin reasons with force (after taking into account all which follows in the note) that Girard was probably familiar with the elements of continued fractions. Simson interprets Girard's reasoning differently.
    For mathematical treatment of problems in golden section, in ordinary or generalized form, see also the papers by C. Thiry56 and R. E. Anderson,57 5 E. Catalan's "Theoremes et Problames de geometrie elementaire'' 58 and Emsmann's program 59containing more than 350 relations and problems.
    In the nineteenth century the literature of golden section is by no means inconsiderable. It includes at least a score of separate pamphlets and books and many times that number of papers. In numerous, voluminous and rather unscientific writings A. Zeising 60 finds golden section the key to all morphology and contends, among other things, that it dominates both architecture and music. A distinctly new line was set under way by  Fechner who applied scientific experimental methods to the study of aesthetic objects.61 He was led to the conclusion that the rectangle of most pleasing proportions was one in which the adjacent sides are in the ratio of parts of a line segment divided in golden section.62 There are some paragraphs on "Golden Section," by J. S. Ames in Dictionary of Philosophy and Psychology 63 edited by J. M. Baldwin. In his article on "The aesthetics of unequal division” 64  P. A. Angler discusses earlier contributions to the aesthetics of golden section, including those by L. Witmer 65 (the chief investigator in the aesthetics of simple forms after Fechner), W. Wundt, 66 and O. Kulpe.67 The subject has been treated still more recently by M. Dessoir 68and J. Volkelt.69
     Sir Theodore Cook discusses 70 golden section from some new points of view in connection with art and anatomy, and the writings of F. X. Pfeifer 71 remind one both in subject matter and style of treatment of Zeising's publications.
     Neikes defined the term golden section for different units (areas, volumes--not alone line segments) such that the smaller part is to the larger as the larger is to the whole. With Piazzi Smyth's work as a basis he applied golden section to an unscientific study of the architecture of the Cheops pyramid. 72

III. THE FIBONACCI SERIES.
    Foremost among mathematicians of his time was Leonardo Pisano (also known as Fibonacci), who flourished in the early part of the thirteenth century. His greatest work is Liber abbaci "a Leonardo filio Bonacci compositus, anno 1202 et correctus ab codera anno 1228." It was first printed in 1857.73
    Among miscellaneous arithmetical problems of the twelfth section is one entitled "How many pairs of rabbits can be produced from a single pair in a year.'' 74  It is supposed (I) that every month each pair begets a new pair which, from the second month on, becomes productive; and (2) that deaths do not occur. From these data it is found that the number of pairs in successive months would be as follows:

(3)   I, 2, 3, 5, 8, 13, 2I, 34, 55, 89, 144, 233, 377-

These numbers follow the law that every term after the second is equal to the sum of the two preceding and form, according to Cantor, the first known recurring series in a mathematical work. The doubtful accuracy of this latter statement has been pointed out by Gunther.75
   The series (3) was well known to Kepler, who discusses and connects it with golden section and growth, in a passage of his "De hive sexangula," 1611.76 Commentaries of Girard and Simson, and the relation of the series to a certain continued fraction, have been noted above. But the literature of the subject is very extensive and reaches out in a number of directions. In what follows un, will be regarded as the (n + 1)st term of what we shall call the Fibonacci series (I); so that uo,  = o,  u1 = u2 = 1, u3  = 2, . . .  For reasons which shall appear later the names Lame series, and Braun or Schimper-Braun series, have been also employed in this connection. Girard observed, l. c., that the three numbers  un, un+1, un+1 77 may be regarded as corresponding to lengths which form an isosceles triangle of which the angle at the vertex is very nearly equal to the angle at the center of the regular pentagon.
    The relation u,  un-1, un+1 - un2, = (- 1)n was stated in 1753 by Simson (l.c.). It was to this relation, and hence to the Fibonacci series that Schlegel78 was led when he sought to generalize the well-known geometrical paradox of dividing a square 8 X 8 into four parts which fitted together form a rectangle 5 X 13 79 Catalan found (1879) the more general relation  un-1-p -pun+1+p- (un+1)2  = (- 1)n-p (up)2, from which may be derived un+12 + un2, = u2n+1, first given, along with many other properties, by Lucas,81  in a paper showing the relation between the Fibonacci series and Pascal's arithmetical triangle. Daniel Bernoulli showed 82 that

 


from this a result given by Catalan readily follows:83
 

A very similar series occurs in a letter written by Euler in 1726.
    Lucas showed the importance of the Fibonacci series in discussions of (a) the decomposition of large numbers into factors and (b) the law of distribution of prime numbers.84 Binet was led to the series in his memoir on linear difference equations (l.c.), and Leger 85 and Finck 86 (and later Lame 87) indicated its application in determining an upper limit to the number of operations made in seeking the greatest common divisor of two integers. Landau evaluated the (summation) series (1/u2n and 1/u2n+1), and found that the first was related to Lambert's series and the second to the theta series.88
    The solution of the problem of determining the convex polyhedra, the number of whose vertices, faces, and edges are in geometrical progression, leads to the Fibonacci series.89
    For further references and mathematical discussions one may consult (I) L'Intermédiaire des mathématiciens, 1899, p. 242; 1900, pp. 172-7, 251; 1902, 92; 1902, p. 4-3; 1913, pp. 50, 51,
147; 1915, pp- 39-40 (see also question 4171, 1915, p. 277); (2) "Sur une generalisation des progressions geometriques," L'Education mathématique, 1914, pp.149-151, 157-158; (3) V. Schlegel, "Series de Lamb superieurs," El progreso matematico, 1894, ano 4, pp. 171-174; (4) T. H. Eagles, Constructive Geometry of Plane Curves, London, 1885, pp. 293-299, 303-304; and (5) L. E. Dickson, History of the Theory of Numbers, vol. I, Washington, 1919, Chapter XVII "Recurring series; Lucas' un, vn."
    As to growths it is particularly in connection with older chapters on leaf arrangement or phyllotaxis that the Fibonacci series comes up. Among the earliest and most important of these are the memoirs of Braun (based on researches of Schimper and himself), 90 and L. et A. Bravais.91 Of later papers there are those by Ellis,92 Dickson,93  Wright,94 Airy,95 Gunther,96 and Ludwig.97  Much that was fanciful and mysterious was swept away by the publication of P. G. Tait's note "On Phyllotaxis.'' 98 Of recent books on the subject the most notable are those by Church,99 Cook,100 and Thompson.101 The first two are beautifully illustrated. The third is a scholarly work, written in an attractive style; it reproduces Tait's discussion in an appreciative manner.


NOTES
1 Most of the following notes appeared in The American Mathematical Monthly, April and May, 1918, but extensive additions, and some corrections, are here introduced.
2 Historical sketches and some of the properties of the curve are given in F. Gomes Teixeira, Traité des courbes spéciales remarquables, tome 2, Coimbre, Imprimerie de l'université, 1909, pp. 76-86, 396-399, etc.; in G. Loria, Spezielle algebraische una transzendente ebene Kurven, Band 2, 2. Auflage, Leipzig, Teubner, 1911, pp. 60 ff.; in Mathematisches Worterbuch... angefangen von G. S. Klugel.. . fortgesetzt von C. B. Mollweide, Leipzig, Band 41, 1823, pp. 429-440.
3 The curve arises in the discussion of a problem in dynamics. For references see the next footnote.
4 Oeuvres de Descartes, tome 2, publiées par C. Adam et P. Tannery. Paris, Cerf, 1898, p. 360; also pp. 232-234; (see Montucla, Histoire des Mathématiques, nouvelle édition, tome 2, Paris, 1799, P. 45). Cf. I. Barrow, Lectiones Geometricae, Londini, 1670, p. 124; or English edition by J. M. Child, London, Open Court, 1916, pp.136-9, 198. From the discussion and figure of Descartes it seems certain that he had no conception of O as an asymptotic point of the spiral. This property of the point was remarked in a letter, dated July 6, 1646, from Toricelli to Robervall (L'Intermédiaire des mathématiciens, 1900, vol. 7, p. 95). See also G. Loria, Atti della accademia dei Lincei, 1897, p. 318.
5  The intrinsic equation smR = K represents a logarithmic spiral when m = - 1, a clothoide when m = 1, a circle when m = 0, the involute of a circle when m = -1/2 and a straight line when m = oo.  Haton de la Goupillière remarked, and Allegret proved (Nouvelles annales de mathématiques, tome 11 (2), 1872, p. 163,) that the logarithmic spiral may be regarded also as a particular case of the spiral sinusoid.
That is, the length of the arc measured from the pole is equal to the length of the tangent drawn at the extremity of the arc and terminated by the line drawn through the pole perpendicular to the radius vector, that is, "the polar tangent." The logarithmic spiral was the first transcendental curve to be rectified.
"Specimen alterum calculi differentialis in dimetienda Spirali Logarithmica Loxodromiis Nautarum . . ," "per J.B.," Acta eruditorum, 1691 pp. 282-283; Opera, tome I, Genevae, 1744, pp. 442-443. Loria's references (l. c., p. 61) to Varignon and Bernoulli are distinctly misleading. In 1675 John Collins used, in this connection, the expression "the spiral line is a logarithmic curve," Correspondence of Scientific Men of the Seventeenth Century, vol. I, 1841, p. 219; [Quoted in full in a later footnote, page 150].
    In more than one place Bernoulli refers to the logarithmic spiral as the ' Spira mirabilis,' e.g., Opera, tome  I, pp. 491, 497, 554; also Acta eruditorum, 1692 and 1693.
P. Nicolas, De Novis Spiralibus, Exereitationes Duae . . . In posteriori autem agitur de alia quadam spirali a prioribus longe diversa, de qua Vvallisius & I/vrenius insignes Geometrae scripserunt; & quae illi non attigere circa  Tangentem hujus spiralis, spatiorum illa contentorum, & curvae ipsius dimensionem absolvuntur. Tolosae, 1693. "Exercitatio II. De spiralibus geometricis" pp. 27-44. Appendix, pp. 45-5I. The following quotation from page 27 may be given: "Esto curva BCDEF cujis sit talis proprietas, ut omnes radii AB, AC, AD, AE, AF constituentes angulos aequales in centro A sine inter se in continua proportione Geometrica. Propter hanc insignen proprietatam curvam BCDEF vocamus Spiralem Geometricam ut distinguatur a Spiralis communi & Archimedea, cujus proprietas est ut radii aequales angulos ad centrum sive principium Spiralis constituentes sese aequaliter excedant, ac proinde servent proportionera Arithmeticam."
9 E. Halley, Philosophical Transactions, 1696. The lengths of segments cut off from a radius vector between successive whorls of the spiral form a geometric progression.
10 A term originating with R. Cotes, Philosophical Transactions, 1714; reprinted after the death of Cotes in his Harmonia Mensurarum, Cantabrigiae, 1722 ("Aequiangula spiralis," p. 19). The term was revived more recently by Whitworth in Messenger of Mathematics, 1861.
11 See two letters of Collins, one undated and the other dated Sept. 30, 1675, in Correspondence of Scientific Men of the Seventeenth Century . . . Vol. 1, Oxford, University Press, 1841, pp. 144, 218-19. The result was first given in print by E. Halley, in Philosophical Transactions, 1696.
    Cf. F. G. M., Exercices de Géométrie Descriptive, 4e éd., Paris, Mame, 1909, pp. 824-6. Chasles showed (Apercu historique, etc.,.. . 2e éd., Paris, 1875, p. 199) that if the logarithmic curve generates a surface by revolving about its asymptote, and if this asymptote is the axis of a helicoidal surface, the two surfaces cut in a skew curve whose orthogonal projection on a plane perpendicular to the asymptote is a logarithmic spiral. See also H. Molins, Mémoires de l'académie des sciences inscriptions et belles-lettres de Toulouse, tome 7 (sem. 2), 1885, p. 293 f.; tome 8, 1886, pp. 426. That the logarithmic spiral is a projection of a certain "elliptic logarithmic spiral" was shown in W. R. Hamilton, Elements of Quaternions, London, 1866, pp. 382-3. For other quaternion discussion of the logarithmic spiral see H. W. L. Hime, The Outlines of Quaternions, London, 1894, pp. 1713.
12 Cf. Turquan, "Démonstrations élémentaires de plusieurs propriétés de la spiral logarithmique," Nouvelles annales de mathématiques, tome 5, 1846, pp- 88-97. "Note" by Terquem on page 97.
13 J. Wallls, Tractatus Duo, 1659, pp. 106-107; also Opera, tome I, 1695, pp. 559-561.
14 Paragraph 9 of an article in Acta eruditorum, May, 1692, entitled "Lineae cycloidales, evolutae, antevolutae, causticae, anti-causticae, peri-causticae. Earurn usus et simplex relatio ad se invicem. Spira mirabills. Aliaque per I.B." Cf. Oeuvres Complètes de Christian Huygens. Tome 10. La Haye, 1905, p. 119. The center of curvature at a point on a logarithmic spiral is the extremity of the polar subnormal of the point.
15 The nth positive pedal of the spiral p = kecFwith respect to the pole is

(J. Edwards, Elementary Treatise on the Differential Calculus, 3d edition, London, Macmillan, 1896, p. 167).
16 Cf., Acta eruditorum, 1706, p. 44. Cf. Acta eruditorum, 1692, p. 212; also Opera, tome 1, Genevae, 1744, P. 502, and p. 30 of "Vita."
17 Cf., L. Isely, "Epigraphes tumulaires de mathématiciens," Bull. de la société des sciences naturelies de Neuchatel, tome 27, 1899, p. 171 . See also W. W. Rupert, Famous Geometrical Theorems and Problems (Heath's Mathematical Monographs, part 4), Boston, 1901, p.99.
18 Book I, proposition 9, and book II, propositions 15 and 16.
19 The hodograph of an equiangular spiral is an equiangular spiral (W. Walton, Collection of Problems in Illustration of the Principles of Theoretical Mechanics, 3d ed., Cambridge, 1876, p. 296). In a chapter on electromagnetic observations in J. C. Maxwell's Treatise on Electricity and Magnetism (vol. 2, Oxford, Clarendon Press, 1873, pp. 336-8) the discussion calls for the investigation of the motion of a body subject to an attraction varying as the distance and to a resistance varying as the velocity. This leads to the reproduction of Tait's application (Proc. Royal Society of Edinburgh, vol. 6, 1867, p. 221 f.) of the principle of the hodograph to investigate this kind of motion by means of the logarithmic spiral.
    "If a particle be describing a logarithmic spiral under the action of a force to the pole, and simultaneously the law of force be altered to the inverse biquadrate and the velocity to v'2/3 X its previous value, the particle will proceed to describe a cardioide." Purkiss, Messenger of Mathematics, vol. 2, 1864. For other results of this type, involving the spiral, see Newton's Principia, first book, sections I-III, with notes and illustrations by P. Frost, London, 1880, p. 203.
20  Also in Christiani Hugenil Zuelechemi . . . Opera Reliqua, tome I, Amstelodami, 1728, pp. 136-288.
21  "Nouvelle formation de spirales," Histoire de l'académie royale des science, année 1704, Paris, 1706, pp. 69-131; see especially pp. 113f.
22   For example: D. C. Jackson and J. P. Jackson, Alternating Currents and Alternating Current Machinery, New edition, New York, 1917, pp. 13-15. The discussion in this connection seems to have originated with C. P. Steinmetz, Trans. Amer. Inst. Electrical Engs., vol. 10, p. 527; Elektrotechnische Zeitschrift, June 20, 1890.
23   I. Stringham, "A classification of logarithmic systems," American Journal of Mathematics, vol. 2, pp. 187-194.
24   Bulletin of the New York Mathematical Society, vol. 2, pp. 164-170, 1893. See also I. Stringham, Uniplanar Algebra, San Francisco, 1893.
25   L. Cremona, Graphical Statics. Translated by T. H. Beare, Oxford, Clarendon Press, 1890, pp. 59-64 Italian edition, Torino, 1874, pp. 39-42. The xylonite logarithmic spiral curve (eight inches in width) sold by Keuffel & Esser Co., New York, furnishes the means for accurately and rapidly drawing the curve. The curvature gradually changing it is peculiarly adapted for fitting to any part of a given curve. It assists in the rapid determination of the center of curvature of a given part of the curve, and, hence, in drawing evolutes and equidistant curves. An eight-page pamphlet by W. Cox (The logarithmic spiral curve and description of its uses,1891) accompanies the instrument. Eugene Dietzgen & Co., Chicago, manufactured a similar celluloid instrument and a ten-page pamphlet descriptive of its use was written by E. M. Scoefeld, and entitled the logarithmic spiral curve (Chicago, 1892).
26  Wien, 1885; especially pp. 40-75.
27  London, 1887, p. 322.
28  "On the Theory of Rolling Curves," Transactions of the Royal Society of Edinburgh,vol. 16, part V, 1849, pp. 519-40. [The Scientific Papers of T. C. Maxwell, edited by W. D. Niven, vol. 1, Cambridge, 1890, pp- 519-40.] Loria, Gomes Teixeira, and Wieleitner seem to be equally ignorant of this paper.
29  JournaI de mathématique pures et appliquées, tome 11 (2), 1866, pp. 329-336.
30  Mathesis, tome 2, 1882, pp. 217-2I9.
31  H. Dittrich, Die logarithmische Spirale, Progr. Breslau, 1872 .
32  F. Klein and R. Fricke, Vorlesungen uber die Theorie der elliptischen Modulfunctionen, Band I, Leipzig, Teubner, 1892 p. 168.
33  G. Holzmuller, Einfuhrung in die Theorie der isogonalen Verwandtschaften und der conformen Abbildung, Leipzig, Teubner, 1882, pp. 65, 238-241; and "Ueber die logarithmische Abbildung und die aus ihr entspringenden Curvensysteme," Zeitschrift fur Mathematik und Physik, Band 16, 1871, pp. 269-289.
34  Mathematische Annalen, Band 4, 1871, P. 77. Cf. Encyklopadie der mathematischen Wissenschaften, Band IlI, Leipzig, 1903, pp. 210, 212; also Clebsch-Lindemann, Vorlesungen uiber Geometrie, Band I, Leipzig, Teubner, 1876, p. 995.
35  This description may be found in T. Olivier, Complements de geometrie descriptive, Paris, 1845, p. 445. See also T. Olivier, Memoires de geornetrie descriptive, Paris, 1851, p. 284.
36  This letter is printed in Correspondence of Scientific Men of the Seventeenth Century, vol. I, Oxford, 1841. The paragraphs of special interest in this connection are as follows: "As to the instrument invented by M. Tschirnhaus for dividing an angle in ratione data, we suppose he gives an angle as geometers do, ready drawn by accident or at pleasure, and then I conceive it an instrument worthy the author: whereas here (so far as I know) we have nothing but the old mechanism, viz. to measure the angle in degrees first, by aid of a sector or opening joint, and then set off the part proportional by aid of an arch or line of chords, which one of the legs may draw after it, which part proportional may be attained by a sliding scale with logcal lines upon it, which may be annexed to the other leg; but here I will a little enlarge on the use of M. Tschirnhaus's invention.
    "We have an instrument called the serpentine line, or, as Oughtred terms it the circles of proportion, in the use whereof, in relation to compound interest, it is often required to divide an angle in ratlone data, or an angle being given to enlarge it in ratlone data. Moreover, conceive the eye at the south pole, projecting the loxodromia or rumb of a ship's course on the earth, on a plane touching the sphere at the north pole, the projected curve will be a spiral line, in which, if the polar rays PE, PD, PC, PjE, [ the figure of the letter is omitted ] make equal angles at the pole P, those rays will be in continual geometrical proportion; and conceiving a circle described upon P as a centre, the equal segments of the arch in the circumference, made by the polar rays, will be an arithmetical progression, suited to a geometrical one; consequently the spiral line is a logarithmic curve and from hence the meridian line of the true sea chart may be demonstrated to be a line of logarithmic tangents, and the spiral line with M. Tschirnhaus's angular instrument, makes the mesolabe [ an instrument for finding mean proportionals between two numbers ], which our late learned Oughtred said was hitherto tenebris obvolutum.
    "To rectify or straighten this spiral, or part of it, as EAE, is all one effect as to draw a touch-line to it, or to find the rumb between two places whose latitudes and difference of longitude are given which to perform in lines is a proposition of great use, and hitherto wanting in navigation, and depends on the quadrature of the hyperbola, as Dr. Barrow, at my instance, proved in his Geometrical Lectures. Moreover such a spiral, being once well described, may serve to take away the use of compasses in Galileus or our Gunter's sector or joint for proportions, all which I thought not impertinent to hint."
37  "Om ratta Formen pa Skepps-Ankrar," Svensk. Vetensk. Academ. nya Handl., 1796, Vol.17, pp. 1-24. Abridged and translated in Annalen der Physik (Gilbert), Band 6, Halle, 1800: "Von der richtigen Form der Schiffsanker," pp. 81-95.
38  W. J. M. Rankine, Manual of Machinery and Millwork, London, 1869, pp. 99-102;
    C. W. MacCord, Kinematics, New York, 1883, pp. 47-50;
    F. Reuleaux, Lehrbuch der Kinematik, Band 2: Die praktischen Beziehungen der Kinematik zu Geometric und Mechanik, Braunschweig, 1900, pp. 473, 542-544;
    P. Schwamb and A. L. Merrill, Elements of Mechanism, New York, 1913, pp. 32-36;
    R. F. McKay, The Theory of Machines, London, 1915, pp. 218-222.
    F. DeR. Furman, "Cam design and construction," American Machinist, vol. 51, pp. 695-698, Oct. 9, 1919.
39  Edinburgh, 1821, p. 438.
40  3 For pictures of the nautilus pompilius see pp. 494, 58I, 582 of D. W. Thompson, On Growth and Form, Cambridge University Press, 1917, and also pp. 57, 457 of T. A. Cook, The Curves of Life, London, Constable, 19I4. This latter work contains many beautiful illustrations and logarithmic spiral forms are specially discussed on pages 60-63, 413-421; another work by the same author, Spirals in Nature and Art, London, Murray, 1903, has some good illustrations.
41  Philosophical Transactions of the Royal Society, London, Vol. 128, 1838, pp. 35 1-370.
42  As early as 1701 Guido Grandi showed, l. c., as already noted, that the orthogonal projection of this spiral on a plane perpendicular to the axis of the cone is a logarithmic spiral. The gauche spiral has been studied by Th. Olivier (who called it the conical logarithmic spiral), Developpements de géométrie descriptive, 1843, pp- 56-76; by P. Serret, Theorie nouvelle géométrique et mecanique des lignes a double courbure, 1860, p.101; etc. A number of results are collected by Gomes Teixiera, l. c., pp. 396-400.
     For other surfaces involving the logarithmic spirals reference should be given to the very interesting pages 132-313 of G. Holzmiiller, Elemente der Stereometrie, Dritter Teil, Leipzig, Goschen, 1902, on logarithmic spiral tubular surfaces and their inverses.
43  This occupies almost the whole of the third volume of Annaes scientificos da academia polytechnica do Porto, Coimbra, 1908. Cf. L'Intermediaire des mathématiciens, 1900, tome 7, p. 40; 1901, tome 8, pp. 167, 314; 1910, tome 17, p.155.
44  A. H. Church, On the Relation of Phyllotaxis to Mechanical Law, London, Williams and Norgate, 1904.
45  In L'Intermediaire des mathématiciens, 1909, and in H. A. Naber, Das Theorem des Pythagoras, Haarlem, Visser, 1908, opposite p. 80.
46 These enunciations are taken from The Thirteen Books of Euclid's Elements translated with introduction and commentary by T. L. Heath, 3 vols., Cambridge, at the University Press, 1908. For statements in connection with our discussion see particularly, Vol. 1, pp.137, 403; Vol. 2, p. 99; Vol. 3, P. 441.
47 The earliest instances which I find of the use of the term golden section are in J. Helmes, "Eine einfachere, auf einer neuen Analyse beruhende Auflosung der sectio aurea, nebst einer kritischen Beleuchtung der gewohnlichen Auflosung und der Betrachtung ihres padagogischen Werthes." Archiv der Mathematik, Grunert, Band 4, 1844, pp. 15-22; in A. Wiegand, Geometrische Lehrsatze und Aufgaben, Band 2, 1. Abtheilung, Halle, 1847, p. 142; and also in A. Wiegand, Der allgemeine goldene Schnitt und sein Zusammenhang mit der harmonischen Theilung. . . Halle, 1849.
    Much negative evidence seems to indicate that the term 'golden section' was originated within the thirty years 1815-1844. For example, it is not mentioned in Klugel-Mollweide's Mathematisches Worterbuch, which contains so many references to the literature of different topics. We do, however, find the following (Erste Abtheilung, vierter Theil, Leipzig, 1823, p. 363): "Die Aufgabe bey Eukleides II, 11, oder VI. 30 ist sonst auch bisweilen sectio divina genannt."
48   Nouvelles annales de mathématiques, Paris, tome 12, 1853, p. 38.
49  Journal de mathématiques pures et appliquées, Paris, tome 3, 1838, p. 98.
50  J. Leslie, Elements of Geometry, geometrical Analysis and Plane Trigonometry, Edinburgh, 1809, p. 66.
51 Divina Proportione opera a tutti gli ingegni perspicaci e curiosi necessaria que ciaseum studioso di philosophia: prospettiva, pictura, sculptura, architectura: musica: e altre matematice . . . Venetiis . . .1509.  Although not printed till 1509 the manuscript of this work was completed in 1497. The geometrical drawings were made by Leonardo da Vinci; cf. G. Libri, Histoire des Sciences math. en Italie, tome 3, Paris, 1840, p. 144, note 2. Another edition of the Latin text "herausgegeben, ubersetzt und erlautert von C. Winterberg" appeared at Vienna (Graser)  1889. Another edition 1896, 6 + 367 pp. A full analysis of Pacioli's work is to be found in A. G. Kiistner, Geschichte der Mathematik.. . Band I, Gottingern, 1796, pp. 417-449- See also M. Cantor, Vorlesungen uber Geschichte der Mathematik, Band 2, 2. Auflage, Leipzig, 1900, pp. 341 ff.,, 347.
52 It has been shown by G. Mancini that parts of Pacioli's Divina Proportione were taken from a Vatican manuscript by Pier della Francesca. See (1) G. Pittarelli, Atti del IV. congresso dei matematici, tomo 3, Roma, 1909;  (2) G. Mancini, "L'opera 'De Corporibus Regularibus' di Pietro Franceschi detto Francesca .usurpata da Fra Luca Pacioli" (con dodici tavole) Reale accademia dei Lincei, 1915. See review by F. Cajori in the American MathematicalMonthly, Vol. 23, 1916, p. 384.  (3) G. B. de Toni, "Intorno al codice sforzesco 'De divina proportione' di Luca Pacioli e i disegni geometrici di qust' opera attributi a Leonardo da Vinci," Modena soc. dei naturalistic e matematici, atti, 134, 1911, pp. 52-79.
53 Exact references to sources, and some quotations from originals, are given in (1) J. Tropfke, Geschichte der Elementar-Mathematik, Band 2, Leipzig, Veit, 1903; (2) F. Sonnenburg, Der goldne Schnitt. Beitrag zur Geschichte der Mathematik und ihre Anwendung. (Progr.), Bonn, 188I. (Not always reliable.) Cf. ftn. 4, P. 155.
54  Annali di scienze matematiche e fisiche (Tortolini), vol. 6, 1855, pp. 307-3-8; also M. Cantor, Vorlesungen uber Getchichte dee Mathematik, vol. 2, 2. ed., 1900, pp. 105-106; see also American Mathematical Monthly, vol. 25, 1918, p.197, and Bulletin of the American Mathematical Society, vol. 15, 1909, p. 408.
55 Les eouvres mathtématiques de Simon Stevln... le tout revu, corrige et augmente par A. Girard. Leyde, 1634, pp. 169-170. The passage in question is reprinted with commentary in G. Maupin, Opinions et Curiotes touchant la Mathematique (deuxieme serie), Paris, 1902, pp. 203-209. It has been discussed also by R. Simson, Philosophical Transactions, 1753, vol. 48, pp. 168-377; see "Reflexions sur la preface d'un memoire de Lagrange intitule 'Solution d'un probleme d'arithmetique" by J. Plana, Memoire della r. accademia d. sdenze di Torino, series 2, vol. 20, Torino, 1863, especially pp. 89-92.
56  C. Thiry, "Quelques proprietes d'une droite partagee en moyenne et extreme raison," Mathesis, 1894, vol. 14, pp. 22-24.
57  "Extension of the medial section problem and derivation of a hyperbolic graph," Proceedings of the Edinburgh Mathematical Society, 1897, Vol. 15, pp. 65-69.
58   6e ed., Paris, 1879, pp. 261-263. Some of these properties are given in the first edition of this work, which was really written by H. C. de La Fremoire, Paris, 1844.
59   D. H. Emsman, Zur sectio aurea. Materialien zu elementaren namentlich durch die Sectio aurea loslichen Constructions-aufgaben, etc. Progr. Stettin, 1874 (Cf.Zeitschrift f. math. und naturw. Unterricht, vol. 5, pp. 289-291)
60  For example (1) Neue Lehre von den Proportionen des menschlichen Korpers aus einem bisher unerkannt gebliebenen, die ganze Natur und Kunst durchdringenden morphologischen Grundgesetze entwickelt, Leipzig, 1854, 457 pp.; particularly pages 133-174;  (2) Aesthetische Forschungen, Frankfort, 1855, pp. 179f. (3) Das Normalverhaltnis der chemischen und morphologischen Proportionen, Leipzig, 1856, 114 pp. and the post-humous work: (4) Der goldene Schnitt, Leipzig, 1884, 28 pp. Cf. S. Guinther, "Adolph Zeising als Mathematiker," Zeltschriftfiir Mathematik und Physik, Historisch-literarische Abtheilung, Band 21, 1876, pp. 157-165.
61   G. T. Fechner, Zur experlmentalen Aesthetik, Leipzig, 1871; also Vorschule der  Aesthetik, Leipzig, 1876, pp. 185f.
62   C. L. A. Kunze speaks of "Rechteck der schonsten Form" in his Lehrbuch der Planimetrie, Weimar, 1839, p. 124. A reference may be given to a recent discussion of "printer's oblong" and "golden oblong" in H. L. Koopman, "Printing page problems with geometric solutions," The Printing Art, Cambridge, Mass, 1911, vol. 16, pp. 353-356.
63   New York, vol. 1, 1901, p. 4-6.
64   Harvard Psychological Studies, vol. 1, 1903, pp. 541-561.
65   L. Witmer, "Zur experimental Aesthetik einfacher riiumlicher Formverhaltnisse" Philosophische Studien, Leipzig, vol. 9, I893, pp. 96-144, 209-263.
66   W. Wundt, Grundzuge der physiologischen Psychologie, Band 2, 4- Auflage, 1893, pp. 240f. (See also Band 3, 6. Auflage, 1911, pp. 136f.).
67  O. Kulpe, Outlines of  Psychology, translated into English by E. P. Titchener, London, 1895, pp. 253-255.
68  M. Dessoir, Aesthetik und allgemeine Kunstuissenschaft in den Grundzuigen dargestellt, Stuttgart, 1906, pp. 124f, 176-177.
69   J. Volkelt, System der Aesthetik, Band 2, Munchen, 1901, pp. 33f.
70   T. A. Cook, The Curves of Life, London, Constable, 1914.
71    (a) "Die Proportion des goldenen Schnittes an den Bliittern und Stengeln der Pflanzen," Zeitschrift fur mathematischen und naturwissenschaftlichen Unterricht, 1885, vol. 15, pp. 325-338; (b) Der goldene  schnitt und dessen  Erscheinungsformen in Mathematik Natur und Kunst, Augsburg, 1885], 3 + 232 pp. + 13 plates. A resume of this work given by O. Willman in Lehrproben und Lehrgange aus der Praxis der Gymnasien und Realschulen, 1892 was the basis of E. C. Ackermann, "The Golden Section," American Mathematical Monthly, 1895, vol. 2, pp. 260--264. Cf. Zeitschrift f. math. und naturwiss. Unterricht, 1887, vol. 18, pp. 44-47, 605-612.
72  H. Neikes, Der goldene Schnitt und ihre Geheimnisse der Cheops Pyramide, Coln, 1907; (reviewed in Jahrbuch uber die Fortschritte der Mathematik, 1907, p. 526). Pages 3-10: "der goldene Schnitt"; pages 11-20: "die Geheimnisse der Cheops Pyramide." C. Piazzi Smyth, Life and Work at the great Pyramids, 1867.
73  Il  liber Abbaci di Leonardo Pisano pubblicato da Baldassare Boncompagni, Roma, MDCCCLVII. For an analysis of this work see M. Cantor, Vorlesungen uber Geschichte der Mathematik, Band II, 3. Auflage, Leipzig, Teubner, 1900, pp. 5-35.
74   Pages 283-284.
75  S. Gunther, Geschichte der Mathematik, L Tell, Lelpzig, Gbschen, 1908, p. 137.
76   J. Kepler, Opera, ed. Frisch, tome 7, pp. 722-3. After discussions of the form of the bees' cells and of the rhombo-dodecahedral form of the seeds of the pomegranite (caused by equalizing pressure) he turns to the structure of flowers whose peculiarities, especially in connection with quincuncial arrangement he looks upon as an emanation of sense of form, and feeling for beauty, from the soul of the plant. He then "unfolds some other reflections" on two regular solids the dodecagon and icosahedron "the former of which is made up entirely of pentagons, the latter of triangles arranged in pentagonal form. The structure of these solids in a form so strikingly pentagonal could not come to pass apart from that proportion which geometers to-day pronounce divine." In discussing this divine proportion he arrives at the series of numbers 1, 1, 2,  3, 5, 8, 13, 21 and concludes: "For we will always have as 5 is to 8 so is 8 to 13, practically, and as 8 is to 13, so is 13 to 21 almost. I think that the seminal faculty is developed in a way analogous to this proportion which perpetuates itself, and so in the flower is displayed a pentagonal standard, so to speak. I let pass all other considerations which might be adduced by the most delightful study to establish this truth."
77   There is a typographical error (13 for 21) in Girard's discussion in this connection.
78   V. Schlegel, "Verallgemeinerung eines geometrischen Paradoxons," Zeitschrift fur Mathematik und Physik, 24. Jahrgang, 1879, pp. 123-128.
79   This paradox was given at least as early as 1868 in Zeitschrift fur Mathematik und Physik, Vol. 13, p. 162. Cf. W. W. R. Ball, Mathematical Recreations and Essays, 5th edition, London, Macmillan, 1911, p. 53; and E. B. Escott, "Geometric Puzzles," Open Court Magazine, vol. 21, 1907, pp. 502-5.
80    E. Catalan, Melanges Mathematiques, tome 2, [Liege, 1887], p. 319.
81   E. Lucas, "Note sur la triangle arithmetique de Pascal et sur la serie de Lame," Nouvelle correspondance mathematique, tome 2, 1876, p. 74.
82   D. Bernoulli, "Observationes de seriebus quae formantur ex additione vel subtractione quacunque terminorum se mutus consequentium," Commentarii academiae scientiarum imperialis Petropolitanae, vol. 3,  1732, p. 90. This memoir was read in September, 1718, but it appears that Bernoulli had the formula in his possession as early as I724 (Cf. Fuss, Correspondance mathematique et physique, St. Petersburg, 1843, vol.2, pp. 189, 193-4, 200-202, 209, 239, 251, 271, 277; see also p. 710). The formula was given also by Euler in 1726 (in an unpublished letter to Daniel Bernoulli). For most of these facts I am indebted to Mr. G. Enestrom. The formula seems to have been discovered independently by J. P. M. Binet, "Memoire sur l'integration des equations lineaires aux diffeences finies d'un ordre quelconque, a coefficients variables," Comptes rendus de l'academie des sciences de Paris, tome 17, 1843, p. 563.
83  Manuel des Candidats a l'ecole Polytechnique, tome 17, Paris, 1857, p. 86.
84   E. Lucas, (a) "Recherches sur plusieurs ouvrages de Leonard de Pise et sur divers es questions d'arithmetique supeieure. Chapter 1. Sur les series recurrentes," Bullettino di bibliografia e di storia delle scienze matematiche e fisiche, tome 10, pp. 129-170, Marzo, 1877; (b) Theorie des fonctions numeriques simplement peiodiques," American Journal of Mathematics, vol. 1, 1878, pp. 184--229, 289-321 [on p. 299 are given the first 61 terms of the Fibonacci series and the factors of every term]; (c) "Sur la theorie des nombres premiers" [dated mai 1876], Atti della r. accademia dellie scienze di Torino, vol. 11, 1875-76, pp. 928-937; (d) "Note sur l'application des series recurrentes a la recherche de la loi de distribution des nombres premiers," Comptes rendus de l'academie des sciences, vol. 82, 1876, pp. 165-167. See also A. Aubry, "Sur divers procedes de factorisation," L'Enseignement mathematique, 1913, especially õõ 11, 16 and 17, pp. 219-223.
85   "Note sur le partage d'une droite en moyenne et extrame, et sur un probleme d'arithmetique," Correspondance mathernatique et physique, vol. 9, 1837, pp. 483-484.
86   Traite Elementaire d'Arithmetique, Paris, 1841; also Nouvelles annales de mathematiques, vol. 1, 1842, p. 354.
87   G. Lame, "Note sur la limite du nombre des divisions dans la recherche du plus grand commun diviseur entre deux nombres entiers." Comptes rendus de l'academie des sciences. tome 19, ~844, pp. 867-870. See also J. P. M. Binet, idem, pp. 939-941.
    Because of results obtained in the above-mentioned memoir the Fibonacci series is frequently called the Lame series. Thompson's statement  (On Growth and Form, p. 643) that the series 2/3, 3/5, 5/8, 8/13, 13/21, ... "is called Lami's series by some, after Father Bernard Lami, a contemporary of Newton's, and one of the co-discoverers of the parallelogram of forces," is incorrect.
88   E. Landau, "Sur la serie des inverses des nombres de Fibonacci," Bulletin de la Societe Mathematique de France, tome, 17, 1899, pp. 298-300.
89   Archiv fur Mathematik und Physik Band 28, 1919, pp. 77-79.
90   A. Braun, "Vergleichende Untersuchung uber die Ordnung der Schuppen an den Tannenzapfen als Einleitung zur Untersuchung der Blatterstellung iiberhaupt," Nova acts acad. Caes Leopoldina, vol. 15, 1830, pp. 199-401.
91   L. et A. Bravais, (I) "Sur la disposition des feuilles curviserites," Ann. des sc. nat., 2e serie, vol. 7, 1837, pp. 42-110; (2) Memoire sur la Disposition geometrique des Feuilles et des Inflorescenses, Paris, 1838.
92   L. Ellis, Mathematical and Other Writings, Cambridge, 1863; "On the theory of vegetable spirals," pp.358-372.
93   Dickson, "On some abnormal cases of pinus pinaster," Transactions of the Royal Society of Edinburgh, vol. 16, 1871, pp. 505-520.
94  C. Wright, "The uses and origin of the arrangements of leaves in plants" (read 1871), Memoirs of the American Academy, Vol 9, part 1, Cambridge, Mass., p. 384f.
95   H. Airy, "On leaf arrangement," Proceedings of the Royal Society of London, vol. 21, 1873, pp. 176-179.
96   S. Gunther, "Das mathematische Grundgesetz im Bau des Pflanzenktrpers," Kosmos, II. Jahrgang, Band 4, 1879 pp- 270-284.
97   F. Ludwig, "Einige wichtige Abschnitte aus der mathematischen Botanik," Zeitschriftfiir mathematischen und naturwiss. Unterricht, Band 1, 1883, p.161f.
98   P. G. Tait, Proc. Royal Society Edinburgh, vol. 7, 1871, pp. 391-41.
99  A. H. Church, On the Relation of Phyllotaxis to Mechanical Laws, London, Willlams and Norgate, 1904. On page 5 Church writes: "The properties of the Schimper-Braun series 1, 2, 3, 5, 8, 13, ..., had long been recognized by mathematicians (Gerhardt, Lame) .... "In Botanisches Centralblatt, Band 68, 1896, F. Ludwig writes (on p. 7) that the numbers of this series "werden vielfach von Botanikern als Braun'sche, von Mathematikern als Gerhardt'sche oder Lame'sche Reihe bezeichnet."
I have not been able to verify that any mathematician used the term Gerhardt series in this connection, or that anyone by the name of Gerhardt wrote about the Fibonacci series. From what has been indicated above it seems certain that Ger-hardt'sche' should be "Girard'sche.'
100   T. A. Cook, The Curves of Life, London, Constable, 1914.
101  D'A. W. Thompson, On Growth and Form, Cambridge: at the University Press, 1917.

NOTES ON THE LOGARITHMIC SPIRAL, GOLDEN SECTION AND THE FIBONACCI SERIES,
by R. C. Archibald

(Note V in DYNAMIC SYMMETRY, by Jay Hambidge,Yale University Press, New Haven 1920:146-157).


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