Animal mechanism : a treatise on terrestrial and aerial locomotion

Front matter

NORTHAMPTON, MASS.

Cibrary.

Shelf.. J No. .1 Vol. —

Chapter I. Of Forces In The Inorganic Kingdom And Among Organised Beings

Matter reveals itself by its properties— When matter acts, we conclude that forces exist — Multiplicity of the forces formerly admitted ; tendency to their reduction to one force in the inorganic kingdom — Indestructibility of force ; its transformations — Vital forces, their multiplicity according to the ancient physiologists — Several vital forces are reduced to physical forces — Of laws in physics and in physiology — General theory of physical forces 5

Chapter II. Transformation Of Physical Forces

To prove the indestructibility of forces, we must know how to measure them — Units of heat and of mechanical work — Thermo-dynamics — Measure of forces in living beings — Successive phases of the transformation of bodies ; successive development of force under this influence — Thermo-dynamics applied to living beings . 13

Chapter III. On Animal Heat

Origin of animal heat — Lavoisier's theory — The perfecting of this theory — Estimates of the forces contained in aliment, and in the secreted products — Difficulty of these estimates — The force vi

Contents

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yielded by alimentary substances is transformed partly into heat and partly into work — Seat of combustion in the organism— Heating of the glands and muscles during tlieir functions— Seat of calorification — Intervention of the causes of cooling — Animal temperature — Automatic regulator of animal temperature . . 19

Chapter IV. Animal Motion

Motion is the most apparent characteristic of life ; it acts on solids, liquids, and gases — Distinction between the motions of organic and animal life — We shall treat of animal motion only — Structure of the muscles — Undulating appearance of the still living fibre— Muscular wave— Shock and myography — Multiplicity of acts of contraction — Intensity of contraction in its relations to the frequency of muscular shocks — Characteristics of fibre at different points of the body 27

Chapter V. Conteaction And Woek Of The Muscles

The function of the nerve— Speed of the nervous agent — Measures of time in physiology — Tetanus and muscular contraction — Theory of contraction — Action of the muscles .... 41

Chapter VI. Of Electricity In Animals

Electricity is produced in almost all organised tissues — Electric currents of the muscles and the nerves — Discharge of electric fishes ; old theories ; demonstration of the electric nature of this phenomenon — Analogies between the discharge of electrical apparatus and the shock of a muscle — Electric tetanus — Rapidity of the nervous agent in the electrical nerves of the torpedo ; duration of its discharge . 49

Chapter VII. Animal Mechanism

Of the forms under which mechanical work presents itself — Every machine must be constructed with a view to the kind of work which it has to perform — Correspondence of the form of muscle with the work which it accomplishes — Theory of Borelli — Specific force of muscles — Of machines ; they only change the

CONTENTS.

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form of work, but do not increase its quality— Necessity of alternate movements in living motive powers — Dynamical energy of animated motors , , . . . . . . .59

CHAPTER VIIL

HARMONY BETWEEN THE ORGAN AND THE FUNCTION.— DEVELOPMENT HYPOTHESIS.

Each muscle of the body presents, in its form, a perfect harmony with the nature of the acts which it has to perform — A similar muscle, in different species of animals, presents differences of form, if the function which it has to fulfil in these different species is not the same — Variety of pectoral muscles in birds, according to their manner of flight— Variety of muscles of the thigh in mammals, according to their mode of locomotion — Was this harmony pre-established ?— Development hypothesis— Eamarck and Darwin. . . , ... .69

Chapter IX. Variability Of The Skeleton

Reasons which have caused the skeleton to be considered the least variable part of the organism — Proofs of the yielding nature of the skeleton during life, under the influence of the slightest pressure, when long continued — Origin of the depressions and projections which are observed in the skeleton —Origin of the articular surfaces — Function rules the organ — Variability of the muscular system 85

Book The Second. Functions: Terrestrial Locomotion

CHAPTEE I.

OF LOCOMOTION IN GENERAL.

Conditions common to all kinds of locomotion— Borelli's comparison — Hypothesis of the reaction of the ground — Classification of the modes of locomotion, according to the nature of the point of resistance, in terrestrial, aquatic, and aerial locomotion — Of the

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CONTENTS.

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partition of muscular force between the point of resistance and the mass of the body — Production of useless work when the point of resistance is movable 102

CHAPTEK II.

TEREESTRIAL LOCOMOTION (BIPEDS).

Choice . of certain types in order to study terrestrial locomotion — Human locomotion — Walking— Pressure exerted on the ground, its duration and intensity — Re-actions on the body during walking— Graphic method of studying them — Vertical oscillations of the body — Horizontal oscillations — Attempt to represent the trajectory of the pubis — Forward translation of the body— Inequalities of its velocity during the instants of a pace 110

CHAPTEE III.

THE DIFFERENT MODES OF PROGRESSION USED BY MAN.

Description of the apparatus for the purpose of studying the various modes of progression used by man — Portable registering apparatus— Experimental apparatus for vertical reactions —Walking — Running — Gallop — Leaping on two feet and hopping on one — Notation of these various methods —Definition of a pace in any of these kinds of locomotion — Synthetic reproduction of the various modes of progression 124

CHAPTER IV.

QUADRUPEDAL LOCOMOTION STUDIED IN THE HORSE. Insufficiency of the senses for the analysis of the paces of the horse — Comparison of Duges — Rhythms of the paces studied by means of the ear — Insufficiency of language to express these rhythms — Musical notation — Notation of the amble, of the walking pace, of the trot — Synoptical table of paces noted according to the definition of each of them by different authors— Instruments intended to determine by the graphic method the rhythms of the various paces, and the re-actions which accompany them . .133

CHAPTER V.

EXPERIMENTS ON THE PACES OF THE HORSE.

Double aim of these experiments ; determination of the movements under the physiological point of view, and of the attitudes with

CONTENTS.

ix

PA.GE

reference to art— Experiments on the trot — Tracings of the pressures of the feet and of the re-actions — Notation of the trot — Piste of the trot— Representation of the trotting horse— Experiments on the walking pace — Notation of this kind of motion ; its varieties — Piste of the walking pace — Representation of a pacing horse 151

CHAPTER VI.

EXPERIMENTS ON THE PACES OF THE HORSE.

{Continued.)

Experiments on the gallop— Notation of the gallop — Re-actions — Bases of support — Piste of the gallop — Representation of a galloping horse in the various times of this pace — Transition, or passage, from one step to the other — Analysis of the paces bymeans of the notation rule — Synthetic reproduction of the different paces of the horse — Modes of walking of various quadrupeds 164

BOOK THE THIED. *

AERIAL LOCOMOTION.

Chapter I. Of The Flight Of Insects

Frequency of the strokes of the wing of insects during flight ; acoustic determination ; graphic determination — Influences which modify the frequency of the movements of the wing — Synchronism of the action of the two wings — Optical determination of the movements of the wing ; its trajectory ; changes in the plane of the wing ; direction of the movement of the wing . . . 180

Chapter II. Mechanism Of The Flight Of Insects

Causes of the movements of the wings of insects — The muscles only give a motion to and fro, the resistance of the air modifies the course of the wing — Artificial representation of the movements of the insect's wing— Of the propulsive efi'ect of the wings of

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CONTENTS.

insects — Construction of an artificial insect which moves horizontally— Change of plane in flight

Chapter III. Of The Flight Of Birds

Conformation of the bird with reference to flight— Structure of the wing, its curves, its muscular apparatus — Muscular force of the bird, rapidity of contraction of its muscles — Form of the bird ; stable equilibrium, conditions favourable to change of plane — Proportion of the surface of the wings to the weight of the body in birds of diff'erent size

Chapter IV. Of The Movements Of The Wing Of The Bird During Flight

Frequency of the movements of the wing — Relative durations of its rise and fall — Electrical determination— My ographical determination—Trajectory of the bird's wing during flight— Construction of the instruments which register this movement — Experiment—Elliptical figure of the trajectory of the point of the wing

Chapter V

OF THE CHANGES IN THE PLANE OF THE BIRD'S WING AT DIFFERENT POINTS IN ITS COURSE.

New determination of the trajectory of the wing — Description of apparatus — Transmission of a movement by the traction of a thread — Instrument and apparatus to suspend the bird — Experiment on the flight of a pigeon — Analysis of the curves — Description of the apparatus intended to give indications of the changes in the plane of the wing during flight — Relation of these changes of plane to the other movements of the wing . 244

CHAPTER VL

RE- ACTIONS OF THE MOVEMENTS OF THE WING ON THE BODY OF THE BIRD.

Re-actions of the movements of the wing — Vertical re -actions in different species ; horizontal re-actions or changes in the rapidity of flight ; simultaneous study of the two orders of re-actions — Theory of the flight of the bird — Passive and active parts of the wing— Reproduction of the mechanism of the flight of the bird . 264

LIST OF ILLUSTEmONS.

APPARATUS FOR EXPERIMENTING ON MOVEMENT.

PAGE

Fig. 2. — Theoretical representation of my ograph . . . . 31

Fig. 3.— Marey's myograph 32

Fig. 7. — Arrangement of a muscular bundle between two myo-

graphical clips 37

Fig. 19. —Experimental shoe, intended to show the pressure of the

foot on the ground, with its duration and its phases . 113

Fig. 42. — Experimental apparatus to show the pressure of the

horse's hoof on the ground 148

Fig. 43. — Apparatus to give the signals of the pressure and rise of

the horse's hoof . . . . . . . .149

Fig. 26. — Apparatus to determine the speed of walking at every

Fig. 27. — Runner provided with the apparatus intended to register

his different paces 126

Fig. 28. — Instrument to register the vertical re-actions during the

various paces 127

• Fig. 44. — Figure to represent a trotting horse, furnished with the different experimental instruments ; the horseman carrying the register of the pace. On the withers and the croup are instruments to show the re-actions . .150

Fig. * 93. — Apparatus for the purpose of experimenting on the contraction of the thoracic muscles of the bird . . .229

Fig. 99. — Buzzard flying, with the apparatus forgiving signals of

the movements of the extremity of its wing . . 241

Fig. 103. — General arrangement of the recording instrument!; a pigeon

attached to it, and conveying signals . . . . 248

Fig. 104. — Suspension of the bird in the apparatus .... 250

Fig. 109. — Apparatus to examine the movements of the wing, and

the changes in its plane . . . . . . . 260

Fig. 21. — Transmission of an oscillatory movement to the registering apparatus 116 xii

LIST OF. ILLUSTRATIONS.

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Fig. 24. — Showing two successive positions of the arm of the instrument, and the corresponding positions of the tracing points of the levers . . . . . . . . 120

Fig. 98. — Elastic point, tracing on a piece of smoked glass . . 239 Fig. 102. — Transmission of a to-and-fro movement by means of a

simple traction thread . . . . . . . 245

ILLUSTRATIVE APPARATUS.

Fig. 1. — Showing the transformation of the electricity of a coil

into mechanical work, heat, light, and chemical action . 10 Fig. 6. — Appearance presented by the waves in a muscular fibre . 36 Fig. 9. — Transformation of heat into work by a strip of india-rubber 39

Of the Flight of Insects.

Fig. 84. — Artificial representation of the movements of the insect's

wing . 199

Fig. 85. — Changes in the plane of the insect's wing . . . . 200

Fig. 87. — Artificial insect, to illustrate its flight .... 202 Fig. 88. — Arrangement of the artificial insect, so as to obtain the

hovering motion or the ascending flight . . . . 205

Of the Hovering of the Bird.

Fig. 90.— Instrument to illustrate the hovering of the bird . .217 Fig. 91. — The same, explaining the upward turn . . . . 218 Fig. 92. — ,, ,, downward ditto . . . 219

AISTATOMY.

Fig. 13.— Skeleton of a flaitiingo (after Alph. Milne-Edwards; the wing is very large, the sternum very short and deep, which indicates the size and the shortness of the pectoral muscles 72

Fig. 14.— Skeleton of a penguin: sternum very long, wing very

short 73

Fig. 15. — Skeleton of the wing and sternum of the sea swallow

(Hirundo marina) ; showing the extreme shortness of the sternum, and the great length of the wing . . . 74 Fig. 89. — Diff'erent curves in the wing of the bird at various parts of

its length 210

Fig. 117. — Active and passive parts of the bird's wing . . . . 276 Fig. 83. —Structure of the insect's wing 196

LIST OF ILLUSTRATIONS.

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Fig. 16. — Muscles of the thigli in man 76

Fig. 17. — Muscles of the thigh of the magot 77

Fig. 18. —Muscles of the thigh of the coaita 78

DETERMINATION'S.

Fig. 8. — Two determinations of the speed of the muscular wave . 38 Fig. 10. — Determination of the speed of the nervous agent in man . 43 Fig. 12. — Measure of the time which elapses between the excitation

of the electric nerve, and the discharge of the torpedo . 58 Fig. 82. — Determination of the direction of the movements in an

Fig. 94. — Experiment to determine by the electric and myographic methods at the same time, the frequency of the movements of the bird's wing, and the relative duration of its elevation and depression ...... 230

Fig. 26. — Determination of the rapidity of walking at various

instants, by means of a chronographic tuning-fork . 122

NOTATIONS.

Fig. 34. — Notation of a tracing of man's mode of walking . . 133 Fig. 35. — Synoptical notation of the four kinds of progression used

Fig. 36. — Notations of the gallop (man) 134

Fig. 37. — (Upper line) notation of a series of jumps on two feet.

(Lower line) notation of hops on right foot . . . 135

Notations of the Paces of the Horse.

Fig. 38. — Notation of a horse's amble 142

Fig. 39. — Notation of the horse's walking pace 143

Fig. 51. — Notation of the walking pace, with predominance of the

lateral pressures . . . . . . . .160

Fig. 45. — Graphic curves and notation of the horse's trot . . . 153

Fig. 40. — Notation of a horse's trot 144

Fig. 46. — Notation of the irregular trot 156

Fig. 41. — Synoptical notations of the paces of the horse, according

to various writers.

No. 1. Amble, according to all writers.

No 2 i -^^^^^^^ amble, according to Merche.

\ High steps, according to Bouley.

( Ordinary step of a pacing horse, according to Magure. No. 3. < Broken amble, according to Bouley.

( Traquenade, according to Lecoq.

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LIST OF ILLUSTRATIONS.

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Ko. 4. Normal walking pace, according to Lecoq.

No. 5. Normal walking pace (Bouley, Vincent and Goiffon

Solleysel, Colin).

No. 6. Normal walking pace, according to Eaabe. No. 7. Disconnected trot (trot de coursier). No. 8. Ordinary trot. (In the figure, it is supposed that the animal trots without leaving the ground, which occurs but rarely. The notation only takes into account the rhythm of the impacts of the feet).

No. 9. Norman pace, from Lecoq.

No. 10. Traquenade, from Merche . • • • • • 146

Fig. 56. — Gallop in three-time 166

Fig. 62. — Notation of the gallop in four-time 170

Fig. 63. — Notation of full gallop ; re-actions of this pace • .171

Fig. 64. — Transition from the walk to the trot . . • • . 174 Fig. 65. — Transition from the trot to the walk . • • .174

Fig. 66. — Transition from the trot to the gallop 174

Fig. 67. — Transition from the gallop to the trot .... 174

Fig. 68. — Notation rule, to represent the different paces . . . 175 Fig. 69. — Notation rule forming the representation of the gallop in

three-time 176

PISTES OR FOOT TRACES OF THE HORSE'S FEET,

Fig. 52. — Piste of the walking pace, after Vincent and GoifFon . . 162 Fig. 53. — Piste of the amble, after Vincent and Goiffon . . .162 Fig. 47. — Piste of the trot according to Vincent and GoifFon . . 157 Fig. .57. — Piste of the short gallop in three-time .... 167 Fig. 58. — Piste of EcUpse^s gallop, from Cornieu. The prints of the

hind-feet are very far before those of the fore-feet . .167

REPRESENTATION OF THE HORSE IN ITS VARIOUS PACES.

Fig. 54.— Representation of the horse at a walking pace . . .163 Fig. 48. — Horse trotting with a low kind of pace . . . . 157 Fig. 49. — Horse at full trot. The dot placed in the notation corresponds with the attitude represented . . . .158 Fig. 59. — Horse galloping in the first time (right foot advancing),

the left foot only on the ground . . . . . 168 Fig. 60. — Horse galloping in the second time (right foot forward) . 169 Fig. 61. — Horse galloping in the third time (right foot forward) . 169

LIST OF ILLUSTRATIONS.

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PAGE

TRACINGS.

Tracings of the Muscles.

Fio. 4. — Character of the shock according to the degree of fatigue

of the muscle . . . . . . . . . 34

Fig. 5. — Successive transformations of the shock of a muscle becoming gradually poisoned by veratrine. Underneath and on the left of the figure are shown the first efi'ects of the poison .35

Fig, 11. — Gradual coalescence of the shocks produced by electric

excitations of increasing frequency 46

Tracings of Human Locomotion.

Fig. 20. — Tracings of the impact and the pressure of the two feet in

our ordinary walk . . . . . . .114

Fig. 22.— Tracings of the oscillations of the body during walking . 117 Fig. 25. — Tracing of the impact and rise of the right foot, furnished by a lever subjected at the same time to 10 vibrations

per second 121

Fig. 29. — Tracing produced by walking upstairs .... 128 Fig. 30. — Tracing produced by running (in man) . . . . 128 Fig. 31. — Man galloping (right foot foremost). Step-curves and reactions. There is an encroachment of one curve over the other, and then a suspension of the body . . .131 Fig. 33. — Series of hops on the right foot. The duration of the time of suspension remains evidently constant, even when that of the pressure of the foot varies . . . 132 Fig. 32. — Leap on two feet at once 131

Tracings of the Locomotion of the Horse.

Fig. 50. — Tracing and notation of the walking pace, with equal

pressures of the feet, both diagonally and laterally . .160

Fig. 45. — Tracing aud notation of the trot 153

Fig. 55. — Tracing and notation of the gallop in three-time . • . 165

Tracings of the Flight of Insects.

Fig. 70. — Showing the frequency of the strokes of the wing of a

drone- fly and a bee .183

Fig. 72.— Graphic tracing of the middle portion of the course of a

bee's wing 189

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PAGE

Fig. 73. — Tracing of the middle zone of a humming-bird moth . 190

Fig. 74. — Tracing of the course of a wasp's wing showing the upper

part of the curve ........ 190

Fig. 75. — Tracing of the course of a wasp's wing : lower loops . 191

Fig. 77. — Tracing obtained from a bee's wing in a plane tangential

to the cylijider 192

Fig. 78 and 79. — Tracing of a wasp's wing, compared with a Wheatstone's rod 192, 193

Fig. 80. — Tracing of the wing of a humming-bird moth (lower

border) 193

Fig. 81. — Tracing of the wing of a tired humming-bird moth . . 194

Tracings of the Flight of Birds.

Fig. 95. — Myographical tracing to determine the frequency of the

strokes of the wing in different species . . . . 232 Fig. 96. — Differences of frequency and of amplitude in the strokes

of a pigeon's wing .234

Fig. 105. — Tracing of different movements of the pigeon's wing . . 253 Fig. 106-107. — Construction of the trajectory of a pigeon's wing, 254, 255 Fig. 110. — Simultaneous tracing of the different movements of a

buzzard's wing 262

Fig. 111. — Inclination of the plane of the wing, with reference to the

axis of the body during flight 263

Fig. 113. — Vertical oscillations of the bird during flight . . . 266 Fig. 114. — Relation of oscillations with muscular acts . . . 268 Fig. 115. — Simultaneous tracing of two kinds of oscillation in the

buzzard 271

TRAJECTORIES.

Fig. 23. — Attempt to illustrate, by means of a metallic wire, the

sinuous trajectory passed through by the pubis . .119 Fig. 71. — Appearance of a wasp the tips of whose wings have been

gilded 187

Fig. 86. — Trajectory of an insect's wing 201

Fig. 100.— Elliptical course of the point of a bird's wing . . . 242 Fig. 76.— Tracing of a vibrating Wheatstone's rod . . . . 191

Fig. 79. — Do. tipped with a wasp's wing 193

Fig. 101. — Ellipse traced by a Wheatstone's red on a revolving

cylinder 243

TEEHESTRIAL AND AERIAL LOCOMOTION.

INTEODUCTION.

Living beings have been frequently and in every age compared to machines, but it is only in the present day that the bearing and the justice of this comparison are fully comprehensible.

No doubt, the physiologists of old discerned levers, pulleys, cordage, pumps, and valves in the animal organism, as in the machine. The working of all this machinery is called Animal Mechanics in a great number of standard treatises. But these passive organs have need of a motor ; it is life, it was said, which set all these mechanisms going, and it was believed that thus there was authoritatively established an inviolable barrier between inanimate and animate machines.

In our time it is at least necessary to seek another basis for such distinctions, because modern engineers have created machines which are much more legitimately to be compared to animated motors ; which, in fact, by means of a little combustible matter which they consume, supply the force requisite to animate a series of organs, and to make them execute the most various operations.

The comparison of animals with machines is not only legitimate, it is also extremely useful from different points of view. It furnishes a valuable means of making the mechanical phenomena which occur in living beings understood, by placing them beside the similar but less generally known phenomena, which are evident in the action of ordinary

ANIMAL MECHANISM.

machines. In the course of this book, we shall frequently borrow from pure mechanics the synthetical demonstrations of the phenomena of animal life. The mechanician, in his turn, may derive useful notions from the study of nature, which will often show him how the most complicated problems may be solved with admirable simplicity.

Animal mechanics is a wide field for exploration. To every function, so to speak, a special machinery is attached. The circulation of the blood, the respiration, &c., may and ought to be treated separately, so that we shall limit this work to the study of one single, essentially mechanical, function, locomotion in the various animals.

It is easy to demonstrate the importance of such a subject as locomotion, which, under its different forms, terrestrial, aquatic, and aerial, has constantly excited interest. Whether man has endeavoured to utilize to the utmost his own motive power, and that of the animals ; whether he has sought to extend his domain, to open a way for himself in the seas, or to rise into the air, it is always from nature that he has drawn his inspirations. We may hope that a deeper knowledge of the different modes of animal locomotion will be a point of departure for fresh investigations, whence further progress will result.

Every scientific research has a powerful attraction in itself; the hope of reaching the truth suffices to sustain those who pursue it, through all their efforts ; the contemplation of the laws of nature has been a great and noble source of enjoyment to those who have discovered them. But to humanity, science is only the means, progress is the aim. If we can show that a study may lead to some useful application, we may induce many to pursue it, who would otherwise merely follow it from afar, with the interest of curiosity only. Without pretending to recapitulate here all that has been gained by the study of nature, we shall endeavour to set forth what may be gained by studying it still further, and with more care.

Terrestrial locomotion, that of man, and of the great mammals, for instance, is very imperfectly understood as yet. If we knew under what conditions the maximum of speed, force.

Introduction

or labour which the living being can furnish, may be obtained, it would put an end to much discussion, and a great deal of conjecture, which is to be regretted. A generation of men would not be condemned to certain military exercises which will be hereafter rejected as useless and ridiculous. One country would not crush its soldiers under an enormous load, while another considers that the best plan is to give them nothing to carry. We should know exactly at what pace an animal does the best service, whether he be required for speed, or for drawing loads ; and we should know what are the conditions of draught best adapted to the utilization of the strength of animals.

It is in this sense that progress is being made ; but if we complain with reason of its slow advance, we must only blame our imperfect notion of the mechanism of locomotion. Let this study be perfected, and then useful applications of it will soon ensue.

Man has been manifestly inspired by nature in the construction of the machinery of navigation. If the hull of the ship is, as it has been justly described, formed on the model of the aquatic fowl, if the sail has been copied from the wing of the swan inflated by the wind, and the oar from its webbed foot as it strikes the water, these are but a small part of nature's loans to art. More than two hundred years ago, Borelli, studying the stability and displacement of fish, traced the plan of a diving- ship constructed upon the same principle as the formidable Monitors which made their appearance in the recent American war.

In modern navigation the dynamic question still leaves several points in obscurity. What form should be given to a ship so as to secure its meeting with the least possible resistance in the water? What propeller should be chosen in order to utilize the force of the machine to the best advantage ? The most competent men in such matters avow that these problems are too complex to admit of the conditions most favourable to the construction of ships being determined by calculation. Must we wait until empiricism, by dint of ruinous guesses, shall have taught us how a problem of which nature offers us such diverse solutions, should be

ANIMAL MECHANISM.

solved? Ingenious constructors have already attempted to imitate the natural propellers ; they have fitted up small boats with machinery which works like the tail of a fish, oscillating with an alternate motion. And it has been found that this apparatus, although still imperfect, already constitutes a powerful propeller, which will perhaps be preferred hereafter to all those which have hitherto been used.

Aerial locomotion has always excited the strongest curiosity among mankind. How frequently has the question been raised, w^hether man must always continue to envy the bird and the insect their wings ; whether he, too, may not one day travel through the air, as he now sails across the ocean. Authorities in science have declared at different periods, as the result of lengthy calculations, that this is a chimerical dream, but how many inventions have we seen realised which have also been pronounced impossible. The truth is, that all intervention by mathematics is premature, so long as the study of nature and experiment have not furnished the precise data which alone can serve as a sound starting point for calculations of this kind.

We shall then attempt to analyse the rapid acts which are produced in the fiight of insects and of birds ; afterwards we shall endeavour to imitate nature, and we shall see, once more, that by seeking inspiration from her we have the best chance of solving the problems which she has solved.

"We may even now affirm, that in the mechanical actions of terrestrial, aquatic, and aerial locomotion, there is nothing which can escape the methods of analysis at our disposal.. Would it be impossible for us to reproduce a phenomenon which we understand ? We will not carry our scepticism so far.