ILLUSTRATIONS 
 
 MECHANICS. 
 
 REV. H. MOSELEY, M.A. F.R.S. 
 
 One of Her Majesty's Inspectors of Schools ; 
 
 LATE PROFESSOR OP NATURAL PHILOSOPHY AND ASTRONOMY IN 
 
 KING'S COLLEGE, LONDON; 
 
 CORRESPONDING MEMBER OF THE NATIONAL INSTITUTE OF FRANCB ; 
 
 AUTHOR OF "THE MECHANICAL PRINCIPLES OF ENGINEERING 
 
 AND ARCHITECTURE." 
 
 
 LONDON 
 LONGMAN, BROWN, GREEN, LONGMANS, & ROBERTS. 
 
 1859 
 
INTRODUCTION. 
 
 THIS work is the first of a series, entitled 
 ILLUSTRATIONS OF SCIENCE, by Professors of 
 King's College, London, to be published at in- 
 tervals of three months, and continued until the 
 circle of the Physical Sciences and the Sciences 
 of Observation is embraced in it. The author 
 has proposed to himself the development of that 
 system of experimental facts and theoretical 
 principles on which the whole superstructure 
 of mechanical art may be considered to rest, 
 and its introduction, urider an available form, 
 to the great business of practical education. 
 To effect this object, and to reconcile, as far 
 as it may be possible, the strictly scientific with 
 the popular and elementary character of the 
 undertaking, a new method has been sought, 
 the nature of which is sufficiently indicated by 
 its title Illustrations of Mechanics. The 
 work consists, in fact, of a series of illustra- 
 tions of the science of mechanics, arranged in 
 A 3 
 
INTRODUCTION. 
 
 the order in which the parts of that science 
 succeed each other, and connected by such. ex- 
 planations only, as may serve to carry the mind 
 on from one principle to another, and enable it 
 to embrace and combine the whole a plan 
 which leaves to the author the selection of such 
 elements only of his science as are capable of 
 popular illustration, and as come within the limits 
 of practical instruction ; and which enables him 
 to exclude from his work all abstract reasoning, 
 and mathematical deduction. 
 
 Throughout, an attempt is made to give to the 
 various illustrations an entirely elementary and 
 practical character; and each illustration form- 
 ing a short distinct article, the subject of which 
 is enunciated at the commencement of it, the 
 work has assumed a broken form, adapted 
 peculiarly, it is conceived, to the purposes of 
 scholastic instruction. 
 
 It is an idea which presents itself to the mind 
 of every man who has children to educate and 
 provide for, which is a constant subject of com- 
 ment and discussion, and which prevails through 
 all classes of society, that a portion of the school 
 life of a boy ought to be devoted to the ac- 
 quisition of those general principles of practical 
 knowledge of which the whole business of his 
 
INTRODUCTION. 
 
 subsequent life is to form a special application ; 
 that there ought, in fact, to be commenced by 
 him at school a common apprenticeship to those 
 great elements of knowledge, on which hang all 
 the questions of interest which are to surround 
 him in nature, and which are destined, under 
 the form of practical science, to take an active 
 share in the profession, trade, manufacture, or 
 art, whatever it may be, which is hereafter to 
 become the occupation of his life. 
 
 It is the object of this work, and of the Series 
 of which it forms part, to promote this great 
 business of PRACTICAL EDUCATION, by supplying 
 to the instructors of youth a system of element- 
 ary science, adapted to the ordinary forms of 
 instruction. No one can doubt that the same 
 capabilities in the scholar, united to the same 
 zeal in the master, which now suffice to carry 
 the elements of a classical education to the very 
 refinements of philological criticism, would be 
 equal to the task of instruction in the nomencla- 
 ture of the physical sciences, their fundamental 
 experiments, their elementary reasonings, and 
 their chief practical results ; nor can it be ques- 
 tioned that the ordinary intelligence of youth, 
 and common diligence on the part of their teach- 
 ers, would enable them to master the secrets of 
 A 4 
 
Vlll INTRODUCTION. 
 
 the more important of the arts, and the chief 
 processes of the manufactures ; and would place 
 within their reach the elements of natural his- 
 tory, the general classification of the animal and 
 vegetable kingdoms of nature, and their various 
 ministries to the uses of man. 
 
 These are elements of a knowledge which is 
 1 of inestimable value in the affairs of life; and 
 the interests of this great commercial and manu- 
 facturing community claim that they should no 
 longer be left to find their way to the young 
 mind (if, indeed, they reach it at all) rather as 
 a relaxation of the graver business of education 
 than as a part of it. 
 
 That instruction which does not unite with all 
 other knowledge the knowledge of those great 
 truths of religion on which rests, as its found- 
 ation, the fabric of human happiness, can at 
 best be considered but 'as a questionable gift. 
 As a work of education, therefore, any treatise 
 which, having for its object the development of 
 principles of natural knowledge, did not point 
 to the great Author of nature, would be an im- 
 perfect work ; and, more than this, such a work, 
 considered in a scientific point of view, would 
 assuredly bear on its face a blemish ; for, were 
 it not an impiety to discuss the infinite mani- 
 
INTRODUCTION. ix 
 
 festation of wisdom and goodness in creation 
 otherwise than with sentiments of reverence to 
 the Creator, and deep humility before him, it 
 could at best be considered but as an affectation 
 and a folly. It is under the influence of this 
 conviction that, in the following work, the laws 
 of the natural world have been taught where 
 the opportunity has been presented with a 
 direct reference to the power, the wisdom, and 
 the goodness of God. 
 
 The illustrations of the mechanical properties 
 of matter and the laws of force are drawn pro- 
 miscuously and almost equally from ART and 
 NATURE. 
 
 It is not by design that examples taken from 
 these distinct sources thus intermingle, but 
 simply because they suggestthemselves as readily 
 from the one source as the other from nature 
 as abundantly as from art. 
 
 An important truth is shadowed forth in this 
 fact. 
 
 There is a RELATION between ART and NA- 
 TURE a relation amounting to more than a 
 resemblance; a relation by which the eye of 
 the practical man may be guided to that God 
 who works with him in every operation of his 
 skill, and mechanical art elevated from a position 
 
X INTRODUCTION. 
 
 which is sometimes unjustly assigned to it among 
 the elements of knowledge. It cannot be mis- 
 placed in this commencement of a work, which 
 has for its object to develop the great principles 
 of natural science, and which bears upon its title 
 the arms and the motto of an institution formed 
 to unite instruction in the precepts of religious 
 knowledge with the elements of human learning, 
 to point out this relation. The following illus- 
 tration will serve the purpose, and will assimi- 
 late with the general method of the work : 
 
 " I take up a work of art, I examine it, I see on 
 it stamped the evidence of the power and skill, 
 the judgment and knowledge, of the maker : 
 there is the evidence of design in it, there is 
 proof of the economy of labour its material is 
 suited for its use> and as little of it as possible is 
 used, and its form is controlled by a perception, 
 however imperfect, of the beauty and regularity 
 of form. These are things, the evidence of 
 which I perceive in the thing itself. It matters 
 not that I saw it not made, that I know not 
 the maker that he has never instructed me in 
 the secret of his art : for centuries he may have 
 been dead, and may have left no record of the 
 manner of his working. 
 
 This matters not, I see plainly the design with 
 
INTRODUCTION. 
 
 which he wrought. The thoughts of his mind 
 rise up before mine as though I were present 
 to them stamped upon it are the traces of in- 
 telligence, power, and skill, which have operated 
 in its formation invisible things no hand 
 any longer works in it no skill has any longer 
 its visible exercise in it no name is inscribed 
 upon it no legend records for me the fact 
 that there wisdom, knowledge, and power, were 
 exercised yet is the existence of these things, 
 and their exercise in that work of art, among 
 the most certain elements of my knowledge : 
 my reason claims for me the admission of these 
 among the most certain of the things that I 
 may know, deduced by no new or unaccustomed 
 operation of my mind, but by processes of 
 thought which I am daily in the habit of ve- 
 rifying. 
 
 Now let me take up a work of nature, and 
 place it beside that thing of art. Evidence such 
 as that which I have found in the artificial 
 thing is to be sought only in the thing itself, 
 and essentially belongs to it. I may seek it then 
 in this work of nature, as in that of art, and it 
 may, or it may not, be found here, as it was 
 found there. By every mark and sign that I 
 judged of that work of art I judge of this of 
 
XII INTRODUCTION 
 
 nature every rule, which I applied to the one, 
 I apply to the other ; and the conclusion which 
 J draw from the one, with a certainty that never, 
 as I know by experience, fails me, I draw with 
 equal certainty from the other. 
 
 Is there in the work of art the evidence of 
 means to an end? I behold the very same 
 evidence in the work of nature. Is there an 
 adaptation of the material, in the one case? 
 there is the like in the other. Is the artificial 
 thing collected and arranged as to all its ele- 
 ments for a specific object, to which each ele- 
 ment is made subordinate? so is the natural 
 thing. Is the contrivance of the one compli- 
 cated, involving many subsidiary contrivances, 
 all having their direction towards an ultimate 
 result ? so is that of the other. Does the work 
 of art manifest an economy of material and of 
 labour in its construction ? there is the like 
 economy apparent in the work of nature. 
 Subject to the adaptation of the form of the 
 artificial thing to its use, and to the economy of 
 its material, and the labour bestowed upon it, 
 is the disposition of its parts governed by a 
 certain perception of beauty and of grace 
 who shall describe the beauty of nature ? 
 
 The only difference is, indeed, this, that in 
 
INTRODUCTION. xiii 
 
 the work of nature all these qualities exist in 
 their infinite perfection in the work of art, in 
 their infinite imperfection. The evidence is per- 
 fectly alike in kind, although it is the evidence 
 of things infinitely remote in degree. 
 
 With whatever certainty, then, I reason of 
 the finite wisdom and power of the artificer from 
 that work of his art, with the same certainty 
 do I reason of the infinite wisdom and power of 
 the eternal God from the works of his hand ; 
 and on this evidence I declare with St. Paul 
 that " the invisible things of him from the be- 
 ginning of the world are manifest, being plainly 
 seen by the things which are made, even his 
 eternal power and Godhead." (Rom. i. 20.)" 
 
 Every work of human art or skill is a thing 
 done by a creature of God ; a creature MADE IN 
 jais OWN IMAGE, and operating upon matter 
 governed by the same laws, which HE, in the 
 beginning, infixed in it, and to which he sub- 
 jected the first operations of his own hands a 
 creature in whom is implanted reason of a like 
 nature with that excellent wisdom by which the 
 heavens were stretched forth living power as 
 that of a worm, and as a vapour that passeth 
 away, but an emanation of Omnipotence a 
 perception of beauty and adaptation akin to 
 
XIV INTRODUCTION. 
 
 that whence flowed the magnificence of the uni- 
 verse and to control these, a volition, whose 
 freedom has its remote analogy and its source 
 in that of the first self-existent and independent 
 Cause. 
 
 It is from this relation between the Author of 
 nature and the being in whom the works of 
 art have their origin that arise those relations, 
 infinitely remote, but distinct, between the things 
 themselves, of which the evidence is every 
 where around us. These are necessary relations : 
 it is not that the works of art are made by any 
 purpose or intention in the resemblance of those 
 of nature, or that there is any unseen influence 
 of nature itself upon art the primary relation 
 is in the causes whence these severally proceed. 
 
 Thus it is possible, that in the infinities of 
 nature, every thing in art may find its type ; 
 this is not, however, necessarily the case, since 
 the causes are infinitely removed, since, more- 
 over, in their operation, these causes are in- 
 dependent, and since nature operates upon 
 materials which are not within the resources of 
 art. 
 
 How full of pride is the thought, that in 
 every exercise of human skill, in each ingenious 
 adaptation, and in each complicated contrivance 
 
INTRODUCTION. XV 
 
 and combination of art, there is included the 
 exercise of a faculty which is akin to the 
 wisdom manifested in creation ! 
 
 And how full of humility is the comparison 
 which, placing the most ingenious and the 
 most perfect of the efforts of human skill by 
 the side of one of the simplest of the works of 
 nature, shows us but one or two rude steps of 
 approach to it. 
 
 How full, too, is it of profit thus to see God 
 in every thing to find him working with us, 
 and in us, in the daily occupations of our hands, 
 wherein we do but reproduce, under different 
 and inferior forms, his own wisdom and creative 
 power. 
 
 A man may thus hold converse with God as 
 intelligibly in art as in nature, and live with 
 him in the workshop, as he may go forth with 
 him in the fields and upon the hills. And 
 whilst he feels himself in those faculties of 
 thought and action, the exercise of which con- 
 stitute his physical being, to be in very deed a 
 creature made in the image of God, he will 
 not fail to be reminded that the resemblance 
 once embraced with these the qualities of his 
 moral being. 
 
 If we conceive space spreading out its dimen- 
 
XVI INTRODUCTION. 
 
 sions infinitely, still through all its interminable 
 fields does science show it to us peopled with 
 matter stars upon stars innumerable a vista 
 in which suns and systems crowd themselves, and 
 to which imagination affixes no limit. If, in like 
 manner, we conceive space to be infinitely di- 
 vided as its dimensions grow before the eye of 
 the mind yet less and less still does it appear a 
 region peopled with the infinite divisions of 
 matter. 
 
 On either side is an abyss an interminable 
 expanse, through which the creative power of 
 God manifests itself, and an unfathomable 
 minuteness. 
 
 it is in this last mentioned region of the in- 
 accessible minuteness of matter that the prin- 
 ciples of the science treated of in the following 
 pages have their origin. Matter is composed 
 of elements, which are inappreciably and infi- 
 nitely minute ; and yet it is within the infinitely 
 minute spaces which separate these elements 
 that the greater number of the forces known to 
 us have their only sensible action. These, 
 including compressibility, extensibility, elas- 
 ticity, strength, capillary attraction and ad- 
 hesion, receive their illustration in the first 
 three chapters of the following work. The 
 
INTRODUCTION. Xvii 
 
 fourth takes up the Science of Equilibrium, 
 or Statics ; applies in numerous examples the 
 fundamental principles of that science, the pa- 
 rallelogram of forces, and the equality of mo- 
 ments ; then passes to the question of stability -, 
 and to the conditions of the resistance of a 
 surface ; traces the operation of each of the 
 mechanical powers under the influence of fric- 
 tion ; and embraces the question of the stability 
 of edifices, piers, walls, arches, and domes. 
 
 The fifth chapter enters upon the Science of 
 Dynamics, Numerous familiar illustrations es- 
 tablish the permanence of the force which 
 accompanies motion show how it may be 
 measured where in a moving body it may be 
 supposed to be collected exhibit the important 
 mechanical properties of the centres of sponta- 
 neous rotation, percussion, and gyration the 
 nature of centrifugal force, and the properties 
 of the principal axes of a body's rotation the 
 accumulation and destruction of motion in a 
 moving body, and the laws of gravitation. 
 
 The last chapter of the work opens with a 
 series of illustrations, the object of which is to 
 make intelligible, under its most general form, 
 the principle of virtual velocities, and to protect 
 practical men against the errors into which, in 
 
XV111 INTRODUCTION. 
 
 the application of this universal principle of 
 mechanics, they are peculiarly liable to fall : it 
 terminates with various illustrations of those 
 general principles which govern the reception, 
 transmission, and application of power by ma- 
 chinery, the measure of dynamical action, and 
 the numerical efficiencies of different agents 
 principles which receive their final application 
 in an estimate of the dynamical action on the 
 moving and working points of a steam engine. 
 The Appendix to the work contains a de- 
 tailed account of the experiments of Messrs. 
 Hodgkinson and Fairbairn upon the mechanical 
 properties of hot and cold blast iron : and an 
 extensive series of tables referred to in the body 
 of the work, and including, 1. Tables of the 
 strength of materials ; 2. Tables of the weights 
 of cubic feet of different kinds of materials; 
 
 3. Tables of the thrusts of semi-circular arches 
 under various circumstances of loading, and 
 of the positions of their points of rupture: 
 
 4. Tables of co-efficients of friction, and of 
 limiting angles of resistance, compiled and cal- 
 culated from the recent experiments of M. 
 Morin. The results of these admirable expe- 
 riments, made at the expense of the French 
 
INTRODUCTION. 
 
 government, are here, for the first time, pub- 
 lished in this country. 
 
 The author has also to acknowledge his ob- 
 ligations to the " Physique" of M. Pouillet, for 
 several valuable illustrations and drawings. 
 
 The articles marked with an asterisk, and the 
 whole of the sixth chapter, are recommended 
 to be omitted on tht first reading. 
 
CONTENTS 
 
 OF 
 
 THE ILLUSTRATIONS OF MECHANICS. 
 
 CHAP. I. 
 
 THE INFINITE MINUTENESS OF THE ELEMENTS OF MATTER. THE 
 
 POROSITY OF MATTER ITS COMPRESSIBILITY ITS ELASTICITY. 
 
 Page 
 
 1. Globules of Blood - 1 
 
 2. Minuteness of the Pores and Scales of the Skin - 2 
 t3. Musk - 3 
 
 4. Dust of the Lycoperdon - 3 
 
 5. Metallic Solutions - - - 4 
 t>. Colours produced by the Attenuation of Transparent 
 
 Bodies - - 4 
 
 7. The Thickness of a Soap Bubble - - 5 
 
 8. Attenuation of the Wings of Insects - 5 
 
 9. The Attenuation of Gold Leaf - 6 
 
 10. The ordinary Process of Gilding - 6 
 
 1 1 . The gilding of Thread for Embroidery - 8 
 
 12. Tenuity of Fibres of Silk - - - 10 
 
 13. The Tenuity of Fibres of Wool - - 10 
 
 14. The Fibre of Cotton - - 10 
 
 15. The Fibre of Flax - - 10 
 
 16. Fibres of the Pine Apple Plant^ - 1 1 
 
 17. Tenuity of the Fibres of a Spider's Thread - 11 
 
 18. Tenuity of Cotton Yarn - - 12 
 
 19. Threads of Glass - - 13 
 
 20. Platinum Wire - - IS 
 
 a 3 
 
CONTENTS. 
 
 THE POROSITY OF MATTER. 
 
 Page 
 
 21. The Porosity of Wood - - - 14 
 
 22. Wood ceases to be buoyant when its Pores are filled 
 
 with Water - - 15 
 
 23. The Porosity of Rocks - 15 
 
 24. The Porosity of Hydrophane - - 15 
 
 25. Porosity of Metals - - - - 16 
 
 COMPRESSIBILITY. 
 
 26. Compressibility of Wood - - 1 6 
 
 27. Compressibility of Aeriform Bodies - - 1 7 
 *28. Compressibility of Water - 17 
 *29. The Compression of Solids by (Ersted's Apparatus 22 
 *30. The Adaptation of (Ersted's Apparatus to High 
 
 Pressures - - - 23 
 
 ELASTICITY. 
 
 31. Marriotte's Experiment - 25 
 
 32. The Elasticity of the Metals - 28 
 
 33. The Law of the Elasticity of Metals - 29 
 
 34. Experiments of S. Gravesande on the Elasticity of 
 
 Wires - ' - - 29 
 
 35. Elasticity of Ivory - - - - 31 
 
 36. Elasticity of Torsion - - 31 
 
 37. Coulomb's Torsion Balance - - 32 
 
 38. The Elasticity of Lead and Pipe-clay - - 34 
 
 39. The Torsion of Bars of Iron - - 35 
 
 40. Elasticity a common Property of Aeriform Bodies, 
 
 Liquids, and Solids - - 35 
 
 41. The Liquefaction of the Gases - - 36 
 
CONTENTS. XX111 
 
 CHAP. II. 
 
 THE. STRENGTH OF MATERIALS. 
 
 THE FORCES PRODUCING EXTENSION OR COMPRESSION. THE 
 
 LIMITS OF ELASTICITY. RUPTURE. THE STRONGEST FORMS 
 
 OF CAST-IRON BEAMS AND COLUMNS. WOOD AS A MATERIAL 
 
 IN THE ARCHITECTURE OF NATURE. THE MECHANICAL PRO- 
 PERTIES OF METALLIC SUBSTANCES AS AFFECTED BY THEIR 
 INTERNAL STRUCTURE. 
 
 Page 
 
 42. The Extensibility of Iron and Wood - 39 
 
 43. The Extensibility of Bar Iron when approaching a 
 
 State of Rupture - - 40 
 
 44. The Volume or Bulk of an Iron Bar, and of a Copper 
 
 Wire, are increased in the act of Extension - 41 
 
 45. The Theoretical Variation in the Diameter of a Solid 
 
 Metallic Cylinder subjected to Extension - - 42 
 
 46. The Limits of Elasticity - - 42 
 
 47. The Elasticity of a Body is not injured when a Set 
 
 is given to it 
 
 48. Malleability - - 44 
 
 49. The Stamping of Metallic Surfaces - - 45 
 
 50. Coining 
 
 51. The Rolling of Metals - - 46 
 
 52. Engraved Steel Plates - 46 
 
 53. Rupture 
 
 54. Tenacity 
 
 55. Resistance to Rupture by Compression - - 51 
 
 56. Influence of the Height of a Prism upon the Resist- 
 
 ance to the crushing of its Material - - 53 
 
 *57. Rule, by Rondelet, for the Strength of Columns of 
 wrought Iron, and of Oak and Deal 
 
 58. A Column of Cast Iron, whose Extremities are 
 
 rounded, will support but One Third the Weight 
 of a similar Column, whose Extremities are flat 55 
 
 59. The strongest Form of a Cast Iron Column - 56 
 
 a * 
 
XXIV CONTENTS. 
 
 Page 
 
 60. The Pressure to which Materials may be subjected 
 
 with Safety in Construction - - 56 
 
 61. Adhesion of the Fibres of Wood to one another - 57 
 
 62. The neutral Axis in a Beam - 58 
 
 63. The Strength of a Beam - 58 
 
 64. To cut a Beam One Half-through, without diminish- 
 
 ing its Strength - 59 
 
 65. The Relation of the Forces necessary to tear Materials 
 
 asunder, and to crush them - - 59 
 
 *66. To make a Beam or Girder of Cast Iron which shall 
 be four Times as strong when turned with one 
 Side, as when turned with the other Side, upwards 60 
 *67. A Wedge, driven out by the Compression of the Rib 62 
 *68. The strongest Form of Section of a Cast Iron Beam 63 
 *69. Rule for the Strength of a Beam, cast on Mr. 
 
 Hodgkinson's Principle - - - 66 
 
 *70. To vary the Section of a Beam at different Dis- 
 tances from the Points of Support, so that for a 
 given Quantity of Material its Form may be the 
 strongest - - - - 67 
 
 71. The Qualities of Wood as a Material of Construction 68 
 *72. The Adaptation of Wood as a Material to the 
 Architecture of Trees in respect to its Distribu- 
 tion - - - - 71 
 
 73. Various Circumstances which affect the Strength of 
 
 Metals, as Materials of Construction - 74 
 
 74. Crystallisation of Bismuth - 74 
 
 75. Saline Crystallisation - - 75 
 
 76. Crystallisation may take place in a Mass which is 
 
 in an imperfect State of Fusion - - 77 
 
 77. The Influence of the various Conditions of Crys- 
 
 tallisation on the cohesive Force of Cast Iron - 78 
 
 78. The Influence of Pressure upon the Solidification 
 
 of Metals ... - 78 
 
 79. Malleable Platinum. - - - 79 
 
 80. Cast Iron ... - 80 
 
 81. The Manufacture of Wrought Iron - 83 
 
CONTENTS. XXV 
 
 Page 
 
 82. The Manufacture of Steel - - - 84 
 
 83. Case-hardening - - 86 
 
 84. Effect of Heat on the Strength of Cast Iron - 87 
 
 85. Permanent Diminution of the Tenacity of Iron 
 
 Wire by heating - - - - 87 
 
 86. Annealing of Cast Iron - - - 88 
 
 87. The different mechanical Properties of Hot and 
 
 Cold Blast Iron - 88 
 
 88. The Tempering of Steel - 90 
 
 89. The Tempering of the Alloy of Copper, called Tam- 
 
 Tam - - - -91 
 
 90. The Annealing of Glass - 92 
 
 91. Prince Rupert's Drops - 92 
 
 92. Mitzcherlich's Experiments on Changes in Crys- 
 
 tallised Forms of Bodies by the Operation of 
 Heat . - - - 93 
 
 CHAP. III. 
 CAPILLARY ATTRACTION, AND ADHESION. 
 
 93. Ascent of Water in Capillary Tubes - - 95 
 
 94. Depression of Mercury in Capillary Tubes - 95 
 
 95. Depression of Water in Capillary Tubes, whose 
 
 Surfaces cannot be wetted - - 95 
 
 96. The Phenomena of Capillary Attraction and Re- 
 
 pulsion are not confined to the internal Surfaces 
 of Tubes, but common to the Surfaces of all 
 Bodies, and only more apparent in these - 96 
 
 97. The Rise of Water between parallel Plates of 
 
 Glass - - 96 
 
 98. The Wick of a Lamp - - 97 
 
 99. An Iron Wick for a Lamp - 97 
 
 100. A Syphon Filter, made with Threads of Cotton - 97 
 
 101. Heavy Bodies made to float by Capillary Repulsion 98 
 
 102. Insects supported on the Surface of Water by 
 
 Capillary Repulsion - - - 98 
 
XXVI CONTENTS. 
 
 Page 
 
 1 03. The Attractions of Capillary Rods, when suspended 
 
 in a Fluid - - 99 
 
 104. The Attraction and Repulsion of floating Bodies- 10O 
 
 105. The Attraction of Needles floating on Water - 10O 
 
 106. Attraction and Repulsion of small Bodies by the 
 
 Sides of Vessels - - - 100 
 
 107. When a Capillary Tube is taken out of the Fluid 
 
 in which it has been plunged, a Portion of the 
 Fluid which remains in it stands at a much 
 greater Height than it stood before - 101 
 
 108. Water will not, under certain Circumstances, find 
 
 its Level in a Capillary Syphon - - 102 
 
 109. To make a Vessel full of Holes, which shall yet 
 
 contain Water - 103 
 
 110. To make a Vessel full of Holes, which shall float - 103 
 
 111. Effects of Capillarity in the Barometer Tube - 103 
 
 112. The Heights to which a Fluid ascends in different 
 
 Capillary Tubes, are greater as their Diameters 
 are less - - - - - 104 
 
 1 13. The Heights to which the same Fluid ascends in 
 
 different Capillary Tubes, do not depend on the 
 Thickness of the Tubes - 104 
 
 114. The Heights to which the same Fluid ascends in 
 
 different Tubes, do not depend upon the Sub- 
 stances out of which the Tube are formed, pro- 
 vided only they be Substances which do not repel 
 the Fluid, or which admit of being wetted by it 105 
 
 115. The Heights to which different Fluids ascend in 
 
 the same Tube, are not the same - -105 
 
 * 1 1 6. The Heights to which the same Fluid ascends in 
 different Capillary Tubes, are inversely pro- 
 portional to the Diameters of the Tubes - 105 
 *117. The Elevation of Water between Plates of Glass 
 
 slightly inclined to one another - 108 
 
 118. Of the Force with which fibrous Substances imbibe 
 Moisture by Capillary Attraction, and thereby 
 increase their Bulk - - 109 
 
CONTENTS. XXvil 
 
 Page 
 
 119. The Theory of Capillary Attraction - -110 
 
 120. Application of Capillary Attraction to Assaying - 113 
 
 121. The Agency of Capillary Attraction in Nature - 115 
 
 122. Endosmose and Exosmose - - - 116 
 
 123. Adhesion of Plates of different Substances to the 
 
 Surfaces of Fluids - - - - 1 1 8 
 
 1 24. Adhesion of a Column of Mercury to the internal 
 
 Surface of a Capillary Tube - - 120 
 
 125. Adhesion of Plates of Glass to one another - 121 
 
 CHAP. IV. 
 STATICS. 
 
 DEFINITIONS. THE EQUILIBRIUM OP THREE PRESSURES. - 
 
 THE EQUILIBRIUM OF ANY NUMBER OF PRESSURES IN THE 
 
 SAME PLANE. THE LEVER. THE WHEEL AND AXLE. 
 
 THE COMPOSITION AND RESOLUTION OF FORCES. THE 
 
 CENTRE OF GRAVITY. THE RESISTANCE OF A SURFACE. 
 
 FRICTION. THE INCLINED PLANE. THE WEDGE. THE 
 
 SCREW. THE EQUILIBRIUM OF BODIES IN CONTACT. 
 
 PIERS. ARCHES. 
 
 126. Equilibrium - - 122 
 
 127. Forces of Pressure, and Forces of Motion - 123 
 
 128. The Relation between Three Pressures in Equi- 
 
 librium. The Parallelogram of Pressures - 1 23 
 
 1 29. The Equilibrium of any Number of Pressures in 
 
 the same Plane. The Principle of the Equality 
 
 of Moments - - 126 
 
 130. The Polygon of Pressures - - - 127 
 
 131. The Lever - 128 
 
 132. Could Archimedes have lifted the World with a 
 
 Lever if he had had a Fulcrum to rest it upon? 132 
 
 133. Two Persons carry a Burden between them by 
 
 means of a Lever or Pole, to find how much of 
 the Weight is borne by each - - 134 
 
XXV111 CONTENTS. 
 
 Page 
 
 134. Method of combining the Efforts of a great Numbei 
 
 of Men to carry a Burden - - - 135 
 
 135. The Wheel and Axle - 136 
 
 136. Modification of the Wheel and Axle, by which 
 
 any Weight can be raised by a given Power - 139 
 
 137. When any number of Pressures acting on a Body, 
 
 in the same Plane, are not in Equilibrium, to 
 apply to it another which shall produce an 
 Equilibrium - - - - 142 
 
 138. The Resultant of any Number of Pressures - 143 
 
 139. The Composition and Resolution of Pressures - 144 
 
 140. The Centre of Gravity - - - 145 
 
 141. To determine the Centre of Gravity of a Body by 
 
 Experiment - - - - 148 
 
 142. The Attitudes of Animals - - - 149 
 
 143. The best Position of the Feet in standing - 150 
 
 144. The Shepherds of the Landes - 152 
 
 145. To cause a Cylinder to roll, by its Weight, a short 
 
 Distance up an inclined Plane - - 152 
 
 *146. Wheeler's Clock - - 153 
 
 *147. To cause a Body, by its own Gravity, to roll con- 
 tinually upwards - - - 155 
 *148. Stable and unstable Equilibrium - - 156 
 *149. That Position of a Body resting upon another, in 
 which its Centre of Gravity is the lowest pos- 
 sible, is a Position of stable Equilibrium; that 
 in which it is the highest possible, one of un- 
 stable Equilibrium - - - 159 
 *150. Every Body, except a Sphere, has at least one 
 Position of stable, and one of unstable, Equi- 
 librium - - - - - 160 
 151. A Body having plane Faces has all its Positions 
 of Equilibrium, on those Faces, Positions of 
 stable Equilibrium ; and all its Positions of 
 Equilibrium, on their Edges, Positions of 
 mixed, and on their Angles, of unstable Equi- 
 librium - - - - 161 
 
CONTENTS XXIX 
 
 Page 
 
 152. A Body's Position is always one of stable Equi- 
 
 librium, when its Centre of Gravity lies beneath 
 the Point on which it is supported - - 163 
 
 153. To construct a Figure which, being placed upon a 
 
 curved Surface, and inclined in any Position, shall, 
 when left to itself, return into its former Position 163 
 
 154. To cause a Body to support itself steadily, on an 
 
 exceedingly small Point - - - 164 
 
 155. A Body having a Portion of its Surface spherical, 
 
 and resting by that Portion of its Surface on a 
 horizontal Plane, has its Equilibrim stable or 
 unstable, according as its Centre of Gravity is 
 beneath or above the Centre of the Sphere, of 
 which that spherical Surface forms Part - 165 
 
 *156. The Stability o. a Body which is suspended from 
 a Point, or a fixed Axis, is greater as the Centre 
 of Gravity of the Body is lower beneath that 
 Point or that Axis - - - 1 67 
 
 157. The Balance - 169 
 
 158. To make a Balance which shall appear true when 
 
 empty, but yet weigh falsely - - 170 
 
 159. To weigh truly with a false Balance - - 171 
 
 160. Borda's Method of weighing truly with a false 
 
 Balance - - - 171 
 * 1 6 1 . Under what Circumstances a Body, supported upon 
 
 a horizontal Plane, is more or less stable - 172 
 
 Walking - 176 
 
 162. The Resistance of a Surface - -178 
 
 163. The Direction of the Resistance of a Surface - 179 
 
 164. The Cone of Resistance - 181 
 
 165. Illustration of the Cone of Resistance in the 
 
 striking of a Hammer - - - 1 82 
 
 1 66. Illustration of the Law of the Resistance of a 
 
 Surface, in the Use of the Crowbar - 183 
 
 167. The mechanical Advantage of any Machine is 
 
 supplied by the Resistances of its Parts 
 
 168. Friction - - - 186 
 
 169. The Friction is proportional to the Pressure - 187 
 
XXX CONTENTS. 
 
 Tagc 
 
 170. Amount of the constant Proportion of the Friction 
 
 to the Pressure in different Substances - 189 
 
 171. The Amount of Friction is independent of the 
 
 Extent of the Surface pressed, provided the 
 whole Amount of the Pressure remain the same, 
 and that the Substance of the Surface pressed 
 is the same - - 190 
 
 172. The Friction of a Body when in a State of con- 
 
 tinuous Motion, bears a constant Ratio to the 
 Pressure upon it, which is the same, whatever 
 may be the Velocity of the Motion - - 191 
 
 173. The Effect of Unguents upon Friction - 191 
 
 174. The Circumstances under which a Body will 
 
 support itself upon an inclined Plane - 1 93 
 
 175. The Circumstances under which a Body may be 
 
 supported upon an inclined Plane - - 194 
 
 176. The moveable inclined Plane - - - 195 
 *177. The Screw - - 196 
 *178. The Wedge - 199 
 *179. The Circumstances under which a Wedge will 
 
 not be forced back by the Tendency of the Sur- 
 faces, between which it is driven to collapse - 201 
 180. Kails - 202 
 
 181. The Circumstances under which an Edifice of 
 
 uncemented Stones is overthrown - - 204 
 
 *182. The Conditions of the Equilibrium of an Edifice 
 
 of uncemented Stones ... 205 
 
 *183. The Line of Resistance in a Pier - 208 
 
 *184. The greatest Height to which a Pier can be built, 
 so as to sustain a given Pressure upon its 
 Summit - - - 209 
 
 *185- The straight Arch, or Plate Bande - - 210 
 
 *186. To find the greatest Height of the Piers, of a 
 given Width, which will support a straight 
 Arch of given Dimensions - - 21 1 
 
 *187. The Arch - - 211 
 
 *188. The Settlement of the Arch - - 214 
 
 189. Pulleys - * . - 215 
 
CONTENTS. XXXJ 
 
 CHAP. V. 
 DYNAMICS. 
 
 THE FORCE OF MOTION. ITS PERMANENCE. THE MEASURE 
 
 OF IT. THE POINT WHERE IT MAY BE SUPPOSED TO BE 
 
 COLLECTED. MOTIONS OF TRANSLATION AND ROTATION, 
 
 INDEPENDENT. THE CENTRE OF GYRATION. THE CENTRE 
 
 OF SPONTANEOUS ROTATION. THE CENTRE OF PERCUSSION. 
 
 THE PRINCIPAL AXES OF ROTATION. THE FORCE OF A 
 
 BODY'S MOTION is NEVER GENERATED OR DESTROYED IN- 
 STANTANEOUSLY. ACCELERATING FORCE. GRAVITATION. 
 
 CAVENDISH'S EXPERIMENTS. DESCENT OF A BODY FREELY 
 
 BY GRAVITY. ATWOOD's MACHINE. DESCENT OF A BODY 
 
 UPON AN INCLINED PLANE AND UPON A CURVE. THE 
 
 CYCLOIDAL PENDULUM. THE SIMPLE PENDULUM. THE 
 
 CENTRE OF OSCILLATION. KATER's PENDULUM. THE COM- 
 PENSATION PENDULUM. 
 
 Page 
 
 190. Certain Laws common to the Operation of all 
 
 Forces 221 
 
 191. Momentum, or the Force of Motion - -222 
 
 192. The Force of a Body's Motion is precisely equiva- 
 
 lent to the Force expended in producing it - 223 
 
 193. There is no Principle of Diminution or Decay in 
 
 the Nature of Motion itself, or in the Nature of 
 the Force of a moving Body - - 225 
 
 1 94. Illustrations of the Permanence of communicated 
 
 Motion 227 
 
 1 95. The Permanence of the Forces of Rotation of the 
 
 Planets, and of their tangential Forces of 
 Motion - 229 
 
 196. Illustrations of the Permanence of the Force of 
 
 Motion - 
 
 197. Of the Force of Motion which tends to overthrow 
 
 a moving Body, the Effect of that will be the. 
 
XXXll CONTENTS. 
 
 Page 
 
 greatest, which exists in the highest Portions 
 of it - - 233 
 
 198. Driving on the Head of a Tool - 234 
 
 199. The breaking of Bodies by Impact - 234 
 9.00. A Jar of the Body - - -236 
 
 201. The Phenomena which attend the sudden Produc- 
 
 tion of Motion, are analagous to those of the 
 sudden Destruction of it - - - 236 
 
 202. The Hammer - - - - 238 
 
 203. If the Causes which tend to destroy the Force of 
 
 a Body's Motion be continually counteracted as 
 
 it moves on, then it will move uniformly - 240 
 
 204. The Tendency of the Force of Motion to Perma- 
 
 nence is a Tendency to Permanence in that 
 particular Direction in which the Body moves, 
 or in which the Force acts - - 240 
 
 205. Illustrations of the Tendency of Motion to Per- 
 
 manence, in respect to its Direction - 242 
 
 206. The Measure of Momentum, or the Force of 
 
 Motion - 242 
 
 207. A Plate of soft Iron may be made, by the Force of 
 
 its Motion, to cut through the hardest Steel - 247 
 
 208. The Art of the Lapidary - - 248 
 
 209. When a Body's Motion is arrested, the whole Force 
 
 with which it moves is made to act upon the 
 Obstacle * - 248 
 
 210. The Impact of Bodies - - - 249 
 
 211. The Recoil of Fire- Arms - - -251 
 
 212. To fire from solid Cannon - 252 
 
 213. The Recoil of a Cannon does not become sensible 
 
 until the Ball has left its Mouth - 253 
 
 214. To determine the initial Velocity of a Cannon Ball 253 
 215^ The Ballistic Pendulum . , - 254 
 216, When a Body moves only with a Motion of trans- 
 lation j. that is, when all the Parts of it move 
 with the same Velocity and in the same Direc- 
 tion, there is a certain Point in it, in which the 
 
CONTENTS. XXX111 
 
 Page 
 
 whole Force of its Motion may be supposed to 
 act. That Point is the Centre of Gravity - 254 
 
 217. The Symmetry of Tools -256 
 
 218. If a Body have an Impulse communicated to it 
 
 whose Direction is not through its Centre of 
 Gravity, then when moving freely by reason of 
 this Impulse, its motion will partly be one of 
 Translation, and partly of Rotation, but subject 
 to this remarkable Law : " That its Motion 
 of Translation will be the same as though the 
 Impulse had been communicated through its 
 Centre of Gravity, and there had thus been no 
 Rotation ; and its Motion of Rotation the same, 
 as though its Centre of Gravity had been fixed, 
 and it had revolved round it thus fixed, so that 
 there could be no Translation " - 257 
 
 219. The Double Motion of the Rotation and Trans- 
 
 lation of the Earth - . 259 
 
 220. To cause a Ball to move forwards a certain Dis- 
 
 tance upon a horizontal Plane, and then, 
 although it meets with no Obstacle, to roll 
 backwards - - 260 
 
 221. The Radius of Gyration - 261 
 
 222. The Force of a Body's Motion depending upon 
 
 its Velocity, it is evident that when the Body is 
 made to revolve a certain Number of Times in 
 a Minute, round a fixed Axis, its force of Motion 
 will be greater, as it revolves at a greater Dis- 
 tance from the Axis, or is connected with it by 
 means of a longer Arm - 262 
 
 223. The Dimensions of the Earth have not diminished 
 
 for the last 2500 Years - - 262 
 
 224. The Compensation Balance Wheel - 263 
 9^5. The Centre of spontaneous Rotation - - 266 
 226. These Facts explain the Ease with which a long 
 
 Pole, or a Ladder, may be balanced on its Ex- 
 b 
 
XXXIV CONTENTS. 
 
 Page 
 
 tremity, and why either of these will be yet 
 more easily balanced if it is loaded at the Top -267 
 
 227. The Centre of Percussion - - - - 268 
 
 228. The Centres of Suspension and Percussion are 
 
 convertible - 269 
 
 229. The Tilt Hammer - - 269 
 
 230. A Body in Motion about a fixed Axis which en- 
 
 counters an Obstacle at a Distance from its Axis, 
 equal to the Radius of Gyration, will expend all 
 the Force of its Motion on the Obstacle. If it en- 
 counter it at any other Point, the Force will be 
 divided between the Obstacle and the fixed Axis 27 1 
 
 231. A Cricket Bat - - 271 
 
 232. Tools of Impact - 272 
 
 233. Centrifugal Force - 273 
 
 234. The Amount of Centrifugal Force . - 273 
 
 235. A Sling - 274 
 
 236. A Man running in a Circle - - - 275 * 
 
 237. The Centrifugal Force of a Body's Motion may 
 
 be supposed to be collected from its different 
 Parts, and made to act through its Centre of 
 Gravity - - 277 
 
 238. It is by reason of the Centrifugal Force that a 
 
 Carriage, rapidly turning a Corner, is liable to 
 
 be overthrown - - - 278 
 
 239. Feats of Horsemanship - - 278 
 
 240. A Glass of Water may be whirled round so as to 
 
 be inverted, without being spilt * - 280 
 
 241. To make a Carriage run in an inverted Position 
 
 without falling - - 280 
 
 242. The Governor - - 281 
 
 243. The Pressure upon the Axis of a revolving Body 283 
 
 244. The principal Axis of a Body's Rotation 283 
 
 245. The Planets rotate about their shortest Diameters 285 
 
 246. Experimental Illustration of the Tendency of a 
 
 Body's Rotation about any other Axis, to pass 
 into one round its shortest principal Axis - 286 
 
CONTENTS. XXXV 
 
 Page 
 
 247. The Force with which a Body moves is never 
 
 generated instantaneously - - 288 
 
 248. If a Guinea be placed upon a Card, and the whole 
 
 balanced on the Tip of the Finger, a sharp Blow 
 struck upon the Edge of this Card will cause it 
 to slip from under the Guinea, and the latter 
 will be left alone on the Finger - - 290 
 
 249. The Effect of Swinging, Riding, &c. on the Circu- 
 
 lation of the Blood ... 290 
 
 250. A Candle fired from a Musket will pierce through 
 
 a thick Board - - - - 291 
 
 251. A Musket Ball passes through a Pane of Glass 
 
 without cracking it 292 
 
 252. The Force with which a Body moves is never 
 
 destroyed instantaneously - - 293 
 
 253. Accumulation and Destruction of the Force of 
 
 Motion in a moving Body. Distinction between 
 Force of Motion and Force of Pressure - 294 
 
 254. Gravitation - - 296 
 
 255. No Force of Motion or Impact can be compared 
 
 with, or measured by, a Weight - - 297 
 
 256. Uniform, accelerated, and retarded, Motion - 298 
 
 257. Velocity - 298 
 
 258. Accelerating Force - 299 
 
 259. The Law of the accelerating Force of Gravity - 299 
 
 260. Gravitation a Force inseparably and universally 
 
 associated with Matter ... 300 
 
 261. The Gravitation of the Bodies around us to the 
 
 great Mass of the Earth, is a sensible Force ; 
 their Gravitation towards one another, almost 
 Insensible - - 302 
 
 262. The Attraction of Mountains - - 303 
 
 263. The Experiments of Cavendish - 305 
 
 264. The Attraction of the Earth would cause all Bodies, 
 
 whether they were light or heavy, to fall to- 
 wards its Surface with the same Rapidity, were 
 "it not for the Resistance of the Air - - 308 
 
 b 2 
 
XXXVI CONTENTS. 
 
 Page 
 265. The Velocity which is communicated to a Body 
 
 falling freely by Gravity ... 309 
 
 *266. Atwood's Machine - - - 310 
 
 267. Descent of a Body by Gravity - - -314 
 
 268. A Body projected downwards or upwards - 315 
 
 269. To find the Depth of a Well by letting a Stone fall 
 
 into it - - 317 
 
 270. Velocity of the Descent of a Body upon an inclined 
 
 Plane - - - - - 317 
 
 271. Velocity of Descent upon a Curve - - 319 
 *272. Time of a Body's Descent upon a Curve - 319 
 
 273. The Cycloid is an isochronous Curve - 322 
 
 274. To make a Pendulum oscillate in a Cycloid - 323 
 
 275. The simple Pendulum ... 325 
 
 276. To determine the Time in which a Pendulum of 
 
 any given Length will perform its Oscillations - 326 
 
 277. To determine what must be the Length of a simple 
 
 Pendulum, so as to beat any given Number of 
 Seconds - - - - - 327 
 
 278* To measure the Force of Gravity at any Place, 
 
 by observing the Beats of a Pendulum - 328 
 
 279. The Force of Gravity diminishes as we approach 
 
 the Equator - - 329 
 
 280. To find the Depth of a Mine by observing the 
 
 Beats of the Pendulum - - - 830 
 
 281. The Centre of Oscillation - - -332 
 
 282. Practical Method of determining the Centres of 
 
 Percussion and Gyration - - - 383 
 
 283. The Pendulum of Borda - - - 335 
 
 284. Borda's Method of Coincidences for observing the 
 
 Time of Oscillation of a Pendulum - - 337 
 
 285. To determine experimentally the Position &f the 
 
 Centre of Oscillation of a Body without knowing 
 the Force of Gravity at the Place of Observation 339 
 
 286. Captain Kater's Pendulum - - 340 
 
 287. Compensation Pendulums ... 342 
 
 288. Harrison's Compensation Pendulum - - 345 
 
CONTENTS. 
 
 CHAP. VI. 
 
 THE RETARDATION OF MOTION. THE PRINCIPLE OF VIRTUAL 
 
 VELOCITIES. THE MEASURE OF THE DYNAMICAL EFFECT OR 
 
 THE ACTION OF AN AGENT. THE DYNAMICAL EFFECTS OF 
 
 DIFFERENT AGENTS. THE MOVING AND WORKING POWERS 
 
 IN A MACHINE. THE MOVING AND WORKING POWERS IN 
 
 ANY MACHINE ARE EQUAL, ABSTRACTION BEING MADE OF THE 
 RESISTANCES WHICH OPPOSE THEMSELVES TO THE MOTIONS 
 
 OF THE PARTS OF THE MACHINE UPON ONE ANOTHER. 
 
 THE MOVING POWER IN A STEAM-ENGINE. THE WORKING 
 
 POWER IN A STEAM-ENGINE. 
 
 Page 
 
 289. The Retardation of a Body's Motion - - 349 
 
 290. The Velocity of a Body's Projection up a Curve 
 
 may be found by observing the Height to which 
 
 it ascends upon it - - - 351 
 
 291. The Depth to which a Cannon or Musket Ball 
 
 enters into a Block of Wood, or a Mass of Earth 
 against which it is fired, varies as the Square of 
 the Velocity with which it impinges upon it - 3.52 
 
 292. The Principle of Virtual Velocities - 353 
 
 293. If any Number of Forces be under any Circum- 
 
 stances in Equilibrium, and to any or all of their 
 Points of Application there be communicated 
 indefinitely small Motions in any Directions; 
 then these Forces, being each multiplied by its 
 corresponding virtual Velocity, and the Sum of 
 these Products being taken in respect to those 
 Forces, the Displacements of whose Points of 
 Application are towards the Directions of their 
 Forces, and the Sum in respect to those whose 
 Displacements are from the Directions of their 
 Forces, the one Sum shall equal the other - 356 
 
 294. Of Machines - - - - - 363 
 
XXXVlll CONTENTS. 
 
 Page 
 
 295. The State of the Motion of a Machine is, at first, 
 
 a State of accelerated Motion - - 364 
 
 296. The Forces operating in a Machine being in 
 
 Equilibrium in every relative Position which 
 the Parts of that Machine can be made to assume, 
 any Momentum or Force of Motion thrown 
 into the Machine will remain in it continually, 
 unimpaired and unaltered - - 364 
 
 297. The Dynamical Effect, or the Amount of the Action 
 
 or Efficiency of any Agent, is measured by the 
 Pressure which it exerts multiplied by the 
 Space through which it exerts it 369 
 
 298. The Dynamical Efficiencies of different Agents - 371 
 
 299. The Dynamical Effect of a Human Agent - 371 
 
 300. The Dynamical Effect of a Horse - 372 
 
 301 . The Power of a living Agent to produce a given 
 
 Dynamical Effect - - 372 
 
 302. The Dynamical Effect of one Pound of Coals - 374 
 
 303. The Dynamical Effect of any Agent operating 
 
 through a Machine which moves with a uniform 
 Motion is the same, whatever that Machine may 
 be, provided only the Resistances opposed to the 
 Motions of the Parts of the Machine by Friction 
 and other opposing Causes be the same - 375 
 
 304. The Dynamical Effect upon the moving Point, or 
 
 the moving Power, in a Steam Engine - 377 
 
 305. The Dynamical Effect upon the working Points or 
 
 the working Power of a Steam Engine - 380 
 
 306. Practical Method of determining the Dynamical 
 
 Effect at any working Point in a Machine, or 
 the working Power operating at that Point - 381 
 
 307. The Theory of the Steam Engine - - 383 
 
CONTENTS. 
 
 xxxix 
 
 APPENDIX. 
 
 Page 
 
 Table 1. -.... . 391 
 
 Table II. - , . - 392 
 
 Table III. - - - . .. -392 
 
 Table IV. , .... 393 
 
 Table V. .... 395 
 
 Table VI. - . . . . 397 
 
 Table VII. - . , ' . ,397 
 
 Table VIII. . . -398 
 
 Table IX. - 398 
 
 Mr. Hodgkinson's Experiments on the mechanical 
 
 Properties of Cast Iron - - - 399 
 
 Table X. - - - - - 406 
 
 Table XI. . . 407 
 
 On the Chemical Composition of Hot and Cold Blast 
 
 Irons, as analysed by Dr. Thompson - - 407 
 
 Table XII. ... . .409 
 
 Table XIII. - - - - - 418 
 
 Table XIV. - -418 
 
 Table XV. . - 419 
 
 Experiments on Friction made at Metz in the Years 
 
 1381, 1832, 1833. M. Morin - 421 
 
 Table XVI. - - 425 
 
 Table XVII. - - - - 430 
 
 Table XVIII. ... - 432 
 
 Table XIX. - ... 435 
 
By the same Author. 
 
 THE MECHANICAL 
 
 PRINCIPLES OF ENGINEERING & ARCHITECTURE. 
 
 New Edition, corrected, with Woodcuts 8vo. 24s. 
 
 THE author has added in this edition 
 articles : first, " On the dynamical sta- 
 bility of floating bodies; " secondly, " On 
 the rolling of a cylinder ;" thirdly, " On 
 the descent of a body upon an inclined 
 plane, when subjected to variations of 
 temperature, which would otherwise rest 
 upon it ;" fourthly, " On the state border- 
 ing upon motion of a body moveabl* about 
 a cylindrical axis of finite dimensions, 
 when acted upon by any number of pres- 
 gures." The conditions of the dynamical 
 stability of floating bodies include those of 
 
 the rolling and pitching motion of ships. 
 The discussion of the rolling motion of a 
 cylinder includes that of the rocking mo- 
 tion to which a locomotive engine is sub- 
 ject when its driving wheels are falsely 
 balanced, and that of the slip of the wheel 
 due to the same cause. The descent of a 
 body upon an inclined plane when sub- 
 jected to variations in temperature, which 
 otherwise would rest upon it, is applied to 
 the explanation of the descent of glaciers. 
 The text has been throughout carefully 
 revised and corrected. 
 
 ASTRO-THEOLOGY. 
 
 Third Edition Just ready. 
 
 London: LONGMAN, BROWN, and CO., Paternoster Row. 
 
ILLUSTRATIONS 
 
 MECHANICS. 
 
 CHAPTER I. 
 
 THE INFINITE MINUTENESS OF THE ELEMENTS OF MAT- 
 TER. THE POROSITY OF MATTER ITS COMPRES- 
 SIBILITY ITS ELASTICITY. 
 
 THE limits of observation are soon passed, and 
 imagination almost as soon wearied, when we seek 
 for the ultimate divisions of matter and its atoms. 
 
 Among the many illustrations which offer them- 
 selves of the extreme minuteness of its elements are 
 the following : 
 
 1. GLOBULES OF BLOOD. 
 
 Blood, when recently taken from the body and 
 examined under the microscope, is seen to be 
 composed of a transparent colourless liquid, called 
 liquor sanguinis, and certain minute globules which 
 float in it and give it its colour. These contain, as 
 there is every reason to believe, the principle of 
 the nourishment of the body, of which the serum is 
 the vehicle. This composition of the blood, which 
 has been found whenever search has been made for 
 
 
2 ILLUSTRATIONS OF MECHANICS. 
 
 it, we may fairly conclude to be an essential part of 
 the animal economy, and to extend through every 
 form of living organisation, even to the lowest. 
 
 Although called globules, they are not of a sphe- 
 rical form, but either cylindrical or lens-shaped, 
 and not unfrequently they are to be seen floating in 
 the serum, packed together thus ^g g in groups. 
 
 In all mammiferous animals they are circular. In 
 birds and fishes their form is elongated. In man 
 each globule has a diameter varying from the 
 two thousandth to the four thousandth of an inch. 
 Now there are creatures "so small, yet visible by 
 the aid of microscopes, that their whole living 
 organisation might be included in the bulk of one 
 globule of human blood ; limbs for motion, for 
 defence, and to provide themselves with food ; 
 organs of sense and of deglutition ; sinews, mus- 
 cles, nerves : nay, a circulating medium blood 
 composed of serum, and having its own globules of 
 blood. In the milky juice which is the blood 
 of certain plants also, as the Euphorbia and the 
 Ficus, may be seen globules, like those of the blood 
 of animals, but greatly less ; they are probably as 
 essential a part of the vegetable as of the animal 
 economy, extending throughout it, and to its minutest 
 forms. 
 
 If the imagination be not yet weaned, it may con- 
 ceive each of these globules divided in respect to the 
 atoms which compose it. Still the minuteness of 
 the elements of matter will never be reached ; for 
 the gelatinous consistency of the globule shows that 
 these its component atoms, are infinite in number. 
 
DUST OF THE LYCOPERDON. 3 
 
 2. CRYSTALS OF THE CELLS OF PLANTS. 
 In certain portions of the cellular texture of many 
 plants, as in the cells of the flower stem of the 
 hyacinth, the bulb of the lily, and of the squill, &c. 
 may be seen by the aid of the microscope., crystals 
 regularly and perfectly formed, composed,*!! is said, 
 of oxalate of lime. The very fact of their; crystal- 
 lisation proves to us (by every analogy ) that each one 
 of these crystals has an infinity of component atoms. 
 Now, in the cuticle of the Scilia maritima, are to be 
 found such crystals, one five thousandth of an inch 
 in length, and one eight thousandth of an inch in 
 their greatest thickness. 
 
 3. MUSK. 
 
 It is said that a grain of musk is capable of per- 
 fuming for several years a chamber twelve feet square 
 without sustaining any sensible diminution of its 
 volume or its weight. But such a chamber contains 
 2,985,984 cubic inches, and each cubic inch contains 
 1000 cubic tenths of inches, making, in all, nearly 
 three billions of cubic tenths of an inch. Now it is 
 probable, indeed almost certain, that each such cubic 
 tenth of an inch of the air of the room contains one 
 or more of the particles of the musk, and that this air 
 has been changed many thousands of times. Imagin- 
 ation recoils before a computation of the number of 
 the particles thus diffused and expended. Yet have 
 they altogether no appreciable weight or magnitude. 
 
 4. DUST OF THE LYCOPERDON. 
 
 The lycoperaon, or puff-ball, is a fungus growing 
 in the form of a tubercle, which, being pressed, bursts, 
 emitting a dust so fine and so light that it floats 
 
4 ILLUSTRATIONS OF MECHANICS. 
 
 through the air with the appearance of smoke. 
 Examined under the microscope, this dust, which 
 is the seed of the plant, appears under the form of 
 globules of an orange colour, perfectly rounded, and 
 in diameter, it is said, about the fiftieth part of a 
 hair; so that, if this calculation be correct, and a 
 globule were taken having the diameter of a hair, it 
 would be one hundred and twenty-five thousand 
 times as great as the seed of the lycoperdon. 
 
 5. METALLIC SOLUTIONS. 
 
 Let one grain of copper be dissolved in nitric 
 acid. A liquid will be obtained of a blue colour ; 
 and if this solution be mingled with three pints of 
 water, the whole will be sensibly coloured. 
 
 Now three pints contain 104 cubical inches, and 
 each linear inch contains at least one hundred 
 equal parts distinguishable by the eye ; each cubical 
 inch contains, then, at least one million of such 
 parts, and the 104? cubical inches of this solution 
 104 millions of such parts : also each of these mi- 
 nute parts of the solution is coloured, otherwise it 
 would not be distinguishable from the rest ; each 
 such part contains then a portion of the nitrate of 
 copper, the colouring substance. Now from each 
 particle of this nitrate, the copper may be preci- 
 pitated in the state of a metallic powder every 
 particle of which is therefore less than the 104 
 millionth of a grain in weight. 
 
 6. COLOURS PRODUCED BY THE ATTENUATION 
 
 OF TRANSPARENT BODIES. 
 The extreme attenuation which may be given to 
 certain forms of matter is a proof of the extreme 
 
ATTENUATION OF THE WINGS OF INSECTS. 6 
 
 minuteness of their elementary particles. In the 
 case of transparent bodies, there is a method of 
 measuring the degree of this attenuation, founded 
 on this principle of optics, " that all transparent 
 bodies become coloured when they are formed into 
 plates, attenuated beyond certain limits, and more- 
 over, that the particular colours, which under these 
 circumstances they show, are dependant upon the 
 degree of their attenuation ; " thus serving as a 
 delicate test and measure of it, so that, knowing the 
 colour, which by being attenuated, a transparent 
 body is made to show, we may know how thin it is. 
 
 7. THE THICKNESS OF A SOAP BUBBLE. 
 
 It is thus that Newton has determined the top, 
 which is the thinnest part, of a soap bubble, to be 
 when colours are first seen in it, the *000,003,937th 
 part, or about the twenty-five-thousandth part of an 
 inch in thickness, and before it bursts to reach an 
 attenuation of at least the four-millionth part of an 
 inch. 
 
 8. ATTENUATION OF THE WINGS OF INSECTS. 
 
 By the same means we know that the transparent 
 wings of certain insects, are not more than the hun- 
 dred-thousandth of an inch in thickness, and that as 
 great an attenuation as this may be given to glass, 
 by blowing it in bubbles, until it bursts like the 
 bubbles of soap. 
 
6 ILLUSTRATIONS OF MECHANICS. 
 
 The property of matter, by which it may be made 
 to receive an extreme degree of attenuation, is of 
 extensive application in the arts. 
 
 9. THE ATTENUATION OF GOLD LEAF. 
 An ounce of gold is equal in bulk to a cube, each 
 ef whose edges is five-twelfths of an inch, or nearly 
 half an inch, in length, so that placed upon a table 
 it would cover nearly one quarter of a square 
 inch of its surface, standing nearly half an inch in 
 height. This cube of gold the gold-beater ex- 
 tends until it covers 146 square feet ; and it may 
 readily be calculated, that to be thus extended from 
 a surface of ^ths of an inch square to one of 146 
 square feet, its thickness must have been reduced 
 from half an inch to the 290,636th part of an inch. 
 Fifteen hundred such leaves of gold placed upon 
 one another, would not equal the thickness of the 
 paper on which this is printed. 
 
 10. THE ORDINARY PROCESS OF GILDING. 
 
 Gilding, according to the process usually adopted 
 in the arts, presents a remarkable example of the 
 minute division, and the attenuation of which gold 
 is capable. 
 
 The following is that process. Gold is dissolved 
 in mercury in the proportion of one part to five or 
 six, by placing the two metals in these proportions 
 in an iron ladle and bringing them to a boiling 
 heat. A half a pound troy of gold, in minute por- 
 tions, may thus be dissolved in six times its weight 
 of mercury in twenty or twenty-five minutes. This 
 solution of gold in mercury is called an amalgam. 
 
ORDINARY PROCESS OF GILDING. 7 
 
 It maybe thickened in its consistency by straining, by 
 means of pressure through a piece of chamois leather 
 through whose pores the mercury, not in actual 
 union with the gold, escapes; or it may be diluted by 
 heating again with more mercury. With this amal- 
 gam, the surface to be gilded, which is usually of 
 copper or brass, is to be covered by means of a 
 brush or otherwise ; but that an intimate cohesion 
 or union of the two may take place, it is found to be 
 necessary first to wash over the surface with a 
 liquid, technically called quick-water, which is made 
 by dissolving about a table-spoonful of mercury 
 into a quart of nitric acid. The effect of washing 
 the surface with this liquid is, to cover Jt with an 
 exceedingly thin amalgam of the metal which forms 
 the surface. Although the amalgam of gold will 
 not unite itself directly with the surface to be gilded, 
 yet it will unite itself with this amalgam of the sur- 
 face, and thus by the adherence to the surface of its 
 own amalgam, and of the gold amalgam to that, 
 both become fixed upon it. 
 
 If now the mercury could be removed, the par- 
 ticles of gold only would remain upon the surface, 
 and the gilding would be complete. The property 
 of mercury by which it is converted into a vapour 
 like water at the temperature at which it boils, makes 
 this an easy process. . 
 
 The various articles thus covered with amalgam 
 of gold have only to be subjected to a powerful 
 heat in a kind of oven of iron specially contrived 
 for that purpose, and the mercury is evaporated, 
 nothing but the gold remaining, and the surfaces 
 being gilded. 
 
8 ILLUSTRATIONS OF MECHANICS. 
 
 A polish is usually given to surfaces thus gilded 
 by rubbing them with a polished mineral known to 
 chemists as black haematite, which is a natural steel. 
 This process is called burnishing. They are usually, 
 moreover, subjected to a chemical process called 
 colouring. 
 
 A perfect and continuous surface of gold is thus 
 placed upon the gilded article, not the minutest 
 aperture or uncovered space is perceivable in it 
 with the most powerful magnifying glass or micro- 
 scope. Nitric acid*, if it be washed with it, will 
 find no aperture by which it may reach and attack 
 the substratum of copper or brass. But what is the 
 thickness of this coating of gold ? It may be spread 
 by the process above described more thinly upon 
 brass than copper; surfaces of brass, when gilded, 
 are said to be similored, and upon these a grain of 
 gold is commonly made to cover about 40 square 
 inches : this being the case, it may readily be cal- 
 culated that the thickness of this coating of gold is 
 about the --th ^ an ^ 
 
 11. THE GILDING OF THREAD FOR EMBROIDERY. 
 
 This process is thus described by Reaumur as 
 practised in his time. A ! cylinder of silver, 360 
 ounces in weight t, is cased with a cylinder of gold 
 at most 6 ounces in weight. J This cylindrical 
 
 * Nitric acid will not attack gold. 
 
 f The weights and measures spoken of in this article are 
 French. 
 
 | A French inch equals fjths of an English inch, and a 
 French ounce -^ths of an English ounce. 
 
GILDING OF THREAD FOR EMBROIDERY. 9 
 
 mass of 366 ounces of metal is then drawn by a 
 powerful force through a series of circular holes 
 in a plate of steel continually diminishing in 
 diameter, until it attains the state of a wire so 
 thin that 202 feet in length weigh but the sixteenth 
 of an ounce: the whole length of the wire into 
 which it is now drawn being 1,182,912 feet, or 
 about 98-J- leagues. This wire is then passed be- 
 tween rollers which in the act of flattening it elon- 
 gate it one-seventh, and its total length thus 
 becomes 112^ leagues. The width of the flattened 
 thread is now -Jth of a line, or -g^th of an inch ; and 
 supposing, with Reaumur, that a cubical foot of 
 gold weighs 21,220 ounces, and a cubical foot of 
 silver 11,523 ounces, it may readily be calculated 
 that the thickness of this gilded thread is very 
 nearly the -j-^gth part of an inch. Now what is 
 the thickness of the plate of gold which envelopes 
 it ? Calculating on the same principles as before, 
 we readily arrive at the conclusion, that the thick- 
 ness of this plate of gold is YTTtT^h ^ an inch. 
 Now gilded threads are made by a process similar 
 to this, in which only -Jd the proportion of gold is 
 used. There is spread over these, therefore, a con- 
 tinuous plate of gold less than the two-millionth 
 part of an inch in thickness. The silver may be 
 taken out of its gold case by plunging the 
 thread in nitric acid, by which the silver will be 
 attacked through the extremities of the gold case 
 and dissolved, whilst the gold will remain un- 
 touched by it. This being done, and the hollow 
 gold case being examined, it is found to be a 
 perfectly continuous plate, and to possess in this 
 
10 ILLUSTRATIONS OF MECHANICS. 
 
 state of extreme attenuation all the sensible and all 
 the chemical properties which belong to the metal. 
 Another but less striking evidence of the minute- 
 ness of the elements of matter is found in the ex- 
 treme tenuity of certain natural and artificial fibres 
 and threads. 
 
 12. TENUITY OF FIBRES OF SILK. 
 
 The thread of the silk-worm is a perfectly smooth 
 cylinder, whose diameter is from the one thousand 
 seven hundredth to the two thousandth part of an 
 inch. 
 
 13. THE TENUITY OF FIBRES OF WOOL. 
 
 Each hair of wool is a cylinder of from the seven 
 hundredth to the two thousandth part of an inch in 
 thickness, covered with what appear to be over- 
 lapping scales, which are laid in the direction in 
 which the hair grows, and the roughness of which 
 we feel when we draw our fingers along it in the 
 opposite direction. 
 
 14. THE FIBRE OF COTTON. 
 
 Under the microscope, each fibre of cotton- wool 
 appears to be composed of two tubular cylinders, at 
 a slight distance from one another, but joined to- 
 gether by a membrane. Its section is somewhat in 
 the form of the figure 8. It is about the thousandth 
 part of an inch in diameter. 
 
 15. THE FIBRE OF FLAX. 
 
 Each fibre of flax is a fasciculus of other fibres, 
 which appear under the microscope jointed and ir- 
 
FIBRES OF A SPIDER^S THREAD. 11 
 
 regular. Some of these have been ascertained to be 
 the two thousand five hundredth part of an inch 
 in diameter. The appearance of the fibre of flax 
 under the microscope is very different from that of 
 cotton. It is by this difference that the fine cere- 
 cloths of the mummies have been determined, by 
 Mr. Bauer, not to be of cotton fabric, but of linen. 
 
 16. FIBRES OF THE PINE APPLE PLANT. 
 
 Some of these have been measured, and are as- 
 certained to be from the five thousandth to the seven 
 thousandth of an inch in diameter. They are per- 
 fectly cylindrical, and when twisted into threads 
 and woven, are said to form cloths of a very beau- 
 tiful texture, and to offer a useful substitute for 
 silk. 
 
 17. TENUITY OF THE FIBRES OF A SPIDER'S 
 THREAD. 
 
 The most remarkable example of the tenuity of a 
 natural fibre is, however, to be sought in the spider's 
 thread, of which two drachms by weight would, it 
 is said, reach from London to Edinburgh. 
 
 It appears from the observations of Reaumur, that 
 the thread of the spider results from the expulsion of 
 a peculiar viscid matter through six teats under the 
 animal's belly, each of which, being pierced by cer- 
 tain minute apertures, not less, probably, than one 
 thousand in number, yields by each of these a separate 
 fibre, which fibres, uniting with one another from 
 each teat and adhering and sometimes the com- 
 
12 ILLUSTRATIONS OF MECHANICS. 
 
 pound threads from different teats thus uniting 
 form those threads which we see composing the web. 
 Now the head of each teat is so small as scarcely to 
 be visible. What then must be the tenuity of the 
 component fibres of the spider's thread, of which 
 more than 1000 spring from the head of each teat? 
 
 It is by reason of the exceeding fineness of many 
 natural threads, that they are made to minister so 
 greatly to the luxury of life under those forms of 
 woven tissues, for which the weavers of India were 
 formerly, and our own manufacturers have been of 
 late, so celebrated. 
 
 18. TENUITY OF COTTON YARN. 
 
 There is a specimen of Dacca muslin in the mu- 
 seum of the India House, of which the yarn, spun 
 by the hand, was ascertained by Sir J. Banks to be 
 so fine, that a weight of it equal to one pound avoir- 
 dupois would extend 115 miles, 2 furlongs, 60 yards. 
 When the muslin made from this Dacca yarn is laid 
 on the grass, and the dew falls upon it, it is said to 
 be no longer visible. The natives, in their meta- 
 phorical language, call it woven air. 
 
 Cotton yarn has been spun by machinery in Eng- 
 land, of which one pound would extend 167 miles ; 
 but this has never been woven. 
 
 There are various methods of drawing artificial 
 threads and wires to an extreme tenuitv. 
 
PLATINUM WIRE. 13 
 
 19. THREADS OF GLASS. 
 
 Glass is artificially drawn out almost to the fine- 
 ness of the fibre of silk. A rod of glass is melted 
 in the middle, in the flame of a blowpipe. One 
 portion is then fastened to a small wheel, which 
 being turned rapidly round, the melted glass is 
 drawn out from the other part of the rod which is 
 still held in the flame. Glass tube has been thus drawn 
 'nit to the fineness of silk, and liquids have afterwards 
 been made to pass through it. It is a remarkable 
 fact, that whatever was the form of the bore of 
 the original tube, the same form is retained in 
 the drawn tube, however great may be its tenuity. 
 
 20. PLATINUM WIRE. 
 
 Wires are used in the arts as fine almost as hairs. 
 There is, however, a mechanical limit fixed to the 
 thinness to which a wire can be drawn, by the force 
 necessary to draw it ; which force, when the wire 
 becomes thin, breaks it. This limit has, however, 
 been greatly passed by a method of art, of which 
 the following is an illustration. Wishing to obtain a 
 wire of extreme tenuity to be used in a micrometer, 
 Dr. Wollaston placed a platinum wire one hun- 
 dredth of an inch in diameter, in the axis of a 
 cylindrical mould one-fifth of an inch in diameter, 
 and cast round it a cylinder of silver. This cylinder 
 he then drew out by the common method, until it 
 became a wire so thin that it would no longer sus- 
 tain the force necessary to draw it. This wire of 
 silver, along the axis of which ran a wire of pla- 
 tinum, he then immersed in boiling nitric acid, 
 by which the silver was dissolved, and a platinum 
 
14? ILLUSTRATIONS OF MECHANICS. 
 
 wire was separated, the three millionth of an in en 
 in diameter ; being an artificial thread of which 
 140 must be placed together to equal in thickness 
 a fibre of the finest silk. 
 
 THE POROSITY OF MATTER. 
 
 All bodies have between their elementary par- 
 ticles or atoms, interstices through which heat pene- 
 trates into them, and into some of them, air, water, 
 and other fluids. These last are said to be POROUS. 
 
 21. THE POROSITY OF WOOD. 
 
 Wood is but a fascicle of tubes permeated when it 
 is growing by the sap. It is a common experiment 
 with the air pump, to make mercury pass through 
 these pores of wood. The mercury being placed in 
 a cup, the bottom of which is a piece of wood cut 
 transversely to the fibre, and this cup being herme- 
 tically fixed upon an aperture in the receiver of an 
 air pump ; when the air is extracted from beneath 
 it in the receiver, the pressure of the external air 
 on the surface of the mercury, no longer balanced 
 by the elasticity of the air within the receiver, 
 presses it with such force as to drive it through the 
 pores of the wood. At the extremity of each pore 
 a minute globule is seen, and these globules, at 
 length, descend in a minute shower of silver. When 
 wood is carbonised, its pores are very easily traced 
 by means of the microscope. Dr. Hook found 
 them extending through the whole length of the 
 wood> and counted in the eighteenth of an inch 
 150 of them ; so that in a piece of charcoal one 
 
POROSITY OF HYDROPHANE. 15 
 
 inch in diameter, there are more than five millions 
 and a half of them. 
 
 22. WOOD CEASES TO BE BUOYANT WHEN ITS 
 PORES ARE FILLED WITH WATER. 
 
 If a piece of wood be subjected to a great pres- 
 sure of water in a hydraulic press, or by sinking it 
 deep in the sea, the water will be driven into its 
 pores, expelling from them the air, and remaining 
 fixed in them by capillary attraction, the wood 
 thus becomes too heavy to float. Being placed in 
 the water, it will sink like lead. Boats used in the 
 whale fishery have been dragged to great depths in 
 the sea by the entanglement of the rope attached to 
 the harpoon with which the whale has been trans- 
 fixed. These, when brought to the surface again, 
 have been found useless, by reason of the water 
 which has been incorporated with them. 
 
 23. THE POROSITY OF ROCKS. 
 
 That many rocks are thus porous, the infiltration 
 into caverns and the formation of stalactites suffi- 
 ciently proves ; and it is thus in winter when, in 
 the act of freezing, the water they have imbibed 
 expands, that their surfaces exfoliate, and they 
 crumble away. 
 
 24. THE POROSITY OF HYDROPHANE. 
 
 Among silicious stones is one called hydrophane, a 
 Kind of agate, whose porosity causes it to present a 
 very remarkable phenomenon. In its ordinary state 
 it is only semi-transparent, but after being plunged 
 in water it takes up about th of its bulk of it, and 
 becomes nearly as transparent as glass. 
 
16 ILLUSTRATIONS OF MECHANICS. 
 
 25. POROSITY OF METALS. 
 
 That metals are porous was proved in 1661 by 
 the academicians of Florence, who submitted a 
 hollow ball of gold filled with water to a great pres- 
 sure, by which the water was made to weep through 
 the pores in the surface of the gold. This experi- 
 ment has often been repeated. 
 
 That all bodies are more or less permeable to 
 heat or porous to fluids, sufficiently accounts for the 
 fact that all bodies are more or less compressible. 
 
 COMPRESSIBILITY. 
 
 In many bodies their compressibility is a property 
 familiar to us. 
 
 A SPONGE, for instance, by compression, gives out 
 the water that it imbibes, and may thus be reduced 
 to one third of its bulk. 
 
 26. COMPRESSIBILITY OF WOOD. 
 
 Wood is compressed by passing it between iron 
 rollers to form the pins or bolts used in ship-build- 
 ing ; it is thus commonly reduced to one half its 
 bulk. 
 
 A CORK immersed 200 feet in the sea, will be so 
 compressed that, instead of rising when left to itself, 
 it will sink. And a bottle of fresh water corked 
 up and sunk a great depth in the sea, will return 
 with the cork still in it as when it descended, but 
 the water will be found to taste of salt. The cork 
 has in fact compressed so as to allow the salt water 
 to mingle with the fresh. Having at the same time, 
 
COMPRESSIBILITY OF WATER. 17 
 
 become heavy, it has sunk in the bottle, and, as the 
 bottle rose again to the surface, it has expanded to 
 its original dimensions, rising and re-occupying; its 
 place in the neck of the bottle. 
 
 27. COMPRESSIBILITY OF AERIFORM BODIES. 
 
 Of all the different forms of matter, the aeriform 
 is that under which it is most compressible. In 
 some recent experiments, a large body of air has 
 been mechanically compressed by GErsted, a Danish 
 philosopher, into the one hundred and tenth part of 
 its original bulk. He used for this purpose pow- 
 erful forcing-pumps originally constructed for com- 
 pressing air into the receivers of certain air-guns 
 belonging to the king of Denmark. It is not only 
 common air that is thus compressible, but all ae'ri- 
 from bodies. Thus, the gas used in cur streets is 
 so compressible that a sufficient quantity may be 
 forced into an iron bottle of comparatively small 
 dimensions, to supply a number of lights for a con- 
 siderable time. The stand which supports a light, 
 being cast hollow, has thus been made the reservoir, 
 whence gas was supplied to it, sufficient to feed 
 the flame for several evenings. A company was a 
 few years ago established for the purpose of selling 
 gas under this compressed form as portable gas. It 
 was commonly sold thus compressed under a pressure 
 of 450lb. on the square inch, into ^th part of its 
 ordinary bulk. 
 
 *28. COMPRESSIBILITY OF WATER. 
 
 The compressibility of water was long disputed. 
 The question has lately, however, been set com- 
 c 
 
18 
 
 ILLUSTRATIONS OF MECHANICS. 
 
 pletely at rest by the experiments of GErsted.* The 
 apparatus used by him was that represented in 
 the accompanying figure ; A B C D is a strong glass 
 cylindrical vessel, having firmly 
 affixed to it at the top a cy- 
 linder of smaller dimensions of 
 metal, AE F B, in which is move- 
 able, by means of a screw, an 
 air-tight piston K. M is a glass 
 bottle, into the neck of which is 
 fixed, by grinding, one extremity 
 of a capillary tube a a, which is 
 open at both ends. The bore of 
 this tube must be extremely fine, 
 and the precise fraction of the 
 contents of the whole bottle, 
 which each inch in length of 
 its bore will hold, must be as- 
 certained with great accuracy. 
 This is done by weighing the 
 quantity of mercury which the 
 bottle will hold, and the quan- 
 tity which an inch of the bore 
 of the tube will hold. What- 
 ever fraction the one weight is 
 of the other, the same is evi- 
 dently the contents of one inch 
 of tiie tube of the content of the bottle. In some of 
 the tubes used by CErsted, each inch in length was 
 found to hold 80 millionths of the contents of the 
 bottle. Let us suppose these to have been the tubes 
 * Transactions of the Royal Society of Sciences at Copen. 
 hqgen, 
 
COMPRESSIBILITY OF WATER. 19 
 
 with which his experiments were made. Let now the 
 bottle and tube be conceived to be filled with water. 
 Any pressure exerted upon this water which will 
 have caused its surface in the tube to descend one 
 inch will have compressed it by 80 millionths of its 
 bulk. Divisions were, however, marked upon a 
 scale annexed to the tube ^th of an inch apart. 
 A depression of the water in the tube through any 
 one of these divisions would therefore indicate a 
 compression of two millionths. But how is this 
 compression to be produced ? The bottle, and its 
 apparatus, are to be introduced into the glass vessel 
 A B C D, the part A E F B having been screwed off 
 to admit them. This vessel is then to be filled 
 with water, and the cylinder AEFB is to be re- 
 placed, its piston K having been first screwed down 
 to H. This piece being firmly fixed, and the pis- 
 ton then screwed back towards its position K, 
 water will be drawn into the vessel by the syphon 
 BP, which communicates with a vessel of water Q. 
 When it is full a cock closes the communication of 
 the syphon with the vessel, and the piston K being 
 screwed back again, or downwards, the pressure 
 begins. 
 
 From the piston and the water in the vessel the 
 pressure is propagated through the tube aa to 
 the water in the bottle ; and the pressure thus pro- 
 duced within and without the bottle is precisely the 
 same.* 
 
 But how is this pressure to be measured ? By 
 this simple contrivance : N is a glass tube, closed 
 
 * By the law of the equal distribution of fluid pressure. 
 See Mechanic! applied to the Arts, Art. 243. 
 C 2 
 
20 ILLUSTRATIONS OF MECHANICS. 
 
 at the top and open at the bottom, and equal divi- 
 sions are marked along it. This tube, being loaded 
 by a rim of lead at the bottom, is immersed in the 
 water of the vessel, in its inverted position, at the 
 same time that the bottle and tube are introduced. 
 And wfien the pressure is applied, the air which it 
 contains is compressed continually into a less and 
 less bulk, the diminution of its bulk being precisely 
 proportional to the pressure.* Thus, by observing 
 the degree to which the air is compressed in this 
 tube, or the height to which the water is raised in 
 it, the pressure which the screw is exerting and the 
 water in the bottle sustaining, is always known. 
 
 Since the whole vessel as well as the tube and 
 bottle are filled with water, a question arises how is 
 the descent of the surface of the water in the tube 
 a a to be distinguished? Some separation must 
 evidently be made between the surface of the water 
 in the tube a a, and the water in the vessel which 
 presses upon it. To produce this separation, when 
 the tube is sunk, care is taken that it shall not be 
 completely full of water; and to keep in theair which 
 thus occupies the top of the tube, and cause it to 
 make a permanent separation between the water 
 within and that without the bottle, a funnel-shaped 
 glass vessel p, open at the bottom and loaded round 
 its lower edge, is inverted over it. This vessel is 
 thus, when the instrument is sunk, nearly filled with 
 air, which by the pressure is made continually to 
 occupy a less and less space, and driven into the 
 cube, so that when the water of the vessel at length 
 
 * By Marriotte's law, afterwards to be explained. 
 
* COMPRESSIBILITY OF WATER. 21 
 
 reaches the top of tne tube and enters it, there 
 intervenes between it and the water already within 
 the tube, a column of compressed air, forming a 
 separation of the two, which may easily be seen 
 without, i i are cork floats, attached by strings for 
 the convenience of removing the apparatus when 
 the experiments are completed. 
 
 The experiments of CErsted, made with this 
 apparatus, not only establish the fact of the com- 
 pressibility of water, the water sinking in the tube 
 about half an inch for each additional pressure of 
 an atmosphere ; but they ascertain its amount by 
 the methods explained above, to be 46 -^ millionths 
 of its bulk for each such additional pressure of one 
 atmosphere or of about 15 pounds the square inch.* 
 Thus for each additional equal pressure the water i& 
 compressed by the same fraction of its bulk. This 
 is a remarkable law, which is found to govern the 
 compression of all other bodies. 
 
 The same method applied by CErsted to the com- 
 pression of water, manifestly enabled him to com- 
 press and measure the compression of any other 
 liquid. For that purpose, he had only to cause that 
 liquid to replace the water in the bottle and tube. 
 Table I., in the Appendix, presents the results thus 
 
 * The pressure of an atmosphere on any surface is a pres- 
 sure equal to that which is exerted upon it by the weight of 
 the air : the pressure of two atmospheres is twice the pressure of 
 the air, and so on. This pressure of the air upon any surface 
 is equivalent to the weight of a column of mercury having a 
 base equal in size to that surface, and a height equal to the 
 height at which the barometer stands. Its mean value is 15 
 pounds to the square inch. 
 
 C 3 
 
22 ILLUSTRATIONS OF MECHANICS. 
 
 obtained. Beneath them are results similarly ob- 
 tained by Messieurs Colladon and Sturm. 
 
 Out of the great and unexplained difference be- 
 tween these results of Colladon and Sturm and 
 those of GErsted, has arisen an interesting discus- 
 sion as to the correction which should be made for 
 the compression of the substance of the tube and 
 bottle by reason of the pressure which they sustain 
 within and without. M. Poisson (Mem. Ac. Sci., 
 1827, 1828,) has arrived at the theoretical conclu- 
 sion, that by this compression the capacity of the 
 bottle is diminished; and he has given a very simple 
 rule for the correction. GErsted denies, however, 
 the accuracy of this correction. He states, indeed, 
 the fact altogether inconsistent with it, that the re- 
 cession of water in the capillary tube is invariably 
 about 1^ millionth*; greater when bottles of lead 
 and tin were used instead of bottles of glass, 
 
 *29. THE COMPRESSION OF SOLIDS BY OERSTED'S 
 APPARATUS. 
 
 The method of CErsted lends itself to direct ex- 
 periments on the compression of solid bodies. To 
 determine the compression of a solid under any 
 given pressure it is placed in the bottle M, the tube 
 being taken out to admit it. The bottle is then 
 filled with water, the tube replaced, and the whole 
 subjected to the pressure of the screw, under the 
 same circumstances as before. The descent of the 
 column of water in the tube shows the joint amount 
 of the compression of the water and the solid. 
 Also, the amount of the compression of the water 
 
OERSTED'S APPARATUS. 23 
 
 is known by the preceding experiments ; that of the 
 solid, then, is easily found. 
 
 * 30. THE ADAPTATION OF OERSTED'S APPARATUS 
 TO HIGH PRESSURES. 
 
 Wishing to try the effect of higher pressure in 
 the compression of air than could with safety be 
 applied to the glass vessel, CErsted replaced it by 
 one of metal* 
 
 Now, however, the vessel being no longer trans- 
 parent, it became necessary to contrive some me- 
 thod by which the pressure produced by the screw, 
 and the degree of compression of the air, might 
 register themselves permanently, so that they might 
 be read off, when the apparatus was taken out of 
 the vessel. 
 
 This permanent registration of the pressure was 
 effected by using, instead of the large tube N, a 
 smaller tube expanded at its closed extremity into a 
 bulb, and having a short column of mercury sus- 
 pended in it, on whose surface floated an index, to 
 which was affixed a hair spring, pressing it against 
 the side of the tube, so that the index would stick at 
 the extreme point to which the mercury might have 
 raised it, when the latter should again recede.* 
 
 By the position of this index, when the appa- 
 ratus was taken out, the extreme pressure which the 
 screw should have produced would evidently be 
 known. To subject the air, on which the experi- 
 ment was to be made, to this pressure, and to 
 measure the amount of its compression, the inge- 
 
 * This contrivance is the same with that in Six's self-regis- 
 tering thermometer. 
 
 c 4 
 
24 
 
 ILLUSTRATIONS OF MECHANICS. 
 
 /g. 2. 
 
 nious and simple apparatus shown in the 
 accompanying figure was used. F G H J 
 [ is an open vessel containing mercury; 
 A B C D E a glass vessel drawn out into a 
 slender, tube E D, which is turned down- 
 wards, as shown in the figure. This 
 vessel, whose only opening is at E, is 
 made to contain the air on which the ex- 
 periment is to be made, ano^ is then sunk 
 ~H in mercury in the position shown in the 
 figure; and in this state the whole is plunged in 
 the receiver, ACDB, of the compressing appa- 
 ratus. (Fig. 1. p* 18.) The pressure being then 
 applied, its effect is to drive the mercury up the 
 tube E D, and into the vessel C B, compressing the 
 air above it, and falling to the bottom of that vessel. 
 When the pressure is withdrawn, only that portion 
 of the mercury which is contained in the tube ED 
 will return, and the volume of that contained in the 
 vessel DCBA being added to the volume of this 
 which was contained in the tube, will equal the 
 volume by which the air was diminished during 
 the experiment, as shown by the maximum pres- 
 sure of the index. CErsted thus compressed air 
 into ^th f *ts original bulk, and measured 
 the pressure, which he found to be just 65 times 
 the ordinary pressure of the atmosphere. Arid 
 in a number of other similar experiments, he 
 found, that by however many times he wished to 
 diminish its bulk, by exactly so many times was it 
 always necessary to increase the pressure upon it, 
 or in other words, that the compression was always 
 proportional to pressure applied ; twice the ordi- 
 nary pressure upon the air producing twice the 
 
MARRIOTTE S EXPERIMENT. 
 
 25 
 
 compression ; three times, thrice the compression ; 
 this relation of the compression to the pressure is 
 called that of perfect elasticity. It is not peculiar 
 to the air, but is common, within certain limits of 
 pressure, to all aerial and solid bodies, and it ap- 
 pears, from the preceding experiments of CErsted, 
 to all liquid bodies. 
 
 ELASTICITY. 
 
 * 31. *MARRIOTTE'S EXPERIMENT. 
 
 The perfect elasticity of air was first proved by 
 fig* 3. Marriotte. The following is (with 
 a slight variation) his experiment. 
 ABC (fig* 3.) is a curved cylindri- 
 cal tube, graduated in equal parts, 
 closed at C and open at A. 
 
 Let mercury be poured into this 
 tube, so as to occupy a portion, H B F, 
 of it, towards the open end A, whilst 
 the rest, FC, contains air. 
 
 Let this tube now be laid flat on a 
 perfectly horizontal table, and let the 
 division which separates the mercury 
 and air be observed. Place it then 
 in an upright position, and again ob- 
 serve the division at which the mer- 
 cury and air are separated ; and more- 
 over, the whole height of the column 
 of mercury above the level of that 
 division.* When the tube was laid 
 
 * Thus, if the division of the air and mercury stand at any 
 point F, in the shorter branch, it is the height of the column 
 H G, which is above the level of F, that is to be measured. 
 
26 ILLUSTRATIONS OF MECHANICS 
 
 flat, the mercury was supported entirely upon 
 its sides, and did not press at all upon the air, so 
 that the space occupied by the air was that which 
 it would occupy out of the tube, or in its natural 
 state, that is, under a pressure equal to that of the 
 barometric column * ; but when it is placed in an 
 upright position, the weight of the whole column 
 of mercury, above the level of the common surface 
 of the air and mercury, presses upon the air. The 
 air is therefore pressed more than in its natural state 
 by the weight of this column; and it is compressed, 
 and the amount of the compression is easily mea- 
 sured by a comparison of the length of the tube 
 which the air now occupies, with that which it 
 occupied when it was laid flat. Now, suppose that 
 the height of the column of mercury above the level 
 of its division with the air, to equal the height of 
 the barometric column at that moment. The na- 
 tural pressure upon the air equalling the weight of 
 this column, and an artificial pressure of the same 
 amount being added to it, the whole pressure upon 
 the air in the tube will be double what it was 
 before. Now it will be found, that under these cir- 
 cumstances, the space occupied by the air will be 
 halved; and if, in like manner, the column of 
 
 * By the pressure of the barometric column is here meant 
 the weight of the column of mercury as it would stand at the 
 time of the experiment in a barometer whose tube had the same 
 diameter with that used in the experiment. The column in 
 the barometer being supported by nothing but the air, is greater 
 as that pressure is greater, and less as it is less ; its weight is 
 exactly equal to the pressure of the external air on a surface 
 equal to the base of the column. 
 
MARRIOTTE S EXPERIMENT, 
 
 mercury in the tube had been made equal to twice 
 the barometric column, so as to triple the whole 
 pressure upon the included air, then the space 
 occupied by it would be reduced to one third ; and 
 generally, it will be found that if the whole pres- 
 sure upon the air be by these means increased in 
 any proportion, the space occupied by it will be 
 diminished in a like proportion. 
 
 Moreover, by inverting the position of the tube, 
 fish 4 as v&fig* 4. we may diminish the pressure 
 / css, upon the included air, instead ot in- 
 creasing it, and this diminution of pres- 
 sure will then just equal the weight of 
 ' the column of mercury, F A, which is 
 suspended beneath the level of the sur- 
 face F, which separates it from the air. 
 So that if this column equal in height 
 one half of the barometric column, then 
 the pressure upon the air will be di- 
 minished one half; if it equal two thirds 
 the barometric column, then the natural 
 pressure will be reduced to one third, &c. 
 Now in these cases it will be found that the 
 space occupied by the air will be doubled, 
 tripled, &c. So that in general, the pres- 
 sure upon air, whether it be more or less 
 condensed than in its natural state, is in- 
 versely proportional to the space it occu- 
 pies. This is called the law of Marriotte.* Inall cases, 
 
 * It is a necessary precaution to the accuracy of this expe- 
 riment, that the air should be perfectly freed from moisture ; 
 the presence of water materially affecting the conditions of its 
 elasticity. To dry it perfectly the tube should be heated, and 
 
28 ILLUSTRATIONS OF MECHANICS. 
 
 the air when released from the pressure applied to 
 it, instantly recovers its original bulk ; the force 
 with which it tends to recover that bulk, being, in 
 fact, that which must be overcome to compress it. 
 
 This property is not peculiar to aeriform bodies. 
 (Ersted has proved it of water and other liquids, 
 enumerated in the table I. in the Appendix. It 
 appears, indeed, that aeriform bodies are but liquids 
 under a diminished state of pressure ; so that by 
 increasing the pressures upon them very greatly, 
 they may be all made to assume a liquid form. 
 
 32. THE ELASTICITY OF THE METALS. 
 
 With the elasticity of metallic bodies every one 
 is conversant. It is a property which, as it belongs 
 to steel, iron, and brass, contributes eminently to 
 the resources -of art, and ministers largely to the 
 uses of society. Were it, indeed, not for this pro- 
 perty, it would be in vain that the metals should 
 be dug out of the earth and elaborated into various 
 utensils. Infinitely more brittle than glass, they 
 would immediately be dashed to pieces by the 
 slight shocks to which every thing is more or less 
 subject ; a shower of hail, or even of rain, would 
 be sufficient to indent * their surfaces, and the im- 
 
 then for several days made to communicate with a vessel con- 
 taining muriate of lime, or some other substance which extracts 
 from the air its moisture. 
 
 * It will be shown in a subsequent part of this work, that 
 the force which accompanies the impact of a body, is in its na- 
 ture infinitely greater than any force of that kind which we call 
 pressure. Now of the class of forces of pressure, are those 
 
ELASTICITY OF WIRE. 29 
 
 pact of the minute particles of dust blown against 
 them by the wind would be sufficient permanently 
 to destroy their polish. 
 
 33. THE LAW OF THE ELASTICITY OF METALS. 
 
 The force with which metals, when extended or 
 compressed, tend to recover their form, that is, the 
 force necessary to keep them extended or com- 
 pressed, is proportional to the amount of the ex- 
 tension or compression they have received. 
 
 Thus double the extension or compression of the 
 same body requires double the force ; triple, triple 
 the force ; quadruple, quadruple the force. Simi- 
 larly, one half the compression or extension, or 
 one third of it, or one fourth, requires one half, one 
 third, or one fourth, the compressing or extending 
 force. 
 
 This is the law which constitutes perfect elasticity, 
 and which has been shown to belong to liquids and 
 gases. It was first accurately proved in respect to 
 metal wires by S. Gravesande. 
 
 34. EXPERIMENTS OF S. GRAVESANDE ON THE 
 ELASTICITY OF WIRES. 
 
 The apparatus used by S. Gravesande is repre- 
 sented in the accompanying figure. The wire, 
 whose elasticity was to be determined, was extended 
 
 cohesions which hold together the particles of solid bodies. 
 These therefore of necessity yield to any force of impact ; and 
 were it not for the force of elasticity by which the displaced 
 particles recover their positions, any such force of impact would 
 produce a permanent indentation. 
 
30 
 
 ILLUSTRATIONS OF MECHANICS. 
 
 fg. 6. 
 
 between the two fixed points A and B. A light 
 scale-pan, C> suspended from a silken thread C H, 
 was hung upon its middle point H ; and to balance 
 this scale-pan, a continuation of the silken thread, 
 which suspended it, passed over a. pulley P, and 
 supported a counterpoise. The pulley P carried an 
 index P G, pointing to equal divisions on a dial 
 plate. Exceedingly small weights were placed in 
 succession, and very gently, in the scale-pan, and 
 the deflexions of the wire produced by these were 
 observed by the motion of the index P G. This 
 deflexion of the wire being thus known, and also 
 the distance A B, in a straight line, between its 
 extremities, its length A H B corresponding to each 
 such deflexion, became known by easy rules of 
 geometry. The difference between this length and 
 its original length was its elongation. It will be 
 observed, that the weight in the scale-pan is not 
 exerted in the direction of the length of the wire; 
 nevertheless it does produce a certain strain or 
 tension in that direction ; now the amount of this 
 strain, exerted in the direction of the length of the 
 wire, can be determined by a very simple rule of 
 mechanics, to be explained hereafter (see Paral- 
 lelogram of Forces) ; and it is this strain or tension 
 which was to be compared with the elongation of 
 
ELASTICITY OF TORSION. 31 
 
 the wire. It resulted from the experiments of 
 Gravesande, that this strain was exactly proportional 
 to the corresponding elongation. 
 
 35. ELASTICITY OF IVORY. 
 
 The elasticity of ivory is sufficiently shown by 
 the impact of billiard-balls. The following expe- 
 riment presents it, however, in a yet more striking 
 form. Let an ivory ball be let fail perpendicularly 
 upon a smooth and hard plane of stone or metal 
 for instance which has been first rubbed over 
 with oil. It will be seen to rebound very nearly to 
 the height from which it has fallen ; the cause 
 which has interfered with its re-ascent exactly to 
 that height being the resistance of the air. Now 
 let the spot where it has struck the plane be exa- 
 mined ; the traces of the impact will be seen in the 
 oil, not at a point only, as would have been the case 
 had the surface of the ball not yielded at the instant 
 of impact, but over a considerable surface, which 
 is greater as the ball is allowed to fall from the 
 greater height. Thus whilst by the rebounding of 
 the ball its elasticity is shown, by this mark on the 
 oil, its compression, or the flattening of its sur- 
 face at the instant of impact, is proved. Balls of 
 wood, stone, glass, and metal, present the same 
 phenomena as those of ivory. 
 
 36. ELASTICITY OF TORSION. 
 
 If a wire be twisted, it will tend to recover its 
 natural state, with a certain force which is called 
 its elasticity of torsion. The law of this force is 
 this, that it is always proportional to the angle 
 
S3 ILLUSTRATIONS OF MECHANICS. 
 
 through which the body has been twisted. This 
 property may be shown to result from that other 
 universal property of bodies by which any portion 
 of them being displaced within certain limits, tends 
 to return to its position with a force proportional 
 to the displacement, and indeed it proves that pro- 
 perty. (See Moseleys Mechanics applied to the 
 Arts, Art. 199., &c.) It exists not only in bodies 
 such as steel, brass, wood, &c., with whose elastic 
 properties we are conversant but more or less in 
 all bodies. 
 
 37. COULOMB'S TORSION BALANCE. 
 
 From the facility with which metal wires and 
 threads of various substances may be twisted, and 
 the perfect regularity and precision with which they 
 tend to return to their former positions with forces 
 proportional to the angles of torsion ; they have come 
 to be used for measuring certain forces too minute 
 to be estimated by the ordinary methods. Coulomb 
 was the first to make this application of the elasti- 
 city of torsion : by means of it, he succeeded in 
 determining, by direct experiment, the laws which 
 govern the variation of magnetic and electric forces ; 
 and it was by means of the torsion balance that 
 Cavendish afterwards detected and measured the 
 almost evanescent attraction of gravitation in balls 
 of lead 
 
 The torsion balance consists of a stand T, sup- 
 porting a hollow vertical rod S T, which, in the 
 balance of Coulomb, was of pewter, that all mag- 
 netic and electric influence might be avoided. On 
 
ELASTICITY OF TORSION. 33 
 
 this rod there are two sliding 
 pieces C A and S P ; the lower 
 of which carries a plate A, with 
 a circle divided like a dial-plate 
 upon it ; and the upper, a piece 
 P, to which the torsion wire 
 or thread is to be fixed. N is 
 a small bar-piece, with a screw 
 which clips the extremity of the 
 wire whose torsion is to be 
 experimented on, to which a 
 
 weight, or an index, or both, may be attached. 
 The following were the principal results obtained 
 
 by Coulomb : 
 
 1. The wire, being loaded with different weights, 
 did not rest in the same position of the index. That 
 is, by adding to the weight borne by the wire, or 
 taking away from it, the index was always made to 
 rest in a different position. 
 
 2. The oscillations of the index were isochronous. 
 That is, when the index, being deflected from its 
 position, was then left to itself, it always returned 
 to that position again in the same time, whether the 
 deflection was great or small ; in the one case mov- 
 ing faster, and in the other slower, precisely in the 
 proportion necessary to preserve this equality of time 
 or isochronism, 
 
 It appears from the theory of dynamics, that 
 this one observed fact is sufficient to establish the 
 principle, that the force with which the wire tends to 
 return is proportional to the angle of torsion, so that, 
 observing the angles through which the index is 
 
34 ILLUSTRATIONS OF MECHANICS. 
 
 twisted by the action of different forces, we can 
 compare the forces: this is the use of the balance. 
 
 The isochronism of the oscillations only obtains, 
 however, within certain limits of torsion ; thus, if an 
 iron wire, so slender that six feet in length weigh 
 but five grains, and nine inches in length, be de- 
 flected, its oscillations will be isochronous so long 
 as they do not exceed half a circumference. But if 
 it be deflected through three circumferences, so as 
 to oscillate at first through six, then the oscillations 
 will be slower by about ^th than before.* 
 
 38. THE ELASTICITY OF LEAD AND PIPE-CLAY. 
 Experiments similar to the above, made with wires 
 of lead and thin cylinders of pipe-clay , showthatthese 
 and many other substances, apparently yielding and 
 inelastic, possess, in reality, elastic properties as per- 
 fect as those of steel. A wire of lead, for example, 
 one fifteenth of an inch thick and ten feet long, 
 suspended as in the experiments of Coulomb, and 
 twisted, being let go, oscillated isochronomly, show- 
 ing that the force was proportional to the angle of 
 torsion, and, therefore, that the elasticity of the sub- 
 stance was perfect. A similar experiment with a 
 thin cylinder of pipe-clay gave the same result. 
 
 * 3. The wire being loaded with different weights, the times of 
 isochronous oscillation were as the square roots of these weights. 
 
 4. The lengths of wire being different, the times were as the 
 square roots of these lengths. 
 
 5. The diameters of the wires being different, the times were 
 as the square roots of these diameters. 
 
 In all these cases the oscillations are supposed to be small 
 enough to be isochronous. 
 
ELASTICITY OF TORSION. 35 
 
 39. THE TORSION OF BARS OF IRON. 
 
 Whilst a wire of small diameter, a few inches or 
 even a few feet long, may be in a degree homo- 
 geneous, this quality is not to be expected in a bar. 
 Thus the conditions of torsion, which in a wire 
 are so simple and uniform, become in a bar com- 
 plicated and anomalous. In the Appendix to this 
 work will be found tables containing the results of 
 experiments on the torsion of bars, made by Mr. 
 Banks, Mr. Dunlop, of Glasgow, and M. Duleau. 
 
 40. ELASTICITY A COMMON PROPERTY OF AERI- 
 FORM BODIES, LIQUIDS, AND SOLIDS. 
 
 That aeriform bodies, liquids, and solids, should 
 possess, in common, the property of elasticity, may 
 appear to us the less singular, if we consider that 
 these are but different forms under which the same 
 body may exist subject to different conditions of 
 heat. 
 
 Steam, for instance, an aeriform * vapour, con- 
 denses into liquid water, a certain abstraction being 
 made of its heat ; and this water, by another reduc- 
 tion of temperature, becomes solid ice. And, to 
 take an example of this process of transition in the 
 
 * Steam, in an entirely uncondensed state, appears strictly 
 under the form of an air : it is perfectly clear, colourless, and 
 transparent, and may be seen in this colourless transparent state 
 in the bubbles of steam which ascend in a vessel of boiling 
 water from the bottom, where they are generated. It is when, 
 coming in contact with the air, the steam begins to condense, 
 that it assumes that cloudy appearance which we usually asso- 
 ciate with our idea of steam. 
 D 2 
 
36 ILLUSTRATIONS OF MECHANICS. 
 
 opposite direction, a solid metal becomes a liquid 
 by a certain addition of heat ; and a yet greater 
 intensity of heat volatalises it. When the partial 
 abstraction of heat is aided by a powerful pressure, 
 a permanent aeriform body or gas may be converted 
 into a liquid. 
 
 41. THE LIQUEFACTION OF THE GASES. 
 
 This interesting experiment was first made by 
 Faraday. Two chemical substances, from which, 
 when brought together, the gas to be liquefied 
 would be liberated (concentrated sulphuric acid and 
 carbonate of ammonia, for instance, when carbonic 
 acid was to be liquefied), were made to occupy op- 
 posite extremities of a bent glass tube, which was 
 then hermetically sealed. * By inclining this tube, 
 the two substances were then brought together, and 
 the gas evolved with immense force; and, being held 
 compressed within the narrow chamber of the tube, 
 was seen to assume a liquid form in the opposite leg 
 of the tube to that in which the two substances 
 mingled. 
 
 In some experiments, the gas to be operated 
 upon was made to occupy a portion of the tube se- 
 parated from the rest by a drop of coloured fluid. 
 In the other portion of the tube, carbonate of am- 
 monia and sulphuric acid were placed in small ex- 
 
 * Care was taken to introduce the acid by means of a long 
 capillary funnel, so as not to wet with it that portion of the 
 tube into which the neutral salt, or other substance from which 
 the gas was to be liberated, was placed. When this precaution 
 was not taken, the disengagement of gas prevented the tube 
 from being effectually sealed. 
 
LIQUEFACTION OF THE GASES. 3? 
 
 panded chambers apart The tube was then sealed, 
 the acid and salt were brought together by in- 
 clining the tube the liberated carbonic acid gas 
 drove the drop of fluid before it, compressing the 
 gas included at the opposite extremity of the tube 
 until it liquefied it. The condensation was assisted 
 by artificially cooling that extremity of the tube 
 where it was to take place. * 
 
 Table II., in the Appendix, states the pressure 
 in atmospheres, and the temperature at which the 
 liquefaction of the gases enumerated in it took 
 place. 
 
 The liquefaction of carbonic acid gas is now pro- 
 duced by means of powerful forcing-pumps.f 
 
 When the pressure is removed, the liquid re- 
 assumes its gaseous form ; and the gas being allowed 
 to escape, the jet, in the act of expanding itself, so 
 depresses its temperature as to congeal at a temper- 
 ature lower than any other known to exist. 
 
 * The specific gravity was measured by introducing, before 
 the tube was sealed, minute bulbs of glass, whose specific gra- 
 vity had been before determined by observing in what fluids of 
 known specific gravity they would float. The degree of pres- 
 sure was measured by a contrivance similar to that used in 
 CErsted's experiments (page 20. ), the tube being here, of course, 
 exceedingly minute, drawn over the blow-pipe. 
 
 f Sir H. Davy found that by a given accession of temper- 
 ature the expansive power of gas in a liquid state was much 
 more increased than by an equal addition of heat to gas in a 
 gaseous state. He found, for instance, that the expansive force 
 of liquid carbonic acid at 1 2^ F. was increased by an accession of 
 20 of temperature from 20 atmospheres to 36. He conceived 
 the idea, that by reason of this property the expansion of the 
 liquefied gases might with advantage be used as a moving power 
 in machinery. 
 
 J> 3 
 
38 ILLUSTRATIONS OF MECHANICS. 
 
 CHAP. II. 
 THE STRENGTH OF MATERIALS. 
 
 THE FORCES PRODUCING EXTENSION OR COMPRESSION. 
 
 THE LIMITS OP ELASTICITY. RUPTURE. THE 
 
 STRONGEST FORMS OF CAST-IRON BEAMS AND COLUMNS. 
 
 WOOD AS A MATERIAL IN THE ARCHITECTURE OF 
 
 NATURE. THE MECHANICAL PROPERTIES OF ME- 
 TALLIC SUBSTANCES AS AFFECTED BY THEIR IN- 
 TERNAL STRUCTURE. 
 
 THERE is no form under which the property of the 
 elasticity of matter offers itself to our notice fraught 
 with more interest or importance than as it affects 
 the strength of the materials of construction. 
 
 All these are necessarily subjected, in the uses to 
 which they are applied, to various degrees of pres- 
 sure ; and it becomes a matter of great importance 
 to know, in the first place, how far they will lengthen 
 themselves under a given strain, or compress 
 themselves under a given thrust * ; in the second 
 place, how far this strain, or thrust, may be carried 
 without rupture. 
 
 With regard to the amount of the extension of 
 materials under given strains, it is to be regretted 
 
 * A bar or a timber is said to suffer a strain when the forces 
 which act upon it tend to lengthen it, and a thrust when they 
 tend to compress it. 
 
FORCES PRODUCING EXTENSION. 39 
 
 that few direct experiments have been made ; and in 
 respect to the amount of their compression under 
 given thrusts (it is believed) none. 
 
 42. THE EXTENSIBILITY OF IRON AND WOOD. 
 
 It appears by the experiments of the engineers of 
 the Pont des Invalids, made with every precaution 
 upon the direct strain of bars of the best wrought 
 iron, that they increase their length by about 82 
 millionths under a load of one ton upon the square 
 inch. 
 
 M. Vicat, from experiments made with a view to 
 the use of iron in the construction of suspension 
 bridges, found that, when formed into bundles 
 firmly bound together, or cables, as they are called, 
 iron wire was much more extensible than bar iron, 
 and that it was the more extensible as it was thinner. 
 Its elongation varied from 85 to 91 millionths for a 
 load of one ton per square inch. 
 
 That a fascicle or bundle of wires, having together 
 a section of one square inch, should be more ex- 
 tensible than an iron bar of the same section, and 
 that such a fascicle should be more extensible as the 
 wires which compose it are thinner, are exceedingly 
 interesting facts, inasmuch as it will hereafter be 
 shown that, under the same circumstances in which 
 iron is thus more extensible, it is stronger. So that, 
 on the whole, we arrive at the conclusion that iron 
 acquires in a remarkable degree that quality which 
 we understand by toughness, by being thus drawn 
 out into wire. 
 
 The elongation of oak is about 14 times greater 
 than that of bar iron under the same load of one 
 D 4 
 
4-0 ILLUSTRATIONS OF MECHANICS. 
 
 ton per square inch. According to Tredgold, bar 
 iron will bear to be elongated by the xjVotii part, 
 or by 714 millionths of itself, without permanent 
 alteration of structure, or injury. Cast iron and 
 brass admit of an extension slightly greater ; but 
 the woods ash, elm, mahogany, fir, oak, and pine, 
 may with safety be extended more than three times 
 as much, according to the experiments of Barlow. 
 Of all the woods, larch and beech appear to admit 
 of the least extension without injury. Tables will 
 be found in the Appendix, containing the results of 
 the experiments from which these conclusions have 
 been drawn. (See Table III.) 
 
 43. THE EXTENSIBILITY OF BAR IRON WHEN AP- 
 PROACHING A STATE OF RUPTURE. 
 
 MM. Minard and Desormes suspended weights 
 to bars of iron varying in section from *12 to 1*63 
 parts of a square inch, until they broke. All the bars 
 were 7*874 English inches in length, and the mean 
 of 25 experiments gave -4 J^th part as the elongation 
 due to a load of 15 tons the square inch, -y^^th to 
 18 tons, 3 ^th to 20 tons, and -^th to 23 tons; 
 25 tons per square inch produced rupture. Thus, 
 whilst approaching a state of rupture, each addi- 
 tional ton weight per square inch produced a much 
 greater elongation of the bar than in the com- 
 mencement of the extension. Then it produced an 
 elongation of but 714 millionths; but when the 
 load, as in these last experiments, is augmented to 
 15 tons per square inch, each additional ton, up to 
 18 tons the square inch, produces an elongation of 
 2,500 millionths ; from that load to 20 tons the square 
 
INCREASE OF VOLUME BY EXTENSION. 41 
 
 inch, of 5,000 millionths ; and from that load again 
 to 23 tons, of 10,000 millionths.* 
 
 Now it cannot be doubted that, before the elon- 
 gation of the bar, all the parts of it were per- 
 fectly elastic. How, then, is this subsequent de- 
 viation from the law of perfect elasticity to be 
 explained ? By the fact, that all the parts of the 
 bar, by reason of their different densities, and the 
 different circumstances of crystallisation to be found 
 even in wrought iron, are not equally extensible, 
 and that the material of the iron has been internally 
 ruptured, and its cohesive power, in many con- 
 cealed parts, destroyed long before it attains a state 
 of actual rupture. 
 
 According to the experiments of M. Lager hj elm, 
 made in Sweden in the year 1826, the most ductile 
 Swedish bar iron elongates the -^^th part, or nearly 
 ^th of itself, before it breaks. 
 
 44. THE VOLUME OR BULK OF AN IRON BAR, AND 
 OF A COPPER WIRE, ARE INCREASED IN THE 
 ACT OF EXTENSION. 
 
 M. Lagerhjelm found that, before it broke, the 
 iron of a bar subjected to extension had diminished 
 its section to the 722th part, and its specific gravity 
 by T J gth part, and therefore increased its bulk by 
 gigth part. 
 
 M. Cagnard de la Tour enclosed a copper wire 
 in a long tube filled with water, and then subjected 
 
 * It is remarkable that the elongation thus produced by 
 each additional ton per square inch, in the state approaching to 
 rupture, varied in these experiments in geometrical progres- 
 sion, each being double of the preceding. 
 
42 ILLUSTRATIONS OF MECHANICS. 
 
 it to extension. Having allowed time for the effect 
 of the heat given out by the extension of the wire 
 to pass away, he found that more water was dis- 
 placed by it after extension than before; showing that 
 its volume had increased in the act of extension. 
 
 45. THE THEORETICAL VARIATION IN THE DIA- 
 METER OF A SOLID METALLIC CYLINDER SUB- 
 JECTED TO EXTENSION. 
 
 M. Poisson has shown theoretically, that if a 
 cylinder an unit in length be uniformly elongated, 
 its diameter will be diminished by one fourth the 
 amount of its elongation ; whence it may be cal- 
 culated that the increase of its volume will equal 
 one half the volume of the elongated part. 
 
 46. THE LIMITS OF ELASTICITY. 
 
 It has been shown that, when displaced, the par- 
 ticles of a body tend to return to the position they 
 before occupied in it, with a force proportional to 
 the amount of the displacement. That this may be 
 the case, the displacement must, however, be con- 
 fined within certain infinitely minute limits. If 
 those limits of displacement be passed, the displaced 
 particle may be wholly separated from the rest of 
 the body in the direction from which it has been 
 moved, and thus a partial rupture may take place ; 
 or, other particles of the body occupying the space 
 which it has left, and through which it has moved, 
 it may take up its position under a new arrangement 
 of particles exactly as it did under the preceding, 
 and enter into precisely the same relation with them 
 as before; so that, in everv respect, the qualities of 
 
LIMITS OF ELASTICITY. 43 
 
 the body shall remain unaltered under this new 
 arrangement of its particles. In this last case it is 
 said to have taken a set, and the phenomenon de- 
 scribed under this name includes all that we un- 
 derstand by ductility and malleability, which terms 
 but imply different ways in which this same pro- 
 perty of taking a set is called into operation. 
 
 47. THE ELASTICITY OF A BODY is NOT INJURED 
 
 WHEN A SET IS GIVEN TO IT. 
 
 Thus, in S. Gravesande's experiments, wire, after 
 it had permanently lengthened, was tried, and found 
 as perfectly elastic as ever. In Coulomb's ex- 
 periments on torsion, wire, which had been twisted 
 so far that it would not return to its former po- 
 sition, was found to retain its elasticity of torsion 
 as perfectly as before. Now, the making of a wire 
 from a bar of metal, or, as it is called, the drawing of 
 it out, is but the gradual producing of a set among 
 its particles ; and, since the wire retains the elastic 
 properties of the bar, we may conclude that these 
 are not affected by the sets which the particles of 
 the bar are successively made to take. 
 
 When beams of iron are so loaded in the middle 
 as to cause them to take a permanent deflexion, or 
 a set, their elasticity is found to remain unimpaired 
 by it ; so that, when again loaded, they tend to re- 
 cover themselves with forces which are, as before, 
 proportional to the deflexion. Whilst some por- 
 tions of the substance of a metallic body are made 
 to take a set, others may, however, be ruptured. 
 Its elasticity may then remain, but its extensibility 
 will be greater, and its strength impaired. . 
 
44 ILLUSTRATIONS OF MECHANICS. 
 
 48. MALLEABILITY. 
 
 The surface of a body always yields to an impact, 
 however slight. If a metallic surface thus yield 
 beyond the limits of elasticity, it takes a set. This 
 property, by which a set is given to metals by impact, 
 is called malleability ; and is that property of matter 
 which, perhaps, more than any other ministers to 
 the uses of society. It gives shape to the tools by 
 which ail other substances are moulded, by which 
 the earth is broken up and cultivated, and by which 
 ships are made, and a communication established 
 between regions separated by the ocean. 
 
 There is no case in which the property of mal- 
 leability exhibits itself more remarkably than in the 
 art of the copper-smith. From a flat plate of copper 
 he beats out a hollow vessel without seam or joint, 
 and of a given shape, contriving, by the skilful use 
 of his hammer, so to move about the particles of the 
 metal, that, although, to give to this flat piece of 
 copper its hollow form, he must of necessity in 
 some places contract its surface, and in others 
 expand it, he causes it yet to retain the same 
 thickness throughout. All this is effected by giving 
 to its parts minute sets, of which, although the result 
 of each is perhaps imperceptible, the aggregate is a 
 displacement which he can carry to any finite extent. 
 Operating thus minutely and by degrees, the sub- 
 stance of the metal becomes soft under his hands, 
 arid he may mould it as though it were clay. There 
 are certain metals, and certain states of the same 
 metal, in which this property of malleability exists 
 in a greater degree than in others. Thus, for in- 
 
STAMPING. 45 
 
 stance, cast iron is not perceptibly malleable (except 
 in a slight degree when annealed): it flies to pieces 
 under the hammer; but when converted into wrought 
 iron it becomes perfectly malleable. 
 
 49. THE STAMPING OF METALLIC SURFACES. 
 
 It is by a property analogous to their mallea- 
 bility that metallic surfaces are stamped. Thus, 
 for instance, in the embossed metallic plates which 
 form the surfaces of plated goods, the pattern is 
 moulded from a steel die, or a block of steel, in 
 which, when it is soft, the pattern is sunk by means 
 of punches, and which is then hardened. Over 
 this die a heavy weight is suspended, which can be 
 made to descend between two upright pieces which 
 guide its descent like the pile driver; the string which 
 suspends this weight passes over a pulley, by means 
 of which it can be raised again. On the under sur- 
 face of the descending weight is fixed a thick plate 
 of lead, and upon the die beneath it is laid the 
 metallic plate to be embossed. The effect of the 
 impact of the weight upon the die is to force the 
 soft substance of the lead, and with it the inter- 
 vening thin plate of metal, into the cavities of the 
 die, where both take at the first impact a partial 
 set ; and, the impact being repeated, eventually the 
 surfaces are made to adapt themselves perfectly to 
 one another, and a complete copy is obtained. 
 
 50. COINING. 
 
 It is by a property analogous to that of mallea- 
 bility that metals are made to take the impression 
 of moulds into which they are stamped. It is thus 
 
46 ILLUSTRATIONS OF MECHANICS, 
 
 that the precious metals are coined. The die, in 
 which is sunk the impression which the metal is to 
 receive, is fixed at the extremity of a powerful screw, 
 which is driven impulsively by an effort of the 
 workman applied to a horizontal arm fixed across 
 the axis of the screw, and carrying, at its extremi- 
 ties, two heavy weights. The metal, thus driven 
 with great force into the cavities of the die, takes 
 there a set, and retains the impressions. 
 
 51. THE ROLLING OF METALS. 
 
 The ductility of metals is most effectually called 
 into operation by rolling them. It is thus that iron 
 and copper plates and bars are made ; and the 
 iron rails used on railroads receive their form by 
 being passed between rollers, in which are cut 
 channels of a corresponding form. According to 
 the experiments of M. Lagerhjelm, rolled bars are 
 nearly of an uniform density, whilst the density of 
 forged bars is extremely variable. Within the 
 limits of elasticity, the forces producing given ex- 
 tensions are the same in the two kinds of bars, but 
 a set is given to the rolled bar sooner than to the 
 forged one. The resistance to rupture appears to 
 be the same in the two cases. 
 
 52. ENGRAVED STEEL PLATES. 
 
 By a like process, duplicates of engraved steel 
 plates are obtained in any number. The steel oi 
 the engraved plate of which the duplicate is sought 
 having been hardened, a cylindrical piece of soft 
 steel is rolled over it under an exceedingly heavy 
 pressure. This pressure causes the soft metal of 
 
RUPTURE. 47 
 
 the roller to be pressed into all the lines of the en- 
 graving, of which it thus receives on its surface a 
 perfect impression in relief. The soft steel of this 
 roller is then hardened, and, thus hardened, it is 
 made, under the same heavy pressure as before, to 
 roll over the surface of the plate of soft steel to 
 which the engraving is to be transferred ; and, in 
 the act of thus rolling over it, it indents it with all 
 the lines which it had itself received from the 
 original plate. There is scarcely any limit to the 
 number of duplicates which can thus be obtained 
 of the same engraved steel plate ; or, therefore, to 
 the number of prints which may be taken from the 
 same engraving. This method is now largely used, 
 and with great advantage, in calico printing. The 
 same pattern being here to be repeated over a large 
 surface, a small but complete portion of this pat- 
 tern is engraved on a block of steel, and thence 
 transferred, by the method described above, to a 
 roller. This roller is then made to traverse, under 
 a heavy pressure, the surface to be engraved, until 
 it has repeated the pattern over as wide a surface 
 as is required. 
 
 53. RUPTURE. 
 
 When the parts of a body are, by any external 
 cause, separated beyond the limits of ductility, the 
 separation becomes permanent; and, if it extend far 
 enough, this separation constitutes a rupture of the 
 mass. 
 
 The rupture of a bar of wood or metal may 
 take place either by a strain or tension in the 
 direction of its length, to which is opposed its 
 
48 ILLUSTRATIONS OF MECHANICS. 
 
 TENACITY ; or by a thrust or compressing force in 
 the direction of its length, to which is opposed its 
 power of resistance to the CRUSHING OF ITS MA- 
 TERIAL ; or, each of these powers of resistance may 
 oppose themselves to its rupture, the one being 
 called into operation on one side of it, and the 
 other on the other side, as in the case of a TRANS- 
 VERSE STRAIN. Or, lastly, the bar may be rup 
 tured by TORSION. 
 
 54}. TENACITY. 
 
 In the Appendix will be found a table of the 
 tenacities of different materials, or the resistances 
 they offer to forces tending to tear them asunder, 
 as these have been determined by the best author- 
 ities, and by the mean results of numerous experi- 
 ments. From this table it will be seen, that of all 
 the materials experimented on, that which has the 
 greatest tenacity, or which requires the greatest 
 strain per square inch to tear it asunder, is thin iron 
 wire a number of pieces of it being placed side 
 by side, and bound together, so as to form what is 
 called a cable of wire. Moreover, that cables of 
 wire thus formed are stronger, as the wires which 
 compose them are thinner. 
 
 The first of the experiments enumerated in this 
 table was made by M. Lame, at St. Petersburg, 
 on wire of the best Russian iron, -j\jth of an inch in 
 diameter. The result is extraordinary. A tenacity 
 of 91 tons on the square inch must be considered 
 as an extreme, and, perhaps, an anomalous^ power 
 of resistance. 
 
 Nevertheless, it results from the experiments of 
 
TENACITY. 49 
 
 Lame and of others, that cables of fine iron wire, of 
 from 3*5 th to ^th of an inch in diameter, may be 
 safely assumed to have the enormous tenacity of 
 60 tons per square inch. 
 
 The experiments of Telford give 40 tons per 
 square inch, for the tenacity of wire ^ th of an inch 
 in diameter. 
 
 It is by reason of this marvellous strength of 
 wire cables, that they have come to be extensively 
 used on the Continent, in the construction of sus- 
 pension bridges. There is a bridge suspended by 
 cables of iron wire at "Pribourg, which is 700 feet in 
 span between the abutments, or 100 feet wider 
 in span than the great catenary of the Menai 
 Bridge.* 
 
 Next in tenacity to cables of fine iron wire, is 
 cast steel, in bars, well tilted or forged.t By the 
 experiments of Renriie (Phil. Trans., 1813), its 
 tenacity appears to be nearly 60 tons on the square 
 inch. Sheer steel, reduced by the hammer, has, 
 on the same authority, a tenacity of 57 tons 
 (Phil. Trans., 1818.) 
 
 Steel, by reason of its great tenacity, has in 
 Germany, where it is manufactured at a compa- 
 
 * The bridge of Fribourg is said however of late to nive 
 become unsafe. If this be the case, it is probably owing not to 
 a want of tenacity in its material to resist the ordinary strain 
 upon it, but to the impulses of vibratory motion to which, from 
 its lightness, it is liable, in high winds, or from the rapid motion 
 of vehicles. 
 
 t It is a remarkable fact that cast steel has its tenacity nearly 
 doubled by being tilted. 
 
 E 
 
50 ILLUSTRATIONS OF MECHANICS. 
 
 ratively small expense been used instead of iron, 
 in the construction of suspension bridges.* 
 
 Russia bar iron (which is perhaps the best) ap- 
 pears, by the experiments of Lame, made at St. 
 Petersburg, with an hydraulic press, in 1826, to have 
 a mean tenacity of about 27 tons on the square 
 inch. Common English, and other bar or wrought 
 irons of an average quality, may be considered to have 
 the mean tenacity of 25^ tons on the square inch.t 
 
 Platinum in wire appears, by the experiments of 
 Morveau (Ann. de Chimie, 25-8.), to have a tena- 
 city a little less than bar iron. 
 
 Silver wire, gun-metal, and forged copper, fol- 
 low next in the order of tenacity, having respectively 
 tenacities of 17, 16 --, and 16 tons, on the square 
 inch. 
 
 Gold wire has (by the experiments of Sickingen, 
 Ann. de Chimie, 25-9.) only one half the tenacity 
 of wrought iron. 
 
 The best grey cast-iron may be taken to have a 
 mean tenacity equal to one third that of Russia bar 
 iron ; that is, equal to nearly 9 tons on the square 
 inch ; whilst the ordinary cast-iron has one third the 
 
 * Steel bridges, in common with wire bridges, are, however, 
 by reason of that very lightness which is the great element of 
 their strength, peculiarly liable to those vibrations which are 
 calculated more than any thing else to try it. 
 
 f- Of the experiments recorded of wrought iron, one by 
 JMuschenbroek gives to it the tenacity of 41 tons on the square 
 inch. This is the highest recorded, and it is a problematical 
 result, The iron used was German bar iron, mark B R. The 
 best Swedish and Russian bar irons have, however, for the most 
 part, exhibited a tenacity of upwards of 30 tons. 
 
RESISTANCE TO RUPTURE BY COMPRESSION. 51 
 
 tenacity of common bar iron, or about 8 J- tons on 
 the square inch. 
 
 Of woods, box has the tenacity of the best cast- 
 iron, and ash that of common cast-iron ; that is, one- 
 third the tenacity of wrought iron. 
 
 Deal, oak, and beech, have about ^th the tena- 
 city of wrought iron, and mahogany -ith. Thus, 
 7 rods of mahogany, taken together ; 5 of deal, oak, 
 or beech ; 3 of box, or of cast-iron ; 2 of gold ; 
 1^ of silver, or copper, have respectively the same 
 tenacity as 1 rod of the same section of wrought 
 iron ; or as a rod of -% ths that section of steel or 
 fine wire cable. 
 
 55. RESISTANCE TO RUPTURE, BY COMPRESSION. 
 
 The results of experiments on this subject are to 
 be found in a parallel column of the same table as 
 the last. 
 
 A cube, whose edges are each -J- of an inch, of 
 the kind of cast-iron known by the natne of gun- 
 metal, requires, according to an experiment of Mr. 
 Reynolds', 10 tons to crush it, or a compressing 
 force of 160 tons on the square inch. No other 
 material on which experiments have been made, 
 exhibits a power of resistance approaching to 
 this. 
 
 From experiments made by Mr. G. Rennie (Phil. 
 Trans., 1818), it appears that horizontal castings of 
 iron, from which cubes were taken of the same di- 
 mensions, offered a resistance equivalent to from 
 62 to 76 tons on the square inch; whilst similar cubes, 
 from vertical castings, resisted crushing with a force 
 2 
 
ILLUSTRATIONS OF MECHANICS. 
 
 of from 70 to 90 tons. The more recent experiments 
 of Mr. Hodgkinson, which have been made with 
 remarkable care, give to the Coedtalon iron, No. 2., 
 a resistance to compression of only 36 tons on the 
 square inch; to the Buffery iron, No. 1., 41 tons; 
 to the Carron, No. 3., 51 tons. Brass offered very 
 nearly the same resistance as horizontal castings of 
 iron. 
 
 Bar iron, according to Rondelet, crushes, with 
 31^ tons on the square inch, with less than one half 
 the pressure which Mr. Rennie found cast-iron to 
 bear; Aberdeen granite, with one sixth; Italian 
 marble, with one seventh ; Portland stone, with om 
 tenth ; brick-work* with from J of a ton to 1 J tons. 
 
 But the most remarkable feature presented by this 
 column of the table, is the small resistance which 
 wood offers to a crushing force, acting in the di- 
 rection of the length of its fibre. Experiments on 
 this subject are somewhat uncertain* and variable 
 in the results they give ; they nevertheless fully 
 establish the fact of the small comparative power of 
 wood to resist a force tending to compress it in the 
 direction of its fibre. 
 
 In every other substance enumerated in the table, 
 it will be seen that the resistance to rupture by com- 
 pression is greater than to rupture by extension ; in 
 wood it is less. A fact on which, as will hereafter 
 be shown, there depend important principles in the 
 'heory of construction. 
 
 * This uncertainty appears to depend upon some unknown 
 condition of the adhesion of the fibres of the wood to one 
 
 another. 
 
WROUGHT IRON COLUMNS. 53 
 
 56. INFLUENCE OF THE HEIGHT OF A PRISM UPON 
 THE RESISTANCE TO THE CRUSHING OF ITS 
 MATERIAL. 
 
 The experiments on which the conclusions stated 
 in the preceding article were founded, were made 
 with cubes of the material. When the cube was 
 converted into a prism of a different height from 
 its width, the results became greatly modified, the 
 strength diminishing as the height increased. Thus, 
 when a cube from a horizontal casting of iron was 
 replaced in succession by prisms having the same 
 base of ^ of an inch square, but each higher than 
 the preceding by -J of an inch, until the last was 
 1 inch, their power of resistance to compression 
 passed from 72 tons per square inch gradually to 
 45 tons. This fact probably accounts for the dif- 
 ference of the results stated in the last article. 
 In all cases when a certain height is passed, rupture 
 takes place by the sliding of one portion of the 
 prism in an oblique section upon the other ; and 
 the angle of this oblique section is, in all cases, the 
 same for the same metaL Extensive and accurate 
 experiments have recently been made, on the much 
 neglected subject of compression, by Mr. Hodgkin- 
 son of Manchester. 
 
 * 57. RULE, BY RONDELET, FOR THE STRENGTH 
 OF COLUMNS OF WROUGHT IRON, AND OF OAK 
 AND DEAL. 
 
 From a great number of experiments on columns 
 of wrought iron, varying from half an inch to an 
 inch square, and from an inch and a half to twenty 
 E 3 
 
54? ILLUSTRATIONS OF MECHANICS. 
 
 feet in length, Rondelet has derived the rule, that 
 the load necessary to compress a cube of wrought 
 iron being assumed to be 512 Ibs. on the square 
 line (or the y^th of a square inch), the loads 
 necessary to bend and break columns of any given 
 square section, which are in length successively 
 27, 54, 81, 108, 135, 162, 189, 216, 24-3 times the 
 side of the square of their section, are respectively 
 256 Ibs., 128 Ibs., 64- Ibs., 32 Ibs., 16 Ibs., 8 Ibs., 4lbs., 
 2 Ibs., 1 lb., upon each square line of section. It 
 will be perceived, that the first numbers are as the 
 arithmetic progression, 1, 2, 3, 4*, 5, 6, 7, 8, 9 ; and 
 the last as the geometric progression, 2 8 , 2", 2 6 , 2 5 , 
 2 4 , 2% 2 2 , 2, 1. 
 
 From similar experiments, made with columns of 
 oak and deal, the same author deduced the rule, 
 that assuming 44 Ibs. per square line to be the 
 load necessary to crush a cube of oak, and 52 Ibs. 
 one of deal, the loads necessary to bend and break 
 columns of any given square section, which are in 
 length successively 12, 24, 36, 48, 60, 72 times the 
 side of the square section, are respectively |th, 9 
 , -Jth, -jVth, ^ 4 th of the force necessary to com- 
 press a cubical piece of the column. 
 
 Rondelet found that a square column of oak or 
 deal began to yield by bending when its height was 
 10 times the side of its section. The weights and 
 measures used by Rondelet, and mentioned in this 
 article, were of the old French system, in which 
 one pound weighs 7,561 English grains troy : and 
 one foot 12-78933 English inches. 
 
CAST-IRON COLUMNS. 55 
 
 58. A COLUMN OF CAST-IRON, WHOSE EXTRE- 
 MITIES ARE ROUNDED, WILL SUPPORT BUT ONE 
 
 THIRD THE WEIGHT OF A SIMILAR COLUMN 
 WHOSE EXTREMITIES ARE FLAT. 
 
 This remarkable fact is one among a great num- 
 ber which have been developed by the recent ex- 
 periments of Mr. Hodgkinson of Manchester. 
 
 Having caused a series of cylindrical columns of 
 cast-iron, of different diameters, to be accurately 
 turned, with their extremities rounded, so as to 
 support an insistent weight by the apex of the 
 rounded end, that is, by a single point in the ex- 
 tremity of the axis ; and having caused another 
 series of columns to be turned, exactly similar and 
 equal to the last, but cut off flat at their extremities, 
 he broke the two series of cylinders by the com- 
 pression of a powerful lever, made to act vertically 
 in the direction of their length, by the intervention 
 of a cylindrical hardened steel bar, acting like a solid 
 piston through a hollow cylinder, which served it as 
 a guide. 
 
 In all these experiments he found the cylinders 
 with the rounded ends to break with a pressure which 
 was scarcely one third that of the cylinders with the 
 flat ends. 
 
 When one end of the cylinder was rounded, and 
 the other flat, the breaking pressure was about two 
 thirds that which broke the cylinder when both ends 
 were flat ; so that, in the three cases, the strengths 
 of columns, equal in every other respect, were as 
 the numbers 1, 2, 3. 
 
56 ILLUSTRATIONS OF MECHANICS. 
 
 59. THE STRONGEST FORM OF A CAST-IRON 
 COLUMN. 
 
 In all Mr. Hodgkinson's experiments, before de- 
 scribed, the cylinder was observed to break in its 
 middle point, indicating that to be the weakest. 
 He commenced, therefore, a series of experiments 
 on columns in which the middle section was in. 
 creased at the expense of the extreme sections, with 
 a view to ascertain that form of the column in 
 which, when breaking in the middle, it should be 
 about to break at every other point ; this being 
 manifestly the strongest form. From these it re- 
 sulted, that the strength of a column of cast-iron, 
 containing a given weight of metal, whether it be 
 solid or hollow, is much greater when it is cast in 
 the form of a double cone ; that is, with its greatest 
 thickness in the middle of its height, and tapering to 
 its extremities, than when cast in any other form. The 
 precise results of these valuable experiments have 
 not been published : we hope, however, to be able 
 to publish them in the Appendix. 
 
 60. THE PRESSURE TO WHICH MATERIALS MAY 
 
 BE SUBJECTED WITH SAFETY IN CONSTRUCTION. 
 
 In the actual practice of construction, materials 
 cannot with safety be subjected to constant strains, or 
 thrusts approaching to those which produce rupture. 
 They are liable to various occasional and accidental 
 pressures ; and to others of a permanent kind, result- 
 ing from settlement, and other causes of which no 
 previous account can be taken, for which allowance 
 must nevertheless be made. 
 
NEUTRAL AXIS IN A BEAM. 57 
 
 The engineer and the architect will therefore in 
 their practice be in a degree guided by the ex- 
 ample Of ACTUAL STRUCTURES. 
 
 From a comparison of numerous examples of 
 these, Navier has deduced the rule, that stone and 
 wood have, in existing structures, with safety, been 
 subjected, the former to ^th the thrust, and the 
 latter to -^th the strain, which breaks them ; and 
 iron, cast or wrought, to ^th. * 
 
 61. ADHESION OF THE FIBRES OF WOOD TO 
 ONE ANOTHER. 
 
 Mr. Barlow found the force necessary to separate 
 two parts of a piece of deal, by causing them to 
 slide upon one another in the direction of the .fibre, 
 to be about 5 cwt. to the square inch; for oak 
 82J cwt. to the square inch was required. When the 
 force was applied in a direction perpendicular to 
 the direction of the fibre, 20| cwt. to the square inch 
 was required for oak, 15 cwt. for poplar, and from 
 8J cwt. to 15 J cwt. for larch. 
 
 * One of the greatest pressures to which the stone of any 
 building is known to have been subjected, is probably that 
 borne by the central column of the Chapter- House at Elgin. 
 It amounts to more than 40,000 Ibs. the square foot : neverthe- 
 less, this stone would certainly bear ten times that pressure, 
 without crushing. It is, however, dangerous to subject stones 
 to any pressure approaching to that at which they crush : one 
 half that pressure causes them to chip ; and the tendency of the 
 overloaded stone to yield increases with the time ; Jth the crush- 
 ing pressure is generally taken as the limit, which should not 
 be exceeded. Telford gives 50,000 Ibs. per square foot as the 
 maximum pressure to which the voussoirs of an arch should be 
 subjected. 
 
58 ILLUSTRATIONS OF MECHANICS. 
 
 62. THE NEUTRAL Axis IN A BEAM. 
 
 Let a beam be supposed to be bent by a weight 
 fig. 7. placed in the middle of it : 
 
 it is clear that the side of 
 the beam nearest to the 
 weight, will, in the act of 
 flexure, be compressed, whilst the opposite side will 
 be extended. 
 
 The point where the extension terminates, and 
 the compression begins, sustains manifestly neither 
 extension nor compression. This point is called 
 the neutral point : or, rather, there are a series of 
 such points across the thickness of the beam, which 
 all lie in an axis, called the neutral axis of the 
 beam. 
 
 Since, throughout its neutral axis, the beam is 
 neither extended nor compressed, its strength is 
 not there at all called into play, and is, in point of 
 fact, of no use ; so that the beam would bear as 
 great a weight if a hole were cut through it along 
 this axis. 
 
 63. THE STRENGTH OF A BEAM. 
 
 What constitutes the strength of a beam is its 
 resistance to extension on one side of its neutral 
 axis, and its resistance to compression on the other. 
 These act on either side of the neutral axis, like 
 antagonist forces at the two extremities of a lever ; 
 if either of them yield the beam will be broken. 
 
RELATION OF TENACITY TO CRUSHING FORCE. 59 
 
 64. To CUT A BEAM ONE HALF THROUGH, 
 
 WITHOUT DIMINISHING ITS STRENGTH. 
 
 It is evident that the resistance of the compressed 
 side of a beam to compression, would not be at all 
 affected by cutting it through, provided it were cut 
 only so far as the compression reached, especially 
 if we could cut it with a saw so thin that none, or 
 scarcely any, of the material should be removed. 
 
 This experiment has actually been made, first by 
 Du Hamel. He found that the strength of a 
 wooden beam was not at all impaired by cutting it 
 one half through on its compressed side and 
 scarcely impaired by cutting it f ths through ; and, 
 by filling up the saw-cut with a harder wood, he 
 found that he could actually strengthen the beam 
 by thus cutting it. 
 
 Barlow found that the compressed portion of a 
 beam extended to about |ths of the depth. Through 
 jths of the depth it might then be cut, without in 
 the least affecting its strength. 
 
 65. THE RELATION OF THE FORCES NECESSARY 
 TO TEAR MATERIALS ASUNDER, AND TO CRUSH 
 THEM. 
 
 If a beam yield either on its compressed side or 
 its extended side, it will be broken. But on which of 
 these is it likely first to yield ? Does the material of 
 which the beam is made yield first to compression 
 or to extension ? And in what proportion does it 
 yield differently to these causes of rupture ? 
 
 In parallel columns of a Table in the Appen- 
 
60 ILLUSTRATIONS OF MECHANICS. 
 
 dix, will be found the forces, reduced to the square 
 inch, which are necessary to tear asunder the ma- 
 terials enumerated, and to crush them. From a 
 comparison of these columns, there will appear the 
 remarkable fact, that whilst the metals require a 
 much greater force to crush them than to tear them 
 asunder, the woods require a much less. 
 
 Experiments on the compression of wood are 
 peculiarly uncertain ; and the numerical results 
 stated in the Table are probably to be received only 
 as distant approximations. Still, the fact remains 
 indisputable, that wood crushes with a force less 
 than that with which it tears asunder ; whereas the 
 metals require a much greater force to crush them 
 than to tear them asunder. Cast-iron seven or 
 eight times as much. 
 
 *66. To MAKE A BEAM OR GIRDER OF CAST-IRON 
 
 WHICH SHALL BE FOUR TIMES AS STRONG 
 WHEN TURNED WITH ONE SlDE, AS WHENTURNED 
 WITH THE OTHER SlDE, UPWARDS. 
 
 A very ingenious experiment was made by Mr. 
 Hodgkinson, of Manchester, to illustrate the fact 
 stated in the preceding article. 
 
 He caused two castings of iron to be made from 
 the same mould 5 feet in length. The form of the 
 fig. 
 
 section was that shown in the figure. It may be 
 described as made up of a large flanch 4 inches in 
 
A CAST-IRON BEAM. 61 
 
 width, along the back of which runs a smaller upright 
 rib 1 inch in height. Mr. Hodgkinson's object 
 was to make one of these castings break, by the 
 extension of this rib, and the other by the com- 
 pression tf \t-, and to compare the load necessary in 
 these two cases to produce fracture. He antici- 
 pated that a greater force would be required to 
 break that casting which yielded by the compression 
 of the rib, than that which yielded by its extension. 
 But how was he to break the one by the extension 
 of this rib, and the other by its compression ? Let 
 the reader imagine these castings to be placed be- 
 tween supports 4 ft. 3 in. apart, the one with the 
 rib upwards, the other with the rib downwards, and 
 both to be loaded in the middle. 
 
 Let us take the case in which the rib is upwards 
 (as shown in the first cut), and therefore com- 
 pressed, and the flanch extended. Were the flanch 
 only of the same size as the rib, and did it exert its 
 strength under similar circumstances, it, being the 
 extended part, might be expected the first to yield : 
 but it is very greatly larger than the rib ; and it 
 was made so greatly larger, that its greater size 
 might make up for its less power of resistance 
 it actually did more than make up for it ; for the 
 casting did not yield by the extension of its lower 
 part, but by the compression of its upper, the 
 rib it broke with a load of 9 cwt. The other 
 casting, placed with the rib downwards, of course 
 fig. 9. 
 
62 ILLUSTRATIONS OF MECHANICS. 
 
 yielded by the extension of that rib ; the extended 
 part being here not only weaker, but smaller than 
 the compressed part. 
 
 This easting broke with 2J cwt. Thus we find 
 that to break the casting by compressing the rib, 
 required nearly four times as great a load as to 
 break it by extending the rib : a result agreeing 
 with the before observed fact, that cast-iron resists 
 compression with greater force than extension. 
 Here, then, was a form of iron beam, which was 
 nearly four times as strong when turned one way 
 as when turned the other : and here was an indi- 
 cation of the fact, that the strength of such a beam 
 may, with the same quantity of material, be pro- 
 digiously influenced by the way in which that ma- 
 terial is distributed. 
 
 *67. A WEDGE, DRIVEN OUT BY THE COMPRES- 
 SION OF THE RIB. 
 
 In the experiment when the rib was uppermost, 
 and it was broken by compression, there started out 
 from it, when in the act of yielding, a wedge, of 
 which the length was four inches, and depth '98 of 
 
 fig 10. 
 
 an inch, and which was exactly of the same form 
 and dimensions in all other experiments with cast- 
 ings from the same mould. The wedge is accu- 
 rately shown in the accompanying cut. 
 
STRONGEST FORM OF AN IRON BEAM. 63 
 
 *68. THE STRONGEST FORM OF SECTION OF A 
 CAST-IRON BEAM. 
 
 What, then, is the best way of distributing the 
 material of a beam ? This was the problem which 
 Mr. Hodgkinson undertook to solve, by the method 
 of experiment; and of which his solution is one of 
 the most important practical results that have been, 
 in modern times, obtained. 
 
 In the first place, let the reader be again re- 
 minded of the fact, that a beam, in bearing a load, 
 sustains it by the resistance of its material to com- 
 pression on one side, and to extension on the other; 
 and that these forces act on opposite sides of its 
 neutral axis, like forces acting at either extremity 
 of a lever, the yielding of either destroying the 
 balance, and breaking the beam. Moreover, let 
 his attention be called to the fact, that the farther 
 these forces are placed from the fulcrum, the greater 
 will be their effect: so that all the forces resisting 
 compression will produce their greatest effect when 
 collected the farthest possible from the neutral 
 point ; and, in like manner, all the iorces resisting 
 extension. Thus, all the material resisting com- 
 pression will produce its greatest effect when col- 
 lected at the top of the beam ; and all the material 
 resisting extension, at the bottom. And thus we are 
 directed to this first general principle of the distri- 
 bution of the material, that it should be collected 
 in two flanches, one at the top, and the other at 
 the bottom of the beam, joined by a comparatively 
 slender rib. This is the first step in the distribu- 
 tion : this is not, however, all ; it does not give 
 the strongest form of beam. 
 
64 ILLUSTRATIONS OF MECHANICS. 
 
 To understand why, let the reader's attention be 
 called to this general principle. 
 
 That form of beam is the strongest, whose ma- 
 terial is so distributed, that at the instant when it is 
 about to break by extension on one side, it is about 
 to break by compression on the other : for if when 
 it is about to break by extension on the one side, it 
 is not about to break by compression on the other, 
 then may some of the material be taken from the 
 compressed side without making that break, and 
 added to the extended side, to prevent that break- 
 ing ; so that now the beam is made to bear the 
 weight which before it would not, or it is strength- 
 ened And this is a general principle. So long as 
 the distribution of the material is not such, as that 
 the compressed and extended sides would yield 
 together, the strongest form of section is not at- 
 tained. 
 
 Now it seems clear that since cast-iron yields to 
 extension sooner than compression, if the upper and 
 lower flanch were of the same size, the lower or 
 extended one would yield first. The compressed 
 side cannot yield at the same time as the extended 
 one, unless it be greatly less than it. On the whole, 
 then, the strongest form of beam will evidently 
 have its lower flanch much larger than its upper. 
 
 But in what proportion ? Mr. Hodgkinson's ex- 
 periments were directed to the determination of 
 this point. He made a series of castings, gradually 
 increasing the lower flanch, at the expense of the 
 upper as shown in the accompanying diagrams; 
 and, as he had anticipated, he found the beams, in 
 this state of transition, to grow stronger and stronger 
 
STRONGEST FORM OF AN IRON BEAM. 65 
 
 fig. 1 1. 
 
 cz 
 
 L) 
 
 J 
 
 No. oi' 
 Experiment. 
 
 Ratio of Surfaces 
 of Compression 
 and Extension. 
 
 Area of Sec- 
 tion in Inches. 
 
 Strength per square 
 Inch or Section in Ibs. 
 
 1 
 
 1 to 1 
 
 2'82 
 
 2368 
 
 2 
 
 1 to 2 
 
 2'87 
 
 2567 
 
 3 
 
 1 to 4 
 
 3-02 
 
 2737 
 
 4 
 
 1 to4j 
 
 3-37 
 
 3183 
 
 5 
 
 1 to 4 
 
 4-50 
 
 3214 
 
 6' 
 
 1 to5j 
 
 5-0 
 
 3346 
 
 7 
 
 1 to 3'2 
 
 4-628 
 
 3246 
 
 8 
 
 1 to 4-3 
 
 5'86 
 
 3317 
 
 9 
 
 1 to 6-1 
 
 6-4 
 
 4075 
 
66 ILLUSTRATIONS OF MECHANICS. 
 
 In the first eight experiments, each beam broke 
 by the tearing asunder of the lower flanch. The 
 distribution by which both would be about to 
 yield together that is, the strongest distribution 
 was not therefore, up to that period, reached. At 
 length, however, in the last experiment, the beam 
 yielded by the crushing of the upper flanch, from 
 which a wedge flew out. 
 
 In this experiment, then, the upper flanch was the 
 weakest. In the one before it, the lower was the 
 weakest. For a form between the two, therefore, the 
 flanches were equal in strength to resist the pressures 
 to which they were severally subjected ; and this was 
 the strongest form. 
 
 In this strongest form the lower flanch had six 
 times the material of the upper. 
 
 In the best forms of girders used before these 
 experiments, there was never attained a strength of 
 more than 2,885 Ibs. the square inch of section. 
 There was therefore, by this form, a gain of 1,190 Ibs. 
 the square inch of section, or |ths of the strength 
 of the beam. 
 
 The great girders cast in Manchester are now 
 commonly cast on this principle ; and there has re- 
 sulted, it is said, a practical economy in the iron 
 of full 25 per cent. 
 
 69.* RULE FOR THE STRENGTH OF A BEAM CAST 
 ON MR. HODGKINSON'S PRINCIPLE. 
 
 From the comparison of a great number of ex- 
 periments, Mr. Hodgkinson has deduced the Fol- 
 lowing rule for the strength of his beams. The 
 dimensions being all taken in inches, multiply the 
 
STRONGEST FORM OF A BEAM. 67 
 
 area of the section of the lower flanch, in inches, by 
 the depth of the beam, and divide the product by the 
 distance between the two points on which the beam is 
 supported. This quotient, multiplied by 536 when the 
 beams are cast erect, and by 514? when they are cast 
 horizontally, will give the breaking weight in cwts. 
 
 70.* To VARY THE SECTION OF A BEAM AT DIF- 
 FERENT DISTANCES FROM THE POINTS OF SUP- 
 PORT, SO THAT FOR A GIVEN QUANTITY OF MA- 
 TERIAL ITS FORM MAY BE THE STRONGEST. 
 
 The strength of a beam, to bear a load, is different 
 according as it is loaded in the centre of its length, 
 or nearer to either of its extremities. It is, for in- 
 stance, evident that a beam will bear a load placed 
 upon it very near to one of its points of support, 
 when it would not bear the same load placed over 
 its middle point. It appears from a mathematical 
 inquiry into this subject, that the effect of a given 
 
 fig. 12. 
 
 A Q O B 
 
 \ I* ~K. | 
 
 load to break the beam, varies when it is placed 
 over different points in it, as the products of the 
 distances of these points from the two points of 
 support of the beam. Thus the effect of a weight 
 placed over the point L, is to the effect of the same 
 weight placed over the point K, as the product of 
 AL by LB is to the product of AK by KB ; A and 
 B being the two points of support. Since, then, the 
 effect of a weight to break a beam is not so great 
 at points nearer to its extremities, as in the middle, 
 the beam need not be so strong any where as at its 
 F 2 
 
68 
 
 ILLUSTRATIONS OF MECHANICS. 
 
 middle point; and, guided by the law stated above, it 
 appears that its strength at different points should vary 
 as the products of the distances of those points from 
 the points of support. Now this difference of strength 
 may be given in two ways ; either by varying the 
 depth of the beam according to this law, or by pre- 
 serving its depth every where the same, and varying' 
 the dimensions of its upper and lower flanches ac- 
 cording to this law. Whether we thus vary the 
 depth of the beam or the dimensions of its flanches, 
 the law in question will give, for the outline, in the 
 one case of the elevation of the beam, in the other 
 of the plan of the flanch, the geometrical curve 
 called a parabola. The cut represents the flanch, 
 according to this form, adopted by Mr. Hodgkinson; 
 the upper and lower flanch were of the same form ; 
 but the dimensions of the latter were six times those 
 of the former. 
 
 71. THE QUALITIES OF WOOD AS A MATERIAL 
 
 OF CONSTRUCTION. 
 
 Such is the form which, guided by experiment 
 and such other resources of science as we possess, 
 we find ourselves led to give to the substance, iron, 
 which, forming part of the solid materials of the 
 earth, and ministering there to some wholly different 
 use, we dig up and apply to our purposes of con- 
 struction. 
 
 
WOOD AS A MATERIAL OF CONSTRUCTION. 69 
 
 Now let us turn to the architecture of trees, and 
 examine Nature's material, and let us consider whe- 
 ther, guided by the light which our efforts to econo- 
 mize this artificial material of construction may 
 have given us, we may not discover, in the material 
 of those stately structures, elaborated in the myste- 
 rious process of vegetation, some feeble traces of 
 that mighty and all-perfect wisdom of which ours, 
 feeble as it is, is yet an emanation. 
 
 And let the principle first of all be stated, as one 
 observable throughout all nature, that creative power, 
 infinite in its development, is infinitely economised 
 in its operation. 
 
 Were wood but as durable as iron and stone, it 
 would supersede their use as a material of construc- 
 tion. 
 
 If other evidence were wanting, the unparalleled 
 boldness of the structures erected with wood would, 
 for itself, speak to the fact. 
 
 What have we to compare with the structures 
 erected in wood ? There is no arch of iron or 
 stone, for example, that approaches to the span of 
 the wooden arches which have been erected by 
 Wiebeking in Germany, or to that arch at Phila- 
 delphia, which, with one vast span of 340 feet, 
 crosses the Sehuylkill. 
 
 The superiority of wood to iron or stone, as a 
 material of construction, results from the extra- 
 ordinary lightness which it unites with its strength. 
 Thus deal has only one fifteenth the weight of 
 cast iron, although it has considerably more than one 
 half the tenacity, and sixteen bars of it would .weigh 
 only the same as one bar of the same dimensions of 
 i 3 
 
70 ILLUSTRATIONS OF MECHANICS. 
 
 wrought iron, although they would have together 
 more than the strength of three. 
 
 Now it is evident that a building erected with a 
 material, however strong, which was in the same 
 proportion heavy, might, and probably would, be a 
 weak building. 
 
 Such a structure, notwithstanding the great 
 strength of its material, might load itself with its 
 own mass to the utmost that it would bear, so that 
 the slightest additional pressure would cause it to 
 yield as it is the last ounce which breaks the 
 camel's back. Many, and memorable, are the in- 
 stances of this weakness in artificial structures. The 
 case of the Brunswick theatre, whose iron roof fell 
 in by the pressure of its own weight, and that of 
 Mr. Maudeslay's manufactory in London, and of 
 the Conservatory at Brighton, are in every body's 
 recollection. 
 
 But wood falls short of other materials in dura- 
 bility* 
 
 The food of living vegetation is extracted from 
 decayed vegetation ; decay was thus, for the great 
 purposes of nature, made its inseparable concomitant. 
 
 This decay which was a necessary property then 
 of timber, as a material of nature's architecture 
 unfitted it for that of man ; who, reserved for im- 
 
 * The recent discovery of Mr. Kyan has, nevertheless, given, 
 it is said, to wood an artificial durability almost equal to that 
 of iron. The great agent in its destruction is a fungus whose 
 ravages we are familiar with under the name of dry rot, and 
 the experiments of Mr. Kyan, confirmed by those of Dutrochet, 
 appear to show that this fungus will not grow in timber 
 steeped in corrosive sublimate. 
 
WOOD AS THE MATERIAL OF TREES. 71 
 
 mortality, and struggling, even here, in an unceas- 
 ing combat with the fleeting and transitory character 
 of all that surrounds him, would construct for him- 
 self an abode whose durability may laugh to scorn 
 the shortness of his tenure ; he digs therefore its 
 material from among those mineral substances out 
 of which the mass of the earth itself is builded up, 
 and whose duration is coeval with it. 
 
 72.* THE ADAPTATION OF WOOD AS A MATERIAL 
 TO THE ARCHITECTURE OF TREES IN RESPECT 
 TO ITS DISTRIBUTION. 
 
 So much for the quality of the material as evi- 
 dencing the infinite skill of the mighty Architect. 
 
 Now for the distribution of it. Can we see, im- 
 perfect as are our faculties, any traces of that per- 
 fect wisdom which governs the distribution of that 
 material ? 
 
 The experimental fact (ascertained with certainty), 
 that its power of resisting extension, when subjected 
 to transverse strain, is so nearly balanced by its 
 power of resisting compression, as to bring its neu- 
 tral point, at the instant of rupture, nearly in the 
 centre of the beam (only one eighth of its thickness 
 from it), is manifest evidence of this. 
 
 To make this appear, let us imagine that this 
 nicely balanced equilibrium had not existed, as, in 
 the case of iron, it does not. Let us suppose, in 
 short, that iron were the material of trees. To give 
 the most economical distribution to its material, a 
 beam of wood must, then, be of that form which we 
 have discovered to be the best for a beam of iron; that 
 is, one side of it must contain six times the material 
 F 4* 
 
72 ILLUSTRATIONS OF MECHANICS. 
 
 that the other side does. But such a beam is only 
 calculated to bear a pressure acting upon one side 
 of it, and to bear it in one particular direction. 
 If fixed, for instance, firmly upright in the earth, 
 and made to be acted upon powerfully by the wind, 
 it might bear it, and would be of the form best cal- 
 culated to bear it when it blew in one direction, but 
 not when it blew in the opposite direction. To 
 make it resist equally a force in either direction, the 
 flanch must evidently be of an equal size on either 
 side : but if you make it thus, all the economy of 
 the distribution of the material is gone. To pre- 
 serve this economy, the relation, of the resistance to 
 compression, to the resistance to extension, must be 
 changed in such a way that an equality of the 
 flanches may constitute the most economical ar- 
 rangement of the material. 
 
 Now in wood precisely this relation appears to 
 obtain. The proximity of the neutral axis to the 
 centre, as determined by Duhamel and others, 
 sufficiently indicates the near equality of the forces 
 resisting extension and compression as they are 
 called into action in the transverse strain of a beam, 
 and renders it extremely probable that in the trans- 
 verse strain of the cylindrical trunk of a tree, 
 whether hollow or solid, this equality becomes 
 absolute. 
 
 Supposing then a beam of wood formed like that 
 of iron, of which we have spoken, and conceiving its 
 flanches to have the form of longitudinal slices of a 
 hollow cylinder, and the circumstances of resistance 
 to be similar to those in a beam of wood ; if these 
 two flanches be very nearly of the same size, when 
 
WOOD AS THE MATERIAL OF TREES. 73 
 
 one is about to yield by extension, the other will 
 be about to yield by compression. This has 
 been before shown to constitute the most economi- 
 cal distribution that can be conceived. To sustain 
 the force of the wind on its opposite sides, a timber, of 
 this form, then, with equal flanches, would have the 
 most perfect form. Let the reader conceive a 
 number of such equal timbers to be placed so 
 that the ribs which join their flanches may all in- 
 tersect along the centre of their length, and their 
 flanches be brought side by side like the staves of a 
 barrel, and let him imagine a hoop to be placed 
 round it; and he will have conceived a structure 
 whose material is of perfect economy in itself, and 
 whose mass is distributed with a perfect economy, 
 so far as these things may be comprehended; he will 
 have embraced a principle which shapes out the 
 bones of every living animal, which distributes the 
 material of the stem of every weed and flower, and 
 of a great family of trees. Surely " God is wise 
 in heart ; He is mighty in wisdom ; He is wonderful 
 in counsel, and excellent in working." 
 
 But it may be asked, If this economy of material 
 be a principle of creative wisdom, why is not every 
 tree made thus with a hollow stem, as are the great 
 tribe of grass-like trees, and the bones of animals ? 
 Let us speak with reverence when we speak of the 
 design by which the vast operations of Providence are 
 directed. It is, however, an unjustifiable conclusion, 
 that, in building up the trees of the forest, there was 
 taken into that far-reaching view the uses to which, 
 in the vast economy of human life, they should 
 
74 ILLUSTRATIONS OF MECHANICS. 
 
 hereafter be placed? How materially would that 
 utility have been impaired had they been but hollow 
 tubes ? And how vast an influence would it have 
 exercised on the destinies of our race, if for this 
 reason large buildings had never been framed to- 
 gether, ships never built ? 
 
 73. VARIOUS CIRCUMSTANCES WHICH AFFECT 
 THE STRENGTH OF METALS, AS MATERIALS 
 OF CONSTRUCTION. 
 
 Of the various circumstances which affect the 
 strength of metallic substances, the most important 
 appear to be those which connect themselves with 
 crystallisation. 
 
 The crystallisation of bismuth and sulphur may be 
 taken as examples of what little is known of the cir- 
 cumstances under which crystallisation takes place 
 generally. 
 
 74. CRYSTALLISATION OF BISMUTH. 
 
 If bismuth, melted in a crucible, be poured into 
 a mould (heated to receive it), and when it has 
 cooled, so that a crust is formed over its surface, if 
 the mould * be inverted, so that the weight of the 
 liquid metal beneath the crust may break through 
 it and run out, the cavity beneath will be found to 
 be surrounded with beautiful crystals of the metal 
 adhering to the crust and to the sides of the vessel. 
 These crystals will be larger as the process of cool- 
 ing takes place more slowly, and as the melted mass 
 
 * The crystals will be very beautifully seen if a test-glass be 
 used for the mould. 
 
SALINE CRYSTALLISATIONS. 75 
 
 is greater. They present to the eye the order and 
 arrangement according to which its parts solidify; 
 which order and arrangement, therefore, charac- 
 terise their solid state. 
 
 A similar experiment may be made with sulphur. 
 M. Mitscherlich, from a melted mass of fifty pounds, 
 slowly cooled for four or five hours, obtained crys- 
 tals half an inch thick. 
 
 75. SALINE CRYSTALLISATIONS. 
 
 The process of crystallisation is much more 
 easily observed in crystals which form, as do vari- 
 ous salts, from aqueous solutions than it is in those 
 which form in the act of cooling from a melted 
 state. It is nevertheless probable that these two pro- 
 cesses of crystallisation are identical in their prin- 
 ciple. If the water in which a salt has been dissolved 
 be slowly evaporated from it, the crystals of the salt 
 will, after the evaporation has passed a certain 
 limit, begin to re-appear in it : each minute crystal 
 will be seen to be terminated by plane surfaces, to 
 have a definite form, and certain inclinations of its 
 planes. As its dimensions increase, which they 
 will be seen to do continually so long as the eva- 
 poration proceeds, it will never lose this definite 
 form or this given inclination of its containing 
 planes. The amount of the increase will, however, 
 probably be different on its different faces, and give 
 rise to certain modifications of its primitive form, 
 which, on examination, will be found to depend upon 
 the different quantities of the fluid mass to which its 
 
76 ILLUSTRATIONS OF MECHANICS. 
 
 different faces are presented, and the different de- 
 grees of facility with which the saline particles of 
 the solution have access to them ; so that by turn- 
 ing the crystal round into different positions in the 
 solution, different faces may be made in succession, 
 to receive the greater increase. The definite cha- 
 racter of this accumulation of the particles of the 
 crystal is strikingly illustrated by the fact, that if a 
 crystal, whilst the process is going on, be taken out 
 of the solution and broken at its angles, so as to 
 make its surfaces rough and uneven, and to destroy 
 the form under which the accumulation of its par- 
 ticles was taking place, and if this broken crystal be 
 then replaced in the solution, the process will re- 
 commence upon it by a restoration of the broken 
 part, a filling-up of the roughness of its faces, and a 
 re-formation of the crystal upon its original model. 
 A yet further evidence of the definite arrangement 
 of the particles of the crystal is found in the cir- 
 cumstances of its cleavage : these it partakes of in 
 common with many crystals, which have never been 
 formed by artificial means, but which are found in 
 nature composing part of the earth's surface, and 
 have probably resulted from igneous fusion. These, 
 which are, some of them, excessively hard, have 
 almost in every case certain particular directions in 
 which they can be divided or cleft, called planes of 
 cleavage. When they are thus cleft, the divided 
 surfaces appear perfectly smooth and even, like the 
 faces of artificial crystals. Every crystallised sub- 
 stance has several of these planes of cleavage, and 
 the sam& substance has always, in every fragment or 
 specimen of it, the same number of them ; and these 
 
CRYSTALLISATION OF WROUGHT IRON. 77 
 
 inclined to one another at the same angles : so that 
 cleaving any such fragment until all its faces are 
 planes of cleavage, the resulting crystal will always, 
 for the same substance, be of the same form. If 
 it be an artificial crystal, this form will be exactly 
 the same as that which it assumes in the solution 
 out of which it crystallises, or one in close alliance 
 with it. 
 
 76. CRYSTALLISATION MAY TAKE PLACE IN A 
 MASS WHICH IS IN AN IMPERFECT STATE OF 
 FUSION. 
 
 Wrought iron is obtained by the forging of masses 
 of cast iron which are heated, but only imperfectly 
 fused. This process, arid others which it undergoes, 
 separate it from the carbon which is combined 
 with it, and from various other impurities under the 
 form of scoriae ; and, as it thus becomes more pure, 
 it becomes more difficult of absolute fusion, and 
 less perfectly fused : nevertheless, cooling from 
 this imperfect state of fusion, it assumes that crystal- 
 lised structure, which is so apparent in wrought iron. 
 It is, perhaps, in consequence of this crystallised 
 structure of wrought iron, that its strength is so 
 greatly modified by drawing it into wire : we have 
 seen that its tenacity may thus be increased from 
 25 \ tons the square inch to from 60 to 90 tons ; that 
 is, it may be tripled. The tenacities of iron, in the 
 three states of cast iron, wrought iron, and fine 
 iron wire, are as the numbers 9 25. 80. 
 
78 ILLUSTRATIONS OF MECHANICS. 
 
 77. THE INFLUENCE OF THE , VARIOUS CONDI- 
 TIONS OF CRYSTALLISATION ON THE COHESIVE 
 FORCE OF CAST IRON. 
 
 This influence is fully shown by an experiment 
 of Mr. G. Rennie. He took a cube of iron, whose 
 edges were each -Jth of an inch, from the centre of 
 a large casting, where the crystals being slowly 
 formed were perfect and large, and plainly seen : he 
 found that it crushed with a pressure of 14401bs. He 
 took a second cube, of the same dimensions, from a 
 small casting, where there was not the same appear- 
 ance of crystallisation, but a close compact grain : 
 tnis crushed with 21 16 Ibs. ; that is, it required half 
 as much force more, to crush it. 
 
 78. THE INFLUENCE OF PRESSURE UPON THE 
 
 SOLIDIFICATION OF METALS. 
 
 The pressure under which the solidification of 
 metals takes place, has an evident influence on their 
 internal structure. Thus, to the strength of a cannon, 
 whether it be cast in a vertical or a horizontal mould 
 that is, whether in the act of cooling it sustains a 
 greater or less weight of superincumbent mate- 
 rial ; and whether the muzzle or the breech be 
 cast upwards, are things of importance to its 
 strength. 
 
 The experiments of Mr. Rennie show, moreover, 
 that bars of metal differ in cohesive power, as they 
 are cast vertically or horizontally. 
 
 Thus a prism, cast horizontally, he found to crush 
 under a load of 9006 Ibs. ; another prism, cast in the 
 same mould, but vertically > required 9328 Ibs. to 
 
MALLEABLE PLATINUM. 79 
 
 crush it ; and, generally, a vertical casting was best 
 adapted to bear a vertical force. 
 
 Mr. Hodgkinson found so great a difference be- 
 tween the strength of iron girders, according as they 
 were cast horizontally or vertically, that he has given 
 different rules for calculating them. In illustration 
 of the same fact it may be mentioned, that great 
 bells are found to be of a different quality of metal 
 at the top of the mould in which they are cast, and 
 at the bottom. 
 
 79. MALLEABLE PLATINUM. 
 
 Platinum was first made known in Europe by 
 Mr. Wood, assay-master at Jamaica, who met with 
 it in its ore in 174-1. It cannot be obtained from the 
 ore in any considerable quantities by direct fusion, 
 as other metals are. The voltaic pile and the oxygen 
 blow-pipe will indeed melt it ; but these can be 
 made to operate only on small portions at a time : 
 to obtain it in larger quantities, under a malleable 
 form, was long a desideratum in science. It is now 
 accomplished as follows : The ore is subjected 
 to the action of nitro-muriatic acid, and the solution 
 precipitated by hydrochlorate of ammonia, under 
 the form of a hydrochlorate of platinum and am- 
 monia ; which, under a high temperature, decom- 
 poses and leaves pure platinum, under the form of 
 a porous friable mass, called (from the resemblance) 
 spongy platinum. 
 
 Under this form it is as far as can be conceived 
 removed from malleability ; and here lay the great 
 difficulty of the process. It is overcome by sub- 
 
80 ILLUSTRATIONS OF MECHANICS. 
 
 jecting the spongy platinum to a high pressure in 
 a mould ; a kind of ingot is thus obtained of suf- 
 ficient tenacity to bear handling. This is then 
 exposed to a high temperature, and carefully 
 forged. Dr. Wollaston was the discoverer of this 
 process, and he has published a detailed account of 
 it in the " Philosophical Transactions " for 1 829. 
 
 80. CAST IRON. 
 
 Of all the causes which affect the mechanical 
 properties of iron, the most remarkable aje those 
 which result from the union with it, in different 
 degrees, of that subtle element, which is called by 
 chemists carbon, and which is the substance so 
 familiarly known to us as charcoal. It is this 
 element, which, in the process of smelting, passes 
 from the charcoal or coke, which is mingled in 
 given proportions with the mass of iron ore in the 
 furnace ; and uniting itself with the pure iron, gives 
 it the properties of a fluid. Run into moulds, and 
 allowed to cool, this compound of carbon and iron 
 becomes cast iron. 
 
 In the melted state, fluidity is that property 
 which the iron receives in a greater or less degree 
 from the greater or less quantity of carbon com- 
 bined with it that is, from the greater or less 
 quantity of coke mingled with it in the furnace, 
 and the better or worse quality of the coke. In 
 the solid state, according as it contains more 
 carbon, cast iron is softer under the file or chisel, 
 and possesses less strength as a material. As it 
 contains less carbon, it is harder, and possesses 
 
MANUFACTURE OF CAST IRON. 81 
 
 more strength as a material. This property of 
 hardness, however, which it acquires as the pro- 
 portion of carbon combined with it is diminished, 
 ultimately passes into brittleness, and, beyond a 
 certain limit, it thus loses its strength as a material, 
 for the ordinary purposes of construction. It is 
 only used in its most highly carbonised state, 
 because in that state, by reason of its fluidity when 
 melted, it may be made to run into the finest and 
 most delicate mouldings, so as to present, when 
 cooled, a minute and perfect reproduction of the 
 model. For castings, on which less minute and ac- 
 curate mouldings are required, iron combined with 
 less carbon is used, because .of its greater strength. 
 Irons of these two qualities, of greater or less car- 
 bonisation, and suited to these distinct purposes, 
 are known to the founders as the irons, No. 1. and 
 No. 2. A third quality of cast iron, known as 
 No. 3., is, in some places, made for castings of great 
 size and strength, with a yet less admixture of car- 
 bon, and possessing less fluidity than No. 2. And 
 there is a fourth quality, called bright iron, yet 
 further without carbon, of an extremely imperfect 
 fluidity when melted, and applicable only to the 
 largest castings. 
 
 There are two qualities of iron mottled and 
 white which are obtained from the furnace with yet 
 less degrees of carbonisation than bright iron. These 
 are, however, so thick when melted, and so brittle 
 when cooled, as to be wholly unfit for the purpose 
 of casting. When combined with carbon in a less 
 proportion than in these qualities, iron does not 
 melt in the furnace, and cannot be separated there 
 from the ore. 
 
82 ILLUSTRATIONS OF MECHANICS. 
 
 To be obtained in this lower state of combina- 
 tion with carbon, it must first be extracted from the 
 ore in the last-mentioned state of cast iron, or in 
 some of the other states before mentioned, and 
 then have more of its carbon, by an independent 
 process, taken from it. Cast iron, thus deprived in 
 a greater or less degree of its carbon (and other 
 foreign ingredients), becomes wrought iron. It is 
 remarkable that iron in this state, united with less 
 and less proportions of carbon, re-acquires rapidly 
 those properties of softness and malleability in its 
 texture, which before, by the deprivation of its 
 carbon, it lost> and soon greatly surpasses them. 
 Whilst its resistance to the file, the chisel, and the 
 hammer, have thus, by de-carbonisation, become 
 less, its tenacity has tripled itself; the brittleness of 
 cast-iron has wholly disappeared from it, and it has 
 become that material, whose union of weight, 
 strength, durability, and hardness, adapt it, above 
 all others, to the various utensils and tools of art; 
 and whose malleability, when heated to a red heat, 
 and the facility with which it is welded, enable us 
 to mould it into any required form upon the anvil, 
 and to frame and unite any number ef distinct 
 portions of it into a continued structure. 
 
 It is the difficulty of melting wrought iron, its 
 extreme softness nevertheless, when brought to a red 
 heat, and that property by which it admits of being 
 welded *, which, as much as its greater toughness 
 and ductility, distinguish it from cast iron. 
 
 * The welding, or joining together of different pieces of iron 
 and steel, is performed by bringing the surfaces to be joined 
 to a temperature bordering upon that of fusion ; a glossy ap- 
 jpearance, like varnish, then appearing upon them, they are spec- 
 
MANUFACTURE OF WROUGHT IRON. 
 
 81. THE MANUFACTURE OF WROUGHT IRON. 
 
 Cast iron is decarbonised, so as to convert it 
 into wrought iron, by exposing it to the action of 
 the air, for a considerable length of time, in a melted 
 state: its carbon combines, in this state, with the oxy- 
 gen of the air, and deserts the fused metal. This pro- 
 cess of fusion is twice undergone : the first time it is 
 called refining the iron. Mingled with the requisite 
 quantity of fuel, the pig-iron is placed in a trough- 
 like furnace, whose sides are of iron plate, and its 
 bottom of masonry, and round whose sides, ex- 
 ternally, a stream of water is made continually to 
 rim. The fuel being lighted, a powerful blast of 
 air is impelled upon it, and the metal having 
 been kept in a state of fusion, with this blast upon 
 it, for not less than two hours, and having lost a 
 large portion of its carbon is run into a long shal- 
 low mould, and cooled : it is then broken into 
 pieces, and carried to a furnace of the kind called 
 a reverberatory furnace, where the powerful flame 
 of a large body of fuel, under combustion, in the 
 grate of the furnace, is made to pass over it, and at 
 length it sinks, in a state of fusion, on what is called 
 the hearth of the furnace ; a space which is wholly 
 separated from the fuel, and open to a free access 
 of air. Here the melted mass is kept in a state of 
 continual motion, and stirred up from the bottom 
 by the workmen, with long iron rods ; a process, 
 which is called puddling. In this state the metal 
 
 dily scraped, placed in contact, and hammered together. Cast 
 
 steel, in common with cast iron, loses the property of welding. 
 
 G 2 
 
84 ILLUSTRATIONS OF MECHANICS. 
 
 may be seen to swell and emit its carbon, gradually 
 losing its fluidity, until at length it can, with the 
 workman's rod, be accumulated into semi-fluid balls 
 of 70 Ibs. or 80 Ibs. in weight ; these he removes 
 from the puddling-furnaee, arid places under the 
 heavy hammer of a forge. After having been v/eli 
 hammered, and received a flattened form from this 
 forge, they are passed between rollers, and con- 
 verted into bars; five or six of which, being piled 
 upon one another and brought to a welding heat in 
 a reverberatory furnace, and again passed through 
 successive pairs of rollers until they are reduced to 
 bars of the required dimensions, the process is 
 finished. 
 
 82. THE MANUFACTURE OF STEEL. 
 
 If cast iron, after having been deprived of its 
 carbon, and other foreign ingredients, and thus 
 brought into the state of wrought iron, be re-car- 
 bonised by a process about to be described, it will 
 not return to the state of cast iron, but to a state in 
 which, admitting of fusion, and re-acquiring more 
 than the hardness of cast iron, it admits of being 
 hammered, forged, and welded like wrought iron ; 
 and, when tempered, becomes equally pliant and 
 yielding, and far more elastic. In this state it is 
 known as STEEL. 
 
 The process by which wrought iron is carbonised 
 into steel, is this : In two long troughs or boxes cf 
 fire-stone, built up on either side of the fire-grate cf 
 a reverberatory furnace, are piled upon one another 
 the bars of iron which are to be subjected to the 
 process of carbonisation : between each layer of 
 
MANUFACTURE OF STEEL. 85 
 
 bars is spread a thick layer of charcoal-powder ; 
 and when the piles are thus completed (usually to 
 the weight of ten or twelve tons), the top of each 
 trough is closely covered over with a bed of sand. 
 The fire of the furnace is then lighted, and the 
 boxes, and their contents, are kept at a red heat 
 for eight or ten days. 
 
 In this heated state the iron attracts, and incor- 
 porates with itself, carbon from the charcoal which 
 surrounds it. If, at the expiration of the time men- 
 tioned, the process is found to have proceeded far 
 enough, by the examination of a bar drawn for that 
 purpose, the furnace is allowed gradually to cool. 
 
 The bars, on examination, are now found greatly 
 to have swelled their dimensions, and to have raised 
 their surfaces every where into blisters ; for which 
 reason the steel, formed by this process (called ce- 
 mentation), is called blistered steel. These bars are 
 then heated to redness, and well forged under a 
 powerful forge -hammer, made to strike with great 
 rapidity, commonly by the action of a water-wheel, 
 and called a tilting hammer : the hollow, vascular 
 texture of the blistered steel is, by this forging, re- 
 duced to a close continuous granular structure, and 
 the metal becomes tilted steel. When the bars of 
 blistered steel are first broken, and then \velded upon 
 the surfaces of one another, and then tilted, then 
 broken and again welded and tilted, and this opera- 
 tion is several times repeated, they become bars of 
 German or shear steel. 
 
 However great the care with which the process 
 of cementation is carried on, the bars of blistered 
 steel which result from this process are never uni- 
 G 3 
 
86 ILLUSTRATIONS OF MECHANICS. 
 
 formly carbonised. To give to steel that uniform 
 quality, which, for cutting instruments, is so desir- 
 able, Mr. Huntsman, of Sheffield, conceived the 
 idea of casting it ; and this process is now commonly 
 pursued. The blistered steel bars are broken into 
 small pieces, and put into crucibles of fine clay, 
 whose mouths are covered with verifiable sand, to 
 prevent the access of the air, and the consequent 
 decarbonisation : the crucibles are then subjected 
 to an intense heat, which, in four or five hours, 
 fuses the included steel. About twenty tons of coals 
 are required thus to fuse one ton of steel; a fact 
 sufficient to account for the high price of cast as 
 compared with other steel. 
 
 An impure and variable kind of steel, called 
 German, or furnace steel, is obtained by carbonisa- 
 tion directly from the ore, or from cast iron. 
 
 It has hitherto been found impossible to convert 
 English bar iron into good steel. The iron used is 
 all Swedish or Russian : it is brought thence, and 
 manufactured into steel at Sheffield, for every market 
 in the world. 
 
 83. CASEHARDENING. 
 
 This is a process for converting the surfaces of 
 wrought iron articles into steel. The manufacture 
 of the articles having been completed, except the 
 polish, they are placed in an iron box, in layers ; a 
 layer of animal carbon (horns, hoofs, skins, or 
 leather, first so burned as to admit of their being 
 reduced to powder) being spread over each : the 
 box being then carefully covered and luted with an 
 equal mixture of clay and sand, is kept at a light 
 
STRENGTH OF IRON AFFECTED BY HEATING. 8? 
 
 red heat for half an hour, and its contents emptied 
 into water. There is thus obtained, over the whole 
 of the articles, a surface of hardened steel, the depth 
 of which depends upon the time during which the 
 process of heating has been carried on : in half an 
 hour, it will, it is said, be extended to a depth some- 
 what less than the thickness of a sixpence. This 
 method is peculiarly applicable to articles which 
 are required to receive a certain degree of external 
 hardness and & polish. It is not applicable to cut- 
 ting instruments. 
 
 84. EFFECT OF HEAT ON THE STRENGTH OF 
 
 CAST IRON. 
 
 Messrs. Fairbairn and Hodgkinson found, in 
 some recent experiments, that when the temperature 
 of cast-iron bars was raised from the freezing point 
 (by covering them with snow) to a blood- red heat, 
 the strength was varied in the proportion of 950 
 to 723. The effect of heat on the strength of iron 
 is not, however, limited to the period during whicri 
 it is subjected to the heat ; in some cases it becomes 
 permanent. 
 
 85. PERMANENT DIMINUTION OF THE TENACITY 
 
 OF IRON WIRE BY HEATING. 
 
 If iron wire, after it has been first drawn out, be 
 put into the fire, heated red hot, and then allowed to 
 cool gradually, it will be found to have acquired 
 great additional flexibility, and to have lost nearly 
 onehalfofits strength; all the extraordinary tenacity 
 acquired by draiuing it is thus lost. The same is 
 true of wires of other metals. 
 
88 . ILLUSTRATIONS OF MECHANICS. 
 
 86. ANNEALING OF CAST IRON. 
 
 It appears, from the experiments of Mr. Watt 
 and Sir J. Hall, that whether a body in the act of 
 cooling from a liquid state shall assume a crystal- 
 lised form or a continuous glassy texture, depends 
 upon whether it be gradually or suddenly cooled. 
 Upon the texture of a metal, as influenced by these 
 circumstances, depend many of its most important 
 mechanical properties ; as, for instance, its hardness 
 and brittleness, or its softness and malleability. The 
 former qualities are given by cooling it rapidly; 
 the latter by cooling it slowly. 
 
 When cast iron has, by too rapidly cooling, ac- 
 quired the quality of hardness, it may, in some 
 degree, be taken from it again by heating it a 
 second time, and cooling it gradually. A number of 
 pieces are piled upon one another, and covered with 
 a heap of turf or cinders ; this is set on fire, and 
 when the iron has acquired a red heat, the fire is 
 allowed to go out of its own accord. Sometimes a 
 gradual cooling is effected by burying the iron, 
 when at red heat, in a heap of dry saw-dust. 
 
 The character of cast-iron is not in any other 
 way altered by this process, which is called anneal- 
 ing, except that it is rendered more malleable. 
 
 87. THE DIFFERENT MECHANICAL PROPERTIES OF 
 HOT AND COLD BLAST IRON. 
 
 It has recently been discovered by Mr. Neilson, 
 of Glasgow, that a prodigious saving of fuel in the 
 smelting of iron from the ore may be effected by 
 
PROPERTIES OF HOT BLAST IRON. 89 
 
 propelling into the furnace, instead of the usual blast 
 of cold air, a similar blast of air previously heated. 
 
 The blast is now commonly heated, before it is 
 propelled into the furnaces of the Clyde Iron- 
 works, to 600 of Fahrenheit ; and the expense of 
 coals to smelt each ton of metal (including those 
 used for heating the blast), averaged,' in 1833, the 
 blast being thus heated, 2 tons 5 cwt. In 1829, 
 when the same furnaces were worked with the cold 
 blast, it averaged 8 tons l^cwt. These coals were, 
 moreover, formerly .converted, at a considerable 
 expense, into coke before they were used ; with the 
 hot blast they are used un-coked. 
 
 From the first it was observed, that there was a 
 difference in the mechanical properties of irons from 
 the same ore melted by the hot and cold blast, and 
 the former were believed to be of inferior quality. 
 Accurate experiments, recently made by Messrs. 
 Hodgkinson and Fairbairn of Manchester, have 
 however shown that this is far from being the case: 
 a Table, contained in the Appendix, gives the results 
 of their experiments. From this, it appears that 
 the absolute strength of some irons, both as regards 
 their tenacity and their powers of resisting compres- 
 sion, is materially increased by the use of the hot 
 blast ; this remark applies indeed to all the irons 
 experimented on, excepting only the Buffery, No. 1. 
 The Carron, No. 3., acquires, by the use of the hot 
 blast, an additional power of resisting compression, 
 amounting to no less than 8 tons on the square 
 inch, and an additional tenacity of 1J tons on the 
 square inch. 
 
 It is to be observed, that hot blast iron possesses 
 
90 ILLUSTRATIONS OF MECHANICS. 
 
 softer qualities under the hammer than cold blast, 
 being of a more yielding and malleable nature. 
 These properties have an analogy to those of an- 
 nealed cast iron ; they, perhaps, connect themselves 
 ultimately with the operation of the same causes. 
 
 88. THE TEMPERING OF STEEL. 
 
 Steel is said to be most hardened when it is raised 
 to the highest temperature which it can receive 
 a white heat and then suddenly cooled by 
 being plunged in mercury or an acid, or into a mass 
 of lead. If, instead of these substances, water or 
 grease be used to cool it, the temper obtained is 
 not so hard : corresponding to every different 
 degree of heat to which the metal is raised, there 
 is a different hardness; but as these are all dif- 
 ferent degrees of red heat, which it is very diffi- 
 cult to distinguish from one another, the workmen 
 avail themselves of a remarkable property by which 
 the metal can be made to lose, to any degree, the 
 hardening which it has acquired, by heating it 
 again to an inferior degree, and allowing it to cool 
 gradually. This is the process to which they have 
 given the name of tempering. Communicating, in 
 the first place, to the steel a hardness above that 
 which they require, they then heat it again over 
 charcoal, and cool it gradually, until sufficient of 
 the hardness is taken from it, or until it is tem- 
 pered to the required degree. This process is 
 facilitated by certain remarkable changes of colour 
 which appear in it as it undergoes this process of a 
 second heating. These colours are, straw-coloured 
 yellow, purple red, violet blue, blue, clear watery blue. 
 
TEMPERING OF STEEL. 91 
 
 The straw-colour indicates the point at which the 
 second heating should be arrested, to obtain the 
 temper of razors and pen-knives : the purple, that 
 for table-knives ; the blue, for watch-springs ; and 
 the commencement of the red, for coach-springs. 
 
 Steel, when it has received the highest degree of 
 hardness, is more brittle than glass ; and thus the 
 dies used in coining, which are of the hardest steel, 
 not unfrequently break by mere atmospheric changes 
 of temperature. 
 
 Cutting instruments, if highly hardened through- 
 out, would be exceedingly liable to break; it is 
 therefore customary not to harden the parts near 
 the handle, or to temper them more than the rest, 
 that the yielding of these may prevent the parts 
 about the point from snapping. 
 
 89. THE TEMPERING OF THE ALLOY OF COPPER 
 CALLED TAM-TAM. 
 
 It is remarkable that copper possesses properties, in 
 respect to its hardening and tempering, which are the 
 opposite of those of cast iron and steel ; when cooled 
 slowly it becomes hard and brittle, but when cooled 
 rapidly soft and malleable. In a yet more remark- 
 able degree is this anomalous property possessed by 
 an alloy, composed of four parts of copper and one 
 of tin called tam-tam, used in the construction of 
 gongs, and other musical instruments. The cir- 
 cumstances under which it becomes malleable have 
 only of late years become known in Europe ; and 
 gongs are now made here nearly as perfect as those 
 of the Chinese. 
 
92 ILLUSTRATIONS OF MECHANICS- 
 
 90. THE ANNEALING OF GLASS. 
 
 Glass admits of being hardened to a very high 
 degree ; and, like steel, and by the same process, it 
 may be made to lose, in any degree, its hardness. 
 
 In the act of cooling, under the hands of the. 
 workman, from the state of fusion in which it is 
 blown, every article of glass becomes irregularly 
 hardened ; and, if taken in that state into use, its 
 brittleness would be so grea.t, that the slightest 
 shock or the slightest change of temperature would 
 be sufficient to break it : it is therefore transferred 
 from the hands of a blower into a large furnace 
 or oven, called the leer ; where it is for some time 
 subjected to an uniform heat, and then allowed gra- 
 dually to cool. It thus becomes tempered. What- 
 ever care may be taken in this process, certain 
 phenomena of the polarisation of light nevertheless 
 show that the same temper cannot be diffused 
 uniformly through any piece of glass, however small. 
 Glass has, however, been so effectually tempered 
 by Mr. Dent, as to admit of being formed into the 
 balance-springs of chronometers. 
 
 91. PRINCE RUPERT'S DROPS. 
 
 This name is given to pieces of glass, which, 
 being let fall into water when in a state of fusion, 
 acquire a long oval form, tapering to a point; 
 which point being afterwards broken off with the 
 lingers, the whole of the drop is thereby made to 
 burst into minute parts. 
 
 These drops present a remarkable instance of the 
 
MITZCHERLICH'S EXPERIMENTS. 93 
 
 irregular tempering of glass. The outside of the 
 drop is suddenly contracted, hardened, and ren- 
 dered brittle, whilst the interior, cooling slowly, 
 retains its elasticity: an equilibrium appears, in 
 the process of cooling, to establish itself between 
 the cohesive force of this external sheet and the 
 elasticity of the mass of glass which it compresses ; 
 to which equilibrium the entireness of the surface 
 about the point or tail of the drop is necessary : the 
 explanation of this last remarkable circumstance is 
 unknown. The fact is however unquestionable : the 
 drop may be struck a sharp blow on the thick part, 
 and even ground down on a cutler's wheel, without 
 breaking; but if even a scratch be made upon it 
 near the point, it will burst into a thousand atoms. 
 
 Many of these drops burst in the water, in the 
 act of making. 
 
 If they be heated, and then gradually cooled or 
 annealed, they lose entirely their property of ex- 
 ploding. 
 
 With these facts manifestly connect themselves 
 the influence of heat upon crystallisation. 
 
 92. MITZCHERLICH'S EXPERIMENTS ON CHANGES 
 IN CRYSTALLISED FORMS OF BODIES BY THE 
 OPERATION OF HEAT. 
 
 M. Mitzcherlich was first led to observe this 
 fact from certain changes in the optical properties 
 of sulphate of lime, at different temperatures. Sub- 
 sequently he ascertained that sulphate of nickel, 
 when exposed to the rays of the sun in summer, in 
 a closed vessel, without any change in its external 
 form or appearance, changed in a few days its 
 
94? ILLUSTRATIONS OF MECHANICS. 
 
 crystalline structure, from the prismatic form to 
 that of the square octohedron; this fact was de- 
 termined by breaking it. An exactly similar change 
 took place in seleniate of zinc, when exposed to the 
 action of the sun on a sheet of paper. 
 
 Crystals of sulphate of zinc and sulphate of mag- 
 nesia, when heated in alcohol, by degrees lose their 
 transparency ; and when broken, are found to be 
 composed of exceedingly small crystals, differing 
 totally from the original crystalline forms of the 
 salts. 
 
ASCENT OF WATER IN CAPILLARY TUBES 
 
 CHAP. III. 
 
 CAPILLARY ATTRACTION, AND ADHESION. 
 
 93. ASCENT OF WATER IN CAPILLARY TUBES. 
 
 IF one extremity of an open glass tube be plunged 
 
 in water, and the tube be held in an up- 
 right position, the surface of the water 
 within will be seen to change from a plane 
 to a concave surface, and to rise above the 
 level of that without it. 
 
 94. DEPRESSION OF MERCURY IN CAPILLARY 
 
 TUBES. 
 
 If an open glass tube be similarly 
 plunged in mercury, the surface of the 
 mercury within it will become convexed, 
 and will sink beneath the surface of that 
 without it. 
 
 95. DEPRESSION OF WATER IN CAPILLARY TUBES 
 
 WHOSE SURFACES CANNOT BE WETTED. 
 
 If the surface of a capillary tube be such that it 
 cannot be wetted; if, for instance, it be covered 
 with a thin coat of oil, so that moisture will not 
 adhere to it, the phenomena which it will present 
 when plunged into water, will be precisely the same 
 
 fig. 15. 
 
96 ILLUSTRATIONS OF MECHANICS. 
 
 as those which are presented by a clean glass tube 
 when plunged in mercury. The water will be re- 
 pelled and depressed all round and within it. 
 
 96. THE PHENOMENA OF CAPILLARY ATTRACTION 
 AND REPULSION ARE NOT CONFINED TO THE 
 INTERNAL SURFACES OF TUBES, BUT COMMON 
 TO THE SURFACES OF ALL BODIES, AND ONLY 
 MORE APPARENT IN THESE. 
 
 Thus, if a fluid be capable of wetting the sides 
 of the vessel which contains it, as well as the tube 
 plunged in it, it will be seen to be raised, and to 
 have become concave all round the sides, and 
 round the outside as well as the inside of the tube ; 
 as in the last figure but one : and, in like manner, 
 if the surface of the vessel and the tube be inca- 
 pable of being wetted by the fluid, this will be 
 depressed all round it, and have its surface convex, 
 and all round the outside, as well as the inside of 
 the tube. This effect is shown in the last figure. 
 
 97. THE RISE OF WATER BETWEEN PARALLEL 
 
 PLATES OF GLA'SS. 
 
 If two plates of glass be kept slightly asunder, 
 and made perfectly parallel to one another by 
 placing between them two pieces of wire, cut from 
 the same length, and if they be then plunged in 
 water, a plate of water will be seen to rise between 
 them. If a tube be taken, whose bore will just 
 admit the wire, or whose diameter equals the dis- 
 tance of the plates, the water will be seen to ascend 
 in this tube precisely to the same height that it does 
 between the plates. 
 
SYPHON FILTER. 97 
 
 98. THE WICK OF A LAMP. 
 
 The wick of a lamp or of a candle feeds the 
 flame by capillary attraction : it is, in point of fact, 
 a fascicle of threads, the surfaces of which, being 
 very nearly in contact, cause the ascent of the oil or 
 melted tallow between them by capillary attrac- 
 tion. 
 
 99. AN IRON WICK FOR A LAMP. 
 
 If a short capillary tube of iron be placed up- 
 right in a reservoir of oil, the oil will ascend in the 
 tube, by capillary attraction, to its top, and may 
 there be lighted. 
 
 100. A SYPHON FILTER, MADE WITH THREADS OF 
 COTTON. 
 
 If a fascicle of threads of cotton, such as forms 
 the wick of a lamp, have one of its extremities im- 
 mersed in a vessel of water, and be then brought 
 over its edge, and be made to hang with its fcther 
 extremity beneath the level of the surface of the 
 water in the vessel, then, by its capillary attraction, 
 the water will ascend into the space between the 
 threads of cotton; and, on the principle of the 
 syphon, it will flow out in drops at the other ex- 
 tremity. If there be any impurities in it, these 
 will be stopped in its progress by the fibres of the 
 cotton. 
 
98 ILLUSTRATIONS OF MECHANICS. 
 
 101. HEAVY BODIES MADE TO FLOAT BY CA- 
 
 PILLARY REPULSION. 
 
 If a small body repelling a fluid, or incapable of 
 being wetted by it, be placed gently upon it as, for 
 instance, a small ball of wax, or a needle rubbed 
 with oil, upon water it will float. The repulsion 
 of the body causes a displacement of the fluid be- 
 neath and all around it ; and since any body, im- 
 mersed in a fluid, is buoyed up with a force equal 
 to the weight of the water it displaces, it follows 
 that this body will float, provided the water it dis- 
 places, and by the weight of which it is buoyed 
 up, equals its own weight ; and it may be made to 
 displace that quantity of water by its repulsion, when 
 otherwise it would not. 
 
 102. INSECTS SUPPORTED ON THE SURFACE OF 
 
 WATER BY CAPILLARY REPULSION. 
 
 It is thus that certain species of insects are sup- 
 ported, and move about freely on the surface of 
 water. Their feet are lubricated with an oily sub- 
 stance, similar perhaps to that which gives a like 
 property to the feathers of aquatic birds. This re- 
 pulsive substance produces, beneath each foot of 
 the animal, a hollow in the surface of the fluid, 
 which is in fact a boat, supporting it; and its 
 body is thus buoyed up on as many such boats as 
 it has feet. 
 
CAPILLARY RODS SUSPENDED IN A FLUID. 99 
 
 103. THE ATTRACTIONS OF CAPILLARY RODS 
 WHEN SUSPENDED IN A FLUID 
 
 If two capillary rods be suspended in a fluid, 
 
 fig. 16. 
 
 parallel to one another, then, 
 so long as they are at such 
 a distance that the disturbed 
 surfaces of the fluid immedi- 
 ately about them do not cross 
 one another, no attraction or 
 repulsion of the rods upon one 
 another will be perceivable ; 
 but when the disturbed sur- 
 faces do thus interfere, such 
 vf5 attraction or repulsion will im- 
 
 mediately become apparent. If 
 both surfaces be raised, or both depressed that is, if 
 both rods attract the fluid or both repel it then 
 the rods will attract one another : but if one sur- 
 face be raised and the other depressed, as in the 
 preceding figure that is, if one of the rods be 
 capable and the other incapable of being wetted by 
 the fluid then they will repel one another. This 
 experiment was made by Haiiy, with two laminae, 
 one of which was of talc, and the other of ivory ; 
 the former of these substances does not admit of 
 being wetted, that is, it is repulsive of water, the 
 latter substance attracts it thus, one depresses the 
 contiguous surface of the water, and the other 
 elevates it; and where the elevation is made to meet 
 the depression, a repulsion is immediately appa- 
 rent. 
 
 H 2 
 
100 ILLUSTRATIONS OF MECHANICS. 
 
 104-. THE ATTRACTION AND REPULSION OF 
 
 FLOATING BODIES. 
 
 Two small balls of pith, or of wood, each of which 
 attracts water, and, therefore, when made to float, 
 fig- 17 elevates it all round the line of 
 
 |i* contact with its surface, being 
 brought near one another, so 
 that the elevations interfere, will 
 attract one another. In like manner, two small balls 
 of wax, or smoked pith balls, which repel water, 
 being made to float in it; or two small iron balls 
 made to float in mercury ; when their depressions 
 interfere, attract one another: but if a pith -ball 
 and a ball of wax, or another pith-ball which has 
 been held over the smoke of a lamp, be made so to 
 approach that the depression of the one interferes 
 with the elevation of the other, they are immediately 
 repelled. 
 
 105. THE ATTRACTION OF NEEDLES FLOATING 
 
 ON WATER. 
 
 If two needles be slightly rubbed with grease, 
 and then placed with care on the surface of the 
 water, they will float upon it, depressing it all around 
 them. If, when thus floating, these needles be made 
 to approach one another, so that their depressions 
 interfere, they will immediately rush into contact. 
 
 106. ATTRACTION AND REPULSION OF % SMALL 
 
 BODIES BY THE SIDES OF VESSELS. 
 
 It is on the principle just stated, that the sides of 
 
 vessels, containing water, attract small bodies, floating 
 
 upon it, when the material of the vessel and of the 
 
 floating body are both capable or both incapable of 
 
CAPILLARY TUBE. 101 
 
 being wetted, and repel them when one is capable 
 of being wetted and the other not. 
 
 In the first case, the attraction takes place when 
 the elevation round the body interferes with the 
 elevation round the sides of the vessel, or the de- 
 pression round the body with the depression round 
 the sides. 
 
 In the second case, the repulsion occurs when the 
 elevation round the one interferes with the depres- 
 sion round the other. 
 
 If the vessel containing the water be brim-full, 
 so that the water stands all round above its edges, 
 then, since its surface will be depressed or convex 
 at its edges, instead of concave, as when it stood be- 
 neath the edges of the vessel, the opposite phenomena 
 will occur. All bodies floating upon this surface 
 will be repelled towards its centre, unless they be 
 in their nature such as cannot be wetted. 
 
 107. WHEN A CAPILLARY TUBE is TAKEN OUT OF 
 THE FLUID IN WHICH IT HAS BEEN PLUNGED, A 
 PORTION OF THE FLUID WHICH REMAINS IN IT 
 STANDS AT A MUCH GREATER HEIGHT THAN IT 
 STOOD BEFORE. 
 
 Thus, if A B be the height of the ca- 
 pillary column when the tube was im- 
 mersed, then, when it is lifted out of the 
 fluid, it will have become A* B', con- 
 siderably greater than AB ; and, if the 
 sides of the tube be very thin, the sus- 
 . pended column, when thus withdrawn 
 f|from the fluid, will even be double that 
 which it was before. 
 
 H S 
 
102 ILLUSTRATIONS OF MECHANICS. 
 
 This greater elevation of the capillary column, 
 when the tube is withdrawn from the fluid, is 
 produced by the drop, which, when it is with- 
 drawn, is suspended from its lower extremity : the 
 force of attraction between that drop and the end 
 of the tube sustains the additional column. In 
 the case of a thin tube, this drop has an extremely 
 convex surface ; in the case of a thick tube, it is 
 spread out and flattened ; it is on this circumstance 
 that the greater elevation in the thin than the thick 
 tube is dependent 
 
 108. WATER WILL NOT, UNDER CERTAIN CIRCUM- 
 STANCES, FIND ITS LEVEL IN A CAPILLARY 
 SYPHON. 
 
 Water being poured into the larger branch of a 
 bent tube (such as that shown in the accompanying 
 figure), will immediately attain the same level in 
 Jig. 19. both branches of the tube until it 
 reaches the extremity of the shorter 
 branch : instead of then flouing out, it 
 will accumulate, and its surface will rise 
 in the longer branch ; whilst, in the 
 shorter, its surface will remain fixed, 
 becoming, at the same time, less and 
 less concave, until, at length, it is per- 
 fectly flat, as in the second figure ; as yet more 
 water is poured in, this flat surface will become con- 
 vex, the water rising in a drop (as in the first figure), 
 until, when this drop has become a hemisphere, the 
 height A* H', having become double of A H, it will 
 burst, and the surface A' will fall a greater or a less 
 distance, according to the thickness of the tube. 
 
CAPILLARITY IN THE BAROMETER TUBE. 103 
 
 109. To MAKE A VESSEL FULL OF HOLES, WHICH 
 
 SHALL YET CONTAIN WATER. 
 
 If a vessel be formed of wire- gauze, of iron or 
 brass, then, the meshes being small, it will hold a 
 certain depth of water ; for to each mesh will be 
 fixed, by capillary attraction, a drop, which, as in 
 the experiment of the tube, will, by the force of its 
 adherence to the mesh, be sufficient to support the 
 weight of the water immediately above it, provided 
 the height of the superincumbent column, that is, 
 the depth of the contained fluid, do not exceed a 
 certain limit,, determined by the smallness of the 
 mesh. 
 
 1 10. To MAKE A VESSEL FULL OF HOLES, WHICH 
 
 SHALL FLOAT. 
 
 If a vessel, constructed of wire-gauze (as above), 
 be immersed in a fluid, the fluid will not enter it, 
 unless it be sunk beyond a certain depth ; for to 
 each mesh will, as before, be made to adhere a small 
 portion of the fluid., which, by the force of its ad- 
 herence, will prevent the rest of the fluid from en- 
 tering by that mesh. 
 
 111. EFFECTS OF CAPILLARITY IN THE BARO- 
 
 METER TUBE. 
 
 The top of the column of mercury suspended in 
 the barometer tube, should, evidently, be a convex 
 surface, glass being repulsive of mercury. The re- 
 mark was, however, made in 1780 (by DonCasbois), 
 that if the mercury be for some time boiled in the 
 mercurial tube before it is hermetically sealed, a 
 H 4 
 
104 ILLUSTRATIONS OF MECHANICS. 
 
 perfect vacuum may be obtained ; instead of being 
 thus convex, the surface will be plane, or even con- 
 cave ; a fact which seemed to indicate an anomalous 
 attraction of the glass, which might interfere with 
 the accuracy of barometric admeasurements. The 
 circumstance has been recently explained by 
 M. Dulong, who has shown that the surface of the 
 mercury is chemically affected by the ebullition, 
 and becomes an oxide whose capillary properties are 
 no longer those of the mercury. 
 
 112. THE HEIGHTS TO WHICH A FLUID ASCENDS 
 IN DIFFERENT CAPILLARY TUBES, ARE GREATER 
 AS THEIR DIAMETERS ARE LESS. 
 
 This will be strikingly seen if a number of capil- 
 lary tubes, having different diameters, be placed 
 side by side in a coloured fluid : the fluid will 
 stand at different heights in all, being highest in 
 those whose diameters are least. 
 
 113. THE HEIGHTS TO WHICH THE SAME FLUID 
 ASCENDS IN DIFFERENT CAPILLARY TUBES, 
 DO NOT DEPEND ON THE THICKNESS OF THE 
 TUBES. 
 
 If tubes be taken of different thicknesses, but the 
 same internal diameter, and partly immersed in the 
 same fluid, the fluid will be seen to stand at the 
 same height in all. 
 
HEIGHT OF CAPILLARY COLUMN. 105 
 
 11 4-. THE HEIGHTS TO WHICH THE SAME FLUID 
 ASCENDS IN DIFFERENT TUBES, DO NOT DEPEND 
 UPON THE SUBSTANCES OUT OF WHICH THE 
 TUBE ARE FORMED, PROVIDED ONLY THEY BE 
 SUBSTANCES WHICH DO NOT REPEL THE FLUID,, 
 OR WHICH ADMIT OF BEING WETTED BY IT. 
 
 It is found, by experiment, that the heights to 
 which the same fluid water, for instance ascends 
 in tubes of glass, iron, lead, tin, wood, &c., are the 
 same, provided the bores of these tubes be all of the 
 same diameter. 
 
 115. THE HEIGHTS TO WHICH DIFFERENT FLUIDS 
 ASCEND IN THE SAME TUBE, ARE NOT THE 
 
 SAME. 
 
 A heavier fluid ascends to a greater height than 
 a lighter ; thus, water ascends to more than twice 
 the height of alcohol, as will be seen by the table 
 of Gay Lussac's experiments. 
 
 116.* THE HEIGHTS TO WHICH THE SAME FLUID 
 ASCENDS IN DIFFERENT CAPILLARY TUBES, ARE 
 INVERSELY PROPORTIONAL TO THE DlAME- 
 TERS OF THE TUBES. 
 
 That is, in whatever proportion the internal dia- 
 meter of any one tube is less than another, precisely 
 in the same proportion is the height of the capillary 
 column in the first greater than the height of that 
 in the other ; so that a tube of J the diameter will 
 have a column of twice the height ; one of ^rd the 
 diameter, a column of 3 times the height, &c. To 
 determine this law accurately, the experiments must 
 
106 ILLUSTRATIONS OF MECHANICS. 
 
 be made with much care, and especially the tubes 
 must be perfectly clean ; for, as we have shown, 
 the attraction may be converted into a repulsion by 
 the slightest covering of any oily substance upon 
 it. In order to get rid of all foreign substances 
 on the surface of the glass, it is found that it must 
 be chemically cleaned ; and that upon this cleaning 
 the precision of all experiments depends. Acids, 
 or alcohol, must be passed through it, according to 
 the nature of the impurities to which it has been 
 liable ; and it must throughout be wetted before- 
 hand \vith the liquid in which the experiment is 
 to be made. 
 
 The most accurate experiments on this subject 
 are those of M. Gay Lussac. The apparatus used by 
 him is represented in the accompanying figure. 
 A B is a tail cylinder of glass, placed on a stand, 
 capable of being levelled by screws; M is the 
 surface of a liquid contained in it ; and M N is a 
 capillary tube, partly immersed in it, and sup- 
 ported by a piece which rests upon the edge of 
 the vessel : beside the glass vessel is a vertical rule, 
 with a divided scale, along which is moveable, by 
 means of a micrometer screw, a small telescope. 
 To measure the height M N, of the column sus- 
 pended in the tube, the telescope is moved until 
 the summit N of the column is seen on the cross 
 wires of the telescope : the tube is then moved a little 
 to the side of the vessel, and the screw K is made 
 to rest, by means of a piece similar to that which 
 supports the tube, upon the edges of the vessel, and 
 turned until its point K just touches the surface of 
 the fluid in the vessel ; then, a little cf the fluid 
 
GAY LUSSAC S EXPERIMENTS. 
 
 fig. *>. 
 
 107 
 
 having been removed with a tube *, the telescope 
 is made to descend until the extremity K of the 
 screw is just seen on its cross wires. The height 
 on the scale at which the telescope before stood, 
 having been observed, and the height at which it 
 now stands, the difference between these two heights 
 is that of the capillary column. The following table 
 contains the result thus obtained by M. Gay 
 Lussac : 
 
 * This is a method commonly used for taking out from a 
 large vase small quantities of a liquid it contains. The tube is 
 plunged into it with both its extremities open, and then drawn 
 out with the upper extremity closed with the finger, the pres- 
 sure of the atmosphere from without keeps a large portion of 
 the fluid, which had entered the tube, from flowing out of it. 
 
108 
 
 ILLUSTRATIONS OF MECHANICS. 
 
 Fluid Expd. 
 on 
 
 Specific 
 Gravity. 
 
 Temp, in de- 
 grees of Cen- 
 tigral Ther. 
 
 Elevation in tubes, whose ' 
 diameters were in millimeters. 
 
 1 "2944 
 
 1 -9038 
 
 10-508 
 
 
 
 
 Milltrs. 
 
 Milltrs. 
 
 Milltrs. 
 
 Water - 
 
 1-000 
 
 8 -5 
 
 23-1634 
 
 15-5861 
 
 
 Alcohol 
 
 0-8196 
 
 8 
 
 9*1823 
 
 6-4012 
 
 
 
 0-8595 
 
 10 
 
 9-301 
 
 
 
 
 0-9415 
 
 8 
 
 9-997 
 
 
 
 
 0-8135 
 
 16 
 
 7-078 
 
 - 
 
 0-3835 
 
 Essence J 
 
 
 
 
 
 
 of Tere-> 
 
 0-8695 
 
 8 
 
 9-8516 
 
 
 
 binthum 3 
 
 
 
 
 
 
 In these experiments the ratio of the diameters 
 of the two first tubes is 1 : 1*474 ; and the ratio of 
 the heights of the capillary columns is, for water. 
 1'486 : 1 ; and for alcohol, 1'434 : 1 ; which results, 
 practically, coincide with the law. 
 
 117.* THE ELEVATION OF WATER BETWEEN 
 PLATES OF GLASS SLIGHTLY INCLINED TO ONE 
 
 ANOTHER. 
 
 If two plates be placed in water^ in the position 
 shown in the figure, the water 
 will rise between them, its sur- 
 face forming a curve, which, 
 on examination, is found to be 
 that which is called, by ma- 
 thematicians the hyperbola. 
 L^ This is easily explained, the 
 greater elevation of the water 
 near the angle of the plates is caused by the less dis- 
 tance of the surfaces of the plates from one another 
 there. This distance of the plates from one another, 
 at different points, is, by geometry, proportional to 
 the distance of those points from the angle of 
 
METHOD OF QUARRYING MILL-STONES. 109 
 
 the plates : now, it follows from the experiment, 
 article 97., and from the last article, that the heights 
 to which the water is raised between the plates at 
 different points, are inversely proportional to the 
 distances of the plates there ; they are therefore in- 
 versely proportional to the distances of these points 
 from the angle. Thus, then, the heights of the dif- 
 ferent points of the surface of the water are in- 
 versely proportional to their distances from the angle; 
 a property of the rectangular hyperbola between 
 the asymptotes. 
 
 118. OF THE FORCE WITH WHICH FIBROUS SUB- 
 STANCES IMBIBE MOISTURE BY CAPILLARY 
 ATTRACTION, .AND THEREBY INCREASE THEIR 
 BULK. 
 
 The following is said to be a method used m 
 France for quarrying, in one piece, the large flat 
 stones which are used as mill-stones.* A whole 
 block of the stone being found, of sufficient dimen- 
 sions, it is hewn into the form of a solid cylinder, 
 several feet in height, and of the diameter required 
 for a mill-stone ; this block is destined to form 
 several mill -stones. To cleave it into them, deep 
 grooves are cut round it, where the divisions should 
 take place ; and into these grooves wedges of wil- 
 low-wood, thoroughly dried in an oven, are firmly 
 driven : these wedges, being sunk to their proper 
 depth, are moistened, or left exposed to the humidity 
 of the night : they take up the moisture by the 
 
 * See Montucla's Philosophical Recreations, Hutton's trans- 
 lation, vol. iv. p. 157. 
 
110 ILLUSTRATIONS OF MECHANICS. 
 
 capillary attraction of their contiguous fibres, and 
 thereby swell out their dimensions with such pro- 
 digious power as to overcome the cohesion of the 
 wide surface of the section of the stone, and 
 divide it. 
 
 Another phenomenon, referable to the same prin- 
 ciple, is, the lifting of great weights, by fasten- 
 ing a cord to them by one of its extremities, and to 
 some firm attachment above them by the other, 
 stretching it tightly, and then wetting it ; the cord 
 will imbibe the moisture by the capillary attraction 
 of its contiguous fibres, and swell out its bulk with 
 prodigious force ; and in the act of swelling it will 
 shorten its length, and lift the weight. 
 
 119. THE THEORY OF CAPILLARY ATTRACTION. 
 
 Matter has been shown to be composed of 
 elements which are inappreciably and infinitely 
 minute. 
 
 It is between these infinitely minute elements 
 that the greater number of the forces known to us 
 have their only sensible action; and there we cannot 
 follow them, to inquire into the law o/'that action. 
 Although these forces which are called molecular 
 forces, and which include among their phenomena 
 extensibility, compressibility, elasticity, the strength 
 of materials, and capillary attraction only thus 
 operate sensibly at insensible distances; yet does 
 their operation result in certain manifest and sen- 
 sible properties of the matter in which they reside. 
 
 Thus, for instance, extensibility, elasticity, and 
 cohesive strength, are sufficiently sensible qualities 
 of matter, although the molecular forces fron? 
 
THEORY OF CAPILLARY ATTRACTION. Ill 
 
 which they result, and into which they ultimately 
 resolve themselves lie hidden, far from our view 
 among the inexhaustible divisions of matter. 
 
 But are there not, it may be asked, in these 
 sensible phenomena, numerous as they are, some 
 indications from which we may reason back to the 
 elementary forces of which they are complicated 
 results ? May we not resolve this complicated 
 manifestation of force into others more simple, 
 these into others, and so on, until the reason has 
 thus followed them where the senses could not, 
 and the eye of science seen their operation be- 
 tween particles of matter, and measured it through 
 infinitesimals of space, where every appliance of 
 physical sight has long lost it ? It is not to be de- 
 spaired of, that this may, in some state of philo- 
 sophy, far advanced beyond that which belongs to 
 it now, be effected. At present we do not approach 
 that state. 
 
 That molecular force whose theory has been 
 most successfully investigated is capillary attrac- 
 tion ; and that theory assumes as its basis, an en- 
 tire ignorance of the law by which one particle of 
 matter attracts another ; it supposes only that this 
 law, whatever it may be, is the same when it is a 
 solid which attracts a fluid, as when it is a fluid 
 which attracts itself; the same law of attraction, 
 but a different intensity of attraction. 
 
 Clairaut was the first, starting from this simple 
 and almost self-evident hypothesis, to prove, by the 
 inexhaustible resources of mathematical analysis, 
 that this one condition was sufficient. If the inten- 
 sity of the attraction of the solid on the fluid was 
 
112 ILLUSTRATIONS OF MECHANICS. 
 
 greater than half that of the fluid on itself, the 
 fluid would elevate itself about the solid if it was 
 less, it would depress itself and if it was equal, it 
 would neither elevate nor depress itself. 
 
 La Place, taking up this theory of Clairaut, 
 and combining with his hypothesis this evident 
 principle, that the unknown law, whatever it may 
 be, causes the attraction to diminish so rapidly, 
 that at sensible distances it becomes insensible ; and, 
 reasoning with admirable ingenuity on the principle, 
 has succeeded in explaining every one of the phe- 
 nomena of capillary attraction which have been de- 
 tailed in this work, with an accuracy which extends 
 even to precise linear admeasurement ; and may be 
 considered as offering one of the most remarkable 
 verifications that theory has ever received from ex- 
 periment. This verification, however, unfortunately 
 indicates to us the fact, that various as the pheno- 
 mena of capillary attraction are, they are none of 
 them sufficient to manifest to us the real law of 
 the force on which they depend. That law is the 
 desideratum, and they leave us in utter ignorance 
 of it. 
 
 It may be mentioned, that some of the principles 
 of the theory of Clairaut and La Place have been 
 impugned by Poisson, in his recent work " On Ca- 
 pillary Attraction ;" but it would seem without 
 sufficient ground. 
 
 * See Professor Challis's valuable report " On the Theory of 
 Capillary Attraction," in the third volume of the "Reports of the 
 British Association of Science." 
 
CUPELLATION. US 
 
 120. APPLICATION OF CAPILLARY ATTRACTION 
 TO ASSAYING. 
 
 There is a very beautiful application of the prin- 
 ciples of capillary attraction in the process by 
 which the precious metals are separated from fo- 
 reign ingredients. The method is used generally 
 in assaying, and is called Cupellation ; we shall 
 describe it, as we have seen it applied to the sepa- 
 ration of gold and silver from the dust which is 
 swept from the shops of working goldsmiths and 
 jewellers.* 
 
 A portion of this dust is mingled with a certain 
 proportion of an oxide of lead (red lead), and a 
 small quantity of flux, and placed in a flat crucible, 
 called a cupel; the material of which is finely 
 powdered bone-ash, made into a paste, and moulded, 
 by pressure, into a circular mould. The cupel, 
 whose bottom and sides are of great comparative 
 thickness, is then placed in a small earthen oven, 
 called a muffle, which is so introduced into the 
 assay furnace, as that a free admission of air shall 
 be allowed to the contents of the cupel. 
 
 The mixture soon enters into a state of fusion ; 
 and the oxide of lead dissolving the foreign ingre- 
 dients, and uniting with itself continually more and 
 more of the oxygen of the air which has admis- 
 sion to it, becomes more and more liquid, until at 
 length it has reached that state of liquidity in 
 which the intensity of the attraction of its particles 
 
 * These sweepings are carefully preserved, they become an 
 article of commerce, and the assaying of them is a separate trade. 
 
 I 
 
114 ILLUSTRATIONS OF MECHANICS. 
 
 for one another, is not so much as double that of 
 its particles for the solid material of the cupel. 
 This limit of liquidity being passed, the whole fluid 
 mass of scoria (composed of the oxide of lead, and 
 the foreign ingredients dissolved in it) passes, by 
 capillary attraction, into the porous material of 
 the cupel, whilst the metallic substances, gold and 
 silver, which have been melted, but have not 
 partaken in the oxidisation of the lead, and have 
 not therefore passed the supposed limits of fluidity 
 remain behind, collected in a globule in the bot- 
 tom of the cupel. When the whole is cooled, the 
 globule of gold and silver is taken out ; and the 
 cupel being broken, the mass of scorea is found 
 collected in a cake, in the substance of the bottom 
 of it, there being no external indication of its pre- 
 sence there.* 
 
 The process, above described, is that used for 
 testing the average quantity of the precious metals 
 in the sweepings, before they are purchased. It is, 
 however, an epitome of the process by which the 
 separation is made on a larger scale ; except that 
 various contrivances are there introduced for econo- 
 mising it. Metallic lead, for instance, is made to 
 supply the place of red oxide of lead ; and, by an 
 ingenious process, this lead, after being used, is 
 separated from the scoria, and made again and 
 again, to serve the purpose of the refiner. The 
 cupel used is made shallow, and of large dimensions ; 
 
 * The silver of the metallic globule, which remains in the 
 cupel, is separated from the gold by solution in nitric acid, and 
 precipitated from this solution by immersing in it bars of copper. 
 
AGENCY OF CAPILLARITY IN NATURE. 115 
 
 air is propelled upon the surface of the liquid mass ; 
 and, as in the process of oxidisation the lead in- 
 creases its volume, a portion of it is allowed to run 
 over the sides of the cupel. 
 
 121. THE AGENCY OF CAPILLARY ATTRACTION 
 IN NATURE. 
 
 Let it not be supposed, that the phenomena of 
 capillary attraction are limited to mere experi- 
 ments in physics, or to its applications in art. Ca- 
 pillarity is one of the most active principles in 
 nature. What is it but this ubiquitous power, 
 which retains in the soil of the earth the moisture 
 necessary to vegetation, ministering it, drop by drop, 
 to the radicles of plants and trees ; conveying it, at 
 one time, beneath the surface, down the slope of 
 a hill, to the valley below, or to some deep-sunken 
 reservoir; thence lifting it up again to quench the 
 thirst of the parched herbage ; checking its progress 
 to the streams, which it would otherwise swell 
 instantaneously to floods floods, whose waters, 
 having uselessly deluged the land, would be lost as 
 uselessly in the ocean. Take away capillary at- 
 traction, or alter it, so that the intensity of the 
 attraction of the solid substances which compose 
 the soil, for water, shall be less instead of more than 
 half the intensity of the attraction of water for 
 itself and the earth must become a desert. Rain 
 would fall upon it as mercury falls upon a piece of 
 glass it would roll off it in drops. There would 
 be no fertilising influence in the shower ; no 
 moisture could reach the parched roots of Dlants 
 and trees ; vegetation would become extinct and 
 i 2 
 
J16 ILLUSTRATIONS OF MECHANICS. 
 
 animal life would gasp itself away in a thick at- 
 mosphere of dust. 
 
 The greater apparent elevation of water, and the 
 greater force of capillarity in its operation in many 
 natural phenomena than in artificial tubes, is to be 
 explained by the extreme proximity of the surfaces 
 between which it there acts. The elevation of 
 water in a tube is inversely proportional to its 
 diameter, or, between two plates, it is inversely pro- 
 portional to the distance of the plates ; thus, if we 
 kept halving the diameter of a tube, or halving the 
 distance between the plates, we should keep doubling 
 the elevation of the fluid. Artificial tubes may 
 thus be made to elevate water to a remarkable 
 height : but nature provides tubes infinitely finer 
 than any that art can reach ; and to the capillary 
 elevation of fluids in them, there seems to be no 
 limit. The same is the case with particles of earth 
 and sand ; the close proximity of their surfaces to 
 .one another, gives them a power of capillary attrac- 
 tion, which is almost without limit. 
 
 122. ENDOSMOSE AND EXOSMOSE. 
 
 M. Dutrochet having introduced into the swim- 
 ming bladder of a carp a thin mucillage,, effectually 
 closed up the aperture by which he introduced it, 
 and placed the bladder in water, found, by weigh- 
 ing it, after it had remained there some time, that 
 its weight had considerably increased : the water 
 in which it was immersed had, in fact, made its way 
 through the substance of the bladder, and mingled 
 itself with the mucilage. 
 
 He then filled the bladder with water, and im- 
 
ENDOSMOSE AND EXOSMOSE. 
 
 117 
 
 mersed it in the thin mucillage, and found that the 
 opposite phenomenon took place. The bladder and 
 its contents lost weight : the water made its way 
 through the substance of the bladder into the mucil- 
 lage. These phenomena he afterwards developed, 
 under a great variety of other circumstances ; and 
 called the first endosmose, and the other, exosmose. 
 His subsequent experiments will best be under- 
 stood from the description of an instrument, which 
 he calls the endosmometer ; and which is represented 
 i n tne accom p an yi n g diagram. It repre- 
 sents two reservoirs an outer, C, and 
 an inner one, A ; which may be of glass : 
 the inner one is open at the bottom, and 
 is supported above the bottom, and away 
 from the sides of the other: a vertical 
 tube, B, is fitted into the top of it by 
 grinding. Over the open bottom of the 
 inner reservoir is stretched, tightly, a 
 membrane of bladder, or there is cemented 
 
 fg. 22. 
 
 
 
 across it a piece of slate, or other porous 
 substance, whose properties are the sub- 
 ject of experiment. 
 
 Now, suppose water to be contained in the ex- 
 terior reservoir, and alcohol in the interior ; the 
 two fluids will then be divided by the partition of 
 bladder, or the porous solid plate forming the bottom 
 of the inner reservoirs. 
 
 This division of the two fluids into two separate 
 chambers will not however be sufficient to prevent 
 them mingling. Through the substance of the par- 
 tition the water will, in a few minutes, be seen to 
 have made its way, by the rising of the alcohol in 
 I 3 
 
118 ILLUSTRATIONS OF MECHANIC^. 
 
 the tube ; and, if the tube be not more than 16 or 18 
 inches in length, in the course of a day the alcohol 
 will have risen to the top of it, and flow over. 
 
 This remarkable fact is hitherto entirely un- 
 explained. Dutrochet is said to have ascertained, 
 by the most delicate experiments, that it is accom- 
 panied by no perceptible traces of change in the 
 electrical state of the substances concerned. 
 
 If one of the vessels contain water and the other 
 gum-water, or acetic acid, or nitric acid, or espe- 
 cially hydrochloric acid, the exosmose will be from 
 the water. 
 
 Besides bladder, numerous other animal, as well 
 as vegetable, substances present similar phenomena 
 of endosmose, and partake of it in common with 
 inorganic substances such as plates of baked earth, 
 of calcined slate, and of clay. 
 
 The extreme elevation of the liquid in the tube, 
 marks the force of the action : that elevation is 
 different for different liquids, and when partitions 
 of different substances are used. 
 
 Dutrochet found water thickened with sugar, in 
 the proportion of one part of sugar to two parts of 
 water, to be productive of a power of endosmose, 
 capable of sustaining the pressure of a column of 
 mercury 127 inches in height. Dutrochet conceived, 
 on no sufficient grounds it would appear, that en- 
 dosmose was the immediate agent in all the pheno- 
 mena of vegetable life. 
 
 123. ADHESION OF PLATES OF DIFFERENT SUB- 
 STANCES TO THE SURFACES OF FLUIDS. 
 If a plate of any substance be brought into con- 
 tact with the surface of a fluid, it will immediately 
 
ADHESION. 
 
 119 
 
 be perceived that an adhesion has taken place be- 
 tween the two, which may be measured by attach- 
 ing the plate, by means of a string, to one extremity 
 of the scale of a balance, and adding weights in 
 the other scale-pan until the adhesion is overcome. 
 The following table contains the results thus ob- 
 tained by M. Gay Lussac, with a plate of glass : 
 
 Fluids experimented on. 
 
 Specific 
 Gravity. 
 
 Tempera- 
 ture. 
 
 Weight necessary to de- 
 tach a circular disc of 
 glass, diameter = 1 18'366 
 millimeters. 
 
 Water - - - - 
 
 1-000 
 
 8'5 
 
 Grammes. 
 59-40 
 
 Alcohol - - - 
 
 0-8196 
 
 8 
 
 31-08 
 
 - - _ 
 
 0.8595 
 
 10 
 
 32-87 
 
 _ . _ 
 
 0-9415 
 
 8 
 
 37-15 
 
 Essence of Tere- ~\ 
 binthum - J 
 
 0-8695 
 
 8 
 
 34-10 
 
 i 
 
 A disc of copper or of any other substance, of 
 the same diameter, and capable of being wetted by 
 the fluid, gives exactly the same result. A circum- 
 stance which is easily understood; for the surface of 
 the plate always brings away with it a thin film of 
 the fluid : it is the adhesion of this film of fluid to 
 the rest which is therefore broken. 
 
 M. Achard by whom an extensive series of ex- 
 periments on this subject was made, and their re- 
 sults published in the Berlin Memoirs for 1776 
 found, by varying the atmospheric pressure, under 
 which his experiments were made, that the results 
 were wholly independent of it. Varying the tem- 
 perature, he found that as it was increased, the ad- 
 hesion uniformly diminished. 
 
 When the substance of the disc is repulsive of the 
 fluid, or incapable of being wetted by it, it is found 
 I 4 
 
120 ILLUSTRATIONS OF MECHANICS. 
 
 that an adhesion of the two still exists ; which is. 
 nevertheless, exceedingly variable, depending on 
 the time during which contact has been allowed to 
 take place. Thus M. G. Lussac found, that to se- 
 parate the disc of glass used before, from the surface 
 of mercury, the weight required increased, with the 
 time of adhesion from 158 to 296 grammes. In 
 this case the adhesion of the fluid to itself is 
 stronger than its adhesion to the plate. 
 
 It is found that, under these circumstances, dif- 
 ferent metals have different forces of adhesion to 
 the surface of mercury. 
 
 M. Guyton de Morveau (Elements de Chymie, 
 1777,) found that the separation of a circular disc 
 of pure gold, one inch in diameter, from the surface 
 of mercury, required a weight of 446 grains ; an 
 equal disc of silver, 429 grains ; a disc of tin, of the 
 same size, 418 grains ; of lead, 397 grains ; of Bis- 
 muth, 372 ; of platina, 282 ; of zinc, 204 ; of copper, 
 142 ; of antimony, 126 ; of iron, 115 ; of Cobalt, 8. 
 
 These forces of adhesion appear to be in the pro- 
 portion of the chemical affinities of mercury to the 
 different metals experimented on ; they were looked 
 upon in that light by M. Guyton. 
 
 124. ADHESION OF A COLUMN OF MERCURY TO 
 THE INTERNAL SURFACE OF A CAPILLARY TUBE. 
 
 The following fact was observed, in 1792, by 
 Huygens: A barometer tube, 70 inches in length, 
 and a few lines in diameter, having been well 
 cleaned with alcohol, filled with mercury, freed 
 from all air, and then carefully inverted, it was 
 
ADHESION. 12J 
 
 seen with amazement, that the column, instead of 
 descending to the barometric height, remained sus- 
 pended, until the tube had been several times slightly 
 shaken, when finally it occupied its proper position 
 of 28 inches. This phenomenon, which occurs under 
 the same circumstances whenever the tube is tho- 
 roughly cleaned, is evidently a result of the ad- 
 herence of the mercury to the tube. 
 
 125. ADHESION OF PLATES OF GLASS TO ONE 
 
 ANOTHER. 
 
 When pieces of plate glass have received their 
 last polish from the hands of the workman, it is 
 customary to clean them, and to place them in a 
 vertical position in the warehouse, somewhat like 
 books on the shelves of a library. In this posi- 
 tion they not unfrequently acquire, in the course 
 of time, an adhesion, which renders it very difficult, 
 and sometimes impossible, to separate them. Three 
 or four plates have been thus absolutely incorpo- 
 rated, so that they might be worked as one piece, 
 and even cut with a diamond, like a single piece. 
 M. Pouillet states that he had seen pieces of glass, 
 thus united, from the royal manufactory of St. 
 Gobin, which adhered as perfectly as though they 
 had been melted together. An exceedingly great 
 mechanical force was applied, to cause them to slip 
 upon one another ; and when at length they yielded, 
 it was found, on examination, that the plates had not 
 separated at their common surfaces, but that the 
 thickness of the glass had been torn away ; so that 
 to the surface of one, still adhered a lamina of the 
 other. 
 
122 ILLUSTRATIONS OF MECHANICS. 
 
 CHAPTER IV. 
 STATICS. 
 
 DEFINITIONS. THE EQUILIBRIUM OF THREE PRESSURES. 
 
 THE EQUILIBRIUM OF ANY NUMBER OF PRESSURES 
 
 IN THE SAME PLANE. THE LEVER. THE WHEEL. 
 
 AND AXLE. THE COMPOSITION AND RESOLUTION OF 
 
 FORCES. THE CENTRE OF GRAVITY. THE RESIST- 
 ANCE OF A SURFACE. FRICTION. THE INCLINED 
 
 PLANE. THE WEDGE. THE SCREW. THE EQUI- 
 LIBRIUM OF BODIES IN CONTACT. PIERS. ARCHES. 
 
 FORCE is that which produces or destroys motion, 
 or which tends to produce or destroy it. 
 
 That which is the subject of motion, or a tend- 
 ency to motion, is MATTER. 
 
 126. EQUILIBRIUM. 
 
 When the tendency of a force to communicate 
 motion does not take effect, it is a thing of daily ex- 
 perience and observation, that there exists some 
 other force or forces, having, one or more of them, 
 an opposite tendency ; which other forces are the 
 causes of the quiescence. That state of a body in 
 which, being acted upon by certain forces, it re- 
 mains at rest, or as it may be termed, the state of 
 its forced rest, is called its state of EQUILIBRIUM : 
 and the forces which constitute that state are said 
 to be forces in equilibrium, or PRESSURES. 
 
EQUILIBRIUM OF THREE PRESSURES. 123 
 
 127. FORCES OF PRESSURE, AND FORCES OF 
 MOTION. 
 
 Pressures are, then, forces whose tendency to 
 produce motion in a body does not take effect; and 
 they are thus distinguished from those in which 
 this tendency does take effect, and which are FORCES 
 OF MOTION. 
 
 The laws which govern the operation of these 
 two great classes of forces are as different as are 
 their phenomena, and the circumstances under 
 which they act. Nevertheless there have been 
 shown to exist certain relations between them, so 
 that the phenomena of either may, in a degree, be 
 deduced from those of the other. 
 
 The general laws which govern the various re- 
 lations of forces of pressure will first be discussed 
 in this work ; and then those of forces of motion. 
 The former discussion belongs to a science, called 
 the science of STATICS, from a Greek word 
 (iorr;p), signifying to stand, or to be in a state of 
 rest; and the latter, to a science called that of 
 DYNAMICS, from a Greek word (cWape), implying 
 force coupled with motion. 
 
 128. THE RELATION BETWEEN THREE PRES- 
 SURES IN EQUILIBRIUM. THEPARALLELOGRAM 
 OF PRESSURES. 
 This fundamental principle of Statics will be 
 
 readily understood from the following experiment: 
 Let a board be made to float on the surface of 
 
 water, in a vessel filled to the brim. 
 
J24< ILLUSTRATIONS OF MECHANICS. 
 
 Let three strings, attached to different points, 
 P, Q, R, in the surface of this board, be made to 
 
 fig. 23. 
 
 pass over pulleys, the heights of which are so ad- 
 justed that the strings may just \\Q flat upon the 
 board : from these strings let different weights be 
 suspended, and let the positions of the pulleys be so 
 adjusted that the board may float free of the sides 
 of the vessel, and that each string may run freely 
 on its pulley. When the whole has, under these 
 circumstances, come to rest, the following remark- 
 able relations will be found to obtain, between the 
 directions of the strings and the magnitudes of the 
 weights attached to them : 
 
 1st. If the directions of the strings A P, B Q, C R, 
 be produced^ they will all meet in the same point, O. 
 
 2d. If in A P produced, O N be measured off, 
 containing as many inches as there are pounds 
 weight in the weight acting on the string AP, 
 and in BQ produced, OM be measured off, con- 
 
THE PARALLELOGRAM OF PRESSURES. 125 
 
 taining as many inches as there are pounds in the 
 weight attached to B Q, and if a parallelogram, 
 ONLM, be then drawn, having OM and ON for 
 two of its adjacent sides, then will the diagonal OL 
 of this parallelogram be exactly in the same straight 
 line with the third string, C R. 
 
 3dly. The number of inches in the diagonal OL 
 of this parallelogram will exactly equal the number of 
 pounds in the weight attached to this third string, 
 CR. 
 
 The weights have been supposed to be measured 
 in pounds^ and the distances in inches. Any other 
 unit of weight, an ounce or an hundred weight 
 might have been used, and instead of inches the 
 distances might have been measured in eighths, or 
 in tenths, or in any other fractions, of an inch, or 
 in feet or yards. The law which governs the equi- 
 librium of weights is manifestly that which governs 
 the equilibrium of any other pressures whatever; for 
 a weight may be taken equivalent to any pressure: 
 moreover, it will be found to be true for any weights 
 whatever. Thus, then, it appears that this law of 
 the parallelogram of pressures is true for any pres- 
 sures, and is a general law. It is usually proved by 
 theory, and is the foundation of the whole theory of 
 statics. 
 
 The board is floated, to neutralise its gravity; 
 which force would introduce other forces into the 
 system, and interfere with the equilibrium of the 
 three, were the board laid upon a table, or only 
 suspended between the pulleys. That the experi- 
 ment may completely succeed, it is necessary that 
 the pulleys should be of the best workmanship, and 
 
126 
 
 ILLUSTRATIONS OF MECHANICS. 
 
 of considerable size, that friction may, as much as 
 possible, be avoided. 
 
 129. THE EQUILIBRIUM OF ANY NUMBER OF PRES- 
 SURES IN THE SAME PLANE. THE PRINCIPLE 
 
 OF THE EQUALITY OF MOMENTS. 
 
 Let us now suppose, that instead of the three 
 pressures applied to the board in the last experi- 
 ment, there were any number, as shown in the 
 accompanying figure. 
 
 The pullies being adjusted as before, and the 
 board having come to rest, the following relation 
 will be found to obtain between the magnitudes 
 and the direction of the pressures : 
 
 If any point M be taken on the board, and per- 
 pendiculars Mm, Mm, &c. be drawn from M on the 
 
 1 2 
 
 directions of all the strings, or on those directions pro- 
 duced, and if the number of inches in the length of 
 each perpendicular be multiplied by the number of 
 
THE POLYGON OF PRESSURES. 127 
 
 pounds in the corresponding weight *, and this 
 product be called the moment of that weight ; then 
 the moments of all the weights being thus taken, 
 and it being observed that some of these weights 
 tend to turn the board in one direction about the 
 point M, and some in the opposite direction, it will be 
 found that the sum of the moments of all those which 
 thus tend to turn it one way, equals the sum of the 
 moments of all those which tend to turn it the other. 
 This principle is called, that of the EQUALITY OF 
 MOMENTS; it maybe deduced from the principle 
 of the parallelogram of forces ; and, like it, it is 
 perfectly general, applied to any number of pressures 
 in the same plane, and to pressures of any kind. 
 The units of weights and measurement have been 
 taken to be pounds and inches, they may be any 
 other units whatever. 
 
 130. THE POLYGON OF PRESSURES. 
 If in the last experiment any point, o, be taken on 
 the board, and if from that point there be drawn a 
 line, o a, parallel to the string P/>,and as many inches 
 
 in length as there are pounds in the weight attached 
 to that string ; if moreover, from the extremity a of 
 this line, a second, a 6, be drawn parallel to the string 
 P , and as many inches long as there are pounds 
 
 2 2 
 
 in the weight attached to that string ; and if from , 
 a third line be similarly drawn, parallel to Pp, and so 
 
 3 5 
 
 on, until a line is drawn parallel to the last string, 
 then will it be found that the line parallel to this 
 
 * That is the weight attached to the string, on which this 
 perpendicular falls. 
 
128 ILLUSTRATIONS OF MECHANICS. 
 
 last string will pass through the point o, from which 
 the first line was drawn, so as to complete a geome- 
 trical figure, called a polygon. Moreover, this last 
 line, c o, so terminating in o, will be found to contain 
 exactly as many inches as there are units (i. e. pounds 
 or ounces) in the last weight. This remarkable 
 principle, called that of the POLYGON OF PRESSURES, 
 and that, last described, of the EQUALITY OF MO- 
 MENTS, are necessary to the equilibrium of any 
 number of pressures acting in the same plane ; and 
 constitute all that is necessary to that equilibrium. 
 
 131. THE LEVER. 
 
 A lever is a rigid bar, moveable about a certain 
 fixed point, called its fulcrum, and acted upon by 
 the resistance of that point and by two other forces 
 applied to other points in it ; one of which it is the 
 use of the lever to overcome by the action of the 
 other. A lever, then, if we put its own weight out 
 of the consideration, is a body acted upon by three 
 forces in the same plane which three forces, when 
 it is on the point of moving, are in equilibrium ; the 
 principle of the equality of moments must then ob- 
 tain in respect to them. Take, then, in every 
 one of the cases represented in the accompanying 
 figures, the fulcrum C of the lever, for the point 
 from which the moments are measured ; the mo- 
 ment about this point of the resistance of the ful- 
 crum will in each case be nothing, because the 
 perpendicular from that point, upon the direction of 
 the resistance which goes through it, is, of neces- 
 sity, nothing : thus, the moment of one of the three 
 forces which act upon the lever being in each case 
 
THE LEVER. 129 
 
 nothing, and the principle of the equality of mo- 
 merits still obtaining, it must obtain in each case 
 in respect to the other two forces. 
 
 The moment about C of the one A, called the 
 power, is then in every 
 case equal to the mo- 
 ment of the other W, 
 called the weight. Thus 
 j$. in the case represented 
 in the first of the 
 accompanying figures, 
 where the power and 
 weight act at opposite 
 
 fig. 26. ^ extremities of the lever, 
 
 and the fulcrum is be- 
 tween them ; and where 
 the weight, (which is 72 
 Ibs.) acts at a distance 
 of one division (repre- 
 senting an inch a foot, 
 a yard, &c.) from C, 
 whilst the power acts at 
 
 Jig. 27. ^t a distance of eight such 
 
 w \\ ?o Divisions ; it follows, by 
 
 the principle of the 
 equality of moments, 
 that, when there is an 
 equilibrium, the first 
 pressure, multiplied by 1, must equal the other 
 multiplied by 8 ; these products being the moments of 
 the two forces. Thus the pressure of the hand P must 
 be such that the number of pounds in it, or equiva- 
 lent to it, being multiplied by 8, shall equal 72 
 
130 ILLUSTRATIONS OF MECHANICS. 
 
 multiplied by 1. Now that this may be the case, it 
 is evident that P must be a force equivalent to 9 
 pounds. In the second figure, the perpendicular 
 distance C A, of A, from the fulcrum is 9 divisions, 
 and that of W, 1 division ; A must then be a force 
 of such a number of pounds, that this number 
 multiplied by 9 shall equal the number of pounds 
 in W multiplied by 1 ; or, W being 72 Ibs., it must 
 equal 72: that this equality may obtain, A must 
 evidently be a force of 8 Ibs. 
 
 In the third figure, the distance of A from the 
 fulcrum is 1 division, and that of W is 8 divisions ; 
 A must then be such that, multiplied by 1, the pro- 
 duct shall equal 72 multiplied by 8 ; an equality to 
 make up which, A must equal no less than 576 Ibs. 
 
 In the two first figures, the power A is nearer to 
 the fulcrum than the weight W to be raised by it ; 
 and, for this reason, a power less than the weight 
 yet has an equal momentum, and makes up the 
 equilibrium. 
 
 In the third figure, the power is nearer to the 
 fulcrum than the weight; to have a momentum 
 equal to that of the weight, it must, therefore, be 
 greater in amount than it. 
 
 In the two first cases, the power is said to act by 
 the intervention of the lever, at a mechanical ad- 
 vantage ; in the last case, at a mechanical disad- 
 vantage. 
 
 Levers, such as those represented in the first 
 figure, in which the weight is on the opposite side 
 of the fulcrum from the power, are said to be of 
 the first class. 
 
 Levers, similar to those in the second figure, in 
 
THE LEVER. 
 
 131 
 
 which the weight is between the fulcrum and the 
 power, are of the second class. 
 
 Levers, like those in the third figure, where the 
 power is applied between the weight and the 
 fulcrum, are of the third class. 
 
 The relation above described manifestly obtains 
 whether the lever be straight, as sUown in the 
 figure, or of any crooked form whatever. 
 
 Of the first class of levers are the hand-spike, the 
 pump handle, the hammer when used to prise up 
 a nail, scissors, shears, nippers, a common poker 
 when used to raise the coals, &c. &c. 
 
 Of the second class, which support the weight 
 between the fulcrum and the power, are the crow- 
 bar, the wheel -barrow, nut- crackers, &c. 
 
 To the third class belong the treddle of a lathe, a 
 pair of tongs, shears such as those used for shear- 
 
 ing sheep, &c. The limbs of locomotion and pre- 
 hension of all animals, are, moreover, levers of this 
 class : the power applied to them is by means of 
 tendons whose direction is near the joint, which is 
 K 2 
 
132 ILLUSTRATIONS OF MECHANICS. 
 
 the fulcrum of each ; and the weight raised at a 
 distance from it, whether it be only the weight of 
 the limb, or something in addition to that weight, 
 which it moves. Thus, applied near the fulcrum of 
 the limb, the muscular force required is enormously 
 great. 
 
 132. COULD ARCHIMEDES HAVE LIFTED THE 
 WORLD WITH A LEVER IF HE HAD HAD A 
 FULCRUM TO REST IT UPON ? 
 
 In reality Archimedes would have had no difficulty 
 In moving the world could he have brought his 
 lever to bear upon it. It rests upon nothing, is sus- 
 pended by nothing, rubs against nothing, and floats 
 in space without being buoyed up. It is perfectly 
 free to move in any direction ; no force would op- 
 pose itself to any attempt which Archimedes might 
 make to move it either upwards or downwards ; 
 the only forces which act upon it its centri- 
 fugal force and that which attracts it to the sun 
 being exactly balanced, and, as it were, neutralised. 
 So that, in point of fact, to move the earth, the me- 
 chanical advantage of a lever is a superfluous thing; 
 it would yield to any, the slightest force, impressed 
 upon it, and Archimedes had only to stamp his foot 
 and the thing was done. These were not, however, 
 the ideas entertained by Archimedes on the subject. 
 His conception of the matter evidently was, that the 
 huge mass of the earth rested upon some other mass 
 based in the infinities of space, towards which other 
 mass it gravitated as does a stone or a rock to the 
 mass of the earth ; and the question which presented 
 
THE LEVER. 133 
 
 itself to his mind was, what, on this supposition, would 
 supply a sufficient force to lift up and overthrow it. 
 This sufficient force he found in his lever, his own 
 arm moving it. " Give me," said he, " a place 
 where I may stand, and I will move the world."* 
 The principle on which his conclusion was founded 
 was undeniable ; the calculation was perfectly cor- 
 rect; but one element was probably omitted from it, 
 it was the time requisite to give so huge a mass any 
 appreciable motion by means of a lever, which 
 should move it with so small a force as that which 
 the arm of Archimedes could supply. 
 
 Taking the diameter of the earth at 7,930 miles, 
 the number of cubic feet in it may be calculated to 
 be 38,434,476,263,828,705,280,000 : and assuming 
 each cubic foot to weigh 300 pounds, which has 
 been assumed as a probable amount t? we shall 
 have for the weight of the earth, in pounds, the 
 number 11,530,342,879,14.8,611,584,000,000. Now, 
 supposing Archimedes to act at the end of his lever 
 with a force of 30 pounds, one arm of it must be 
 384,344,762,638,287,052,800,000 times longer than 
 the other, that he may move this mass with it. And, 
 one arm of the lever being this number of times 
 longer than the other, when it was made to turn 
 round its fujcrum, the end of that longer arm must 
 move exactly this number of times faster, or farther, 
 than the end of the other: so that, whilst the end of the 
 shorter arm was moving one inch, the end of the longer 
 arm must move 384,344,762,638,287,052,800,000 
 
 * Aof /mot Trou yroa xW TOV xoyfAov Kkwicroo. 
 f Hutton's "Mathematical Recreations," vol. ii. p. 14. 
 K 3 
 
 1.4 
 
134 ILLUSTRATIONS OF MECHANICS. 
 
 inches ; and conversely, when Archimedes had made 
 the end of the lever to which he applied his arm 
 move this immense number of inches, he would only 
 have prised up the earth, to which the other end 
 was applied, one inch. 
 
 Now, a man pulling with a force of 30 pounds, 
 and moving the object which he pulls at the rate of 
 10,000 feet an hour, can work continually for 
 from eight to ten hours a day, and this is all 
 that he can accomplish. Each day, then, Archi- 
 medes could, at the utmost, move his end of his 
 lever 100,000 feet, or 1,200,000 inches; and hence 
 it may thus readily be calculated, that to move it 
 384,344,762,638,287,052,800,000 inches, or to move 
 the other end that is, the earth one inch, would 
 require the continual labour of Archimedes for 
 8,774,994,580,737 CENTURIES. 
 
 133. Two PERSONS CARRY A BURDEN BETWEEN 
 
 THEM BY MEANS OF A LEVER OR POLE, TO FIND 
 HOW MUCH OF THE WEIGHT IS BORNE BY EACH. 
 
 Three -forces are in equilibrium on such a pole: 
 the burden borne, and the two forces which bear it. 
 Therefore, by the principle of the equality of mo- 
 ments, if we take any point in it, and take the mo- 
 ments of these forces, severally, about *that point, 
 the sum of the moments of those which tend to turn 
 it one way about it, must equal the sum of those 
 which tend to turn it the other. Take either extre- 
 mity for the point. The moment of the force at 
 that extremity will then be nothing, since the per- 
 pendicular upon it will be nothing. The moments 
 
THE LEVER. 185 
 
 of the two other forces must therefore be equal. 
 Thus, if C be the burden, and A the force with 
 
 fig. 29. 
 
 which the pole is supported at the extremity A, 
 then A multiplied by AB must equal C multiplied 
 by CB; and the value of A being found so as to 
 make up this equality, will be the true force ex- 
 erted at A. In the same manner the force at B 
 may be found. 
 
 134-. METHOD OF COMBINING THE EFFORTS OF A 
 GREAT NUMBER OF MEN TO CARRY A BURDEN. 
 
 The following method is said * to have been used 
 in Constantinople for raising and carrying the 
 heaviest burdens, such as cannons, mortars, and 
 enormous stones ; and the rapidity with which they 
 were thus transported from one place to another is 
 stated to have been truly surprising. 
 
 A B is a bar of sufficient strength to sustain the 
 
 whole weight of the load P, which is attached to its 
 middle point ; C D and E F are cross bars fixed to 
 this, near its extremities ; and to the extremities of 
 
 * Hutton's " Mathematical Recreations," vol. ii. p 8. 
 K 4 
 
136 ILLUSTRATIONS OF iMECHANICS. 
 
 these cross bars are affixed others, a b, cd, ef, yh; 
 to these last again, in like manner, others, whose 
 extremities are borne upon the shoulders of the 
 men who are to carry the load. Supposing one 
 man's shoulder to support each extremity of these 
 last mentioned bars, the whole number, whose effort 
 will be combined to lift and carry the weight, will 
 be 16. If other cross bars had been fixed in like 
 manner to the extremities of these, the united 
 effort of 32 men might have been applied. If other 
 cross bars had been fixed yet again to the extre- 
 mities of these, 64 men might have united their 
 strength to the task ; and so on for any number. 
 If the bar AB, which supports the weight, carry it 
 suspended accurately from its middle point; and if 
 the point where each cross bar is fixed to the one 
 preceding it in the series be exactly half way be- 
 tween the points where the two which follow it are 
 affixed to it then the weight will be equally dis- 
 tributed between all the bearers. A small deviation 
 from this rule will produce great inequality in the 
 distribution, and it would be easy to adapt this 
 distribution, according to the principles explained 
 in the last article, so that each should have any 
 given share of the load. The inconvenience of the 
 method is the increase of the load by the weights of 
 the additional cross pieces. 
 
 135. THE WHEEL AND AXLE. 
 
 If we imagine a wheel moveable about a fixed 
 axis O, and having cut in it two circular grooves 
 A and B, whose centres are in O, and if we con- 
 
THE WHEEL AND AXLE. 137 
 
 ceive strings, AW and B P, to be wound round 
 these grooves, carrying at their extremities,? and W, 
 weights which just balance one another, then shall 
 we have a system of three forces acting in the 
 same plane and in equilibrium, and therefore subject 
 to the law of the equality of moments. 
 
 These three forces are the 
 weights P and W acting in 
 tne directions B P and A W, 
 ! and the resistance of the fixed 
 axis O. Now let us take O 
 for the point from which we 
 measure the moments; the 
 moment of one of the three 
 r J -.. forces the resistance of the 
 axis will then vanish; for 
 the perpendicular from O 
 r- > , upon this resistance, which 
 
 acts through O, is manifestly 
 
 nothing, and therefore the product of the resistance 
 by this perpendicular is nothing. The only mo- 
 merits which remain are those of P and W. These, 
 therefore, by the principle of the equality of mo- 
 ments, are equal. 
 
 Now the perpendicular from O, upon the di- 
 rection of W, is O A, and that upon the direction 
 of P is O B. The number of Ibs. or cwts. in W 
 multiplied by the number of inches in O A, being 
 its moment, is then equal to the number of Ibs. or 
 cwts. in P multiplied by the number of inches in 
 O P, being its moment ; and this is the relation 
 which must exist between P and W, so that they 
 may be in equilibrium. If, for instance, O A were 
 
138 ILLUSTRATIONS OF MECHANICS. 
 
 3 inches, and O B 1 1 inches, and if W were 132 Ibs., 
 then would the moment of W be 3 times 132 or 
 396; and that P must be such that, being at 11 
 inches distance, it may have the same moment, 
 396, that W has. P must therefore be 36 Ibs., be- 
 cause 11 times 36 is 396. 
 
 By diminishing the distance O A at which the 
 weight W is applied, in comparison with the distance 
 O B at which the power P is applied, we may by 
 this contrivance balance ever so great a weight by 
 ever so small a power. 
 
 In the actual use of the wheel and axle, it is 
 customary not to apply the weight to be raised in 
 the plane of the same circle to which the power is 
 applied, but to widen the circular groove A into a 
 cylinder of considerable length, as shown by the 
 dotted lines in the figure, and to cause the string 
 which carries W to wind round this cylinder. It 
 is evident that, applied any where to the circum- 
 ference of this cylinder, which is supposed to be 
 solid and rigid, it will produce exactly the same 
 effect as though it were applied at A, and will have 
 the same relation to the power P as though it were 
 applied there. 
 
 Moreover, it is customary in the use of this in- 
 strument, not to apply the power P as shown in 
 the figure, by means of a circle and a cord winding 
 round it. 
 
 The power is usually the effort; of a workman, 
 and is applied by means of an arm fixed to the 
 cylinder, and carrying at its extremity a handle. 
 The instrument then becomes the WINDLASS. The 
 power applied to it in this case by the workman is 
 
THE WHEEL AND AXLE. 139 
 
 not the same througliout each revolution. The 
 direction in which he pulls or pushes the handle 
 varies continually, and the perpendicular upon this 
 direction from O varies therefore continually ; so 
 that, unless the force which he exerts continually 
 vary in amount, its moment cannot remain the same, 
 so as to equal to or a little exceed the moment of 
 the weight which does always remain the same. 
 Of this necessary variation of his effort dependent 
 upon the direction in which it is made, the workman 
 is perfectly conscious. 
 
 If the cylinder is placed vertically, instead of 
 horizontally, and the force is applied by means of a 
 number of bars fixed horizontally, like radii, in its 
 upper extremity, which are pushed forwards by the 
 workmen, the instrument becomes the CAPSTAIN, 
 whose principal use is to elevate the heavy anchors 
 on ship-board.* 
 
 136. MODIFICATION OF THE WHEEL AND AXLE, 
 BY WHICH ANY WEIGHT CAN BE RAISED BY A 
 GIVEN POWER. 
 
 A limit is in practice fixed to the weight which a 
 given power will raise on the common wheel and 
 axle by the insufficiency of the cylinder, when its 
 radius O A is diminished beyond a certain limit to 
 bear the weight suspended from it. It is in di- 
 minishing this radius O A of the cylinder, or in- 
 creasing the distance O B at which the power is 
 
 * See " Mechanics applied to Arts," page 84. 
 
140 
 
 ILLUSTRATIONS OF MECHANICS. 
 
 Jig. 32. 
 
 applied, that we increase the weight which a given 
 power will raise; and both these methods of in- 
 creasing it become, beyond a certain limit, imprac- 
 ticable. There is another form of the wheel and 
 
 axle, of admirable in- 
 genuity, which com- 
 pletely removes the dif- 
 culty. Instead of one 
 cylinder, two of different 
 diameters, as shown by 
 the dotted lines on the 
 figure, are fixed toge- 
 ther,and moveable upon 
 the same axle. The 
 two ends of the cord 
 which support the 
 weight,are wound in op- 
 posite directions round 
 these cylinders, and this 
 cord passes round a 
 
 moveable pulley which carries the weight suspended 
 from it. It is evident that, thus applied to the cy- 
 linder, the equal tensions of the two strings which 
 support the weight will produce the same effect as 
 though they were applied in the same plane as the 
 power at B and C. Suppose them to be applied 
 there ; this plane will then be acted upon by four 
 forces, the power P. at A, the equal tensions of the 
 two strings which carry the weight at B and C, and 
 the resistance of the axis at O. Now these are 
 forces in equilibrium ; the principle of the equality 
 of moments obtains then in respect to them ; and 
 taking O for the point from which the moments are 
 
THE WHEEL AND AXLE. 141 
 
 measured, since the moment of the resistance of the 
 axis vanishes there, it follows that the moments of 
 P, and the tension at C, which tend to turn the 
 system one way about O, are, together, equal to the 
 moment of the tension at B. which tends to turn it 
 the other way. The moment of P then must be 
 such that, being added to the moment of C, it shall 
 make up a sum equal to the moment of B, or, in 
 other words, the moment of P must equal the dif- 
 ference of the moments of B and C ; and the less 
 the difference of the moments of B and C, the less 
 need P be, to produce this equality and balance the 
 system. Now the actual forces at B and C are equal, 
 for the strings B D and CD support the weight 
 equally ; the difference of their moments depends 
 then entirely upon the difference of their distances 
 O B and O C from O, or upon the difference of the 
 radii of the circles to which they are applied, or 
 upon the difference of the diameters of the two cy- 
 linders ; the less the difference of these diameters, 
 the less the power required to maintain the weight 
 in equilibrium, and to move it. Thus, by making 
 the two cylinders more nearly of the same diameter, 
 we can diminish the power necessary to raise any 
 given weight, or increase the weight which any 
 given power will raise, without limit. 
 
H2 ILLUSTRATIONS OF MECHANICS. 
 
 137. WHEN ANY NUMBER OF PRESSURES ACTING 
 ON A BODY, IN THE SAME PLANE, ARE NOT IN 
 EQUILIBRIUM, TO APPLY TO IT ANOTHER WHICH 
 SHALL PRODUCE AN EQUILIBRIUM. 
 
 If we know all the pressures in a system of pres- 
 sures in equilibrium, excepting one, we can, from 
 the principles of the equality of moments and the 
 polygon of pressures (see articles 129. and 130.), 
 determine what that one must be ; for that one 
 must be applied at such a distance, and of such 
 a magnitude, as to make up the deficiency in 
 the equality of moments, and in such a direction 
 that it may complete the polygon of pressures. 
 Thus, then, to find a pressure which will cause any 
 number of other pressures to be in equilibrium, we 
 have only to take any point, and tfience estimate 
 the moments of all the other forces, and find how 
 much is necessary to make up the equality of their 
 sums, as explained in article 129. We shall thus 
 know what must be the moment of the required 
 force. Drawing then the polygon of pressures, as 
 was also described, this polygon will be complete, 
 all but one side, and we shall know the magnitude 
 and direction of that side. The number of inches 
 in its magnitude will tell us the number of pounds 
 in the required force. Also, we before have found 
 its moment, and we now know its amount; we can, 
 therefore, tell what must be its perpendicular 
 distance from the point we have assumed. More- 
 over, its direction is parallel to the last side of the 
 polygon. These two facts guide us to the exact 
 
THE RESULTANT. 14-3 
 
 position where it must be applied, so that thus it is 
 fully determined. 
 
 188. THE RESULTANT OF ANY NUMBER OF 
 PRESSURES. 
 
 The resultant of any number of forces acting 
 upon a body is that force which would singly pro- 
 duce the same effect, as to the equilibrium or motion 
 of the body, that they do conjointly. Now let us 
 imagine any number of forces to be in equilibrium, 
 and of these let us take all except one particular 
 force, and let us consider what is their resultant. 
 It is that force which would produce the same 
 effect singly that they do conjointly. But what 
 effect do they produce ? They just balance the one 
 remaining force : this is their effect. Any force, 
 therefore, which would balance the one remaining 
 force would produce the same effect that they do. 
 But a force exactly opposite to that one remaining 
 force, and equal to it, would balance it. That force 
 is then the resultant we want. 
 
 And to find the resultant of any number of forces, 
 we must first find a single force which will produce 
 an equilibrium with them. Having found this, \ve 
 know the resultant ; for it is equal and opposite to 
 this force. 
 
 Thus, for instance, if two forces act upon & point 
 in directions inclined to one another, and we ish 
 to find their resultant, we must, in the first place, 
 find the third force which will produce an equi- 
 librium with these two. This we may do at once 
 
144 ILLUSTRATIONS OF MECHANICS. 
 
 by the parallelogram of forces. The third force 
 in question is the diagonal of that parallelogram. 
 The resultant required is equal and opposite to that 
 third force. And so, in every other case, to find the 
 resultant force of any number of forces, we must 
 examine what these want of the conditions which 
 make up an equilibrium, and then find a force which 
 would just make up these conditions. A force equal 
 and opposite to this will be the resultant. 
 
 139. THE COMPOSITION AND RESOLUTION OF 
 PRESSURES. 
 
 The forces of which any other is the resultant 
 are called the components of that resultant. Since 
 the resultant force produces the same effect singly 
 that all its components do conjointly, we shall not 
 at all affect the conditions of the equilibrium of a 
 body acted on by any forces, if we conceive cer- 
 tain of its forces to be taken away, and their re- 
 sultant put in their place: if it was in equilibrium 
 before, it will be in equilibrium now, and under pre- 
 cisely the same circumstances. 
 
 This putting of a single resultant in the place of 
 any number of component forces is called com- 
 pounding them. The process is that of the COMPO- 
 SITION OF FORCES. Conversely, we may find a num- 
 ber or group of forces which shall be such as, if we 
 found their resultant, would have for it a particular 
 given force. These forces would then, conjointly, 
 produce the same effect which that does singly. 
 This group of forces might then be substituted for 
 
71IE CENTRE OF GRAVITY. 145 
 
 that ONE without affecting the conditions of the 
 equilibrium. 
 
 The process of thus substituting an equivalent 
 set of forces for a single one, is called that of the 
 RESOLUTION OF FORCES ; the single force being 
 said to be resolved into the others. 
 
 14-0. THE CENTRE OF GRAVITY. 
 
 Of all forces, that whose operation we are 
 most conversant with is GRAVITY: it operates, 
 under various modifications, in every thing around 
 us, and in every part of that thing. Every 
 material substance is thus acted upon by as many 
 separate pressures of gravity as we may imagine it 
 divided into parts. We can lay hold of nothing 
 which is not a body acted upon by a system of 
 gravitating forces infinite in number. Of these, 
 this is the characteristic property, that their direc- 
 tions all tend accurately to one point the centre 
 of the earth which is so distant that, although 
 they thus in reality meet when continually pro- 
 duced, yet are they, as to all practical consider- 
 ations, parallel, by reason of the great distance 
 (nearly 4000 miles) of the point in which they meet. 
 
 These forces of gravity, thus infinite in number, 
 acting upon the different points of any body which 
 we take up, have always a resultant; that is, a 
 single force may always be found acting in a 
 certain direction, which shall singly produce the 
 same effect that they do conjointly. This force is 
 equal and opposite to the single force which would 
 produce an equilibrium in the body on which they 
 act, that is, which would support it. 
 
 L 
 
146 ILLUSTRATIONS OF MECHANICS. 
 
 As we turn a body about, the direction through 
 it of the forces of gravity which act upon it will be 
 continually changed ; at one time they wift traverse 
 it lengthwise, at another they will traverse it across, 
 at another diagonally; in short, every new position 
 will cause them to traverse it in a new direction ; 
 and by turning it completely round, we shall 
 cause them to traverse it in an infinity of different 
 directions. In each position they will have a re- 
 sultant. Thus they will have an infinity of different 
 resultants ; and their resultants will traverse the 
 body, as it is turned round, in an infinity of different 
 directions. 
 
 Now there is this remarkable relation (easily 
 determined by geometry) between the directions of 
 these different resultants through the body, that 
 they all pass througfi the same point in it : that 
 point is called the CENTRE OF GRAVITY. This is, 
 I say, a remarkable relation ; it might have been 
 otherwise. Under other laws of force, and other con- 
 ditions of equilibrium, dependent upon these, the 
 properties of that point would have had no ex- 
 istence. 
 
 It is difficult nay, it is impossible to conceive 
 the amount of change, the confusion of all the great 
 elements of nature, which this one simple circum- 
 stance would have been sufficient to introduce. 
 
 Take away this one property of matter, which 
 determines in every mass a single point through 
 which the resultant of the gravitations of its parts, 
 in all its proportions, passes, and the fabric of the 
 universe would reel from its very foundations ; the 
 order arid uniformity of the vast machine would 
 
THE CENTRE OF GRAVITY. 14-7 
 
 cease; the cycles which bring back its mighty mo- 
 tions in their appointed seasons would be broken; 
 and, to bear its part in the universal wreck, each 
 organised and existing thing on the earth's surface 
 would have the stability of the form under which 
 it exists converted into one of the greatest con- 
 ceivable instability. An upright position of the 
 human body would be impossible ; no vehicle could 
 move without being overthrown; and four-footed 
 animals, when they sought to walk, would but 
 totter on, from one fall to another. 
 
 The centre of gravity of a body is then that 
 point through which the resultant of the gravities 
 of its parts passes, in every position in which we 
 turn the body. This resultant, producing the same 
 effect as do the gravities of the parts, evidently acts 
 in a vertical direction ; for the effect of the gra- 
 vities of the parts is in a vertical direction. The 
 resultant is evidently equal in amount to the weight 
 of the body ; for, by the definition of a resultant, 
 it is equal to the single force which would support 
 the body. Thus, then, we shall, in reality, conceive 
 this resultant to act alone, through the centre of 
 gravity, and in its proper vertical direction, if we 
 conceive all the gravity or weight to be extracted, 
 by some chemical process, from the different parts 
 of the body among which it is diffused, and collected 
 and condensed into this one single point its centre 
 of gravity ; and were it possible to make this change, 
 all the conditions of the equilibrium of the body, bo 
 fr as they are affected by its weight, would reinam 
 unaltered. 
 
14-8 ILLUSTRATIONS OF MECHANICS. 
 
 141. TO DETERMINE THE CENTRE OF GRAVITY 
 OF A BODY BY EXPERIMENT. 
 
 It is evidently through its centre of gravity that 
 any force which is intended to support a body must 
 be made to pass ; and, conversely, any sufficient 
 single force which is made to act through the centre 
 of gravity vertically, or in a direction opposite to 
 the weight of the body, would support it. To Jpe 
 sufficient^ this single force must, as has before been 
 shown, equal the weight of the body. Thus, if the 
 centre of gravity of a body of any shape, however 
 irregular, were found; and if the resistance of the 
 finest point that can be conceived, that of a needle, 
 for instance, were applied, so that its direction 
 'should be vertical and accurately through the centre 
 of gravity of the body, it would support it. 
 
 It would perhaps be impossible practically thus 
 to cause the resistance of a point accurately to pass 
 through a body's centre of gravity. If, however, a 
 body be suspended by a single point from a string^ 
 it will of itself fall into such a position, that the 
 direction of the tension of the string on that point 
 shall be through the centre of gravity ; and having 
 assumed that position, it will be supported. This, 
 in fact, furnishes us with a very easy practical way of 
 determining the position of a body's centre of gra- 
 vity. We have only to suspend it by a string from 
 any point in its surface, and, waiting until it rests, to 
 mark, by some means, what would be the direction 
 of the line of the string through the body, if it 
 were produced ; then, hanging it from some other 
 
THE ATTITUDES OF ANIMALS. 149 
 
 point in the body's surface, to observe in like 
 manner the line of the string's direction through 
 the body, when suspended from that point. Both 
 these lines pass (by what has before been said) 
 through the centre of gravity. But the only point 
 through which they both pass is that in which they 
 intersect. Their intersection is therefore the body's 
 centre of gravity. 
 
 14-2. THE ATTITUDES OF ANIMALS. 
 
 When a body afters its form it changes^ the same 
 time, the position of its centre of gravity. The 
 accompanying diagram presents an illustration of 
 this fact in the attitudes of a bird. The line drawn 
 
 jig. 33. 
 
 from R directs the eye to the position of the centre 
 of gravity when the bird is standing, being then 
 immediately above his foot. When he swims, the 
 only alteration in his position is the elevation of 
 his legs, accompanied by a corresponding elevation 
 L 3 
 
150 ILLUSTRATIONS OF MECHANICS. 
 
 of his centre of gravity, whose position is now 
 shown by the line from N. When he walks, his 
 head is thrown a little forwards, and his legs 
 are alternately raised, but not so much as in 
 swimming : a more forward and a somewhat lower 
 position must therefore be assigned to his centre of 
 gravity, pointed to by the line from M. When he 
 flies, his neck is thrown forwards and depressed ; 
 his centre of gravity therefore advances and sinks, 
 as shown by the line from V. 
 
 No heavy body can evidently be supported upon 
 a surface on which it is placed, unless the vertical 
 resistance of some point with which it is in contact 
 passes through its centre of gravity. 
 
 14-3. THE BEST POSITION OF THE FEET IN 
 
 STANDING. 
 
 The human body has a different position of its 
 centre of gravity, corresponding to each different 
 attitude. All are, however, subject to this condition, 
 that the centre of gravity shall remain vertically over 
 some point or another in the base of the feet.* This 
 base of the feet, or pedestal of the body, has for its 
 boundaries, to the right and left, the outer edges of 
 the soles of the feet ; and before and behind, lines 
 joining the toes and the heels. In the accompany- 
 ing diagram it is represented by the trapezium 
 A B C D. The attitudes of the body may evidently 
 be varied, so as not to destroy the equilibrium, with 
 
 * For a variety of illustrations of this subject, the reader is 
 referred to the author's treatise on " Mechanics applied to the 
 Arts," p. 33, &c. 
 
STANDING. 151 
 
 the greatest facility and with the fewest precautions, 
 when this base of the feet is the largest. Thus the 
 securest position of the feet in standing is that which 
 causes the pedestal to cover the greatest surface, or 
 the figure A B C D, as shown in the diagram to have 
 the greatest area. Supposing the 
 c P heels to be placed in a given posi- 
 
 tion, and the feet turned round 
 upon them, this greatest area will 
 be found not in a parallel position 
 of the feet, but in an inclined 
 position, like that shown in the 
 figure. 
 
 Thus we see a sufficient reason for the military 
 custom of causing soldiers on drill to stand with 
 their toes turned out. 
 
 If the outside points A and B of the heels could 
 be brought accurately to coincide with one another, 
 then, when the heels thus touched, it is found (by 
 a mathematical discussion of the subject) that this 
 greatest base would be obtained when the feet were 
 turned each half-way round, or when they made with 
 one another a right angle. As these points can never, 
 nowever, coincide, but must always be distant by at 
 least double the width of the heel, it is certain that 
 the feet never should be turned apart so far as a 
 right angle. 
 
 If the distance A B of the outer edges of the 
 heels exactly equals the length A D of the foot, the 
 inclination of the feet to one another should equal 
 sixty degrees. Or, imagining the lines D A and C B 
 to be produced so as to meet in E, their inclination 
 L 4 
 
152 ILLUSTRATIONS OF MECHANICS. 
 
 should be such as to make the triangle C E D an 
 equilateral triangle. 
 
 144. THE SHEPHERDS OF THE LANDES. 
 
 The vast plains of the Landes, in the south-west 
 of France, are covered with a loose sandy soil, and 
 overgrown with thick furze ; moreover, during some 
 months of the year, they are in many places flooded. 
 This wild region, nevertheless, yields pasture to sheep. 
 To traverse it in search of their flocks, the shepherds 
 have, from time immemorial, adopted the singular 
 custom of mounting fhemselves on high stilts. They 
 are said on these to travel over the loose sand as 
 through the water, with steps of eight or ten feet in 
 length, and with the speed at which a horse trots. 
 This is a remarkable instance of the power which 
 the body possesses of varying its attitude so as to 
 fix the position of its centre of gravity over the base 
 which supports it, even when that base is, as here, 
 of greatly less dimensions than the natural base of 
 the foot, and the body elevated upon it greatly above 
 its natural position. 
 
 145. To CAUSE A CYLINDER TO ROLL, BY ITS 
 WEIGHT, A SHORT DISTANCE UP AN INCLINED 
 PLANE. 
 
 If the vertical from the centre of gravity of a 
 body do not pass through the base on which it rests, 
 but have a direction on either side, then the body 
 will turn over towards that side. The accompany- 
 ing figure is intended to represent a cylinder placed 
 upon an inclined plane. If this cylinder were not 
 
WHEELER'S CLOCK 153 
 
 loaded on one side, its centre of gravity would be in 
 its axis A ; and the vertical A L, from its centre of 
 
 Jig.35. 
 
 gravity, would evidently fall below the point B, 
 where it rests upon the plane ; so that, when left to 
 itself, it would roll downwards. But by loading it 
 near its surface at F (by pouring lead in a groove 
 parallel to its axis), the position of its centre of 
 gravity G may be moved, so that the vertical G K 
 from it shall not be below, but above the point of 
 support B It will then roll for a short time up the 
 inclined plane instead of down it, until, by the 
 descent of F, the line G K is made to pass through 
 the point of contact B of the cylinder with the 
 plane. It will then rest. 
 
 * 146. WHEELER'S CLOCK. 
 
 An ingenious person of the name of Wheeler, 
 some years ago, conceived the idea of constructing 
 a clock to be contained in a cylinder ; the prin- 
 ciple of whose motion should be, the tendency of 
 the cylinder, when placed upon an inclined plane, to 
 roll down it. 
 
 Let the cylinder represented in the last article 
 be imagined to be hollow ; and the weight F 
 moveable in it round the axis A, by means of an 
 arm A G F, to the extremity of which it is fixed : 
 
154} ILLUSTRATIONS OF MECHANICS. 
 
 imagine that with this arm is connected a train 
 of wheels similar to those of a watch, terminat- 
 ing in a scapement and balance-wheel, and giving 
 motion to a hand moveable on the end of the 
 axis, and showing hours on the extremity of the 
 cylinder, which has its circumference divided 
 like the face of a clock. The arm A G F being 
 turned, motion is given to all this train of wheels, 
 which motion is checked and regulated by the 
 balance-wheel. But how is the arm to be turned ? 
 Thus : conceive the cylinder to be placed upon an 
 inclined plane ; it will seek for itself the position in 
 which the vertical G K, from its centre of gravity, 
 passes through B ; in this position F will not coin- 
 cide with B, being balanced about that point by the 
 weight of the cylinder itself, and its wheels, which 
 are so contrived that their common centre of gra- 
 vity shall be in the axis A : the position of the equi- 
 librium of the cylinder is then in an inclined posi- 
 tion of the arm A F. Now, the axis being supported, 
 and the arm A F inclined, it is evident that the 
 weight F at its extremity tends to turn the arm 
 about A ; and being unopposed, except by the 
 friction of the train of wheels connected with it, 
 may readily have its size so adjusted that it shall, 
 under these circumstances, turn the arm, and give 
 motion to the wheels : but as the arm thus turns, 
 the weight F descends, and the vertical G K, which 
 before passed through the point of support B, now 
 falls below it : the cylinder will now, therefore , roll 
 down the plane. Now, as it rolls down the plane, it 
 elevates the weight F again, so as to place the arm AF 
 in the same inclined position as before, and give it the 
 
A BODY ROLLING UPWARDS. 155 
 
 same leverage precisely, to turn the wheels : thus the 
 descent is continued, and the same power is con- 
 tinually supplied to give motion to the works of 
 the clock, whilst the scapement and balance-wheel 
 give a yet further uniformity to the motion which, 
 the proper adjustments being made, may be re- 
 gulated to keep any required time. The motion of 
 the clock will only be stopped when it has rolled 
 completely down the plane : that it may be begun 
 again, the cylinder must be placed again at the top 
 of the plane. 
 
 * 147. To CAUSE A BODY, BY ITS OWN GRAVITY, 
 
 TO ROLL CONTINUALLY UPWARDS. 
 
 Let a double cone, such as that shown in the 
 figure, be made of wood ; and let there be formed 
 
 fig. 36. 
 
 two inclined planes of boards of wood *, which, 
 meeting at their bases, afterwards diverge from one 
 another at an angle, as shown in the figure. If 
 the double cone be placed between these planes, 
 so as to rest equally upon each of them at the bot- 
 tom, it will immediately put itself in motion and 
 roll up them. 
 
 This apparent paradox is easily explained. The 
 centre of gravity of the double cone is in the 
 middle of the line joining its two extremities. Now., 
 when it is placed between the two inclined planes, 
 
 * Pieces of string will answer the purpose. 
 
156 ILLUSTRATIONS OF MECHANICS. 
 
 the points on which it rests are, of necessity, nearer 
 to the lowest point of the planes than this line is : 
 the vertical from the centre of gravity is, therefore, 
 on that side of the points of support which is to- 
 wards the highest points of the inclined planes ; it 
 is in that direction, therefore, that the body has a 
 tendency to roll. It does not, however, in reality 
 ascend, although it appears to do so : the points on 
 which it is supported, continually approach its ex- 
 tremities ; so that, although by reason of the inclina- 
 tion of the planes, the points of support ascend, yet, 
 for the above-mentioned reason, the thicker part of 
 the mass between them, descends; and it is necessary 
 to the success of the experiment, that, for this last 
 reason, the centre of gravity of the mass should de- 
 scend more than for the other it ascends. That this 
 may be the case, the inclination of each plane must 
 be such, that a distance equal to the length of the 
 double cone being measured along it, its corre- 
 sponding height shall be somewhat less than half 
 the diameter of the double cone in the middle. 
 
 * 148. STABLE AND UNSTABLE EQUILIBRIUM. 
 
 When a body, being slightly moved out of any 
 position in which it rests upon another body, tends 
 to return to it ; and being left to itself, will roll 
 back of its own accord into it that position is said 
 to be one of STABLE EQUILIBRIUM : when the 
 body will not thus retufn to its previous position, 
 that position is said to be one of UNSTABLE EQUI- 
 LIBRIUM. 
 
 Since the whole of the weight of a body may 
 be conceived to be collected in its centre of gravity, 
 
STABILITY. 157 
 
 without affecting the conditions of its equilibrium, 
 it is evident that if it be supported by the resistance 
 of a single point, that single point must be either 
 immediately above, or immediately beneath, or 
 actually in, its centre of gravity ; and if it be sup- 
 ported, not upon a point, but upon an extended 
 surface or base, or beneath such a surface, then 
 must the centre of gravity be either directly above 
 some point in that base or surface, or directly beneath 
 some such point. If the position of a body, which 
 thus rests, be so changed, that its point, or surface of 
 support, shall no longer lie vertically above, or ver- 
 tically beneath, its centre of gravity then, no verti- 
 cal supporting force acting upwards, through its 
 centre of gravity, and the whole weight or gravity 
 acting downwards through it (or, rather, acting as 
 though it so acted), it is manifest that the centre 
 of gravity will have a tendency to descend, and that, 
 if the body be left to itself, its centre of gravity 
 will descend. It is possible that, moving a body 
 from its position of equilibrium, we may, at the 
 same time, so alter the position of its point of sup- 
 port, that it shall still remain directly beneath, or 
 above, its centre of gravity. Thus, if a sphere 
 rest upon a horizontal plane, and we roll it out of 
 the position in which it rests into some other, we 
 shall, in the act of rolling it, so alter the position 01 
 its point of support, that it shall still be beneath its 
 centre of gravity; for the centre of gravity is in 
 the centre of the sphere ; and the perpendicular to a 
 plane, on which a sphere rests, drawn from the 
 point where it rests upon it, necessarily goes through 
 its centre. Thus, into whatever position we roll a 
 
158 ILLUSTRATIONS OF MECHANICS. 
 
 sphei'e on a horizontal plane, the vertical, from the 
 point on which it rests, passes through its centre of 
 gravity ; and the centre of gravity is vertically 
 above the point of support. When a body, being 
 moved more or less from its position of equilibrium, 
 will rest in any of the positions in which it is 
 placed, and is indifferent to any particular position, 
 its equilibrium is said to be one of INDIFFERENCE. 
 
 This state of indifferent equilibrium is, however, 
 one of exceedingly rare occurrence, even in respect 
 to slight deflections of a body from its position of 
 rest ; and no other body besides a sphere, or a 
 body resting on a spherical surface, and having its 
 centre of gravity at the centre of that spherical 
 surface, can thus be indifferent to all the positions 
 in which it may be placed. Every solid body, with 
 the exception above stated, tends to return to the 
 position of equilibrium out of which it has been 
 moved, or to recede from it ; and if left to itself, it 
 will spontaneously either return to that position, 
 or roll farther from it. 
 
 The centre of gravity being moved from under, or 
 from over, its point of support, and being unsup- 
 ported, of necessity descends. The question then, 
 whether a body's position of equilibrium be stable 
 or unstable depends upon this other ; will the de- 
 scent of its centre of gravity, when the body is thus 
 left to itself, bring it into its former position, or de- 
 flect it farther from it? In the first case the equili- 
 brium is STABLE, and in the other, UNSTABLE. 
 
 Now, if the centre of gravity of the body be 
 elevated in the act of deflecting it from its position 
 of equilibrium, it is evident that it must be de- 
 
STABILITY. 159 
 
 pressed to be returned to it ; and, conversely, that 
 depressing itself, it will return to it. In this case, 
 then, the position out of which it was disturbed 
 was stable. But if, in the act of deflecting it from 
 its position of equilibrium, the centre of gravity of 
 the body be depressed, then, to return of its own 
 accord, the centre of gravity (where all the weight 
 acts downwards, and which is unsupported) must 
 elevate itself. This is impossible. The centre of 
 gravity descends ; and the body continues, therefore, 
 in this case, to deflect more and more from its 
 former position of equilibrium, which was, there- 
 fore, unstable. Thus, then, when the position in 
 which a body rests is such that, being deflected 
 from it, its centre of gravity ascends, that position 
 is one of STABLE equilibrium ; when the body 
 being thus deflected, its centre of gravity descends, 
 the position of equilibrium is UNSTABLE. 
 
 * 149. THAT POSITION OF A BODY RESTING UPON 
 ANOTHER IN WHICH ITS CENTRE OF GRAVITY 
 IS THE LOWEST POSSIBLE, IS A POSITION OF 
 STABLE EQUILIBRIUM ; THAT IN WHICH IT is 
 THE HIGHEST POSSIBLE, ONE OF UNSTABLE 
 EQUILIBRIUM. 
 
 If the centre of gravity of the body descends 
 when you deflect it from its position of rest, in any 
 direction, it is evident that the height of the centre 
 of gravity, in that position, is greater than in any 
 of the positions into which you deflect it; its po- 
 sition of UNSTABLE EQUILIBRIUM corresponds, then, 
 by what is stated in the last article, to that position 
 in which, being placed, its centre of gravity is highest 
 
160 ILLUSTRATIONS OF MECHANICS. 
 
 in respect to the adjacent positions. If, on the con- 
 trary, the centre of gravity rises when you deflect 
 it from its position of rest in any direction, then is 
 it in that position lower than in any of the others. 
 A body's position of STABLE EQUILIBRIUM corre- 
 sponds, then, to the lowest position of its centre of 
 gravity in respect to the adjacent positions of the 
 body. 
 
 *150. EVERYBODY, EXCEPT A SPHERE, HAS AT 
 LEAST ONE POSITION OF STABLE, AND ONE 
 OF UNSTABLE, EQUILIBRIUM. 
 
 For if a body be made to turn round on the sur- 
 face on which it rests, its centre of gravity will not 
 continually ascend or continually descend ; there 
 must be a certain position of the body, after it has 
 passed which, its centre of gravity, from ascending, 
 begins to descend; and another in which, from de- 
 scending, it begins to ascend. The first is a position 
 of unstable, and the last of stable, equilibrium. 
 
 In a sphere, the centre of gravity (which is the 
 centre of the sphere) continues always at the same 
 height as you roll it. There is, therefore, no position 
 either of stable or unstable equilibrium. Every 
 position in a sphere is one of indifferent equilibrium. 
 
 A body's position of equilibrium may be stable in 
 respect to a deflexion in one direction, and unstable 
 in respect to a deflexion in another. It is then said 
 to be a position of MIXED equilibrium. 
 
STABILITY. 161 
 
 151. A BODY HAVING PLANE FACES HAS ALL 
 ITS POSITIONS OF EQUILIBRIUM, ON THOSE 
 FACES, POSITIONS OF STABLE EQUILIBRIUM ; 
 AND ALL ITS POSITIONS OF EQUILIBRIUM, ON 
 THEIR EDGES, POSITIONS OF MIXED, AND ON 
 THEIR ANGLES, OF UNSTABLE EQUILIBRIUM. 
 
 For it is evident, that if a body rest upon a plane 
 face L M N O, and be inclined from its position of 
 equilibrium, it must turn 
 upon one of the edges of 
 that face as M N, so that 
 its centre of gravity G 
 must ascend * in a circle 
 round some point K in that 
 edge. 
 
 The centre of gravity thus ascending, when the 
 body is deflected round either edge of the plane on 
 which it rests, it follows that, when resting on that 
 plane, its centre of gravity is lowest, in respect to 
 
 * It will be observed that the position upon, the face L M N O 
 being supposed to be one of equilibrium, the centre of gravity 
 G is vertically over that face, so that the line G K is inclined 
 downwards towards the face, and must be elevated in turning 
 the body upon the edge M N. There is an exception in the 
 case in which the point G is vertically over the edge. M N on 
 which the body is turned ; the position is in that case unstable* 
 in respect to a deflexion of the body round that edge, and stable 
 in respect to a deflexion round every other. A position of 
 equilibrium of this kind is said to be one of MIXED equilibrium, 
 being stable one way and unstable the other. 
 M 
 
162 ILLUSTRATIONS OF MECHANICS. 
 
 the adjacent positions, and, therefore, that its 
 position upon it is one of STABLE equilibrium. 
 
 If, now, the body be turned upon its edge M N, 
 and be placed in such a position that the vertical 
 G H, through the centre of 
 gravity, shall pass accurately 
 through some point H in 
 that edge this, too, will be 
 a position of equilibrium, 
 for the centre of gravity 
 will be supported ; but it 
 will be a position of MIXED 
 
 equilibrium, that is, a position from which the 
 body being deflected in certain directions would 
 tend to return, and being deflected in others would 
 not, so as to be in respect to the first directions 
 of deflexion stable, and in respect to the others 
 unstable. For the centre of gravity G being verti- 
 cally over the edge M N, it is clear that when the 
 body is turned round that edge either way, the 
 centre of gravity will be depressed; so that over the 
 edge it is in its highest position, and the equilibrium 
 is, in respect to deflexions round the edge, unstable. 
 But if, instead of being turned round the .edge, the 
 body be lifted so as to turn about either of its extremi- 
 ties M or N, then its centre of gravity will be raised, 
 so that, in respect to those deflexions, it is in its low- 
 est position, and the equilibrium is stable. Thus the 
 equilibrium about either edge is, in certain direc- 
 tions, stable, and in others, unstable ; or it is MIXED. 
 By reasoning precisely similar to the above, it is 
 evident that, if the body can be made to rest on 
 
STABILITY. 163 
 
 either of its angles A, B, C, D, &c., so that the 
 centre of gravity shall be vertically over that angle, 
 the position will be one of unstable equilibrium. 
 
 152. A BODY'S POSITION is ALWAYS ONE OF 
 STABLE EQUILIBRIUM, WHEN ITS CENTRE OF 
 GRAVITY LIES BENEATH THE POINT ON WHICH 
 IT IS SUPPORTED. 
 
 Thus, if the body represented in the last figure, 
 instead of resting on a plane, had been suspended 
 from a fixed point by either of its angles, or if it 
 had been hung upon axis X Y passing through it 
 above its centre of gravity, then it is clear that, de- 
 flecting its position, from that in which it rests with 
 its centre of gravity vertically beneath the point of 
 support, the centre of gravity will be raised: the 
 position in which it rested was, therefore, a stable 
 position. 
 
 153. To CONSTRUCT A FlGURE WHICH, BEING 
 PLACED UPON A CURVED SURFACE, AND IN- 
 CLINED IN ANY POSITION, SHALL, WHEN LEFT 
 TO ITSELF, RETURN INTO ITS FORMER POSITION. 
 
 The accompanying cut represents a figure of any 
 light substance, to which are attached, so as to hang 
 beneath it, two heavy balls. The feet of the figure 
 are fixed upon a piece of wood, the lower surface 
 of which is curved, and this curved surface rests 
 loosely upon a small table which is supported by a 
 stand. If this figure be ever so far inclined in any 
 M 2 
 
164 
 
 ILLUSTRATIONS OF MECHANICS. 
 
 direction, it will immediately recover its position 
 when left to itself, and with the greater force as it is 
 more inclined. The explan- 
 ation is as follows. The effect 
 of the weight of the balls, 
 which weight is much greater 
 than that of the figure, is to 
 bring the centre of gravity of 
 the whole greatly beneath the 
 point on which it rests. This 
 being the case, it is evident 
 that, in whatever direction the 
 figure is made to incline in 
 F respect to its point of sup- 
 port the centre of gravity of 
 the whole will be made to 
 rise. In the position in which it rests, the centre of 
 gravity is therefore in its lowest point, and the equi- 
 librium is stable. 
 
 154. To CAUSE A BODY TO SUPPORT ITSELF STEA- 
 DILY, ON AN EXCEEDINGLY SMALL POINT. 
 
 A body may be made to support itself steadily on 
 an exceedingly small point, if it be so loaded that 
 its centre of gravity may be beneath this point. 
 This is strikingly illustrated in the following very 
 simple experiment. 
 
 On opposite sides of a cork towards the top, let 
 two forks be stuck, inclining downwards, and let 
 the edge of the bottom of the cork be then made 
 just to rest on the edge of a wine-glass, which 
 must be held, if necessary, to prevent it from 
 
STABILITY. 165 
 
 falling. The cork may be brought, by pushing 
 it gently sidewise, to rest upon so small a portion 
 of the glass that it shall seem scarcely to touch it ; 
 and yet the whole will support itself steadily upon 
 it : if slightly moved, it will return to its position, 
 and the glass may be raised without causing it to 
 fall. By the weight of the handles of the forks, 
 the centre of gravity is brought far below the point 
 of 'support ; hence the steadiness of the equilibrium, 
 and the facility with which it may be brought 
 about, on so small a point. 
 
 155. A BODY HAVING A PORTION OF ITS SURFACE 
 SPHERICAL, AND RESTING BY THAT PORTION 
 OF ITS SURFACE ON A HORIZONTAL PLANE, 
 HAS ITS EQUILIBRIUM STABLE OR UNSTABLE, 
 ACCORDING AS ITS CENTRE OF GRAVITY IS BE- 
 NEATH OR ABOVE THE CENTRE OF THE SPHERE 
 OF WHICH THAT SPHERICAL SURFACE FORMS 
 PART. 
 
 The figure in the woodcut is supported on a solid 
 base whose curved surface B A K is part of the 
 surface of a sphere having its centre in C. The 
 common centre of gravity of the 
 figure, and the mass which sup- 
 jfig- 40. /^.ZjA ports it, is G. On whatever point 
 D the spherical surface B A K 
 rests on the horizontal plane, the 
 vertical through its point of sup- 
 port passes through C (by a geo- 
 metrical property of the sphere); 
 when it rests then on A, A C is 
 the vertical, and this vertical passes then through 
 M 3 
 
166 ILLUSTRATIONS OF MECHANICS. 
 
 its centre of gravity G. When it rests on A, there- 
 fore, the figure is in equilibrium. Now, when G is 
 beneath C, this position is one of stable equilibrium ; 
 when G is above C, it is one of unstable equilibrium. 
 For, let the figure be placed in the inclined position 
 shown in the cut, so as to rest on D, and draw 
 through G the vertical G H to the horizontal plane, 
 then is G H the height of the centre of gravity in 
 the present inclined position of the figure. But, in 
 the upright position of the figure, when it rested 
 on A, the height of its centre of gravity was A G. 
 
 Now G H is greater than A G if G be, as in the 
 figure, beneath C* ; but if G were above C, as, for 
 instance, at E, then G H would be less than G A. 
 In the first case, the centre of gravity is raised, 
 then, by deflecting the body from its position of 
 equilibrium; in the second case, it is depressed. 
 In the one case, then, the equilibrium is stable ; and 
 in the other, unstable. 
 
 The same conclusion may yet more easily be 
 drawn from the consideration, that when the centre 
 of gravity is at G, the whole weight acting on that 
 side of the point of support D, which is towards 
 the former position of the body, tends to bring it 
 back to it : and that when it is at E, this weight, 
 acting on that side of D which is from its former 
 position, tends to deflect it yet farther from that 
 position. 
 
 A very ingenious toy is constructed on this prin- 
 ciple. A hemisphere (or half-sphere) is rounded 
 
 * For by Euclid, Proposition 7, Book iii., Gh, which is 
 only part of G H, is greater than G A ; much more, then, is G H 
 greater than it. 
 
STABILITY. 167 
 
 of some very heavy substance, lead, for instance; 
 (the half of a leaden bullet will answer the pur- 
 pose). On this is fixed a figure cut out of some 
 very light substance, such as the pith of the elder 
 tree. This figure, if placed on the table, and 
 inclined ever so much in any direction, will always 
 regain its upright position. The explanation is 
 contained in the principle stated above : the centre 
 of gravity of the whole figure is beneath the centre 
 of the spherical base ; for the centre of gravity of 
 the hemispherical ball is evidently within its mass, 
 and therefore below the centre of the sphere of 
 which it would form a part ; and the weight of the 
 figure placed upon it is so small, that it is not suffi- 
 cient to raise the centre of gravity of the whole 
 above that point, as it would do if it were heavy. 
 
 In this manner were constructed the French toys 
 called Prussians. The figures represented soldiers : 
 they were formed into battalions, and being made to 
 fall down by drawing a rod over them, they imme- 
 diately started up again as soon as it was removed. 
 
 Screens of the same form have since been in- 
 vented, which always rise up of themselves when 
 they happen to be pressed down. 
 
 *156. THE STABILITY OF A BODY WHICH is SUS- 
 PENDED FROM A POINT, OR A FIXED AXIS, IS 
 GREATER AS THE CENTRE OF GRAVITY OF THE 
 
 BODY is LOWER BENEATH THAT POINT OR 
 
 THAT AXIS. 
 
 Suppose the body represented in the accompa- 
 nying figure to be supported upon a point at C, or 
 
 M 4 
 
168 ILLUSTRATIONS OF MECHANICS. 
 
 fig- 41. upon a fixed axis passing through it at 
 that point. Let C Z be the vertical 
 through C, and let the body be de- 
 flected from its ordinary position of 
 equilibrium by the action of the force 
 j P, so that its centre of gravity G may 
 
 occupy the position shown in the 
 
 figure, instead of resting suspended beneath C in the 
 vertical C Z. 
 
 The body being thus held in equilibrium by the 
 action of the weight in G by the force P, and by 
 the resistance of the axis C, it follows, by the prin- 
 ciple of the equality of moments, that, if we take C 
 for the point from which we measure the moments of 
 these forces, that of the last-mentioned force vanish- 
 ing, those oi the two others will be equal; that is, the 
 product of P by C P will be equal to that, of the 
 weight of the body, by C M ; C P and C M being 
 respectively perpendiculars upon these forces from 
 C. Now, supposing P to be applied always at the 
 same perpendicular distance from C, or C P always 
 to be the same, it follows, from this equality, that P 
 must be greater according as the product of the 
 weight of the mass by C M is greater ; or that, for 
 bodies of the same weight, it must be greater as C M 
 is greater. Now C M is equal to G N, and G N 
 would evidently be greater if G were lower upon 
 the line C G ; or if the centre of gravity were lower 
 beneath the point of suspension in that position of 
 the body in which it rests of itself. Thus, then, the 
 force P necessary to deflect the body from the po- 
 sition in which it rests of its own accord, into any in- 
 clination to that position, is greater as the centre of 
 
THE BALANCE. 
 
 169 
 
 Jig. 42. 
 
 gravity is lower. The body is therefore more stable 
 as the centre of gravity is lower. 
 
 157. THE BALANCE. 
 
 It is for this reason, that in the construction of 
 delicate balances, whose degree of stability is re- 
 quired to be the least possible, that they may turn 
 with the least possible difference of the weights in 
 the scale-pans, precautions are taken by means of 
 which the centre of gravity, G, 
 of the whole moveable portion of 
 the balance,, including the beam, 
 the scale-pans, and the weights 
 they contain, shall lie, in every 
 case, at an exceedingly small dis- 
 tance beneath the point of sus- 
 pension, or fulcrum of the ba- 
 lance F. By making the scale-pans equal in 
 weight, suspending them at equal distances, F P 
 and F Q, from the fulcrum and from points lying at 
 the extremities of a line, P Q, passing through the 
 fulcrum F, their centre of gravity, and that of the 
 weights they contain when equal, is brought, so 
 that it would exactly coincide with the fulcrum if 
 the beam did not bend; and it would then only be 
 the centre of gravity of the beam itself, which 
 would lie beneath the fulcrum, and produce the 
 stability of the balance, bringing it back from its 
 deflections so as to vibrate it. The beam, however, 
 in reality always bends, whatever may be the care 
 taken to give it rigidity. And thus the centre of 
 gravity of the weights in the scale-pans, as well as 
 that of the beam itself, is brought beneath the ful- 
 
170 ILLUSTRATIONS OF MECHANICS. 
 
 crum ; and this depression is greater as the objects 
 weighed are heavier. The best balances are those 
 made by Mr. Robinson; every precaution which 
 science may suggest to ensure the accuracy of these 
 balances, is taken in their construction and their 
 adjustment.* 
 
 158. To MAKE A BALANCE WHICH SHALL APPEAR 
 
 TRUE WHEN EMPTY, BUT YET WEIGH FALSELY. 
 
 Let a balance be constructed with unequal arms, 
 and let scale-pans be suspended from them of un- 
 equal weights, so adjusted that they shall just equi- 
 poise one another, and make the beam to rest in a 
 horizontal position. This balance will appear a just 
 one when the scale-pans are empty, but it will not 
 weigh truly ; for any weights put in its scale-pans 
 will be suspended at different distances from the 
 fulcrum. They cannot, therefore, balance one 
 another when they are equal that suspended from 
 the shorter arm must be greater than the other. 
 The weights used being then put into this scale, 
 and the commodities to be weighed into the other, 
 the balance, appearing to be true, will weigh short 
 weight. 
 
 The deception is easily detected by changing the 
 scales in which weights and the things weighed are 
 placed. If the balance be false, the equilibrium will 
 then no longer exist. 
 
 * For a more complete discussion of the theory of the balance, 
 the reader is referred to the author's treatise on " Mechanics 
 applied to the Arts," p. 68. 
 
THE BALANCE. 171 
 
 159. To WEIGH TRULY WITH A FALSE BALANCE. 
 
 Let the article to be weighed be placed in either 
 scale-pan, and let the weight necessary to balance 
 it in the other be found ; place it then in the other 
 scale, and let the weight necessary to balance it be 
 found as before ; take then the product of these 
 two false weights : the square root of this product 
 will be the true weight. Thus, if in one scale the 
 article weigh 14< ounces, and in the other 16, taking 
 the product of them we have is 224 ; the square root 
 of this product is 14f, which is the true weight in 
 ounces. 
 
 160. BORDA'S METHOD OF WEIGHING TRULY WITH 
 
 A FALSE BALANCE. 
 
 A much simpler method than the above, of weigh- 
 ing truly with a false balance, has been conceived 
 by Borda, and may be considered as in all cases 
 the most certain way of ascertaining the weight of 
 any substance. Let the thing to be weighed be 
 placed in either scale of the balance, and any heavy 
 but minute substance leaden filings, for instance 
 accumulated in the other, until they precisely balance 
 it ; let the article to be weighed be now removed, 
 and, in the scale which contained it, let weights be 
 introduced until an equilibrium is again accurately 
 produced. These weights will give the true weight 
 of the body, independent of any error in the ba- 
 lance ; for their weight produces exactly the same 
 effect that its weight did balancing identically the 
 same load in the opposite scale. 
 
172 ILLUSTRATIONS OF MECHANICS. 
 
 * 161. UNDER WHAT CIRCUMSTANCES A BODY, 
 SUPPORTED UPON A HORIZONTAL PLANE, IS 
 
 MORE OR LESS STABLE. 
 
 In order that a body which rests upon another, 
 and is therefore in a stable position of equilibrium, 
 may be overthrown, it must be made to pass from 
 that position of stable into one of unstable equi- 
 librium. Thus, for, instance,, to be overthrown, the 
 body A B C D, from the stable position shown in 
 the first of the accompanying figures, must be made 
 to revolve into, and slightly beyond, the unstable 
 position shown in the second. 
 
 fig.44. 
 
 KB 
 
 The body is more or less stable in its position 
 represented in the first figure, according as the re. 
 volution it must receive to bring it into the position 
 represented in the second figure is greater or less, 
 and according as the force required to produce this 
 revolution is greater or less. Now, by this revolu- 
 tion, the line G A, in the first figure (G represent- 
 ing the centre of gravity) is made to pass, from an 
 inclined to a vertical position, in the second figure; 
 so that G may be vertically above the angular point 
 A, on which the body turns. But that G A (Jig A3.) 
 may revolve into a vertical position, it must revolve 
 
STABILITY. 173 
 
 through the angle G A M, which angle is equal to 
 the angle A G K. The revolution, then, which the 
 body must receive before it will fall over of itself, 
 is greater or less, according as the angle A G K is 
 greater or less. Now the angle A G K is greater 
 according as G is lower, and according as A K 
 is greater. For it is evident, that if G had been 
 higher than it is as, for instance, at H then the 
 angle G A M, or its equal A G K, would have been 
 less than it is : moreover, if AK had been less than 
 it is, hen, also, it is evident that A G K would have 
 been less. 
 
 Thus, then, we see one reason why it is that, as 
 a body's centre of gravity is lower, and its base 
 wider, it is more difficult to overthrow it the body 
 requiring, according as these conditions obtain, to 
 be turned farther before it will pass into a position 
 (one of unstable equilibrium) from which it will 
 fall over of itself. 
 
 The amount of the revolution which must thus 
 be given to a body, by the application of a sufficient 
 force, before it can be overthrown, is not, however, 
 the only element on which the degree of its stability 
 depends. Another is the amount of the force ne- 
 cessary to produce this revolution. 
 
 The amount of the force depends upon the weight 
 of the body, and the distance of the vertical G K 
 through its centre of gravity from the point A, round 
 which it is to be made to turn. 
 
 To make this appear, let us suppose that the force 
 intended to turn it is applied at C in a horizontal 
 direction ; in which direction the line C M is drawn, 
 meeting the vertical line A M in M. Suppose this 
 
174? ILLUSTRATIONS OF MECHANICS. 
 
 force C to be just upon the point of causing the 
 body to turn on A, and very slightly to have raised 
 it, so that the forces which act upon it are exactly 
 in equilibrium; imagine, moreover, all the weight 
 of the body to be collected in G, an allowable sup- 
 position : the weight acting in G, the force acting 
 at C, and the resistance of the surface on which 
 the body rests acting at A, then, are the forces 
 in equilibrium. There must then obtain between 
 them the relation of the equality of moments. (See 
 art. 129.) If, then, from the point A perpendicu- 
 lars be drawn upon the directions of the force at 
 C, and the weight through G, then the products of 
 the lengths of these perpendiculars, by the numbers 
 of cwts., or pounds, or ounces, in their correspond- 
 ing forces, must be equal.* Now these perpendiculars 
 are evidently A M and A K. When the force C is 
 just then upon the point of turning the body, it is a 
 force equivalent to such a number of pounds, that 
 this number of pounds being multiplied by the num- 
 ber of inches in A M, the product shall equal the 
 number of pounds' weight in the body multiplied by 
 the number of inches in A K. And the first pro- 
 duct must be greater according as the last is greater; 
 so that supposing the force C to be applied always 
 at the same height, that force itself must be greater 
 according as the last of the above mentioned pro- 
 
 * The perpendicular from A upon the resistance acting 
 through that point, is of course nothing or of no length ; the 
 moment of this resistance is therefore nothing : thus this third 
 force vanishes from the relation of the equality of moments, 
 when we measure them from A, for which reason it is that A 
 above all other points is selected to measure them from. 
 
STABILITY. 175 
 
 ducts is greater, and this last product is greater 
 according as A K is greater. Thus, then, the force 
 necessary to turn the body is greater according as 
 the distance of the vertical through its centre of 
 gravity from the point on which it is to turn is 
 greater or less. The amount of this force has, 
 however, nothing to do with the height of the centre 
 of gravity ; thus it is the same in the figure, how- 
 ever high G may be, provided it remains in the line 
 KH, 
 
 So far, then, as the stability of the body is depend- 
 ant upon the force necessary first to move it, it is 
 independent of the height of the centre of gravity ; 
 so far as it is dependant upon the amount of the 
 deflexion which will be sufficient to overthrow it, it 
 is dependant upon that height. 
 
 It is because a slight deflexion will overthrow a 
 body when loaded high, that it is then of little sta- 
 bility, not because a less force will then move it. 
 As great a force is necessary at first to move a high 
 body as a low one, but a less deflexion will over- 
 throw it. Thus, when a body is of necessity sub- 
 jected to certain deflexions, it should never be 
 loaded high ; a coach, for instance, which is of ne- 
 cessity deflected by the irregularity of the road, if 
 it be loaded high, may be brought by some of 
 these deflexions into, and beyond, its position of un- 
 stable equilibrium, and overthrown ; whereas a tower, 
 or a spire as high as that of Salisbury cathedral, 
 stands firmly on its base. 
 
 If the vertical through the centre of gravity of a 
 body do not pass through the middle of its base, the 
 body is more stable to resist a force in one direction 
 
176 ILLUSTRATIONS OF MECHANICS. 
 
 than another. Thus in the figure the point K not 
 being in the middle of the base A, it is evident 
 from what has been said, that the body is more 
 stable in respect to a force tending to turn it about 
 A, than to one tending to turn it about B. There 
 are structures whose centres of gravity are over 
 points thus greatly nearer to one side of the base 
 than the other, so as in one direction to possess but 
 a slight degree of stability, which, by reason of their 
 great weight, stand nevertheless firmly. Such are 
 the hanging towers of Pisa and Bologna. 
 
 WALKING. 
 
 In the act of walking, the centre of gravity is 
 raised, alternately, over the legs. The motion some- 
 what resembling that of a pair of open compasses, 
 made to rest alternately on their points ; the centre 
 of gravity is over the fork of the legs, and may be 
 imagined to be over the angle of the compasses. If, 
 as the compasses are thus made to travel forwards, 
 resting on their alternate points, these points are 
 not placed in the same straight line, but alternately 
 to the right and left of it, then the centre of gravity 
 will describe a series of arcs to the right and left, 
 and it will not be carried so far forwards, by a cer- 
 tain number of steps, as though these were made in 
 the same right line ; this corresponds to that .un- 
 gainly motion in walking, which is called waddling. 
 It is remarkable how nearly the footsteps of a person 
 who walks well, are in the same straight line, as 
 may be seen especially, if we trace them in the 
 snow ; this is, moreover, remarkably the case with 
 
WALKING. 177 
 
 animals, horses for instance, and especially it is the 
 case with birds, whose centres of gravity being for 
 the most part very high, in comparison with the 
 dimensions of their feet, they are taught instinctively 
 to avoid those deflexions of their bodies to the right 
 and left, by which they might be overthrown. 
 
 Taking the width of a man's foot at about three 
 inches, and giving him an average stature, it may 
 be calculated that a deflexion of his body of less 
 than two degrees would, when he rests on either 
 foot, be sufficient to overthrow him. How justly 
 regulated then must be the effort which he makes at 
 every step, to transfer his centre of gravity from 
 above one of his feet to above the other, that his 
 position may be kept within this narrow limit ! Put 
 upon his shoulders a burden, and you will raise his 
 centre of gravity, and greatly increase the difficulty 
 he will experience in balancing himself; yet how 
 firmly and securely does he tread ! A man carrying 
 a burden as heavy as himself, and inclining his 
 position as he steps on each foot, only half a degree 
 to the right or left of the position in which he would 
 rest on that foot, would be overthrown. 
 
 At each step the centre of gravity is raised and 
 made to revolve over the foot. It is this raising of 
 the centre of gravity, in which the whole weight of 
 the body may be supposed to be collected, which 
 constitutes the great effort of walking. It has been 
 calculated that at every step the centre of gravity 
 is raised a perpendicular height equal to about one 
 eleventh the length of the step ; so that a person who 
 walks eleven miles, raises his centre of gravity and 
 therefore the whole weight of his body, a succession of 
 
178 ILLUSTRATIONS OF MECHANICS. 
 
 lifts, equivalent to the direct raising of it, one mile. 
 If six men, weighing each 182 Ibs., and a boy of 
 half that weight, walk at the rate of eleven miles in 
 three hours, the aggregate of their labour, while thus 
 walking, will be about equal to one horse's power ; 
 as the amount of a horse's power is usually estimated. 
 
 162. THE RESISTANCE OF A SURFACE. 
 
 RESISTANCE is a force which is lodged, like 
 gravity, universally in matter. When it presents 
 itself under the form of a pressure, or as one of a 
 system of forces producing equilibrium, its charac- 
 teristic property is this, that, at each point of its 
 application, it is supplied precisely in that quantity 
 and degree in which it is necessary, that motion may 
 not be produced there, and in neither more nor less 
 than that degree. It is by reason of this property 
 of resistance, adapting its energies, as it were to the 
 demand made upon them, that an infinite variety 
 can (within certain limits) be introduced among the 
 remainder of a system of pressures, of which a 
 resistance is one, without, nevertheless, disturbing 
 their equilibrium. 
 
 This property of supplying a force precisely equal 
 of the amount required to counteract the tendency 
 to motion is, however, in every case of resistance 
 known to us, confined within limits, more or less 
 extensive indeed, but yet definite and fixed. 
 
 Airs and liquids supply no resistance of this kind 
 at all, or none that is appreciable, the bodies we call 
 soft, but little; and all solid bodies, are subject to 
 this law, that they yield, by reason of their elasticity, 
 
THE RESISTANCE OF A SURFACE. 179 
 
 more or less, but for the most part inappreciably, to 
 every pressure, and that there are certain limits 
 beyond which they resist no longer (or in other 
 words do not supply that resistance which is neces- 
 sary to prevent motion) ; motion then takes place, 
 the structure of their parts is destroyed, and they 
 crush, or break, or fly in pieces these being all 
 but so many terms used to express the insufficiency 
 of their resisting power to supply the pressure ne- 
 cessary to equilibrium. 
 
 These remarks apply only to the magnitude or 
 amount of the force by which the surfaces of solid 
 bodies resist its direction is another question. 
 
 163. THE DIRECTION OF THE RESISTANCE OF A 
 SURFACE. 
 
 The direction in which a solid body resists was, 
 when the theory of statics was first discussed, taken, 
 hypothetically, to be a direction perpendicular to 
 the surface of the resisting body. 
 
 It is difficult to assign any better reason for this 
 hypothesis, than that desire to simplify the con- 
 ditions of a question which is natural and, perhaps, 
 necessary to the first discussion of it. The same 
 reason does not, however, sufficiently account for 
 the preservation of it. An abundance of examples 
 will suggest themselves to every one, showing that 
 the hypothesis is in no case true. Did the surface 
 of the earth, for instance, on which we tread, resist 
 only in a perpendicular direction, although we might 
 stand, the first step we made would infallibly bring 
 us to the ground ; and, as to stretching our legs as 
 
180 ILLUSTRATIONS OF MECHANICS. 
 
 we do when we walk rapidly, inclining them at a 
 considerable angle, and trusting to the resistance of 
 the ground to counteract their oblique pressure 
 upon it, it would be madness. 
 
 Resisting only in a perpendicular direction, the 
 surface on which we trod could not possibly supply 
 any opposite force to the oblique pressure which 
 each leg in its turn would exert upon it and fall- 
 ing, where we fell we must lie, unless some immove- 
 able obstacle were at hand, by clinging to which we 
 might regain an upright position ; for to rise by the 
 usual method, supported by our hands and knees, 
 would be impracticable every effort which we so 
 made would be accompanied by an oblique thrust 
 or pressure, and no such oblique pressure would 
 be counteracted : at every effort our hands and 
 knees would slip from under us ; and it would be 
 an equally useless task to attempt to rise ourselves, 
 and to trust to the assistance of others. In short, 
 under a state of things like this, to live the life of 
 locomotive creatures on the solid surface of the 
 earth would scarcely be possible ; and had it existed 
 from the beginning, we cannot but believe that all 
 animated being, other than that which peopled the 
 air or the seas, would have been rooted into the 
 ground. 
 
 Whilst it is thus certain that the resistances of 
 the surfaces of solid bodies are not confined to their 
 perpendicular directions, it is by no means certain 
 what that particular law is, by which, in each differ- 
 ent body, the direction of its resistance is really 
 governed. Nevertheless, the following is a very 
 near approximation to that law. 
 
THE CONE OF RESISTANCE. 181 
 
 164?. THE CONE OF RESISTANCE. 
 
 The resistance of the surface of a solid body is 
 equally exerted in every direction which does not 
 make with the perpendicular an angle greater than 
 a certain angle, called the limiting angle of resist- 
 ance, which is always the same for the same surface, 
 but different for different surfaces. 
 
 Or, perhaps, this law will be better understood 
 under this other form of it. 
 
 If a cone be imagined to be taken, as in the ac- 
 companying diagram, having its axis A Q perpen- 
 dicular to any point Q of the surface of a solid body, 
 and having its angle, at the vertex, dependent, by a 
 very simple law, upon its friction with the surface 
 
 of another body pressed upon it at Q ; then the 
 pressure will be resisted, provided its direction be 
 any where within the surface of the cone, as, for 
 instance, in the direction of the arrow P Q ; and it 
 will not be resisted if its direction be any where 
 without it. 
 
 N 3 
 
182 ILLUSTRATIONS OF MECHANICS. 
 
 The remarkable feature of this law is this, that 
 it is true whatever may be the amount of the force 
 P Q, within the limits of abrasion. Whether this 
 force be great or small, it will be resisted if its 
 direction be within this cone ; and if it be without 
 the cone, it will not be resisted. 
 
 The angle of the cone of resistance is dependant, 
 by a very simple law*, upon this property of the 
 friction of two surfaces in contact, that the amount 
 of the friction is always proportional to the per- 
 pendicular pressure which produces it. We shall 
 hereafter speak fully of this property. It will be 
 here sufficient to state, that the angle of the cone 
 being dependant upon the friction, there must always 
 be the same cone of resistance for the same sur- 
 faces of contact, and different cones of resistance 
 for different surfaces, inasmuch as the friction is 
 the same for the same surface, and different for 
 different surfaces. 
 
 165. ILLUSTRATION OF THE CONE OF RESISTANCE 
 IN THE STRIKING OF A HAMMER. 
 
 If a hammer be struck upon a polished mass of 
 metal an ANVIL for instance, it will be found that 
 there are an infinity of directions, besides the per- 
 pendicular one, in which the blow being given, it 
 will be resisted ; and this, however strong the blow 
 may be, even although it were given with a sledge 
 
 * The vertical angle of the cone is twice that whose tangent 
 is the constant ratio of the friction to the pressure. See " Me- 
 chanics applied to Arts," p. 43. ; also p. 47., where is a table of 
 the angles of the cones of resistance for different surfaces. 
 
THE CONE OF RESISTANCE. 183 
 
 hammer. If now, the direction of the stroke be 
 continually, and very gradually, inclined farther and 
 farther from the perpendicular, it will be found 
 that there is a certain inclination up to which the 
 resistance will continue perfect, and that after this 
 inclination is passed, every blow will slip. The 
 surface of the cone is, in point of fact, beyond that 
 inclination, passed, and the principle above stated, is 
 fully illustrated. 
 
 166. ILLUSTRATION OF THE LAW OF THE RE- 
 SISTANCE OF A SURFACE, IN THE USE OF THE 
 CROWBAR. 
 
 There is scarcely any use to which the mechanical 
 powers are put, in which the principle of resistance 
 stated in the last article, does not find its application. 
 In the crowbar, for instance, represented in the 
 annexed engraving by A B, and used to lift up the 
 
 fig. 46. 
 
 mass G. Motion cannot take place, or the mass 
 be lifted, except by its surface sliding on the surface 
 of the lever, at the point where it rests on it, and 
 in order that the two surfaces may thus slip upon 
 one another, the direction M w, in which they are 
 pressed upon one another, must be without the 
 N 4* 
 
184- ILLUSTRATIONS OF MECHANICS. 
 
 surface of the cone of resistance. When by the 
 action of the force P, which moves the lever and 
 the resistance of the point A, on which it rests, the 
 direction of the pressure M n, is made to assume a 
 direction just without the surface of this cone ; the 
 surfaces begin to slip, and the mass to be elevated. 
 Knowing the friction of the surfaces, we know 
 what is the cone of resistance at M. Thus we 
 know what must be the direction of M n, when 
 motion is about to take place, and knowing this 
 direction, we know the perpendicular A m. Knowing 
 then A m and A P, we can compare the pressure 
 of the lever in the direction M n with the force P, 
 since by the principle of the equality of moments, 
 the moments of these two forces about A, are equal. 
 Proceeding then to the point N, and observing 
 that, by the same principle, the moments about that 
 point, of the force in M n, and of the weight of the 
 mass supposed to be collected in G, are equal, we 
 can determine a relation between this weight and 
 the force in M n. Knowing then a relation between 
 the weight of the mass and the pressure of the 
 lever upon it in M n, and knowing also a relation 
 between this last pressure and the force P, we can 
 determine a relation between P and the weight of 
 the mass, when motion is about to take place ; that 
 is, we can determine what force P, is necessary to 
 raise the weight, when in any position. This problem 
 is, however, a complicated one, and requires, to its 
 complete solution, the application of considerable 
 mathematical knowledge. It is merely described 
 here, that the nature of such investigations may be 
 presented to the mind of the reader. There are 
 
RESISTANCE IN A MACHINE. 185 
 
 other considerations, which yet more complicate 
 the problem. It may be, that before P attains that 
 amount which is thus shown to be necessary to lift 
 the mass, it may produce a pressure upon the ex- 
 tremity A of the crowbar, whose direction i& 
 without the cone of resistance at that point, so that 
 it may cause it to slip ; or it may, before it reaches 
 this limit, produce a pressure on N, in a direction 
 without the cone of resistance at that point, so as 
 to cause that point to slip. In either case, the 
 elevation of the mass will be arrested. 
 
 167. THE MECHANICAL ADVANTAGE OF ANY 
 MACHINE is SUPPLIED BY THE RESISTANCES 
 OF ITS PARTS. 
 
 Of all the different forms of force, that under 
 which it most directly connects itself with practical 
 mechanics., and with the operation of machinery 
 that without which no machine can act, and which 
 every machine is indeed but a contrivance for 
 applying, is RESISTANCE. 
 
 The resistances of the axles of its wheels, the 
 fulcra of its levers, and of the various surfaces by 
 which its parts move in contact with one another, 
 are in point of fact but so many pressures, which it 
 borrows, and which are made to co-operate in the 
 effect it produces. 
 
 That which is known to us in a machine, by 
 the name of a mechanical advantage, is no other 
 than a contrivance by which we are enabled to avail 
 ourselves of the resistance of some surface or sur- 
 faces entering into the construction of the machine 
 
186 ILLUSTRATIONS OF MECHANICS. 
 
 we use. Thus in the lever, the resistance of the 
 fulcrum aids in supporting the weight ; and by just 
 so much as it resists, diminishes the pressure which 
 must be made to act upon it, before the weight can 
 be put in motion, or the work done. So too of the 
 wheel and axle, it is but a contrivance by which the 
 resistances of the points on which the axle turns, 
 are made to contribute to the force which must be 
 used before the weight can be raised, and which 
 must be kept up during the whole time that it is in 
 the act of being raised. And the inclined plane is 
 but an instrument whereby the resistance of the 
 surface of the plane is made to supply a certain 
 portion of that pressure which would be necessary 
 to raise the weight, directly, through a distance equal 
 to the height of the plane. 
 
 Such in practical mechanics, and in the operation 
 of machinery, is the essential part which belongs to 
 the resistances of points of support. 
 
 168. FRICTION. 
 
 When a body is pressed upon the surface of 
 another, it is moved along that surface with dif- 
 ficulty. If you attempt to cause one of the surfaces 
 to slide on the other, a certain force opposes itself 
 to the effort, which is found to be greater, as they 
 are pressed together with greater force. By ren- 
 dering the surfaces of contact more smooth, or by 
 interposing unguents between them, the amount of 
 this resistance, called friction, may be greatly di- 
 minished, but it can never be altogether got rid of. 
 
 The principal experiments which have been made 
 
FRICTION. 187 
 
 upon friction, have reference ; First, To the pro- 
 portion in which the friction increases with the 
 pressure, on the same surface. Secondly, To the 
 variation of the amount of friction, produced by the 
 same pressure, upon equal surfaces of different sub- 
 stances. Thirdly, To its relation to the size of the 
 surface of contact, the pressure being the same. 
 Fourthly, To the influence of the time in which the 
 bodies have been in contact, on the amount of the 
 friction ; and especially to the distinction between 
 the friction which resists the first motion of a body 
 from rest, and that which opposes itself to its 
 motion during the continuance of that motion. 
 The principal experiments on this subject have 
 been made by Coulomb*, Professor Vince, Mr. G. 
 Renniet, M. A. Morin. J The following are among 
 the principal results of these experiments ; a more 
 detailed statement of them is contained in tables in 
 the Appendix. 
 
 169. THE FRICTION is PROPORTIONAL TO THE 
 PRESSURE. 
 
 Thus, if the surface of one body be pressed upon 
 that of another with a certain force, and if that force 
 be then doubled, the friction will be doubled ; if the 
 force pressing them together be tripled, the friction 
 will be tripled, &c. &c. 
 
 Thus, for instance, a piece of cast-iron having a 
 plane surface of 44 square inches was laid, by Mr. 
 
 * Mem. des Sav. Strangers, 1781. 
 t Philosophical Transactions, 1829. 
 $ Mems. de PInstitute, 1833. 
 
188 ILLUSTRATIONS OF MECHANICS. 
 
 Rennie, upon another larger plane surface of the 
 same metal, and loaded with a weight, which, together 
 with its own, amounted to 24 Ibs. ; and it was found 
 that a force applied to it, parallel to the surface, by 
 means of a string and pulley, just moved it, when it 
 amounted to 3 Ib. 3 oz. 
 
 It was then loaded, so as to be similarly pressed, 
 with twice the first weight, or with 48 Ibs. and a force 
 of 6 Ibs. 8 oz. was then required; indicating a friction 
 in the last case, or under double the pressure, which 
 only differed by 2 oz. from twice the former friction. 
 
 When, again, the surfaces were made to press upon 
 one another with a weight of 36 Ibs., being 1-J times 
 the first pressure, the force required to move the 
 body, that is its friction, was found to be 4 Ibs. 14? oz., 
 differing by only l^oz. from 1^ times the first friction. 
 
 By similar means, a piece of black beech, which had 
 a surface of two inches square, being pressed upon 
 another with a force of one hundred weight, was 
 found to have a friction of 15 Ibs. 5 oz. ; and, being 
 pressed with a force of three hundred weight, to have 
 a friction of 45 Ibs. 3 oz., differing from triple its 
 previous friction, by no more than 12 ounces. A 
 piece of Norway oak, of the same size, being pressed 
 upon another with a weight of one hundred weight, 
 exhibited a friction of 14 Ibs. 5 oz. ; whilst, under a 
 pressure of four hundred weight, its friction became 
 56 Ibs. 7 oz., differing from four times its former 
 friction, by only 13 oz. 
 
 This rule is however only an approximate one, 
 from which the actual friction varies but little, in the 
 case of hard metals, for pressures less than 32 Ibs. 
 upon the square inch ; but from which there is a 
 
FRICTION. 189 
 
 rapid deviation, for pressures exceeding that limit. 
 For woods, the limit is somewhat higher ; but, within 
 this limit, the results are more irregular than in the 
 case of metals. 
 
 170. AMOUNT OF THE CONSTANT PROPORTION OF 
 THE FRICTION TO THE PRESSURE IN DIFFERENT 
 SUBSTANCES. 
 
 An extensive table of the results which have been 
 obtained on this subject will be found in the Appendix. 
 The following may be mentioned as general conclu- 
 sions : 1st, That the ratio of the friction to the 
 pressure in all hard metals is, for pressures less than 
 32 Ibs. on the square inch, nearly the same. For all 
 these metals, the friction is very little different from 
 one sixth of the pressure. 
 
 2d. The friction of the soft metals is greater than 
 that of the hard ones. 
 
 3d. The same relation obtains in respect to the 
 friction of the soft woods and the hard ones. Thus 
 two surfaces of yellow deal being pressed together, 
 exhibited a friction equal to more than one third the 
 pressure, whilst the friction of two surfaces of red 
 teak was scarcely more than one ninth of the pres- 
 sure. These were the two extreme ratios in the case 
 of woods. 
 
 Whether the fibres of the two surfaces of wood 
 be parallel or perpendicular, materially affects the 
 amount of friction, and whether they be wet or dry. 
 
 Thus, when one surface of oak was pressed upon 
 another, the fibres being parallel, the ratio of the 
 friction to the pressure was from -60 to '65 ; when 
 
190 ILLUSTRATIONS OF MECHANICS. 
 
 the surfaces were so placed in contact that their 
 fibres were perpendicular, the ratio sank to *54?; and 
 when, the fibres remaining thus perpendicular, the 
 surfaces were wetted it rose again to ?!. It is a 
 practical fact of some importance, that the friction 
 of surfaces of wood upon one another is thus so 
 considerably increased by wetting them. 
 
 171. THE AMOUNT OF FRICTION is INDEPENDENT 
 OF THE EXTENT OF THE SURFACE PRESSED, PRO- 
 VIDED THE WHOLE AMOUNT OF THE PRESSURE 
 REMAIN THE SAME, AND THAT THE SUBSTANCE 
 
 OF THE SURFACE PRESSED is THE SAME. 
 
 This is an important property of friction, which 
 has been established by numerous experiments. 
 By increasing the surface which supports the pres- 
 sure, you diminish the amount of pressure upon 
 every point of it, and you thus so diminish the 
 friction upon every point, that although there are 
 more points which rub, their aggregate amount of 
 friction is only the same as before. 
 
 Thus, in one of the experiments of Mr. Rennie, 
 a piece of cast iron, when laid upon its flat side, 
 which had a surface of 44 square inches, and 
 loaded, so as to press upon another surface of cast- 
 iron, with a force of 141bs., required a force 21bs. 
 4 oz. to make it slide : when placed upon its edge, 
 which had a surface only of 6 J square inches, and 
 subjected to the same pressure, 21bs. 2oz., were 
 found sufficient to move it. The friction, in the 
 one case, was then 34 ounces, and in the other, 36. 
 
FRICTION. 191 
 
 172, THE FRICTION OF A BODY WHEN IN A STATE 
 OP CONTINUOUS MOTION, BEARS A CONSTANT 
 RATIO TO THE PRESSURE UPON IT, WHICH is 
 
 THE SAME, WHATEVER MAY BE THE VELOCITY 
 OF THE MOTION. 
 
 This fact results from the experiments of M. 
 Morin, made in the years 1831, 1832, at Metz, on 
 a very extensive scale, and under the sanction of 
 the French government. 
 
 The force with which the cord was, at any time, 
 pulling the various bodies which it put in motion (and 
 whose friction this force was always equal to), was 
 estimated by the deflexions of a steel spring to which 
 it was attached ; and these deflexions were made by a 
 very ingenious contrivance, to register themselves, at 
 every period of the motion. The principal facts 
 resulting from them were those stated at the head 
 of this article, that the friction in this case of con- 
 continued motion, as well as when the body is 
 moved from a state of rest, is always the same frac- 
 tion of the pressure, however great (within certain 
 limits), or however small, that pressure may be ; 
 and moreover, that its amount is wholly independent 
 of the velocity with which the body is moving, 
 being of the nature of that force which is called by 
 mathematicians, a uniformly retarding force. 
 
 173. THE EFFECT OF UNGUENTS UPON FRICTION. 
 
 The general effect of unguents upon friction is, 
 as it is well known, materially to diminish it. It is, 
 however, important to observe, that in doing this, 
 
192 ILLUSTRATIONS OF MECHANICS. 
 
 they entirely destroy the constant ratio which, with- 
 out unguents, friction is found to bear to the pres- 
 sure. 
 
 Thus, Mr. Rennie found that the friction of an 
 axle of yellow brass upon a collar of cast iron 
 was, without unguents, in every case about -|th the 
 pressure. * When the surfaces were oiled, this 
 ratio became under a pressure of \ cwt., only ^th ; 
 but when the pressure was increased to 1 1 cwt., it 
 rose to ^th. 
 
 Axles of yellow brass, moving in collars of cast 
 iron appear, from the experiments of Mr. Rennie, 
 to exhibit when used with unguents, the least 
 amount of friction ; and that unguent, which is 
 best adapted to them, appears to be tallow. With 
 this unguent, the mean result of his experiments 
 gives for a pressure of from 1 cwt. to 5 cwt., a fric- 
 tion of somewhat less than ^th the pressure. With 
 soft soap it becomes -g^th. It is a remarkable fact, 
 that while, with the softer unguents, such as oil, 
 hog's lard, &c., the ratio of the friction to the pres- 
 sure increases with the pressure ; with the harder 
 unguents, soft soap, tallow, and anti-attrition com- 
 position, it diminishes. 
 
 The question of time does not appear to have 
 been sufficiently attended to, in these and other ex- 
 periments on friction, and the subject is one in 
 which much probably yet remains to be learned. 
 
 * The axle was revolving during the experiment over 4J 
 inches of surface, in 90 seconds. 
 
THE INCLINED PLANE. 193 
 
 174. THE CIRCUMSTANCES UNDER WHICH A BODY 
 WILL SUPPORT ITSELF UPON AN INCLINED 
 PLANE. 
 
 Let the weight of a body, resting upon the in- 
 clined plane, represented in the accompanying 
 fig. 47. figure, be supposed to be collected 
 
 in its centre of gravity G, and 
 draw the vertical line G N ; it is 
 j j ^^ in the direction of this line that 
 
 A the whole weight of the body will 
 act,, and it is in the direction of this line, there- 
 fore, that the body may be supposed to be pressed 
 upon the plane at L. If, then, this direction lie 
 without the cone of resistance at L, the body will 
 slip down the plane ; if it lie within it, it will not. 
 Now, if L K be drawn perpendicular to the sur- 
 face of the plane, from L, it will be the axis of the 
 cone of resistance at that point, and the direction 
 of G L will be within or without the cone, accord- 
 ing as the angle K L G, is less or greater than one 
 half the angle at the vertex of the cone. But the 
 angle CAB is equal to the angle K L G ; the 
 body will therefore rest, of its own accord, or slip 
 upon the inclined plane, according as the inclin- 
 ation C A B of the plane is less or greater than half 
 the angle of its cone of resistance : and conversely, 
 the inclination of the plane just equals half the 
 angle of the cone of resistance, when it is such, 
 that the body begins to slip upon it. Half the 
 angle of the cone of resistance, is called the limit* 
 ing angle of resistance, being that inclination of the 
 
191' ILLUSTRATIONS OF MECHANICS. 
 
 pressure to the perpendicular, which first, in any 
 case, causes the body to slip. It is thus that the 
 limiting angle of resistance, has, in respect to a 
 great number of substances, been determined. 
 Their surfaces having been made perfectly plane, 
 have been placed upon one another, and then both 
 bodies have been made to rest on an inclined 
 plane ; this inclined plane being moveable, so as 
 to admit of receiving a greater or less inclination. 
 It has then been gradually elevated, until the 
 bodies were first observed to slip, and the angle of 
 elevation, or, as it is called, the slipping angle, being 
 observed, the limiting angle of resistance became 
 known. 
 
 A table in the Appendix contains the results of 
 experiments thus made by Mr. G. Rennie. 
 
 175. THE CIRCUMSTANCES UNDER WHICH A BODY 
 MAY BE SUPPORTED UPON AN INCLINED PLANE. 
 
 Let the supporting force, be applied by means 
 of a string D P, passing over a pulley D, and sup- 
 porting a weight W. Let the direction, D P of 
 
 48. 
 
 this string be produced, so as to meet the vertical 
 G L, through the centre of gravity in M. Mea- 
 
THE MOVEABLE INCLINED PLANE. 
 
 sure off M K, containing as many equal parts as 
 there are pounds or ounces in the weight W ; 
 and M L, containing as many as there are in 
 the body to be supported. Complete then the 
 parallelogram M K N L, and draw its diagonal 
 M N. Then, by the principle of the parallelogram 
 of forces, it is in the direction of this line M N that 
 the resultant of the force P, and of the weight of 
 the body to be supported, will act ; it is, therefore, 
 in the direction of this line that the body will be 
 pressed upon the plane. If this 
 direction be within the surface 
 of the cone of resistance, this 
 pressure will be counteracted by 
 the resistance of the plane, and 
 the body will rest ; if it be with- 
 out it, it will not, and the body 
 will move. The direction of its 
 motion, whether it be up or down 
 the plane, depends upon the di- 
 c A rection of the line M N ; whether 
 
 i it be, as in the first figure, up- 
 
 wards, in respect to the surface of the plane, or as 
 in the second, downwards.* 
 
 49 
 
 176. THE MOVEABLE INCLINED PLANE. 
 
 Suppose an inclined plane A B C to be free to 
 move along the surface on which its base B C rests, 
 and let it be pressed along it by a force P, acting 
 in a direction perpendicular to its back until it en- 
 
 * See " Mechanics applied to the Arts," p. 49. 
 o 2 
 
196 ILLUSTRATIONS OF MECHANICS. 
 
 counters and presses against a mass M, which resists 
 its farther progress. The pressure of the surface 
 fig. 50. o f the plane upon M is pro- 
 
 duced by the resistance Q, of 
 the mass on which the base 
 P of the plane rests, and by the 
 pressure P on its back, and it 
 is equal to the resultant of 
 ^ these two pressures. Suppose 
 
 that P is so increased, as to make it sufficient just 
 to overcome the resistance of the mass M, and to 
 cause the two surfaces to slide upon one another ; 
 the direction of this resultant pressure of the plane 
 upon the mass must then be just without the cone 
 of resistance, and in the same direction must be the 
 opposite pressure R of the mass M, upon the plane. 
 We know, then, what must be the direction of the 
 pressure, whatever may be its amount, which causes 
 the resistance to yield.* The amount of this force 
 is dependent upon the nature of the resistance of 
 the mass. In the cases about to be described, in 
 which the moveable inclined plane is used under 
 the forms of the screw and the wedge, the resistance 
 commonly results from the cohesion of the parts of 
 the mass, which must be overcome, before the plane 
 can move. 
 
 * 177. THE SCREW. 
 
 The surface of the plane, above and below that 
 part of it M (see the last figure), on which the 
 
 * See "Mechanics applied to the Arts," p. 52. 
 
THE SCREW. 197 
 
 mass rests, has nothing to do with its equilibrium 
 upon that part, or with its pressure exerted upon it 
 thus, for instance, the parts A M and B M above and 
 below M, might in any way be altered, provided 
 only that part were left on which M rests, without 
 at all affecting the circumstances of the equilibrium 
 or the pressure. These, manifestly, only concern 
 themselves with that portion of the plane on which 
 the body is actually resting. Imagine, then, that, 
 in the preceding figure, these two portions of the 
 plane, A M and B M, are twisted round so as to 
 convert the base B C into a circle, and make the 
 two points B and C to meet; 
 fig. 51. tne pi ane w i}[ then assume the 
 
 form represented in the accom- 
 panying figure, and the circum- 
 stances under which it exerts 
 its pressure upon the mass M 
 will be precisely those of the 
 thread of a screw. 
 
 The thread of a screw is, in point of fact, the 
 surface of an inclined plane wound round a cylin- 
 der. It is pressed against the resisting mass M, 
 which it is intended to move, by the leverage of a 
 screw-driver, a winch, or an arm % which, giving to the 
 screw a tendency to turn upon its axis, communi- 
 cates to its surface a pressure P, which is parallel 
 to its base, and therefore perpendicular to the back 
 of the inclined plane, from the curving of which it 
 may be supposed to result. The resistance Q per- 
 pendicular to the base of the plane, or parallel to 
 the axis of the screw, is supplied by the resistance 
 of the mass on which the extremity of that axis 
 o 3 
 
198 ILLUSTRATIONS CF MECHANICS. 
 
 turns. If this resistance be not sufficient to supply 
 the requisite force to move the mass M, then the 
 point of the mass on which the extremity of the 
 axis turns yields, and the screw enters into the mass. 
 Of this kind are the screws used by carpenters, and 
 tools, such as gimblets and augurs, which make their 
 way into timber by means of the screws at their ex- 
 tremities. In all these, it is necessary that the depth 
 of their thread, and the distance of their conse- 
 cutive threads, should be enough to cause the fibre 
 of the wood, which represents the mass M, to oppose 
 such a resistance as shall not be overcome, before 
 the mass on which the extremity of the axis of the 
 screw turns yields. Such screws should, therefore, 
 have deep and distant threads. 
 
 The use of the common carpenter's screw is, 
 commonly, to oppose itself to any force which may 
 tend to tear asunder the pieces of timber which it 
 screws together. To this tendency the adhesion of 
 the fibres of the wood, which it receives between 
 its threads, and the strength or tenacity of the 
 screw itself, oppose themselves. If either of these 
 fail, the attachment is broken, in the first case by 
 the tearing out of the screw, in the other by the 
 tearing of it asunder. 
 
 Now it is evident that the greatest economy of the 
 material of the screw will be attained, when these two 
 liabilities to failure are just alike, so that the screw 
 is exactly upon the point of being torn asunder 
 when it is on the point of being torn out; for 
 any strength beyond this will not prevent rupture, 
 nor have any tendency to prevent it, or to increase 
 the strength. Screws are now commonly made 
 
THE WEDGE. 
 
 199 
 
 with reference to this proportion. With square 
 threads, the inclination of the thread is about 7, 
 and with angular threads about 3J. The depth of 
 the thread is usually made equal to about half the 
 distance between two threads.* 
 
 *178. THE WEDGE. 
 
 The wedge is a double moveable inclined plane, 
 presenting two faces to two resistances to be over- 
 jig. 52. come. In the accompanying figure, the 
 points Q and Q', are supposed to be the re- 
 sisting points upon the wedge represented 
 in it ; and P is the direction of the force 
 acting upon the back of the wedge, to 
 drive it ; and may be supposed to include 
 the weight of the wedge. These three 
 forces are in equilibrium. Moreover when 
 the force P is on the point of driving 
 the wedge, so that the points Q and Q' of 
 it are on the point of slipping upon the 
 resisting surfaces, then the resistances at those 
 points, have their directions accurately in the sur- 
 faces of the cones of resistance there. These di- 
 rections Qn and Q! n are therefore known. If 
 therefore the force P be known, the amounts of the 
 resistances may be determined by the principle of 
 the parallelogram of forces. And conversely, if 
 the amounts of the resistance Q and Q' be known, 
 
 * For a more complete discussion of the theory of the screw, 
 the reader is referred to the " Mechanics applied to the Arts," 
 p. 99. 
 
 o 4 
 
200 ILLUSTRATIONS OF MECHANICS. 
 
 the amount of the force P necessary to overcome 
 them, will be known. 
 
 By applying the principle of the parallelogram 
 of forces to this case, it will become evident, that 
 the sum of the forces Q and Q' is always essentially 
 greater than P ; and that in the case in which the 
 angle Q n Q' is greater than a right angle, each of 
 these forces by which the wedge acts, from its two 
 sides, upon the two resistances, is greater than the 
 force P, by which it is impelled. This case occurs, 
 when the vertical angle of the cone of resistance, 
 and the vertical angle of the wedge, are together 
 less than a right angle. 
 
 The great practical advantages in the use of 
 the wedge, are, however, these, that it admits of 
 being driven by impact^ and that when its vertical 
 angle is small enough, it retains every new position, 
 between the resisting surfaces, into which it is 
 driven. 
 
 It is especially the first of these properties which 
 gives to the wedge its marvellous power. It will be 
 shown in a subsequent part of this work, that any 
 force of impact is infinitely great, as compared with 
 any force of pressure. Now the resistances of the 
 surfaces Q and Q' are of the nature of forces of 
 pressure, they necessarily therefore yield to any 
 force of impact communicated to the wedge ; and 
 it is a second and scarcely a less useful property of 
 the wedge, that every such yielding and separation 
 of the surfaces between which it acts, it takes ad- 
 vantage of, and renders permanent. 
 
THE WEDGE. 201 
 
 * 179. THE CIRCUMSTANCES UNDER WHICH A 
 WEDGE WILL NOT BE FORCED BACK BY THE 
 TENDENCY OF THE SURFACES BETWEEN WHICH 
 IT IS DRIVEN TO COLLAPSE. 
 
 Suppose the wedge to be in contact with the 
 surfaces between which it is driven, at a great 
 fig. 53. number of points. Let P and P' 
 
 be the pressures with which two 
 of those points similarly situated, 
 on its opposite faces, tend to col- 
 lapse, and to drive back the wedge. 
 The pressure P, being propagated 
 p through the mass of the wedge, 
 will press the opposite face A B 
 upon the surface with which it is 
 in contact at Q ; and the pressure 
 P', the face AC upon Q'. If, then, the directions 
 PQ and P'Q' be without the surfaces of the cones 
 of resistance at those points, the wedge will be 
 driven back ; if they be within the cones of resist- 
 ance, the forces PQ and P'Q' will be wholly sus- 
 tained by the resistances at Q and Q', and the 
 wedge will retain its position. The tendency of 
 each surface to collapse being supposed to be 
 exerted in a direction perpendicular to that surface, 
 so that the forces P and P' are respectively per- 
 pendicular to the faces AC and AB of the wedge*, 
 
 * It will be observed that the wedge being no longer sup- 
 posed to be on the point of being driven either way, the forces 
 P and P' have no longer their directions necessarily upon the 
 surfaces of the cones of resistance. 
 
202 ILLUSTRATIONS OF MECHANICS. 
 
 it may easily be shown (see Mech. app. to the Arts, 
 p. 55.) that the directions of PQ and P'Q' will be 
 within the cones of resistance, and that these forces 
 will not therefore expel the wedge, provided its ver- 
 tical angle A be less than the limiting angle of resist- 
 ance, or less than half the vertical angle of the cone 
 of resistance. 
 
 A wedge will be of little or no use unless it be 
 made, subject to this law. Thus, for instance, 
 adopting the experiments of Mr. Rennie (which 
 those of M. Morin do not however sanction), it 
 appears that a wedge of oak to be driven into oak, 
 and to keep any position into which it is driven, 
 should not have a vertical angle of more than 8 
 Adopting, however, the experiments of M. Morin, 
 we may assign to it a vertical angle of 31. It is 
 greatly to be regretted that no experiments have 
 been made, in this country, on a sufficiently exten- 
 sive scale, or with sufficient precautions for accu- 
 racy, to enable us to pronounce on these opposite 
 results. 
 
 *180. NAILS. 
 
 When the angle of a wedge is equal to its limit- 
 ing angle of resistance, in respect to the surfaces 
 between which it is driven, the tendency of these 
 surfaces to collapse will be. upon the point of 
 expelling it ; when it is less than this limiting angle, 
 the application of a certain force will become 
 necessary to expel it, it must be drawn back The 
 directions of PQ and P'Q' being within the cones 
 of resistance at Q and Q', a force must act upwards 
 
NAILS. 203 
 
 at A ; or, which is the same thing, it must be 
 applied to draw up fhe back of the wedge CB, so 
 as, combining with the pressures P and P', to give 
 them a more oblique direction at Q and Q x , and 
 bring them there without the cones of resistance. 
 
 The smaller is the angle A of the wedge, the 
 further will the directions of P and P' be within 
 the cones of resistance; and the greater will be 
 the force requisite to bring them without the 
 cones, and to extract the wedge. Of this class 
 of wedges, with exceedingly small vertical angles, 
 are NAILS. 
 
 A table will be found in the appendix containing 
 the results of experiments, made by Mr. Sevan, on 
 the forces necessary to extract nails of different 
 sizes, driven into different substances. It is evident 
 that the length of the nail will greatly increase the 
 force necessary to extract it, increasing rapidly the 
 number of points P, by which it sustains the pressure 
 of the surfaces into which it is driven. 
 
 Nails, as well as screws, are made with the 
 greatest economy of their material, when they are 
 made of such a thickness, that the force necessary 
 to tear them asunder is exactly equal to that ne- 
 cessary to draw them. Any additional thickness 
 would evidently have no effect in preventing the 
 separation of the pieces of wood which they fix 
 together, and would therefore be useless. 
 
204< ILLUSTRATIONS OF MECHANICS. 
 
 181. THE CIRCUMSTANCES UNDER WHICH AN 
 
 EDIFICE OF UNCEMENTED STONES is OVER- 
 THROWN. 
 
 An edifice built up with uncemented stones may 
 fall, either by the turning of some of its stones 
 on the edges of one another, or by their slipping 
 upon one another. 
 
 These two cases are represented in the accom- 
 panying cuts. In the first, an arch is seen to be 
 fig 54., fig- 55. 
 
 1J 
 
 falling by the turning of its voussoirs or arch-stones, 
 at the crown, upon the upper edges of one another, 
 and of those at the haunches, upon their lower edges. 
 In the second figure, an arch falls by the sliding of 
 the arch-stones near the abutment downwards, and 
 by the sliding of those near the crown upwards. 
 
 The last case is of rare occurrence ; such is, for 
 the most part, the friction of the surfaces of the 
 stones used in construction, that their slipping upon 
 one another is a contingency against which few, if 
 any, precautions need be taken.* 
 
 It is by the turning of certain of its component 
 masses upon the edges of others, that an edifice for 
 
 * The question or the slipping of the voussoirs upon one 
 another, was a few years ago considered to involve the whole 
 question of the stability of the arch. 
 
THE EQUILIBRIUM OF AN EDIFICE. 205 
 
 the most part shows symptoms of failure. An ex- 
 ample presents itself in the dome of St. Peter's at 
 Rome. The walls of that mighty structure have 
 in many places yielded under the outward thrust 
 which they have to bear. Numerous cracks are 
 apparent in them, and they have especially opened 
 on the outside, about the 
 haunches V V, and on the 
 inside, about the springing 
 D D of the dome. To coun- 
 teract this tendency of the 
 walls to turn at 'the haunches 
 on their internal, and at the 
 base on their external edges. 
 Vanvitelli caused, in the year 
 
 174-8, immense girdles of iron 
 
 r.'v'y \i v;| to be placed round the 
 
 haunches of the dome at V V ; to which others^ of 
 great strength, have since been added. It is by a 
 similar contrivance that Sir Christopher Wren has 
 strengthened the dome of St. Paul's. 
 
 * 182. THE CONDITIONS OF THE EQUILIBRIUM OF 
 AN EDIFICE OF UNCEMENTED STONES. 
 
 Let the extreme stone A D, of an edifice of un- 
 cemented stones be supposed, as in the accompanying 
 figure, to have, impressed upon it, any given force 
 P. Besides this force P, the stone is acted upon by 
 gravity, which may be supposed to be collected in its 
 centre of gravity Let the resultant (art. 138.), 
 of these two forces be imagined to be taken. This 
 resultant will represent the whole force by which the 
 
206 ILLUSTRATIONS OF MECHANICS. 
 
 first stone is pressed upon the second. If this result- 
 ant have its direction anywhere within the edges, of 
 fig. 57. the joint or surface of contact, of the first 
 stone with the second, the one will rest 
 upon the other ; if not, it will turn over 
 upon it. Let it be supposed to rest upon 
 it, and let us proceed to consider the con- 
 ditions of the equilibrium of the second 
 stone. This second stone may be con- ' 
 sidered to have its upper surface acted 
 upon by the resultant force just spoken 
 of, and this to be the only force pressing 
 it downwards, besides its own weight 
 collected in its centre of gravity. If then 
 a second resultant be taken, being that of two forces, 
 of which the first resultant is one, and the weight 
 of the second stone the other, then this second re- 
 sultant will be that force by which the second stone 
 may be supposed to be pressed upon the third. If 
 its direction lie within the edges of the joint of the 
 second and third stones, the second will rest upon 
 the third ; if not, the superstructure will turn upon 
 the third stone. Similarly, if a third resultant be 
 imagined to be taken, being that of two forces, of 
 which one is the second resultant and the other the 
 weight of the third stone, then this third resultant 
 will be that force by which the third stone is pressed 
 upon the fourth ; and the conditions of the equi- 
 librium of this third stone are, that this resultant 
 shall have its direction within the edges of the joint 
 of the third and fourth stones ; and so on of the 
 rest. 
 
 Thus then the great condition, that the structure. 
 
THE EQUILIBRIUM OF AN EDIFICE. 207 
 
 shall not be overthrown by the turning over of any 
 one of its stones upon the edge of the subjacent 
 stone, is included in this that none of the re- 
 sultants spoken of above, shall have its direction 
 beyond the edges of the surface, by which the 
 stone, to which it corresponds, touches the sub- 
 jacent stone. Now let us suppose that the inter- 
 sections of all these resultants, with the planes of 
 the joints of the successive stones, are, by some 
 mathematical investigation, found ; and let a line be 
 imagined to be drawn, passing through all these 
 points of intersection. That line is called THE 
 LINE OF RESISTANCE. It is a curved line, whose 
 form may be completely determined in every case, 
 by the methods of analysis.* 
 
 If this curve, so determined, be found to have 
 its direction anywhere beyond the joints of the 
 stones, that is, if at any of those joints the curve 
 passes without the mass of the stone, the edifice 
 will, at that joint, be overthrown. If the curve 
 nowhere lie without the mass of the edifice, it will 
 nowhere be overthrown by the turning of its stones. 
 
 That none of them may slip, or that the second 
 condition may be satisfied, it is further necessary, 
 that none of the resultants spoken of in the com- 
 mencement of the article, should have its direction 
 without the cone of resistance of its corresponding 
 
 * For the analytical discussion of this curve, and of all the 
 facts stated in this and the following articles, the reader is re- 
 ferred to a paper by the author, in the third volume of the 
 " Cambridge Philosophical Transactions," part 3.; and to a 
 a second, in the sixth volume, part 3. The theory stated above 
 was for the first time given in the former of these papers. 
 
208 ILLUSTRATIONS OF MECHANICS. 
 
 joint. These two conditions include all that is re- 
 quired to the equilibrium. 
 
 * 183. THE LINE OF RESISTANCE IN A PIER. 
 
 In an upright pier or ivall, the line of resistance 
 is the geometrical curve called the hyperbola.* 
 The position and magnitude of this hyperbola may 
 readily be determined by the following construc- 
 tion. Resolve the force P (see the last figure), which 
 acts upon the summit of the pier, into two others, 
 by the method explained in article 139, one of 
 which two is in a vertical, and the other in a hori- 
 zontal, direction, 
 
 Calculate the height of a mass, which being of 
 the same substance, and the same thickness as the 
 pier, shall have a weight equal to the vertical force 
 of these two, and let this height be A T. Calculate, 
 in like manner, the height of a mass whose weight 
 shall equal the horizontal force, and let this height 
 be A S. t Take B, the centre of the width of the 
 pier, and set off B K, equal to A S. Draw then 
 the vertical K C. C will be the centre of the 
 hyperbola, and the vertical C K E will be its asymp- 
 tote. Now the curve of an hyperbola always ap- 
 proaches, but never touches, its asymptote. The 
 curve of resistance always then approaches, but 
 never touches, the line C E ; and if this line lie, as 
 in the figure, within the mass of the sphere, then 
 the line of resistance, never passing the line C E, 
 
 * Memoir on theory of equilibrium of bodies in contact, 
 " Cambridge Philosophical Transactions, vol. vi. part 3. 
 
 f The dotted lines in the figure represent the two imaginary 
 masses here spoken of. 
 
THE PIER. 
 
 209 
 
 can never cut the outward surface of the pier ; and 
 however tall it may be, the pier can never be over- 
 thrown by the action of this force. Moreover 
 (and this is a remarkable feature of the theory), the 
 pier will bear this insistent pressure P, wherever, 
 in A K, it is applied parallel to its present direction ; 
 the position of the centre of the hyperbola C, not 
 being changed by any alteration in the point of ap- 
 plication of that pressure, but only in its magnitude. 
 
 *184. THE GREATEST HEIGHT TO WHICH A PIER 
 CAN BE BUILT, SO AS TO SUSTAIN A GIVEN 
 PRESSURE UPON ITS SUMMIT. 
 
 If A S be greater than half the width of the 
 pier, or if K lie beyond D, then there will be 
 some point in the outward surface or extrados of 
 the pier, where the line of resistance will cut it ; 
 and there will, therefore, be a certain height beyond 
 which the pier cannot be carried, without being 
 overthrown. This height is thus readily determined. 
 Let P be, as before, the point where the insistent 
 pressure intersects the summit of the 
 ^ pier, and let A S, and A T, and B K, 
 be taken as before ; join U K, and 
 through P draw P Z, parallel to U K. 
 Z will be the point where the line of 
 resistance cuts the extrados, and will 
 indicate the greatest height to which 
 the pier can be carried, without being 
 overthrown ; or, if it be carried higher, 
 then is this the point to which an in- 
 clined buttress should be built to sup- 
 port it. 
 
 p 
 
 fig- 58. 
 
210 ILLUSTRATIONS OF MECHANICS. 
 
 185. THE STRAIGHT ARCH, OR PLATE BANDE. 
 
 If stones be placed side by side, horizontally, and 
 supported at their extremities, as in the accom- 
 fig. 59. panying figure, they con- 
 
 stitute a straight arch or 
 plate bande. If such a 
 structure be supposed to 
 rest by its inferior angle 
 A, at either extremity, 
 ^against an immoveable 
 abutment, the following construction will determine 
 the direction and amount of its pressure upon 
 that abutment. Divide its length into two equal 
 parts in I, and divide ID again into two equal 
 parts in L ; join A to L ; AL will be the direc- 
 tion of the pressure. Take DF equal to AL ; the 
 imaginary mass DC, shown by the dotted lines, 
 having the same width and thickness with the 
 straight arch, and half the length, and being of the 
 same material, will then have its weight exactly 
 equal to the amount of the whole pressure A upon 
 the abutment. If DE be taken equal to DL, 
 the weight of the mass DH will equal the hori- 
 zontal portion of the force A, or the outward 
 thrust.* 
 
 * For the analytical formulae on which this construction, 
 and that in the next article, are grounded, the reader is re- 
 ferred to the paper on the equilibrium of bodies in contact 
 before alluded to. 
 
THE ARCH. 
 
 211 
 
 *186. To FIND THE GREATEST HEIGHT OF THE 
 
 PIERS, OF A GIVEN WIDTH, WHICH WILL SUP- 
 PORT A\ STRAIGHT ARCH OF GIVEN DIMENSIONS. 
 
 Let A I B be the straight arch to be supported, 
 
 and A K the given width of the piers. 
 
 Divide A B into two equal parts in C : upon 
 jig. 60. A C describe a semicircle, 
 
 and measure off A D equal 
 to A K, so as to cut the 
 circumference of this semi- 
 circle in D : produce AD, 
 and let it intersect the 
 vertical line through C in 
 E : measure off E F equal 
 to A I, and A G equal to 
 A B : join D F, and draw 
 GH parallel toDF; then 
 
 A H will be the extreme height of the pier. Being 
 
 of any less height, it will stand firmly ; being of any 
 
 greater, it will be overthrown. 
 
 *187. THE ARCH. 
 
 The most useful and the most interesting appli- 
 cation of the theory of the line of resistance, is that 
 which may be made of it, to the conditions of the 
 equilibrium of the arch. Any detailed discussion, 
 of a subject of so much difficulty, is, however, be- 
 yond the scope of this treatise.* It may, however, 
 
 * The reader is referred to the author's memoir in the Cam- 
 bridge Philosophical Transactions, vol. vi. part 3., and to his 
 elementary treatise on " Mechanics applied to the Arts," article 
 185. 
 
 p 2 
 
 A 
 
 C 
 
 
 
 
 B 
 
 
 ;\ ^ 
 
 V V O' 
 
 
 
 
 
 
 3\ | 
 
 
 
 
 
 
 Ef 
 
 s 
 
 
 
 
 
 i 
 
 
 ? 
 
 
 
 
 A 
 
 
 
 \l 
 
 
 
 ,/ 1 
 
 
 
 
 
 
 11 ! 
 
 
 
 
 
212 
 
 ILLUSTRATIONS OF MECHANICS. 
 
 be stated as a general condition of the line of re- 
 sistance in the arch, that it touches the intrados, 
 or inner surface of the arch, on both sides at its 
 haunches ; and that afterwards at lower points, it 
 cuts the extrados, or outer surface of the arch. If 
 some resistance, of an abutment or pier, be not op- 
 posed at this last point to the pressure, the whole 
 of which acts there, the arch will be overthrown. 
 If it be supported there by a pier, the line of 
 resistance passes into the pier, and assumes a new 
 character and direction ; that direction having a 
 general tendency towards the back or outer surface 
 of the pier. If by reason of the comparatively 
 small height of the pier, the line of resistance does 
 not any where reach the back of the pier, but inter- 
 sects its base, then the pier will stand. If on the 
 contrary the height be, as in the accompanying 
 
 fig. 61. 
 
 figure, so great, as to cause the 
 line of resistance to cut the back of 
 the pier at some point above its 
 base, then the pier will be over- 
 thrown, and the arch will fall. 
 When the arch falls, the line of 
 resistance is made to cut the in- 
 trados at the points m m in the 
 haunches, where before it touched 
 it. These points are called the 
 AB points of rupture. The line of re- 
 sistance, thus cutting the intrados of the arch at 
 m, m, the direction of the whole pressure is made, at 
 those points, to act beyond the joints of the stones 
 there ; so that it causes the stones there to turn upon 
 their lower edges, opening at their upper edges in 
 
THE ARCH, 213 
 
 the extrados at n and n. Besides touching the in- 
 trados at the haunches m m, it is another general 
 characteristic of the line of resistance, in the state 
 of the equilibrium of the arch, that it touches the 
 extrados over the crown at N, and that when the 
 arch is falling, it is made to cut the extrados there : 
 so that the pressure, there also, acting beyond the 
 fig. 62. 
 
 joints of the stone, causes them to turn, but in this 
 case on their superior, instead of their inferior 
 edges. The arch then opens at the crown, at its 
 intrados in M ; and thus it falls, separating itself 
 into four distinct parts. These are the general 
 conditions under which an arch may be understood 
 to fall, by the too great height, or insufficient weight 
 of its piers, in respect to the load it bears on its 
 crown. There is yet; however, another condition 
 which may bring about its overthrow. It may be 
 so overloaded about its haunches as entirely to alter 
 the direction of its line of resistance ; to flatten this 
 line at the top and give it two elbows on either side 
 of the crown ; so as to cause it to cut the intrados 
 instead of the extrados at the crown, and the extrados 
 at two points, a short distance on either side of the 
 crown ; the points where it touches the intrados, being 
 by this process thrown much lower down upon the 
 arch. 
 
 The arch will in this case fail, as shown in the 
 p 3 
 
2H ILLUSTRATIONS OF MECHANICS. 
 
 accompanying figure, by the rising of its crown, and 
 
 the falling in of its sides. The great art of arch 
 
 jig- 63. 
 
 building consists in so loading the arch as to secure 
 it against either of these contingencies. It is one of 
 the most important and the most difficult problems 
 of practical mechanics. 
 
 *188. THE SETTLEMENT OF THE ARCH. 
 
 Whilst the stones of an arch are being placed 
 together, they are supported upon a frame of wood, 
 whose upper surface is of the exact form of the 
 arch to be constructed. This frame, called a centre, 
 is supported upon wedges, and it is not until its 
 removal, by the knocking away of these wedges^ 
 that the arch stones are allowed to bear upon one 
 another. This process of removing the centre is 
 called striking it. From a very early period in 
 arch building, it was observed that, after the striking 
 of the centre, when the whole pressure of the arch 
 was, for the first, thrown upon the stones which com- 
 pose it, certain motions took place among them. 
 To ascertain what these motions were, at the bridge 
 of Nogent sur Seine, Perronet caused three straight 
 lines to be cut in the stones, upon the face of the 
 arch, before the striking of the centre, one hori- 
 zontally above the crown, and two others, equally 
 
PULLEYS. 
 
 215 
 
 inclined to it, on either side, beginning from the 
 
 abutments. These lines are represented by the 
 
 straight lines in the accompanying figure. After 
 
 fig. 64. 
 
 the striking of the centres they altered their forms, 
 and became the curved lines which are seen cross- 
 ing the others. 
 
 The curvature of these lines plainly shows, that, 
 after the striking of the centres, the arch stones 
 above the crown, and from the crown for some dis- 
 tance towards the haunches, descended; but that 
 beyond a certain point in the haunches, and from 
 thence to the abutments, they ascended. These 
 points where the pressure of the arch changes from 
 a pressure downwards, in respect to the faces of 
 the voussoirs, to a pressure upwards, correspond to 
 the points of rupture spoken of in a preceding 
 article. 
 
 189. PULLEYS. 
 
 The accompanying cut represents the different 
 systems of pulleys which are commonly used. The 
 pulley may be described as a circle of wood or iron, 
 moveable round an axis which passes through the 
 centre of the circle, and having a groove in its cir- 
 cumference or edge, round which is wound the 
 string whose tension it is the use of the pulley to 
 p 4< > 
 
216 ILLUSTRATIONS OF MECHANICS. 
 
 direct and apply. This string passing round one 
 half of the pulley (as in the first of the above 
 
 fig. 65. 
 
 figures), and fixing itself upon it by friction, so 
 that it cannot be moved without turning the pulley, 
 it is evident that its effect, upon each side of 
 the pulley, must be the same as though it were 
 actually fastened to its circumference at the points 
 where it leaves it. Now, these points are equidistant 
 from the axis of the pulley, and that forces, applied, 
 at equal perpendicular distances from an axis, may 
 balance one another about it, it is manifestly ne- 
 cessary (by the principle of the equality of moments), 
 that they should be equal. Thus then it appears 
 that the two weights which balance themselves on 
 the first of the pulleys shown in the figure, must be 
 
PULLEYS. 217 
 
 equal.* When the two equal weights P thus ba- 
 lance themselves on this pulley, which is called the 
 fixed pulley, it is evident that the hook from which 
 it is suspended, must sustain a pressure equal to 
 2 P, together with the weight of the pulley itself. 
 
 The second system shown in the figure, is com- 
 posed of two pulleys, one of which is fixed and the 
 other moveable,and this last supports from its centre, 
 the weight to be raised. The same string passes round 
 both pulleys, and supports the power P. By the 
 same reasoning as in the last case, it appears that this 
 string must, on both sides of both pulleys, sustain 
 a tension equal to P, so that at the point where, 
 ultimately, one extremity is fastened to the beam, a 
 force P, would hold it. Thus it is clear, that the 
 moveable pulley and its appended weight are sup- 
 ported, on either side, by a force equal to P : now 
 these together, will just support a weight equal 
 to 2 P. Thus, then, in this system, called that of 
 the single moveable pulley, a power can be made 
 to support a weight which, including the moveable 
 pulley, is equal to twice that power; or a given 
 weight can be raised by the effort of a force equi- 
 valent to but little more than half that weight. 
 
 In the third system, called the Spanish Barton, 
 there are two moveable pulleys, and one fixed. 
 There are moreover two strings, one of which carries 
 the power, and passing round that moveable pulley 
 which carries the weight, is ultimately attached to 
 
 * The effects of the friction on the axle, and the rigidity of 
 the cord, are not here considered. These, however, greatly 
 influence the result in practice. 
 
218 ILLUSTRATIONS OF MECHANICS. 
 
 the beam. The other string suspends the two 
 moveable pulleys, passing over the fixed one. The 
 first string, having the power P suspended from it, 
 acts exactly as in the last described system of the 
 single moveable pulley, and thus it sustains, of the 
 weight to be raised, a portion equal to 2 P. But 
 this string, passing over the first moveable pulley, 
 produces a tension in the string which suspends 
 that pulley, equal to twice its own amount, or 
 to 2 P, and this tension is ultimately applied to the 
 last pulley, supporting an additional portion of the 
 weight equal to 2 P. Thus on the whole, in this 
 system, a given power will support four times its 
 weight, or a given weight may be raised by a power 
 equal to a little more than one fourth that weight. 
 
 In the fourth system there are as many different 
 strings as moveable pulleys. The first, having a 
 weight P suspended from it, produces a tension of 
 2 P on the second string, which holds down the first 
 moveable pulley. A tension of 2 P thus being pro- 
 duced upon the second string, this going round the 
 second moveable pulley, draws it upwards with a 
 force equal to 4 P, and produces a tension of that 
 amount in the third string ; this third string, in like 
 manner, drawing up the third pulley with a force 
 equal to 8 P, produces that tension in the fourth 
 string, so that, ultimately, the last pulley and weight, 
 are supported by a force equal to 16 P. Or, by this 
 system, a given weight could be raised by a little 
 more than one sixteenth of that weight. Had there 
 been a fifth moveable pulley, but one thirty-second 
 of its amount would have been required to raise 
 
PULLEYS. 219 
 
 the weight. If there had been six, but one sixty- 
 fourth. 
 
 In the fifth system there are as many strings as 
 pulleys. The first string, carrying the power P, 
 supports a portion of the weight equal to P, and 
 produces a tension in the second string equal to 2 P. 
 This second string supports, by this tension, a portion 
 of the weight equal to 2 P, and produces a tension 
 in the third string equal to 4 P. The tension of 4 P, 
 in the third string, causes that string to support a 
 portion of the weight equal to 4 P, and to produce 
 in the fourth string a tension equal to 8 P ; this last 
 tension, again, supports a portion of the weight, 
 equal to 8 P. Thus, then, the four strings support 
 portions of the weight, respectively equal to P, 2 P, 
 4 P, 8 P ; and thus, together, they support a weight 
 equal to 15 P. Had there been a fifth pulley in the 
 system, it would have supported an additional portion 
 of the weight, equal to 16 P ; and the whole weight 
 supported would have been 31 P. 
 
 In the sixth and last system, the same string passes 
 round all the pulleys, and its tension is the same 
 throughout. Thus the weight is borne by six dis- 
 tinct and equal tensions, which together will bear a 
 pressure equal to six times any one of them ; so that 
 by this system a given power will support and raise 
 nearly six times its weight. Had there been another 
 pulley in each block, the weight raised would have 
 been eight times the power. 
 
 This last system, although each additional pulley 
 does not give, in it, the same additional amount of 
 power as in the others, is yet much more convenient 
 
220 ILLUSTRATIONS OF MECHANICS. 
 
 in practice. In the other systems, whilst they raise 
 the weight a given height, the pulleys move through 
 different distances, and unless the strings be very 
 long and the pulleys very wide apart at first, they 
 soon become encumbered with one another. In the 
 last system, the pulleys approach one another only 
 by as much as the weight is raised. 
 
221 
 
 CHAPTER V. 
 DYNAMICS. 
 
 THE FORCE OF MOTION ITS PERMANENCE THE MEA- 
 SURE OF IT THE POINT WHERE IT MAY BE SUP- 
 POSED TO BE COLLECTED. MOTIONS OF TRANSLA- 
 TION AND ROTATION, INDEPENDENT. THE CENTRE 
 
 OF GYRATION. THE CENTRE OF SPONTANEOUS 
 
 ROTATION. THE CENTRE OF PERCUSSION. THE 
 
 PRINCIPAL AXES OF ROTATION. THE FORCE OF A 
 
 BODY'S MOTION is NEVER GENERATED OR DESTROYED 
 
 INSTANTANEOUSLY. ACCELERATING FORCE. GRA- 
 VITATION. CAVENDISH'S EXPERIMENTS. DESCENT 
 
 OF A BODY FREELY BY GRAVITY. ATWOOD*S MA- 
 CHINE. DESCENT OF A BODY UPON AN INCLINED 
 
 PLANE AND UPON A CURVE. THE CYCLOIDAL PEN- 
 DULUM. THE SIMPLE PENDULUM. THE CENTRE 
 
 OF OSCILLATION. KATER^S PENDULUM THE COM- 
 PENSATION PENDULUM. 
 
 190. CERTAIN LAWS COMMON TO THE OPERATION 
 OF ALL FORCES. 
 
 THE force, of which we trace the existence in the 
 material substances around us, is presented under a 
 variety of different forms and different circum- 
 stances 
 
 Thus we find it in the descent of all bodies to- 
 wards the centre of the earth it is then called 
 GRAVITY ; we discover it a pervading principle 
 in the material world, under another form, and 
 
222 ILLUSTRATIONS OF MECHANICS. 
 
 call it ELECTRICITY ; related to this is force 
 under yet another form, which we call MAGNETISM; 
 and there are forces of Adherence, of Attraction, 
 and Repulsion, between the material particles of 
 which all bodies are made up, which are known 
 under the names of CAPILLARY ATTRACTION, CO- 
 HESION, and CHEMICAL AFFINITY. 
 
 Whether the forces which we thus distinguish, by 
 reason of certain differences, in the manner and cir- 
 cumstances of their action, be or be not, different 
 modes of action of the same principle of force, 
 whether they be of the same family, or flow from 
 the same fountain or source, we know not. This, 
 however, we certainly know, that there are LAWS 
 of force which are common to all. 
 
 The development of these LAWS, as they regard 
 the equilibrium of bodies, constitutes the science of 
 STATICS ; as it regards their motion, it is the science 
 of DYNAMICS. We have now then to inquire what 
 are the laws which govern the MOTIONS of material 
 bodies, and what relation exists between these and 
 the FORCES in which they originate. 
 
 191. MOMENTUM, OR THE FORCE OF MOTION. 
 
 It is a matter of continual observation that a 
 moving body becomes, by reason of its motion, ca- 
 pable of communicating motion to another body, or 
 of destroying motion in that body. 
 
 Now that which causes or destroys motion is (by 
 our definition, page 122.) FORCE. 
 
 A body, in the act of changing its place, pos- 
 sesses therefore a principle of force, co-existent with 
 its motion, and dependant upon it. It is a force 
 
THE FORCE OF MOTION. 223 
 
 wholly distinct and different from the force of pres- 
 sure, which belongs to the state of the body's rest. 
 Thus, for instance, the force with which a stone, 
 falling to the ground, strikes it, is wholly distinct 
 and different from that with which, resting upon 
 the ground, it presses it ; the one has wholly ceased, 
 and has been destroyed, before the other begins 
 to operate. 
 
 The force which thus exists in every moving body, 
 which co-exists with its motion, and is dependant 
 for its existence upon its motion, is called its FORCE 
 OF MOTION, or, more frequently, its MOMENTUM. 
 
 192. THE FORCE OF A BODY'S MOTION is PRE- 
 CISELY EQUIVALENT TO THE FORCE EXPENDED 
 IN PRODUCING IT. 
 
 This force of the body's motion is a result of the 
 force which first gave it motion an effect of that 
 cause and the effect and cause are equivalent 
 the force of motion in the body, and the force ex- 
 pended in producing it, are equal things. It is as 
 though a transfer of the principle of force were 
 made from the moving thing into the thing moved ; 
 thus, for instance, if a ball be put in motion by the 
 recoil of a spring, the force with which the spring 
 recoils is not lost, it is but transferred to the ball ; 
 and the ball is then ready to bring precisely the 
 same quantity of force into operation on any other 
 object which it encounters, as the spring did on it. 
 So, too, if the ball were put in motion by the hand, 
 the force expended in the production of its motion 
 will not be lost ; it will only be transferred from the 
 hand to the ball, and the ball will be ready to re- 
 
224- ILLUSTRATIONS OF MECHANICS. 
 
 produce the whole of it, and to cause it to operate on 
 the first obstacle which it encounters. 
 
 It is, of course, here supposed that there is no 
 opposition to the free motion of the ball arising from 
 the resistance of the air, friction, gravity, or any 
 other of the causes which interfere with the motions 
 of bodies on the earth's surface. 
 
 If there be such causes of retardation, their oper- 
 ation will continually destroy a portion of that 
 force of motion in the ball, which was, nevertheless, 
 originally, precisely equal to the force expended in 
 putting it in motion. 
 
 Thus, a billiard ball continually loses a portion of 
 the force with which it was originally struck by 
 reason of the friction of the baize, and the resist- 
 ance of the air which, to move, it must continually 
 displace; and, by this continual destruction of its force 
 of motion, it may eventually be deprived of the 
 whole of it, in which case it is said (improperly) to 
 rest of itself. The same is true of a bowl, which 
 continually loses the force of its motion, as it rolls 
 over the turf ; and of a cannon ball., which, by rea- 
 son of the resistance of the air, and frequent im- 
 pacts, perhaps, on the ground, loses continually the 
 force of its motion, until it becomes, at length, what 
 is called a spent ball. In all these cases, at the 
 commencement of its motion, before any opposing 
 causes came into operation, the force of the body's 
 motion was precisely equal to that expended in pro- 
 ducing it; and it would have been found the longer 
 to retain it, as these causes were more and more 
 completely removed. Thus a smooth ball, rolled 
 over the grass, soon stops ; rolled over the cloth of 
 
THE FIRST LAW OF MOTION. 225 
 
 a billiard table, its motion, and force of motion, are 
 longer continued ; on a smooth plank, or iron plate, 
 yet longer; on a level sheet of ice it suffers but 
 little retardation ; and, if the surface of the ice be 
 continuous, and perfectly smooth, and no wind op- 
 pose the motion of the ball, it will lose very little 
 of its force of motion for a great distance. Thus 
 then we see, that, as the causes of the destruction of 
 a body's motion, and force of motion, are more and 
 more taken away, these approach more to the con- 
 dition of permanence ; and from this we conclude, 
 that if they were completely removed, that condition 
 of permanence would be absolutely attained ; so that 
 if there were no causes of retardation EXTERNAL TO 
 ITSELF, a body's motion and force of motion would 
 continue for ever ; hence the following law. 
 
 193. THERE is NO PRINCIPLE OF DIMINUTION OR 
 DECAY IN THE NATURE OF MOTION ITSELF, 
 OR IN THE NATURE OF THE FORCE OF A 
 MOVING BODY.* 
 This principle is commonly known as the FIRST 
 
 LAW OF MOTION. 
 
 The difficulty of conceiving or admitting it, lies 
 in this, that we observe all those forces of motion 
 which are produced around us, continually to di- 
 minish , and eventually to become extinct, as it 
 appears to us, of themselves. Our own bodies 
 when we have moved them, do not of themselves 
 move on ; fresh efforts must be continually made : 
 our carriages require the continual draught of the 
 horses, and even if we put them on a smooth road 
 * The body is here supposed not to be endued with vital power, 
 Q 
 
226 ILLUSTRATIONS OF MECHANICS. 
 
 of iron, there is required a force continually to 
 impel them : we move a stone with our foot, and 
 but a few steps further on we find it at rest. It is 
 a most wise provision of Providence by which the 
 natural tendency of all these forces to permanence, 
 is thus continually destroyed. Without it the world 
 would scarcely be habitable. Were there no fric- 
 tion to check the superfluous force which we give to 
 our bodies at every step, our state of existence 
 would become one of incessant and involuntary 
 motion : every thing we touched would, from 
 that instant, become an ever-moving body ; every 
 thing not rooted in the earth, would be a sport of 
 the winds, and men would soon desert the land, to 
 dwell on the sea, as the more stable element. Could 
 we diminish the resistance of FRICTION and the 
 AIR to any conceivable extent, and if it were found 
 that, as we diminished these, the motion of a moving 
 body approached continually to a state of per- 
 manence, so that, by thus diminishing the causes 
 of retardation, we could make the motion to differ 
 from a permanent motion, by as little as we chose, 
 this first law of motion would be completely proved. 
 For if there were any sensible diminution of the 
 force communicated to the mass, arising from a 
 failure in its own energies, and independent of the 
 resistances opposed to it, then that diminution would 
 be apparent and sensible when the resistances were 
 so far diminished as to be insensible. 
 
 Unfortunately, however, we cannot diminish the 
 resistances of friction and the air beyond certain 
 limits. As an absolute demonstration, this method 
 therefore fails. Nevertheless, the fact that, dimi- 
 
THE FIRST LAW OF MOTION. 227 
 
 nishing the resistances to motion as far as we can, we 
 find it continually approximating to a state of per- 
 manence, renders it in a high degree probable, that, 
 if we could carry this diminution on indefinitely, mo- 
 tion would approach indefinitely to a state of perma- 
 nence, and that if these resistances could be abso- 
 lutely destroyed, it would become permanent. 
 
 194. ILLUSTRATIONS OF THE PERMANENCE OF 
 COMMUNICATED MOTION. 
 
 It is in the case of a revolving body that we can 
 most effectually diminish these resistances to mo- 
 tion, by causing it to be supported and to turn on 
 a very small surface, as compared with the di- 
 mensions of the body itself ; as for instance, a large 
 wheel round a slender axle, or a large spinning-top 
 on a fine point, by which contrivance the resistance 
 of friction is made to act at a great mechanical dis- 
 advantage, as compared with the force of the body's 
 rotation ; and we may, further, remove the resistance 
 of the air, almost to any degree we choose, by 
 placing the revolving body under the receiver of an 
 air-pump. 
 
 Now, if we thus remove the air from the receiver 
 of an air-pump, and then, without re-admitting it, 
 by some mechanical contrivance, put rapidly in 
 motion under the receiver, a large wheel with an ex- 
 ceedingly small axis, or, better, a large spinning-top 
 with a fine hard point ; we shall find that motion, 
 which would, under other circumstances, soon cease, 
 lasting, apparently unaltered, for hours. And a pen- 
 dulum, delicately suspended on knife edges, and 
 having thus yet greatly less friction to contend with 
 Q 2 
 
228 ILLUSTRATIONS OF MECHANICS. 
 
 than either the axis of a wheel or the point of a top, 
 when once a motion has been given to it, will retain 
 the force of that motion, and continue to oscillate 
 with it for more than a day. Mr. Roberts of Man- 
 chester, is said to have constructed a body which 
 is of such a form and so truly balanced upon a fine 
 point, that, having put it in motion round that 
 point, it would not lose the force o f its motion, but 
 continue to spin with it for 43 minutes. These are 
 all proofs of a tendency to the permanence of mo- 
 tion, and the force of motion which accompanies it, 
 when causes of retardation from without are more 
 or less removed ; that is, of its tendency to absolute 
 permanence, so far as any cause within itself is 
 concerned. It does not die or diminish of itself; 
 there is within it no principle of death or decay,- 
 to cease, it must be operated upon by causes ex- 
 ternal to itself. The proofs hitherto given show, 
 however, only the probability of this truth. It is 
 probable that, since when we continually diminish 
 the external causes of a body's retardation, its mo- 
 tion approaches to a state of permanence, if we 
 were completely to take away those causes of re- 
 tardation, that permanence of motion would be 
 completely attained. But we cannot take away 
 these causes of retardation we cannot completely 
 take away friction and the resistance of the air : 
 we can therefore only speak of what would pro- 
 bably happen if these were completely removed. 
 
 To complete the proof, we must look out for 
 some case of motion, in which there is no friction, 
 and no resistance of the air. Such a motion we 
 cannot find on or near the earth's surface, but we 
 do find it in the heavens. 
 
THE FIRST LAW OF MOTION. 229 
 
 195. THE PERMANENCE OF THE FORCES OF 
 ROTATION OF THE PLANETS, AND OF THEIR 
 TANGENTIAL FORCES OF MOTION. 
 
 The PLANETS all roll in their orbits round the 
 SUN, and their SATELLITES each round its primary 
 planet, without friction, and unopposed by the re- 
 sistance of any fluid atmosphere ; and the motion 
 first communicated to them, the velocity of their 
 first projection, remains, in accordance with the 
 first law of nature, unabated, permanent') from year 
 to year, from century to century. It has remained 
 the same from the period when they first went forth 
 into space, at the mandate of God, to fulfil the de- 
 signs of his providence, and it will remain the same 
 until time is swallowed up and lost in eternity. 
 That force by reason of which each planet moves 
 not directly towards the sun which attracts it, but 
 always nearly at right angles to that direction, is a 
 force the principle of which resides within the pla- 
 net itself: there is no external force to draw it from 
 the path which it has a continual tendency to take 
 towards the sun. The force, whatever it is, which 
 produces this effect, does not emanate from without, 
 but from within the planet itself; it is the force of 
 its motion. 
 
 Were it not, then, in its own nature permanent, 
 but suchj that although unopposed, it would yet 
 gradually, of itself, lose its original vigour and 
 energy, then this force of motion in the planets, be- 
 coming from year to year gradually less, would 
 continually be more and more controlled by the 
 Q 3 
 
230 ILLUSTRATIONS OF MECHANICS. 
 
 attractive power of the sun, so that, from year to 
 year, their orbits would alter their forms, becoming 
 continually ellipses more elongated, until at length 
 the deflecting force of motion in each planet being 
 extinct^ each elliptic orbit would resolve itself into a 
 straight line, and each planet fall directly towards 
 the central sun. 
 
 Now the very contrary of all this we know, by 
 direct observation, to be the real state of things. 
 There is no elongation of the orbit of any planet 
 arising from any such cause. There is no alter- 
 ation whatever in the orbits of any of the planets, 
 except a slight one arising out of the influence 
 of their mutual attractions ; an alteration which of 
 necessity returns perpetually in a cycle, and which, 
 far from indicating an ultimate destruction of the 
 existing system, supplies the most striking evidence 
 of its permanence. 
 
 This is not, however, the only proof of the first 
 law of motion which astronomy offers to us. In 
 the system of the satellites of Jupiter, for instance, 
 the astronomer beholds a beautiful epitome and 
 model, of the great system of the universe. To dis- 
 believe the revolutions of those satellites, he must 
 disbelieve the direct evidence of his senses : and he 
 finds their revolutions from month to month, and 
 year to year, to be same, and the same as they were 
 observed by other astronomers to be, two centuries 
 ago ; the effect of the primaeval impulse, in which 
 the motion of each had its origin, remains then in 
 it unabated unaltered from the beginning. The 
 earth, too, rotates daily upon its axis by reason 
 of a first impulse, given when the foundations of 
 
THE FIRST LAW OF MOTION. 231 
 
 the universe were laid, and not since renewed. No 
 hand is now upon it; no cause now operates to 
 turn it : it turns of itself, with its own innate force 
 the force given to it when it first came into the 
 existing state of its being, the force of its motion. 
 A question then arises Does the effect of that 
 impulse, the force of that original motion, remain 
 unabated, unimpaired, to this day, or does it not? 
 We have before us the evidence of 2000 years, 
 and we thus know with certainty that the earth 
 turns upon its axis now precisely in the same time 
 that it did then : not the slightest appreciable frac- 
 tion of the original force of its motion has in the 
 intervening period disappeared. But this is not all. 
 On this principle of the permanence of commu- 
 nicated force are grounded all the calculations of 
 physical astronomy: these apply to all the phenu- 
 mena of the heavens ; they enter, for instance, 
 largely into the calculation of eclipses, into those 
 calculations of the positions of the moon in refer- 
 ence to certain of the fixed stars by which the 
 navigator determines his longitude, and guides the 
 course of his ship ; and into an infinite variety of 
 others which are every day submitted to the test of 
 observation, and every day verified. Were motion 
 not governed by this law, every one of these calcu- 
 lations would be false. That they are true is in 
 itself, therefore, a sufficient proof of it. Such is the 
 direct evidence of the permanence of unopposed 
 force of motion. 
 
 The indirect manifestation of the existence of 
 the same principle in the things around us, is not 
 less remarkable. 
 
 Q 4 
 
232 ILLUSTRATIONS OF MECHANICS. 
 
 196. ILLUSTRATIONS OF THE PERMANENCE OF 
 THE FORCE OF MOTION. 
 
 There is scarcely any case of motion in which it 
 Cannot easily be traced. The flying of the dust 
 out of a carpet on one side, which is struck on the 
 other, is but an effect of the force of motion com- 
 municated to the dust, in common with the carpet, 
 by the blow, and an indication of its tendency to 
 permanence. 
 
 When a man rides in a carriage or on horseback, 
 with his motion, a force of motion is impressed 
 upon him, which he does not indeed perceive, as 
 long as his carriage or his horse moves with him ; 
 but which, if they be suddenly stopped, may throw 
 him from his seat. 
 
 If he stands upright in a boat, as it approaches 
 the shore, however slowly it may be moving, he 
 will be in great danger of falling if it suddenly 
 ground, because the motion which he before par- 
 took of, in common with the boat, has a tendency 
 to permanence. 
 
 When a man JUMPS FROM A CARRIAGE in mo- 
 tion, unless, in the act of reaching the ground, he 
 commence running, with a velocity at least equal to 
 that of the carriage, he will certainly fall ; for the 
 force with which he was moving, in common with 
 the carriage, will remain in the upper part of 
 his body, whilst in his feet it will be arrested by 
 contact with the ground. 
 
 It is by reason of this tendency to permanence 
 in the force of communicated motion, that a RACE 
 HORSE, whatever efforts he makes to stop himself, 
 
THE FIRST LAW OF MOTION. 233 
 
 cannot be brought up, until he has long passed the 
 goal ; that a man LEAPS farthest when he runs to 
 make his leap ; and that in a SHIP WHICH STRIKES 
 when under sail, upon a rock, every thing is dashed 
 forwards. 
 
 197. OF THE FORCE OF MOTION WHICH TENDS 
 TO OVERTHROW A MOVING BODY, THE EFFECT 
 OF THAT WILL BE THE GREATEST, WHICH EXISTS 
 IN THE HIGHEST PORTIONS OF IT. 
 
 Because, there, the force acts with the greatest 
 leverage, or at the greatest distance from the point or 
 edge about which the whole is to be made to turn, in 
 the act of being overthrown ; and for this reason it is, 
 that a tall person would be much more liable to 
 fall, by reason of such a shock, than a short one : 
 thus also, when a vessel strikes on a rock, a high 
 mast is more likely to go by the board, than a short 
 one supposing its strength to be only the same. 
 It is not uncommon for vessels thus striking to lose 
 all their masts at once. It is for a similar reason, 
 that a man, jumping from a carriage in motion, is 
 in great danger of falling, the force of the motion, 
 existing alike in all parts of his body, is suddenly 
 arrested in his feet, whilst it carries forward the 
 upper portion of his body, and with it his centre 
 of gravity, beyond the limits of its natural pedestal ; 
 all that he can do to avoid this, is to run in the 
 direction in which his body is thus carried forwards, 
 so as to bring his feet beneath it ; otherwise it will 
 leave them behind. 
 
234? ILLUSTRATIONS OF MECHANICS. 
 
 198. DRIVING ON THE HEAD OF A TOOL. 
 
 Another illustration of the permanence of the 
 force of motion may be found in that very common 
 expedient of practical mechanics, by which, when 
 they require the iron portion of one of their tools 
 to fix itself into or upon the wooden parts of it, 
 they put both in motion, and suddenly stop that part 
 into or upon which the other is to be driven. Thus, 
 to drive the head of a hammer firmly upon its 
 handle, they place it loosely upon it, then strike 
 the end of the handle upon the bench, arresting 
 suddenly its motion by the intervention of the 
 bench, by which means the force of motion in the 
 iron head is made to take effect upon the handle, 
 and the two are fixed together. The same expe- 
 dient serves to drive a chisel into its handle ; the 
 handle is suddenly stopped, and by its acquired 
 force of motion, the iron of the chisel drives itself 
 into it. 
 
 199. THE BREAKING OF BODIES BY IMPACT. 
 
 The force of motion exists in every particle of a 
 moving body hence, when such a body is to be 
 brought absolutely to rest, the force of motion 
 must be destroyed in every particle of it. 
 
 Now if a moving body be thus brought to rest 
 by encountering an immoveable obstacle, the motion 
 and force of motion in those parts of it immedi- 
 ately in contact with the obstacle will be destroyed 
 at once.* 
 
 * This expression is used relatively ; it will be shown here- 
 after that force of motion can never be destroyed at once, accord- 
 ing to the accurate meaning of that term. 
 
THE FIRST LAW OF MOTION. 235 
 
 The parts of the body immediately behind them 
 retaining, however, the force of their motion, will 
 press directly on the first those behind these, on 
 them ; and so of the rest, until the momentum of 
 each, in succession, is destroyed by the resistance 
 of those before it.* 
 
 Of the parts which do not lie immediately be- 
 hind the point or points of impact, each would, of 
 necessity, at the instant of impact, separate itself 
 from the rest by reason of its own proper force of 
 motion, and move onwards, as do the particles of 
 a mass of water dashed against an obstacle, were it 
 not for that force, common to all solid bodies, which 
 is called cohesion. If, moreover, the momentum of 
 any one part of the solid be such, as the cohesion 
 of that part to the rest is not sufficient to counter- 
 act, that part will separate from the rest, and a 
 piece is then said to break out of it. 
 
 Sometimes the pressure which the destruction of 
 the force of motion, in some interior portion of the 
 body, produces in this way, overcomes the cohesion, 
 and destroys the internal structure of that portion 
 of the body, without affecting its external form and 
 appearance. Thus, a stone after it has been se- 
 veral times struck against another, although there 
 
 * This entire destruction of the motion will not in reality ob- 
 tain until after several oscillations of each particle for a certain 
 distance on either side of its ultimate position of rest, to which 
 it will continually be brought back by the elasticity of the mass, 
 and carried through it by its acquired momentum ; this last 
 becoming, however, at every oscillation less, it will eventually 
 rest. It is from these oscillations of the particles of bodies about 
 their ultimate positions of equilibrium, that certain bodies 
 become sonorous when struck. 
 
236 ILLUSTRATIONS OF MECHANICS. 
 
 be no external appearance of injury, will afterwards 
 yield to a blow which would not before have 
 broken it. 
 
 200. A JAR OF THE BODY. 
 
 The sudden destruction of motion in the human 
 body, is attended by effects analogous to these. 
 Thus, a person walking carelessly, if he meet with 
 some unevenness of the surface, and his heel come 
 first in contact with the ground, will experience a 
 very painful sensation of the kind called a jar; 
 which is, in point of fact, but the indication his 
 whole nervous system gives, of an unnatural pressure 
 of the different solid portions of his body upon 
 one another ; resulting from a sudden destruction of 
 the force of motion, first in his heel, then, by 
 pressure upon that, in the bones of his leg, then in 
 the successive vertebrae of his back, and lastly in 
 his head each of these having in succession its 
 proper force of motion destroyed, by pressure upon 
 that below it in the series. Of the same nature is 
 the shock which a man feels whose seat is suddenly 
 taken from under him, and it is thus that a man is 
 stunned or perhaps crushed to pieces, who falls 
 upon his legs from a great height. 
 
 201. THE PHENOMENA WHICH ATTEND THE 
 SUDDEN PRODUCTION OF MOTION, ARE ANA- 
 LOGOUS TO THOSE OF THE SUDDEN DESTRUCTION 
 OF IT. 
 
 Thus, if a man be standing upright in a boat, 
 which is suddenly pushed off from the shore, he 
 
THE FIRST LAW OF MOTION. 237 
 
 will probably fall, in the direction from which the 
 boat is movin-g. 
 
 And the reason is this: When the boat first 
 moves, a certain force of motion is communicated 
 to his feet which are in contact with it, and cannot 
 slip along it, whilst no such force exists in, or is 
 propagated to, the upper portions of his body.* 
 Thus, then, his legs will be carried forwards by this 
 force of motion, ;whilst his body retains its position, 
 until by this relative displacement, the centre of 
 gravity of the body is brought beyond the base of 
 the feet, and he falls. 
 
 It is in the same way, that a sharp blow on a 
 man's feet will strike them from under him ; they 
 receiving a motion in which the upper portion of 
 his body does not partake. 
 
 Analogous to the process by which a body is 
 broken in pieces when it is made to impinge upon 
 an immoveable obstacle, is that by which it is 
 broken, when, being itself immoveable, another body 
 is made to impinge upon it. These are all instances 
 of the permanence of the force of motion, once 
 communicated to a body, except it be counteracted 
 by the operation of some force from without. 
 
 Examples like these might readily be multiplied : 
 no person, however, will be disposed to doubt the 
 tendency of communicated motion, and the force ot 
 communicated motion to permanence, who has en- 
 deavoured to stop himself when running, or seen a 
 
 * Since he stands upright the force of motion in a horizontal 
 direction could not propagate itself to the upper portion of his 
 body without propagating itself in a direction at right angles 
 to that in which it acts, which is mechanically impossible. 
 
^8 ILLUSTRATIONS OF MECHANICS. 
 
 race-horse pulled up at the goal, or a skater by 
 trusting to the mere impulse of a communicated 
 motion, glide rapidly over fifty or sixty yards of 
 the surface of the ice, or a loaded carriage de- 
 scend a hill, and by the mere tendency to per- 
 manence of the force of motion communicated to 
 it in its descent, ascend a considerable distance up 
 the next hill, with scarcely any traction of the 
 horses ; or who has seen a pendulum, by the mere 
 tendency to permanence of the force of motion 
 which it acquires in the descending arc of its os- 
 cillation, complete its ascending arc against the force 
 of gravity ; which arc it does not terminate until, 
 by the continual operation of that force of gravity, 
 its force of motion is entirely destroyed, and it falls 
 back, to re-acquire it in a second descent. 
 
 202. THE HAMMER. 
 
 The principle of the permanence oj the force of 
 communicated motion, so far as any cause within the 
 moving body itself is concerned that is of its ab- 
 solute permanence, except in so far as it is counter- 
 acted by some external and opposite force whilst 
 it lies at the very foundation of all just views of the 
 theory, is sufficiently shown, by the above examples, 
 to be a most important element in the practice of 
 mechanics. What is it, in fact, but this which con- 
 stitutes the giant force of impact, and makes the 
 HAMMER a weapon more powerful than any other 
 irresistible in moulding and submitting the various 
 objects around him to the uses and purposes of man. 
 There is no machine comparable to the hammer* 
 
THE FIRST LAW OF MOTION. 239 
 
 The force of heat, indeed, insinuates itself between 
 the pores and interstices of bodies, and operating 
 there, separately, upon their particles, breaks them 
 up in detail but the hammer encounters the ac- 
 cumulated force of their cohesion and overcomes it. 
 The hardest rocks and the most unyielding metals 
 submit to it. If man reigns over inanimate matter, 
 shapes out the face of the earth to his use or to his 
 humour, and puts the impress of his skill and his 
 labour upon the whole face of nature ; it is chiefly 
 with the aid which this mighty force of impact gives 
 him. It is this that clears away for him the trees 
 of the forest that shapes for him the materials of 
 his dwelling that beats out for him the instruments 
 of tillage that digs and hoes up the earth, that 
 after having cut for him his corn, threshes it, and 
 crushes it into flour, that tames for him his cattle, 
 shapes and binds together his waggons and carts, 
 and makes his roads; in short there is no use of 
 society for which this force of impact does not 
 labour, and there is no operation of it which does 
 not manifest this tendency of communicated force 
 of motion to permanence. 
 
 Were there no tendency to permanence in the 
 force of motion which his hammer acquires in its 
 descent, its power on the substance which the arti- 
 ficer seeks to shape out, would only be the same as 
 though he were to lay it gently down upon it ; its 
 impact would be no greater force than the pressure 
 of its weight. So far is this, however, from being 
 the case, that, as it is well known to the workman, a 
 slight blow from the lightest hammer is sufficient to 
 abrade a surface, which the direct pressure of a ton 
 
240 ILLUSTRATIONS OF MECHANICS. 
 
 weight would not make to yield. There is no force 
 in nature comparable to that of impact. 
 
 203. IF THE CAUSES WHICH TEND TO DESTROY 
 THE FORCE OF A BODY'S MOTION BE CONTINU- 
 ALLY COUNTERACTED AS IT MOVES ON, THEN 
 IT WILL MOVE UNIFORMLY. 
 
 Thus, if the friction which would otherwise gradu- 
 ally destroy the motion of a CARRIAGE, be continually 
 neutralised, by the traction of the horses, it will roll 
 on uniformly. The friction of the road would not 
 instantly, and at once, destroy the force with which 
 the carriage moved, if left to itself, but by little and 
 little. This friction is, therefore, at any instant, less 
 than the force of the carriage's motion, and to over- 
 come it, requires less effort of the horses, than to 
 communicate, at first, its motion to the carriage. 
 It is the permanence of this originally communi- 
 cative force of motion which causes the carriage to 
 move on, although the horses, at every instant, 
 exert a much less force than that necessary to 
 move it from rest. 
 
 204. THE TENDENCY OF THE FORCE OF MOTION 
 TO PERMANENCE is A TENDENCY TO PERMA- 
 NENCE IN THAT PARTICULAR DIRECTION IN 
 WHICH THE BODY MOVES, OR IN WHICH THE 
 FORCE ACTS. 
 
 Thus, if a man run rapidly, and, without at 
 all abating his speed, so as to diminish the actual 
 force of his motion, attempt to alter suddenly the 
 
THE FIRST LAW OF MOTION. 241 
 
 direction in which he runs, he will find that he has 
 a considerable force to resist and destroy before he 
 can do this a force tending to carry him straight 
 forwards in the path in which he was before moving 
 and the force which he thus has to counteract, 
 he will find to be greater or less, as his turn is 
 more or less abrupt, and the previous force of his 
 motion greater or less. If he wish to turn directly at 
 right angles to his former path, he will find that he 
 must destroy absolutely all the force of his previous 
 motion an effort which is therefore precisely the 
 same as though he were brought to a complete 
 stand still ; and if he has to proceed with the same 
 speed in his new path, he will have to reproduce 
 all this force of motion in that path. 
 In the same manner, if his new path 
 be in any way backwards, or making 
 an angle less than a right angle, with 
 the path in which he has been run- 
 ning, as, for instance, if it be repre- 
 sented by B P, in the figure, A B being his previous 
 direction, then, as before, all the force of his motion 
 in A B must be destroyed, and reproduced in B P ; 
 and in point of fact, the quantity of force which he 
 must destroy to take up a new direction is the same, 
 whatever that direction may be, provided that it 
 Ues within the right angle ABC; being the whole 
 force of the motion in A B. 
 
242 ILLUSTRATIONS OF MECHANICS. 
 
 205. ILLUSTRATIONS OF THE TENDENCY OF MO- 
 TION TO PERMANENCE, IN RESPECT TO ITS DI- 
 RECTION. 
 
 A man whose HORSE STARTS when he is riding 
 rapidly, falls over his head, because his motion 
 tends to permanence, in the direction in which he 
 was moving. A SHRAPNEL SHELL, when it bursts, 
 although, if it were at rest, it would scatter the 
 bullets with which it is filled in all directions, being 
 in motion, gives to each a force of motion, which, 
 operating conjointly with, and modifying the forces, 
 whose tendency is to disperse them, throws them 
 all more or less forwards. 
 
 COURSING derives all its interest from the 
 doubling of the hare, which finds a protection from 
 the greater swiftness of the greyhound, in continu- 
 ally changing the direction of its motion a change 
 which the latter is less able to make, by reason of 
 his greater weight and greater swiftness producing 
 a greater force of motion, and the greater length of 
 his legs rendering him the less able to check it. 
 Independent of these causes, the principle furnishes, 
 moreover, a protection to the pursued from the 
 pursuer. The former may thus be made always to 
 pass the point where the latter turns to him, unex- 
 pectedly. 
 
 206. THE MEASURE OF MOMENTUM, OR THE 
 
 FORCE OF MOTION. 
 
 Force being that which produces motion in a 
 body, it is easy to conceive that that force must 
 
THE MEASURE OF MOMENTUM. 243 
 
 be double, which produces twice the motion in that 
 body, that triple, which produces three times the 
 motion, that quadruple, which produces four times 
 the motion, and so on ; in short, that the force 
 which produces motion in the body must be exactly 
 proportional to the motion which that body receives. 
 Now this force producing the motion, has been 
 shown to be exactly equal to the force which the 
 body receives^ with its motion, and which accom- 
 panies it *, the force, in fact, of its motion, or its 
 momentum. The force of a body's motion, or its 
 momentum is then doubled when the body's velocity 
 is doubled, tripled when its velocity is tripled, &c. ; 
 and by however many times you increase its velo- 
 city, or by however many times you make it less 
 than it was, by so many times exactly do you in- 
 crease or make less its momentum. 
 
 Thus, in the same body, or in equal bodies, the 
 momentum is proportional to the velocity. But 
 how shall we compare the momenta of unequal 
 bodies? Let the force of gravity be imagined to 
 be extinguished, and let it be conceived that I have 
 the power of propelling a number of equal balls 
 with the same forces, so that they shall have exactly 
 the same velocities. Let them all be propelled at 
 the same instant, from different points, but in 
 parallel directions, so as to form a flight of balls, 
 ail directed one way. 
 
 It is evident that these balls, all moving parallel 
 
 * It is, in fact, as though a transfer of the force took place 
 from the moving body to the body moved ; as though it were 
 poured into it like water from one vase into another. 
 R 2 
 
244- ILLUSTRATIONS OF MECHANICS. 
 
 to one another, with the same velocity, and towards 
 one direction, will retain exactly the same relative 
 distances ; each ball will remain always at the same 
 distance from the neighbouring balls, not at all 
 altering its position amongst them as they all 
 move forward together ; so that if it be conceived 
 that I could throw over these balls some hidden 
 spell or power of resistance, which, without adding 
 to their mass, should bind them altogether: if, for 
 instance, I could freeze them into one continuous 
 mass ; then in the act of thus uniting them, since 
 they had before no tendency to separate, I should 
 not add to, or take away from, the force with which 
 any one of them was moving ; and the aggregate 
 force of their motion in this united state would be 
 the same as it was in their separate and divided 
 state. 
 
 But what was this aggregate of their forces of 
 motion, when they moved separately ? 
 
 Their masses were all equal, and all moved with 
 the same velocity ; they moved, therefore, each with 
 the same force of motion, and the aggregate of 
 their force of motion was as many times the force 
 of motion of one, as there were bodies. The aggre- 
 gate of their forces of motion now that they are 
 united, is therefore as many times the force of 
 motion of one of the component bodies of the 
 mass, as there are such bodies. Thus, if there were 
 twice the number of the same component bodies in 
 the mass, or if it were of twice the size, then would 
 the aggregate force of motion in it be twice what 
 jt was before ; if it were three times the size, its 
 force of motion would be thrice, and so on. 
 
THE MEASURE OF MOMENTUM. 24-5 
 
 Thus, then, if there be two masses, one of which 
 contains double the quantity of matter that the 
 other does, and they both move with the same 
 velocity, then the one will have double the force of 
 motion of the other ; if the one have triple the 
 mass of the other, it will have triple the force of 
 motion, and so on. 
 
 On the whole, then, it appears that when equal 
 bodies move with different velocities, their forces of 
 motion are proportional to their velocities ; and that 
 when unequal bodies move with the same velocity, 
 their forces of motion are proportional to their 
 masses. From this it follows, by a well known 
 principle of proportion, that when the masses of 
 the bodies, and their velocities, are both unequal, 
 their forces of motion are proportional to the pro- 
 ducts of their masses by their velocities. Thus, if 
 there be two bodies, one of whose masses is repre- 
 sented by the number 12, and the other by the 
 number 8, and the first have a velocity of 3 feet 
 per second, and the other a velocity of 9 feet, 
 then the force of motion in the first would be to 
 that in the second as 12 multiplied by 3, to 8 mul- 
 tiplied by 9, or as 36 to 72. So that the lesser body 
 by reason of its greater velocity, would have no 
 less than twice the force of motion that the greater 
 has, or move with twice the force that it does. 
 The mass of a body is proportional to its weight : 
 thus then, the force of its motion is proportional to 
 its weight, multiplied by its velocity. Thus, if 
 there be a cannon ball of 20 Ibs. weight, which flies 
 with a velocity of 1200* feet per second, and a 
 
 * The velocity of a cannon ball when it leaves the mouth of 
 R3 
 
246 ILLUSTRATIONS OF MECHANICS. 
 
 ship of 100 tons weight, which moves through the 
 water at the rate 6^ feet per minute, it may easily 
 be calculated that the force of motion in the ship, 
 moving thus so slowly that its motion would 
 scarcely be perceptible, would yet a little more than 
 equal the force of motion in the cannon ball, thus 
 flying with its swiftest motion, and bearing with it 
 its most destructive force. The velocity of the ball 
 is 72,000 feet per minute, and its weight being 
 20 Ibs. its force of motion is represented by the 
 number 1,4-40,000. The velocity of the ship is 6| 
 feet per minute, and its weight being 224,000 Ibs. 
 its force of motion is 1,456,000. 
 
 Great force of motion may be thrown into a 
 small body or a large one ; in the former case it 
 will give great velocity, in the latter, little velocity. 
 Conversely, if great velocity be thrown into a small 
 body, although small itself, it will have great force 
 of motion ; and if small velocity be given to a great 
 body, notwithstanding the smallness of the velocity, 
 the force of the motion will be very great. 
 
 This fact of the dependance of the force of a 
 body's motion, partly upon the velocity with which 
 it moves, and partly upon its weight, is one of which 
 almost every case of motion presents an illustration. 
 A LARGE SHIP moving so slowly that it can scarcely 
 be seen to move, yet by the great amount of the 
 motion distributed through its great mass, crushes 
 to pieces any obstacle that intervenes between it 
 
 the cannon, varies from 1600 to 2000 feet per second, it loses 
 by the resistance of the air about 800 feet in the first 1500 feet 
 of its flight. 
 
THE CONCENTRATION OF MOMENTUM. 24-7 
 
 and the shore. A CANNON BALL of comparatively 
 small dimensions, by reason of the great velocity 
 of its motion, bears with it a force which, after 
 struggling in a fierce and unceasing contest with 
 the air in its path, and again and again striking and 
 rebounding from the surface of the earth or the 
 water over which it flies, hurls destruction on some 
 spot which may be miles distant from the cannon's 
 mouth. 
 
 If a blow were struck by a sledge hammer on a 
 THIN PLATE laid on a man's chest, the force of 
 motion transferred to the plate would, by reason of 
 its small weight, give it a great velocity, and it 
 would probably be driven into the man's body. But 
 if the same blow had been struck on an ANVIL 
 laid in like manner upon his chest, it would scarcely 
 have been felt, for the same force of motion diluted 
 over the great mass of the anvil, would produce in 
 it a velocity as greatly less than that in the plate, as 
 its weight was greater. Whilst force of motion 
 may thus be so diluted, by diffusing it through a large 
 body, as to produce no sensible effect, it may on the 
 contrary be so condensed in a small body as to 
 become irresistible in its action. 
 
 207. A PLATE OF SOFT IRON MAY BE MADE, BY 
 THE FORCE OF ITS MOTION, TO CUT THROUGH 
 THE HARDEST STEEL. 
 
 If a circular plate of soft iron be made to revolve 
 with great rapidity, the force of motion in each 
 particle on its circumference will become so great, 
 that if a piece of hard steel a steel fie for instance 
 
248 ILLUSTRATIONS OF MECHANICS. 
 
 be held against it, the particles of this hard cohe- 
 sive substance will be driven away by those of the 
 soft iron, and it will be cut through as by a knife. 
 
 208. THE ART OF THE LAPIDARY. 
 
 The lapidary, by means of a crank, moved by his 
 foot like a lathe, causes a horizontal rod or tool, 
 with a small circular disc or button of soft iron at its 
 extremity, to revolve rapidly round its axis. On this 
 soft iron disc, thus revolving, a mixture of fine emery 
 and water, and in a certain stage of the engraving, 
 diamond dust is continually dropping. The fine angu- 
 lar particles of this powder fixing themselves in the 
 interstices, it would seem, of the iron, are swept 
 round by it with great velocity and driven against 
 the surface of the stone which is to be engraved, and 
 which is held against the tool by the lapidary. It is 
 thus cut with ease and engraved. 
 
 209. WHEN A BODY'S MOTION is ARRESTED THE 
 WHOLE FORCE WITH WHICH IT MOVES is MADE 
 
 TO ACT UPON THE OBSTACLE. 
 
 Thus the effect, to crush an obstacle, is propor- 
 tionate to the force of motion in the moving body. 
 A heavy ship y although it moves but slowly, would 
 break down an obstacle, against which a boat might 
 dash with violence without injuring it. On the 
 other hand, a heavy mass of some cwts. may be 
 slowly allowed to descend upon the surface of a 
 table without indenting it, whilst the blow of ever 
 so slight a hammer would be sufficient to abrade 
 an equal surface to that on which the other rests. 
 
IMPACT. 249 
 
 210. THE IMPACT OF BODIES. 
 
 If one body, moving with a certain force of 
 motion inpinges upon another at rest, but free to 
 move, it transfers to it a portion of its own force of 
 motion ; so that in the two together there is afterwards 
 as much of this force as there was before in the one, 
 and the force of motion thus being, as it were, di- 
 luted through a larger mass, the actual motion of 
 each body, must be in the same proportion less. If 
 the two bodies after impact move on together, so as 
 both to have the same motion or velocity, then the 
 force of motion being the same now in the two that 
 it was in the one, the product of the velocity now, 
 by the quantity of matter in the two, must equal 
 the product of the velocity before by the quantity of 
 matter in the one. Thus if the bodies weigh re- 
 spectively nine and eleven pounds, and the first 
 impinge upon the other with a velocity of seven feet 
 per second, or with a force of motion represented 
 by the number 63 ; then, after impact, this force of 
 motion being distributed through the two bodies, 
 having together a mass of 20 pounds, the common 
 velocity of this mass must be such, that its product 
 by 20 shall equal 63 ; that is it must be 3^ 5 ^ eet P er 
 second. 
 
 If a body in motion overtake another, also in 
 motion in the same direction, and carry it along 
 with it, then the force of motion in the two, after 
 impact, will equal the sum of their two forces of 
 motion before impact. Thus, if, in the last example, 
 the second body had been moving with a velocity 
 of 5 feet per second, so as to have a force of nio- 
 
250 ILLUSTRATIONS OF MECHANICS. 
 
 tion represented by 55, then, before impact, the sum 
 of the forces of motion of the two would be repre- 
 sented by 118 ; and all this they will have after im- 
 pact ; only then it will be so distributed that they 
 shall have a common velocity ; this common velocity 
 must then be such that, multiplied by the sum of 
 their weights, or 20 pounds, it may equal 118. 
 Their common velocity must then be 5 T % feet per 
 second. 
 
 If, instead of one of the bodies overtaking the 
 other, they had met ; that which had the least force 
 of motion would have destroyed the whole force 
 of the motion of the other, losing, at the same 
 time, itself, as much as it destroyed; so that, on 
 the whole, after impact, there would only be, in 
 the two, a force of motion equal to the difference of 
 what was in the two before. Thus, taking the last 
 example, and supposing the balls to move in oppo- 
 site directions, and to meet ; since before impact, 
 their forces of motion were represented by 63 and 
 55, afterwards their remaining force of motion will 
 be represented by the difference of these numbers, 
 or by 8. 
 
 This remaining excess of the force of motion in 
 the one body, will carry along with it the other, 
 distributing itself equally through the two. Thus, 
 then, the two whose united weight is 20 pounds, will 
 after impact move with such a velocity that their 
 force of motion is 8 ; this velocity must then be 2-| 
 feet per second.* 
 
 * The whole of the conclusions in this article depend upon 
 the supposition of the entire absence of elasticity in the im- 
 pinging body ; the condition of elasticity greatly modifies them. 
 
THE RECOIL OF FIRE-ARMS. 251 
 
 211. THE RECOIL OF FIRE-ARMS. 
 
 The elasticity of an elastic fluid, such as the air 
 or a gas, exerts itself equally in all directions. 
 Thus, in the discharge of a cannon, which is but an 
 effect of the elasticity of the gas liberated by set- 
 ting fire to the gunpowder, this elasticity is made 
 to act equally towards either side, and towards the 
 muzzle and breech, of the cannon. The cannon 
 does not move sideways, although an immense force 
 is thus made to act sideways upon it, because the 
 gas, expanding equally in all directions, acts with 
 equal expansive forces on its two sides, and in 
 opposite directions, so that these two equal and 
 opposite forces neutralise one another, unless the 
 strength of the cannon yields to either of them, 
 and it bursts. The two forces acting towards the 
 sides of the cannon being thus neutralised, there 
 remain only those which act towards the muzzle 
 and breech. These two would counteract and neu- 
 tralise one another, if the mouth of the cannon were 
 completely and effectually secured, but it is not; 
 the ball and the wadding, however firmly driven, 
 yield; the expansive forces of the gas towards the 
 muzzle and breech do not counteract and neutralise 
 one another, as do the other two ; both of them 
 take effect; and, being equal, they produce an 
 equal effect; the one upon the ball, and the other 
 upon the cannon. Thus the cannon and the ball 
 
 The subject, however, under this form can only be discussed in 
 theoretical treatises, to which the reader is referred for further 
 information. 
 
252 ILLUSTRATIONS OF MECHANICS. 
 
 receive from the explosion equal* forces of motion; 
 the one backwards, and the other forwards. The 
 former is the force of the recoil. 
 
 If the weight of the cannon were only equal to 
 that of the ball, having the same force of motion, it 
 would have the same velocity that the ball has, and 
 the two would, in fact, fly, in opposite directions, 
 equal distances. But the cannon is greatly heavier 
 than the ball ; the same force of motion in it, pro- 
 duces, therefore, greatly less velocity, and the less as 
 this disproportion is greater. Thus a light gun 
 recoils greatly more than a heavy one. The effect 
 of the recoil of the guns of a ship of war falls ulti- 
 mately on the vessel herself. Thus a broadside 
 causes her to heel towards the opposite side, and if 
 she is chased, guns fired from her stern will acce- 
 lerate her flight. 
 
 212. To FIRE FROM SOLID CANNON. 
 
 It has been proposed to replace cannon balls, by 
 pieces of iron with cylindrical apertures cast in 
 them, and cannons by solid cylinders of iron, on 
 which these apertures fit. A cartridge being placed 
 in this aperture, and the aperture then fitted on 
 the solid cylinder, the cartridge would be fired 
 through a touch-hole, and the missile thrown off by 
 its recoil. The force of motion produced in this 
 missile would certainly be the same as though the 
 cartridge were exploded in a cannon, loaded with 
 a ball of equal weight. The idea is exceedingly 
 
 * The cannon is here supposed to run without friction upon 
 the wheels of its carriage. 
 
THE VELOCITY OF A CANNON BALL. 253 
 
 ingenious, and the method presents advantages well 
 worthy of consideration. 
 
 213. THE RECOIL OF A CANNON DOES NOT BE- 
 COME SENSIBLE UNTIL THE BALL HAS LEFT ITS 
 
 MOUTH. 
 
 This was first proved in an experiment made at 
 Rochelle, in 1667, by order of the Cardinal de 
 Richelieu. 
 
 A cannon was fixed in a horizontal position at 
 the end of a long vertical shaft or rod, moveable 
 freely about an axis, at its other extremity. The 
 ball fired from it under these circumstances struck 
 the object towards which it was directed, precisely 
 as it would have done if the cannon had been fixed, 
 showing that there was no sensible alteration of its 
 position until the ball was discharged from it. 
 
 214-. To DETERMINE THE INITIAL VELOCITY OF 
 A CANNON BALL. 
 
 It is evident that, by observing the velocity com- 
 municated to the cannon in the first instant of its 
 recoil, the velocity with which the ball leaves it 
 may be determined. For the weight of cannon, 
 multiplied by the initial velocity of its recoil, re- 
 presents its force of motion when the ball leaves 
 it, which is equal to the ball's force of motion. 
 And dividing the ball's force of motion by its 
 weight, we evidently get its velocity. This method 
 was used by Mr. Robins, and the results are given 
 in his treatise on Gunnery. To determine the 
 initial velocity of the recoil, it is only required to 
 observe the height through which the cannon when 
 
254 ILLUSTRATIONS OF MECHANICS. 
 
 suspended, as described in the last article, is made 
 by its discharge to oscillate. The velocity is that 
 which a body would acquire by fallingfreely through 
 that height, and is therefore easily determinate, as 
 will be shown in a subsequent part of this work. 
 
 215. THE BALLISTIC PENDULUM. 
 
 To determine the velocity of a cannon ball di- 
 rectly, it is fired into a heavy mass of wood, sus- 
 pended from a long iron bar. The height to which 
 this mass is by the blow made to oscillate, is shown 
 by an index on a wooden arc, which forms part of 
 the apparatus, and determines the velocity with 
 which the mass first began to move, when its force 
 of motion was equal to that with which the ball 
 struck it. From this consideration the latter is 
 easily calculated, and the force of motion of the ball 
 heing thus known, as also its weight, its velocity is 
 at once ascertained by dividing the former of these 
 by the latter. There is sometimes used a simple 
 contrivance by which the pendulum is made itself 
 to register the height of its oscillation. 
 
 216. WHEN A BODY MOVES ONLY WITH A MO- 
 TION OF TRANSLATION ; THAT is, WHEN ALL 
 THE PARTS OF IT MOVE WITH THE SAME VE- 
 LOCITY AND IN THE SAME DIRECTION, THERE 
 
 IS A CERTAIN POINT IN IT, IN WHICH THE 
 WHOLE FORCE OF ITS MOTION MAY BE SUP- 
 POSED TO ACT. THAT POINT is THE CENTRE 
 OF GRAVITY. 
 
 If all the parts of a body move with the same 
 velocity, or if it move only with a motion of trans- 
 
THE CENTRE OF MOMENTUM. 255 
 
 lation and do not rotate, as for instance, a ball 
 which flies through the air without turning round, 
 or a heavy mass which falls to the earth without 
 turning upon itself, its force of motion will be 
 distributed through its parts in proportion to their 
 weights; for the velocities of all the parts being the 
 same, it is evident that the quantities of the force 
 of motion in the different parts, must be propor- 
 tional to these weights. Now, the forces of motion 
 are by supposition all parallel to one another, as 
 the weights are, and it has been shown that they 
 are all proportional to the weights ; they are there- 
 fore a system of forces distributed through the 
 body precisely as the weights of its parts are, and 
 acting upon it precisely as they do. At whatever 
 point then a single force would sustain the one 
 system of forces, it would sustain the other : that 
 is, a single force would support all the forces of 
 motion of the parts of the body at the same point, 
 where it would support all their weights, or at its 
 centre of gravity; and therefore all these forces of 
 motion produce the same effect, as a single force of 
 motion equal to their sum would do, if it were made 
 to act through that point. 
 
 THE CONVERSE OF THE PROPOSITION STATED 
 IN THIS ARTICLE IS ALSO TRUE,, THAT IS, 
 " IF THE FORCE OF A BODY'S MOTION BE THE 
 SAME AS THOUGH IT ALL ACTED THROUGH ITS 
 CENTRE OF GRAVITY, THEN IT WILL MOVE ONLY 
 WITH A MOTION OF TRANSLATION, OR IT 
 WILL NOT ROTATE AS IT MOVES ON." From 
 
 this it follows, that a solid body, descending freely 
 and exclusively by the action of gravity, will de- 
 
256 ILLUSTRATIONS OF MECHANICS. 
 
 scend with a motion of translation only, and will 
 not turn upon itself, or rotate as it descends ; for 
 the force of such body's motion being the ag- 
 gregate of the gravitations of its parts, must evi- 
 dently have its direction through the body's centre 
 of gravity. Thus too, a body to which its force of 
 motion is communicated by an impulse through its 
 centre of gravity ^ will move, only with a motion of 
 translation, and will not rotate. 
 
 217. THE SYMMETRY OF TOOLS. 
 
 It is for the reason assigned in the last article, 
 that tools, especially those of impact, are made sym- 
 metrical, about a certain plane passing through those 
 points or surfaces by which, and parallel to the di- 
 rection in which, they act. Thus, for instance, an 
 axe, acting by its edge, is made symmetrical, about a 
 plane passing through its edge : the handle of a 
 chisel is symmetrical, about a plane in like manner 
 passing through its edge ; the heavy stone-mason's 
 chisel is symmetrical, about the end by which it 
 acts ; and a nail about its point. A hammer is 
 symmetrical, about a plane passing through its 
 striking surface, and the same is true of a cricket- 
 bat, forge-hammer, a pile-driver, &c. 
 
 None of these tools would strike straight, if this 
 symmetry were not observed. 
 
 The reason of this is, that the centres of gravity 
 of all those bodies (and all others), are in their 
 planes of symmetry ; their forces of motion pro- 
 ducing the same effects as though they acted only 
 in their centres of gravity, produce the same effect 
 
MOTIONS OF TRANSLATION ON ROTATION. 257 
 
 as though they acted, therefore, exclusively in these 
 planes of symmetry ; that is immediately over the 
 cutting or striking point, or line, or surface of the 
 tool. If their centres of gravity were on either 
 side of these striking parts of the tools, that is, 
 if the planes of symmetry did not go through the 
 striking parts, the force of motion of the tool would 
 produce an effect as though it acted on one side or 
 other of the striking part. It would therefore de- 
 flect the direction of the blow, and its effect. 
 
 218, IF A BODY HAVE AN IMPULSE COMMUNI- 
 CATED TO IT WHOSE DIRECTION is NOT THROUGH 
 ITS CENTRE OF GRAVITY, THEN WHEN MOVING 
 FREELY BY REASON OF THIS IMPULSE, ITS Mo- 
 TION WILL PARTLY BE ONE OF TRANSLATION, 
 AND PARTLY OF ROTATION, BUT SUBJECT TO 
 THIS REMARKABLE LAW : " THAT ITS MOTION 
 OF TRANSLATION WILL BE THE SAME AS THOUGH 
 THE IMPULSE HAD BEEN COMMUNICATED 
 THROUGH ITS CENTRE OF GRAVITY, AND THERE 
 HAD THUS BEEN NO ROTATION ; AND ITS Mo- 
 TION OF ROTATION THE SAME, AS THOUGH 
 ITS CENTRE OF GRAVITY HAD BEEN FIXED, 
 AND IT HAD REVOLVED ROUND IT THUS FIXED, 
 SO THAT THERE COULD BE NO TRANSLATION. 
 
 These remarkable properties are proved by 
 analysis, and can only here be enunciated. (See 
 Pratt 's Mechanical Philosophy, pp. 458, 459.) The 
 following are illustrations of them : 
 
 CHAIN SHOT. Two cannon balls fastened to- 
 
258 ILLUSTRATIONS OF MECHANICS. 
 
 gether by a strong chain, and fired from the 
 same cannon, are found to constitute a fearfully 
 destructive missile, sweepng down at once whole 
 ranks of men. This is easily explained. The 
 impulse which the balls receive on leaving the 
 cannon does not, except by a rare accident, pass 
 through the common centre of gravity of the 
 two balls and the chain ; by the property stated 
 at the head of this article, two motions are there- 
 fore of necessity communicated to the system, 
 one of rotation about its centre of gravity, and 
 the other, of translation ; and these do not in- 
 terfere with one another ; so that the balls fly for- 
 wards as far as though they did not revolve*; and 
 they revolve as they would do if they did not fly 
 forwards. The rotation thus produced in the balls 
 causes them to recede from one another, by what is 
 called centrifugal force, (art. 233.) and distends the 
 chain. Thus, as they fly forwards, they sweep con- 
 tinually with a rapid revolution over a circle, whose 
 diameter is equal to the length of the chain, in- 
 creased by the diameters of the two shot ; and over 
 the whole of this space they carry destruction with 
 them. 
 
 Double headed shot, instead of being joined by a 
 chain, are connected by a strong iron bar. The 
 theory of their motion is the same. These have 
 now, we believe, superseded chain shot. 
 
 * The effect of the resistance of the air is not here taken 
 into account. 
 
ROTATION AND TRANSLATION OF THE EARTH. 259 
 
 219. THE DOUBLE MOTION OF THE ROTATION 
 AND TRANSLATION OF THE EARTH. 
 
 Every analogy of nature points to an ecomony of 
 creative power. Reasoning, therefore, on the two 
 motions of rotation and translation, which we find 
 in the earth's mass, and of which the origin is to 
 be traced to the period when it first moved through 
 space, in the path in which it now moves, and God 
 t( divided the day from the night ;" we must come 
 to the conclusion that the mighty event of this 
 epoch, was the result of a single impulse of that 
 single impulse which having its direction, not through 
 the centre of gravity of the earth, would have been 
 sufficient to produce the amount of these two mo- 
 tions of rotation and translation which we find the 
 earth to have. 
 
 The distance from its centre, where this single 
 impulse must have been communicated, has been 
 calculated by John Bernouilh. Supposing the earth's 
 mass to be homogeneous, he finds it to be 165th part 
 of the radius, or about 24 miles from its centre. 
 Similar calculations applied to the other planets,, 
 give, for the distances from their centres, at which the 
 single forces which have given them their existing 
 motions of rotation and translation must have been 
 struck for Mars ^-f^ths of his radius, for Jupiter 
 s, for the Moon yj-^ths. 
 
 s 2 
 
260 ILLUSTRATIONS OF MECHANICS. 
 
 220. To CAUSE A BALL TO MOVE FORWARDS A 
 CERTAIN DISTANCE UPON A HORIZONTAL PLANE, 
 AND THEN, ALTHOUGH IT MEETS WITH NO 
 OBSTACLE, TO ROLL BACKWARDS. 
 
 This remarkable effect will be produced, if the 
 ball, lying on a perfectly horizontal table over 
 which a cloth is tightly stretched, be struck down- 
 wards, not through its centre, but on that side of 
 it which is from the direction in which the ball is 
 first to move. To explain this, let it be observed, 
 that the ball in being thus struck, not through its 
 centre of gravity, but on one side of it, receives 
 two motions, (art. 2 18.), one of rotation, and the other 
 of translation, the latter being the result of the dis- 
 placement of the ball sideways, by the descent of the 
 hand, and the former the direct effect of the im- 
 pulse moreover that" the direction of the bodies 
 rotation, is the opposite of that which it would 
 have, if it rolled in the direction in which it is 
 actually transferred by its motion of translation, so 
 that it in fact slides forwards, rotating as though it 
 would roll back. To both these motions, of sliding 
 and rotation, the friction of the table opposes itself, 
 and whichever of the two is destroyed by it first, 
 will leave the other to take effect alone. Now, it 
 is pretty evident, that since the rotation is the effect 
 of the direct blow, whilst the translation is only 
 that of the indirect displacement : the force of the 
 former must, if the blow be properly struck, be 
 much greater than the latter; so that the body's 
 
1HE RADIUS OF GYRATION. 261 
 
 force of translation will be destroyed by the friction 
 much sooner than its force of rotation is destroyed. 
 The ball's motion of translation, or sliding motion, 
 being thus destroyed, and its force of rotation re- 
 maining, whose direction is backwards, it will evi- 
 dently roll back. 
 
 221. THE RADIUS OF GYRATION. 
 
 When the motions of all the parts of a body are 
 not equal and parallel, the resultant of all their 
 forces of motion passes no longer through the body's 
 centre of gravity. If the motion be round a fixed 
 axis, so that all these parts describe circles about 
 that axis, their velocities, and therefore their several 
 forces of motion, will be proportional to their seve- 
 ral distances from it. All these forces of motion 
 will produce the same dynamical effect as would be 
 produced if the whole weight of the body were col- 
 lected in a certain point, whose distance from the 
 axis is called the radius of gyration. 
 
 If a straight line be made to revolve about its 
 centre, its radius of gyration is equal to half its 
 length, divided by the square root of 3. 
 
 If a cylinder be put in motion about its axis, its 
 radius of gyration will equal its radius, divided by 
 the square root of 2. 
 
 If a circular plate be put in motion round one of 
 its diameters, its radius of gyration will equal one- 
 half the radius of the circle. 
 
 If a sphere be put in motion about one of its 
 tangents, its radius of gyration will equal its radius 
 multiplied by the square root of the fraction f . 
 s 3 
 
262 ILLUSTRATIONS OF MECHANICS. 
 
 222. THE FORCE OF A BODY'S MOTION DEPEND- 
 ING UPON ITS VELOCITY, IT is EVIDENT THAT 
 WHEN THE BODY IS MADE TO REVOLVE A 
 CERTAIN NUMBER OF TIMES IN A MINUTE, 
 ROUND A FIXED AXIS, ITS FORCE OF MOTION 
 WILL BE GREATER, AS IT REVOLVES AT A 
 GREATER DISTANCE FROM THE AXIS, OR IS 
 CONNECTED WITH IT BY MEANS OF A LONGER 
 ARM. 
 
 If, for instance, an axis be revolving a certain 
 number of times per minute, and two equal balls be 
 connected with it by arms of unequal lengths, then 
 that which is attached to the longer arm will have 
 the greater velocity, and therefore the greater force 
 of motion. 
 
 From this it follows, that by varying the distances 
 of the parts of a revolving body from its axis of 
 revolution, we may vary greatly the force of its 
 motion, provided we do not vary the velocity of its 
 revolution ; and conversely, that if we vary the 
 distances of a body's parts from the axis of rotation, 
 not varying its force of motion, we of necessity vary 
 its velocity. 
 
 223. THE DIMENSIONS OF THE EARTH HAVE NOT 
 DIMINISHED FOR THE LAST 2500 YEARS. 
 
 For, no obstacle being opposed to the force of 
 motion with which the earth rotates, that force must 
 be the same now that it always was. But if by the 
 contraction of the earth's mass, its parts are brought 
 now nearer to the axis about which it rotates, than 
 
THE COMPENSATION BALANCE WHEEL. 263 
 
 they were formerly, it is clear that these, revolving 
 at a less distance, must, to have the same force of 
 motion, revolve faster. So that if the earth's dimen- 
 sions had contracted, the day would now be shorter 
 than it was. Now we have observations which show, 
 that the day is now, precisely of the same length that 
 it was, 2500 years ago. None of that diminution 
 of bulk from the cooling of its mass, of which geo- 
 logists speak, can therefore have taken place, with 
 any perceptible influence within that period. 
 
 224. THE COMPENSATION BALANCE WHEEL. 
 
 The balance wheel of a watch is that which sup- 
 plies in it, by the isochronism of its vibrations, the 
 place of a pendulum. It receives by means of a con- 
 trivance connected with it, called the 'scapement 
 successive impulsive motions, through a train of 
 wheels from the main spring of the watch ; and after 
 each impulse it is brought back by a fine hair spring, 
 which is fixed to the axis about which it turns, and 
 may be seen coiled round in its centre. The escape- 
 ment is so contrived, that no second impulse can be 
 given to the balance wheel, until it has vibrated back 
 into the position where it received its first impulse, 
 and until this second impulse is given, the watch 
 cannot go on. Thus ultimately the whole regularity 
 of the motion of the watch is made to be dependant 
 upon the regularity of the vibrations of this balance 
 wheel. Now it was found by theory and confirmed 
 by experiment, that the vibrations of a body to 
 which a spring was attached, as in the case of this 
 wheel, were performed in the same time, however 
 s 4 
 
264 ILLUSTRATIONS OF MECHANICS. 
 
 great they were, within certain limits, that is, how- 
 ever great, or however short, the distance through 
 which the body was drawn back from its position of 
 rest before it was left to itself, it yet returned to its 
 position of rest in the same time ; the greater force 
 of the spring, when farther uncoiled, exactly making 
 up, for the greater space through which it had to 
 move the wheel ; so that in the watch, whether the 
 impulse given to the balance wheel was small or 
 great, it would yet vibrate back, always in the same 
 time. Thus then, whatever irregularity there might 
 be in the action of the main spring of the watch, 
 and in the working of the train which connected it 
 with the balance wheel, so that this should receive 
 at one time a more violent impulse than at another; 
 yet none of this irregularity would find its way into 
 the actual going of the watch, governed as it was 
 by the duration of the vibrations of the balance 
 wheel, which, under all these circumstances of irre- 
 gularity, would yet be of equal duration.* The 
 actual length of each one of the isochronous vibra- 
 tions of the balance wheel, is dependant first upon 
 the length of the spring, and secondly on the dimen- 
 sions of the balance wheel ; the force of the spring 
 being dependent upon the former cause, and the 
 velocity of the motion, communicated to it at each 
 impulse and to be destroyed by the action of the 
 spring, on the latter. By varying either of these 
 elements, the time of the vibrations may be varied 
 as we like. That which is usually varied is the 
 
 * To this isochronism of the vibrations of the balance 
 wheel, it is necessary that the length of the spring should be so 
 great, that at each vibration it should not be greatly uncoiled. 
 
THE COMPENSATION BALANCE WHEEL. 265 
 
 length of the spring, of which, more or less, is set 
 free to vibrate, by a contrivance which is generally 
 visible, and which is easily understood. 
 
 Both the length of the spring, and the dimensions 
 of the balance wheel, are however, of themselves, 
 made to vary, by variations in the 
 Jiff* 6 i . temperature ; and both these are, for 
 the reasons we have stated, causes 
 of error in the going of the watch. 
 The compensation balance wheel is a 
 contrivance, by which they are made 
 to compensate one another. It consists of a wheel, 
 to two extremities, A and B, of a diameter of which, 
 are fixed two curved arms or branches, A C and B D* 
 These curved arms, A C and B D, are each formed of 
 two bands of metal, soldered together ; one of which 
 bands, that forming the convex surface, is of brass, 
 and the other, forming the concave surface, of steel. 
 Now, an increase, of the same degree of tempera- 
 ture, causes brass to expand much more than steel ; 
 any increase of temperature will therefore, cause 
 the outside surfaces of these two arms to lengthen, 
 much more, than the inner surfaces of them ; this 
 can only happen by the curling up, as it were, of 
 each arm, at that extremity which is tree to move. 
 Thus, by the turning in of these extremities of 
 the arms, the material of the wheel is brought 
 nearer to the centre, about which it revolves ; an 
 alteration in its form, which produces an immediate 
 change in the time of its vibrations, compensating 
 for the elongation of the arm A B, and the in- 
 creased length and diminished elasticity of the hair 
 spring. 
 
266 ILLUSTRATIONS OF MECHANICS. 
 
 225. THE CENTRE OF SPONTANEOUS ROTATION. 
 
 If a force be made to act impulsively, on a body 
 at rest, but free to move in any direction; in the in- 
 stant of impact, certain forces of motion will be 
 communicated to all its parts. If the blow be 
 struck through the centre of gravity, it has been 
 before shown, that all these forces of motion will 
 be equal an& parallel ; but if it be not struck through 
 the centre of gravity, the body will move, partly 
 with a motion of translation, in which all its parts 
 partake equally, and partly with a motion of rotation 
 about its centre of gravity. Whilst, by this rotation, 
 some of the parts of the body have a tendency to 
 be carried backwards; by the motion of translation, 
 these, and all the other parts of the body, are car- 
 ried with a direction forwards. And if this rota- 
 tion of any of the parts backwards, exceed their 
 motion of translation forwards, then, whilst the rest 
 move forwards, these parts will actually move back- 
 wards in space ; a feet which any body may verify, 
 who strikes near one of its extremities, a piece of 
 wood, lying on a smooth surface, or floating in 
 water. Tracing the different parts of the body, 
 from those which thus move forwards after the 
 blow, to those which move backwards, we shall 
 evidently arrive at a point, or rather an axis, where 
 the motion passes, from the one direction to the 
 other; which point does not, therefore, move, either 
 forwards or backwards ; this axis is called the axis 
 of spontaneous rotation. It is that about which the 
 body tends, in the first instant of its motion, of its 
 
THE CENTRE OF SPONTANEOUS ROTATION. 267 
 
 own accord, to revolve. An analytical expression for 
 the position of the axis of spontaneous rotation is 
 easily found.* From this expression, it results that 
 the axis is more remote from the point where the 
 disturbing force is applied, as the centre of gravity 
 is more distant from that point, and as the great mass 
 of the body is more distant from its centre of gra- 
 vity. Thus, in a body of considerable length, from 
 the point of application of the disturbing force, it 
 is more distant than in a shorter body ; and in a 
 body, the greater part of whose mass is collected 
 near its extremity, it is more distant than in one 
 whose mass is uniform, or which is lightest at its 
 extremity. 
 
 226. THESE FACTS EXPLAIN THE EASE WITH 
 
 WHICH A LONG POLE OR A LADDER MAY BE 
 BALANCED ON ITS EXTREMITY, AND WHY 
 EITHER OF THESE WILL BE YET MORE EASILY 
 BALANCED IF IT IS LOADED AT THE TOP. 
 
 The axis of spontaneous rotation is that line in 
 the body, which rests when it is slightly displaced, 
 or made to revolve through a small angle. If the 
 action of the disturbing force be continued* after 
 this small angle is passed, this axis will be carried 
 forward with the rest of the body. Now, in balanc- 
 ing a body on its extremity, when we move its lower 
 extremity to preserve the equilibrium, we cause 
 the whole to revolve about its axis of spontaneous 
 
 * A practical method of determining the centre of spon- 
 taneous rotation will be given when we come to speak of the 
 centre of oscillation. 
 
268 ILLUSTRATIONS OF MECHANICS. 
 
 rotation ; and the higher this axis is, the farther we 
 can move the lower extremity, without inclining the 
 body round its axis, beyond this small limiting 
 angle ; so that in fact, when the axis of spontaneous 
 rotation is very high above us, it remains stationary, 
 or nearly so, notwithstanding that we give consider- 
 able motion to the lower extremity of the body, 
 thus greatly facilitating the efforts we make to pre- 
 serve its equilibrium. 
 
 Thus is explained that common feat of posture 
 masters, by which they balance a long ladder, with 
 the lowest stave resting on their chins ; they even 
 move about with it thus balanced ; and have been 
 known to do it, carrying one of their companions 
 at the top. 
 
 227. THE CENTRE OF PERCUSSION. 
 
 Since a blow communicates no motion, and there- 
 fore no force of motion to those particles of a body 
 which lie in its axis of spontaneous rotation (in re- 
 ference to that blow), it is evident that if a fixed 
 axis were made actually to pass through the body, 
 where its axis of spontaneous rotation passes through 
 it, the blow would communicate no tendency to 
 move, and therefore no percussion, to that axis. Now 
 the position of the axis of spontaneous rotation is 
 evidently dependant on the place in the body where 
 the blow is struck ; and, since the blow may be 
 struck in an infinite number of different places, the 
 axis of spontaneous rotation may be made to occupy 
 an infinite number of different positions, and thus 
 may be made to coincide, in one of these positions. 
 
THE TILT HAMMER. 269 
 
 with a particular axis, before determined upon : a 
 particular point of impact thus becoming necessary 
 to cause the axis of spontaneous rotation to coincide 
 with a particular axis. 
 
 This point of impact is called the centre of per- 
 cussion, in respect to that particular axis. If the 
 body be suspended from that axis, and struck upon 
 that point, there will be no re-percussion on the 
 axis, and it is the only point in the body possessing 
 this property. The ballistic pendulum (art. 215.) 
 presents an application of this principle. If the 
 ball strike the mass against which it is fired at any 
 other point than its centre of percussion, the blow* 
 will tend to tear away the axis. 
 
 228 THE CENTRES OF SUSPENSION AND PER- 
 CUSSION ARE CONVERTIBLE. 
 
 If the centre of percussion of a body about a 
 certain axis be found, and that axis be then changed 
 for one passing through what was its centre of per- 
 cussion, then its new centre of percussion will be 
 in what was before its axis of rotation, so that the 
 two are convertible. This is a very remarkable pro- 
 perty, of which many important applications may 
 be made, as will hereafter be shown. 
 
 229. THE TILT HAMMER. 
 
 The tilt hammer is that used in the forging of 
 steel (see art. 82.). It is of great weight, and is 
 fixed to a strong arm, commonly a beam of wood, 
 of considerable length, near whose opposite ex- 
 tremity, is a horizontal iron axis moveable in collars, 
 
270 ILLUSTRATIONS OF MECHANICS. 
 
 which are firmly bound down to a solid mass of 
 iron and masonry, deeply imbedded in the earth. 
 The hammer is raised by the action of a wheel 
 commonly turned by water-power, on the circum- 
 ference of which are fixed at equal intervals cogs, 
 which are made to strike on a projection of the 
 extremity of the arm of the hammer. The hammer 
 is thus made rapidly to rise and fall, and a rapid 
 series of impulses is given to the bar of steel which 
 is placed on an anvil beneath it. The expense of 
 erecting and maintaining one of these hammers is 
 exceedingly great : they are extremely liable to 
 * break their axes, and to tear away their collars. 
 That there might be no percussion upon the axis, 
 when the hammer receives the blow which lifts it, 
 it would be necessary (see art. 227.) that this blow 
 should be struck at a distance equal to that of the 
 centre of percussion of the hammer. That there 
 should be no re-percussion upon the axis and col- 
 lars, when the hammer gives its blow to the steel, it 
 is necessary (see art. 230.) that it should give this 
 blow at the same distance from the axis as it re- 
 ceives it. The hammer might be made of such 
 different forms, as to satisfy these conditions under 
 a great variety of different circumstances, and some 
 of these might be such, as not at all to interfere 
 with the usual method of working it. Some prac- 
 tical knowledge of the expediency of such an ar- 
 rangement appears to have been arrived at by the 
 workmen, and there is professed to be much skill 
 exercised in the erecting of these hammers. In 
 reality, however, they appear all, to expend a large 
 portion of the power which raises them, in beating 
 
A CRICKET BAT. 271 
 
 about their axes, and in perpetual efforts to tear 
 away their collars. 
 
 230. A BODY IN MOTION ABOUT A FIXED Axis 
 WHICH ENCOUNTERS AN OBSTACLE AT ITS 
 CENTRE OF PERCUSSION, WILL EXPEND ALL 
 THE FORCE OF ITS MOTION ON THE OBSTACLE. 
 
 IF IT ENCOUNTER IT AT ANY OTHER POINT, 
 THE FORCE WILL BE DIVIDED BETWEEN THE 
 
 OBSTACLE AND THE FIXED Axis. 
 
 The resultant of the forces of motion of a revolv- 
 ing body passes through its centre of percussion ; the 
 whole of these forces may therefore be supposed to 
 be collected there. If the obstacle be not encoun- 
 tered at the centre of percussion, this collected force 
 will evidently act both upon the fixed axis and upon 
 the obstacle., and will be divided between them, on 
 the principle of a weight supported between two 
 props. Thus also it appears that a body revolving 
 round a fixed axis, and encountering an obstacle at 
 its centre of percussion, does not in the act of im- 
 pact produce any impulse or repercussion upon its 
 axis. 
 
 231. A CRICKET BAT. 
 
 A cricket bat, when the ball is struck by it, may be 
 considered to be revolving round an axis near the 
 shoulder of the player. The whole force of its motion 
 will therefore expend itself on the ball, only when the 
 latter is struck at the point in it, which is the centre 
 of percussion of the system, made up of the bat and 
 
272 TOOLS OF IMPACT. 
 
 the arms of the player ; and it is at this point, and 
 at this point only, that when the ball is struck, there 
 will be no reaction of the blow upon his shoulder. 
 Expert players soon learn to know about what point 
 of a bat they thus strike most effectually, and in this 
 consists a great secret of good batting. 
 
 232. TOOLS OF IMPACT. 
 
 In speaking of its centre of gravity as the point 
 where the impact of a hammer, &c. may be consi- 
 dered to take place, we have supposed all its parts 
 to move with the same velocity. In reality they do 
 not. In the act of impact the instrument is turning 
 round an axis, the points more distant from which 
 are revolving more rapidly than those nearer to it. 
 The point in which all the force of its motion may 
 be supposed to be collected, is in reality its centre 
 of percussion. Thus a carpenter's mallet (in which 
 the parts farther from the handle evidently move 
 faster than those nearer to it, so that the greater 
 portion of the force of motion is collected about 
 the end of it), if he strike with it at its middle point 
 will not produce its greatest effect, and will sting 
 his ,hand ; the true point is the centre of percussion. 
 A similar remark applies to the use of the forge or 
 tilt hammer. 
 
CENTRIFUGAL FORCE. 273 
 
 233. CENTRIFUGAL FORCE. 
 
 The tendency of the force of a body's motion is 
 to carry it forwards, in the same straight line in 
 which at any time it is moving (art. 204.) ; if there- 
 fore it do not continue to move in that line, there 
 must be some force or another controlling that 
 tendency, and deflecting it from that path. That 
 portion of the body's force of motion which is 
 subdued by this deflecting force, is called its cen- 
 trifugal force. 
 
 It is subdued by an effort perpendicular to the 
 straight direction in which the body has a tendency 
 at each instant to move, that is, to the tangent to 
 the curve in which the body is moving : its di- 
 rection is therefore perpendicular to the tangent, at 
 the point where the body is at that instant moving ; 
 that is, it is perpendicular to the line of the curve 
 itself at that point. 
 
 234. THE AMOUNT OF CENTRIFUGAL FORCE. 
 
 The centrifugal force being equal to that which 
 subdues the force of a body's motion from a straight 
 to a curvilinear direction, must manifestly be greater 
 as the force of the motion is greater, and less as it 
 is less.* 
 
 There is a striking example of the effect of high 
 
 * It varies at the square of the angular velocity, and as the 
 radius of curvature conjointly; thus, if a represent the an- 
 gular velocity, and R the radius of curvature, the centrifugal 
 is represented by <*2 R. 
 
 T 
 
274 ILLUSTRATIONS OF MECHANICS. 
 
 velocities in increasing the amount of centrifugal 
 force in the frequent rupture of the GRINDING 
 STONES used for the grinding of cutlery. These, 
 although the stone which composes them is of 
 great cohesive power, yet by reason of the rapid 
 revolution which is given to them, are often shat- 
 tered to pieces by the centrifugal force which 
 results from it. Large fragments of the stone have 
 been known to be carried through the roof of a 
 building and hurled to a considerable distance from 
 the spot where it was worked. The wreck pro- 
 duced by the disruption of one of these stones, re- 
 sembles nothing more than the bursting of the 
 boiler of a steam-engine. 
 
 235. A SLING. 
 
 If a stone is whirled rapidly round, at every in- 
 stant of its circular motion, it tends to continue to 
 move in the straight line, in which, during that 
 instant, it may be considered to be moving ; of 
 straight lines similar to which, the whole circum- 
 ference of the circle may be considered to be made 
 up, and of which any one, being produced, is a 
 tangent to the circle. To keep it from moving in 
 that path, a certain other force must be combined 
 every where with the force of its motion, the two 
 together having a different direction from either 
 separately. That other force is supplied by the 
 tension of the string. This tension of the string 
 is thus a force necessary to keep the body from 
 moving in that straight line in which it continually 
 tends to move, and if the tension of the string be 
 taken away, it will move in that line. Thus when 
 
A SLING. 275 
 
 the string is unloosed, at the instant wnen the stone 
 is ascending in one of its gyrations, it ceases, at 
 once, to move in a curved line ; and by reason of 
 the tendency to permanence of its force of motion, 
 pursues the right line which is a tangent to the curve 
 at the point in which at the instant of its release 
 it was moving ; and this right line in which it was 
 moving, being directed upwards, it describes the 
 same sort of curve as a stone thrown upwards by 
 the hand, or a ball fired upwards from the mouth 
 of a cannon- The mechanical advantage of using 
 a sling, rather than the hand, is this, that by the 
 interposition of the sling, it is possible to communi- 
 cate to the stone a very rapid motion, and a pro- 
 portionately great force of motion, with a compara- 
 tively small and slow motion of the hand ; whereas, 
 to throw the same stone from the hand itself, you 
 must necessarily give to the hand at the instant 
 when it discharges the stone, a motion as great as 
 the stone is to have, and a force of motion much 
 greater. Thus, by means of the sling, you produce 
 the required force of motion in the stone with much 
 less effort than would otherwise be necessary. 
 
 236. A MAN RUNNING IN A CIRCLE. 
 
 Illustrations of the fact that if a body move in a 
 curved line, some other force than that of its mo- 
 tion, or than any force in the direction of its 
 motion, must act upon it, might be multiplied almost 
 without number. 
 
 Let us take the following: If a man runs in a 
 curved line, he becomes at once conscious that a 
 certain muscular effort of that foot which is on 
 T 2 
 
276 ILLUSTRATIONS OF MECHANICS. 
 
 the convex side of his path, greater than that of the 
 other, is necessary. Thus it is that,, in respect to 
 the lower portion of his body, the force requisite 
 to deflect it from the rectilinear path, in which it 
 every where tends to proceed, is supplied. But, 
 the upper portion of his body has also a certain 
 force of motion tending to carry it forward in the 
 same right line, and some other force must com- 
 bine with this, in order to produce its deflection 
 from that line, otherwise, although his legs might 
 accurately enough proceed in the curve, the upper 
 portion of his body would pass off in a tangent 
 to it, and thus the man would be overthrown. He 
 might supply this deflecting force to the upper 
 portion of his body, by a direct muscular effort, 
 propagated from the base of the feet. But he is 
 taught instinctively to economise his muscular ef- 
 forts, and does so by every conceivable means ; and 
 in this case he does it, by causing the weight of 
 the upper portion of his body, to become the de- 
 flecting force required for its curvilinear motion. 
 He inclines his body inwards, so that its centre of 
 gravity is brought beyond the base of his feet. 
 Thus the weight of his body tending to cause him 
 to fall over inwards, constitutes every where a force 
 at right angles to the direction in which he moves, 
 acting inwards ; and this force, combining with the 
 force of his motion, deflects it from the rectilinear 
 direction, and causes it to move continually in the 
 same curve, into which by a slight muscular effort, 
 he causes his feet, and the lower portion of his body 
 to move. By this arrangement, it is wonderful with 
 how small an exertion he is able to deflect himself 
 
THE CENTRE OF CENTRIFUGAL FORCE, 277 
 
 from a straight path, and move in a curve even of 
 the greatest curvature. By a most perfect and 
 beautiful adjustment, he causes his body to incline 
 just so far as is necessary to supply the requisite 
 deflecting or centripetal force, as it is called ; the 
 ninety of which adjustment will be understood 
 when we consider that for every variation, even the 
 slightest, in his forward motion, and therefore in 
 the force of his forward motion, there must be a 
 corresponding adjustment of his inclination. 
 
 237- THE CENTRIFUGAL FORCE OF A BODY'S 
 MOTION MAY BE SUPPOSED TO BE COLLECTED 
 
 FROM ITS DIFFERENT PARTS, AND MADE TO 
 ACT THROUGH ITS CENTRE OF GRAVITY. 
 
 For if it be supposed to be moving in a straight 
 line, all its parts moving with the same velocity and 
 in parallel directions, it has before been shown 
 (art. 216.) that the whole force of its motion may 
 be supposed to act through its centre of gravity : 
 any force therefore, which is to control this rec- 
 tilinear force of motion without causing the body 
 to turn round upon itself, must be made to act 
 through this same point. Now opposite to this 
 force thus necessary to deflect the body, is mani- 
 festly its centrifugal force. The centrifugal force 
 acts then, as though it acted, through the centre of 
 gravity. 
 
278 ILLUSTRATIONS OF MECHANICS. 
 
 238. IT IS BY REASON OF THE CENTRIFUGAL 
 
 FORCE THAT A CARRIAGE, RAPIDLY TURNING 
 A CORNER, IS LIABLE TO BE OVERTHROWN. 
 
 This force, acting horizontally as though it acjed 
 at its centre of gravity, and being greater as the 
 velocity is greater and the deflexion greater, or the 
 turn sharper, may be sufficient to overbalance the 
 weight, which acts as though it acted vertically at 
 the same point, and especially this will be likely to 
 be the case, as the centre of gravity is higher. It 
 is for the same reason that a horseman who gallops 
 rapidly round a sharp corner, is liable to be un. 
 seated. It has been objected that the high velocities 
 given to railroad carriages might produce sufficient 
 centrifugal force on certain curves, to overthrow 
 them. It is easy, however, to show satisfactorily by 
 calculation, that this cannot be the case on any of 
 the curves, or with any of the velocities, con- 
 templated. The only danger which the centrifugal 
 force might produce, is that of the carriages running 
 off the rails, and this seems to be obviated by the 
 conical form which is given to the surfaces of their 
 wheels. 
 
 239. FEATS OF HORSEMANSHIP. 
 
 The horseman who would ride in a straight line, 
 standing upon his saddle, must so alter the position 
 of his body, with each motion of the horse, as to keep 
 the centre of gravity of his body, continually over 
 the narrow base of his feet. This is probably an im- 
 practicable task. If, however, instead of riding in a 
 
FEATS OF HORSEMANSHIP. 279 
 
 straight line, he rides in a curve, a new force is lent 
 to him to support his weight, acting too as if it 
 acted at the same point where his weight may be 
 supposed to act, viz. his centre of gravity ; this new 
 force is his centrifugal force. His centre of gravity 
 has now no longer any occasion to be brought over 
 the base of his feet, another horizontal force joins 
 in supporting it, and poised between the horizontal 
 force and the resistance of his feet, its equilibrium 
 is easily found. To the action of the centrifugal 
 force, which would otherwise overthrow him out- 
 wards, the horseman slightly opposes the weight of 
 his body by leaning inwards : and does he find his in- 
 clination too great, he urges on his horse, and his 
 centrifugal force, thus increased, raises him up again. 
 By thus varying his velocity and the inclination of 
 his body, the conditions of his equilibrium are placed 
 completely under his control, and' he can perform 
 a thousand evolutions, that, moving in a straight 
 line, he could not ; he can leap upon his horse,, 
 stand upon his head or his hands, whilst he is per- 
 forming his gyrations, or jump from his horse upon 
 the ground, and running to accompany its motion, 
 vault again upon his saddle : the conditions of his 
 stability, and even the force of his gravity appear 
 to be mastered. There is in fact given to him a 
 third invisible power, by the act of his revolution, 
 which is a certain modification of the force of his 
 onward motion ; this acts with him in all the evo- 
 lutions he makes, and is the secret of all his feats. 
 
280 ILLUSTRATIONS OF MECHANICS. 
 
 240. A GLASS OF WATER MAY BE WHIRLED 
 
 ROUND SO AS TO BE INVERTED, WITHOUT BEING 
 SPILT. 
 
 This is a well-known feat. A tumbler of water is 
 usually placed in a wide wooden hoop, in the circum- 
 ference of which is a handle, round which it may be 
 turned. The hoop is then whirled rapidly round in 
 a vertical direction ; the centrifugal force is sufficient 
 to prevent the glass from falling from the hoop, and 
 the water from the glass. Instead of being placed 
 in the hoop, the glass may be tied to a string. 
 
 241. To MAKE A CARRIAGE RUN IN AN INVERTED 
 
 POSITION WITHOUT FALLING. 
 
 Let a bar of iron be turned round so as to form 
 a circle, as shown in the accompanying figure, the 
 
 two ends being brought out into two inclined planes, 
 and the two curved portions of the bar being made 
 to lie a small distance apart at the point where they 
 pass one another. This bar being now placed with 
 the curved portion of it in a vertical position as 
 shown in the cut, let a small heavy carriage be 
 placed at one of its extremities, with wheels, on the 
 outside of which are Blanches, to keep it, as it rolls, 
 
THE GOVERNOR. 
 
 281 
 
 upon the bar. Descending the inclined plane, this 
 carriage will ascend the curve, and if the point from 
 which it has descended be high enough, the velocity 
 it will have acquired will cause it to ascend,, in the 
 direction of the arrow, to the top of the curve, ana 
 give to it sufficient centrifugal force at that point, to 
 overcome its gravity, and cause it to run on in 
 that inverted position without falling. It will thus 
 descend in safety on the opposite branch of the 
 Curve, and will again be brought to rest as it ascends 
 the opposite inclined plane towards the other extre- 
 mity of the bar. This ingenious illustration of the 
 effect of centrifugal force was devised by Mr. Roberts 
 of Manchester. 
 
 242. THE GOVERNOR. 
 
 This instrument, long used for the regulation of 
 mill- work, is jnost generally known by the beautiful 
 A application which Mr. 
 
 Watt has made of it, to 
 the steam-engine. It 
 consists of two heavy 
 balls B B, suspended from 
 crooked levers, B E F, 
 which turn upon a com- 
 mon axis, at E. The ex- 
 tremities F, of these, are 
 " jointed to short bars FH, 
 which last at their op- 
 posite extremities are 
 also jointed upon a move- 
 able piece, D H, which slides upon the upright 
 rod, A C. This slide, by means of two shoulders 
 
282 ILLUSTRATIONS OF MECHANICS. 
 
 worked upon it, carries with it the extremity of a 
 lever H K, whose opposite extremity acts to open 
 or close a valve in the pipe which conveys the 
 steam from the boiler of the steam engine, to the 
 cylinder; or, when the governor is used in the water- 
 mill, it acts to raise or fall the sluice, which admits 
 the water to the wheel ; so that in either case the 
 motion of this lever governs the moving power of 
 the machine. 
 
 Now, the action of the balls is such, as to cause 
 the machine itself thus to govern and control the 
 power which moves it, so as itself to temper and 
 equalise its own action ; for the shaft A C is con- 
 nected, by means of the wheel W, and the cord 
 which passes round it, with the working part of the 
 engine, by which the wheel and shaft are made to 
 revolve, carrying with them the balls B. As the 
 engine moves faster, these balls therefore revolve 
 quicker, and their centrifugal force is greater ; this 
 centrifugal force, causing them to fly farther apart, 
 causes them at the same time, to rise, causing the 
 levers B E to revolve about E, and the points F, 
 therefore to descend. These bring with them the slide 
 D H, and the extremity H of the lever H K, by 
 which means the steam valve, on which this lever acts, 
 is more closed, less steam is admitted to the cylinder, 
 and the machine slackens its action, and corrects its 
 too rapid motion. An opposite action of the go- 
 vernor opens the valve, and throws more power into 
 the engine when its action is too slow. 
 
THE PRINCIPAL AXIS OF ROTATION. 283 
 
 243. THE PRESSURE UPON THE Axis OF A RE- 
 
 VOLVING BODY. 
 
 When a body revolves round a fixed axis, the 
 parts of it, situated at different distances from that 
 axis, having different velocities, have different cen- 
 trifugal forces ; and a yet greater difference in the 
 centrifugal forces of different parts is introduced, 
 if they have different weights. These centrifugal 
 forces act all directly from the axis; since all the 
 parts of the body are describing circles round it. 
 If the axis pass through the mass of the body, to 
 the centrifugal force of each part, there is that of 
 some other, on the opposite side of the axis, opposed. 
 It is a possible case, that all these opposite centri- 
 fugal forces may exactly balance one another : there 
 will then be no pressure upon the axis. The general 
 case is, however, that they will not thus balance 
 one another, and that a certain residuum of force 
 will have to be borne by the axis itself, constituting 
 the pressure upon it. 
 
 244. THE PRINCIPAL Axis OF A BODY'S ROTA- 
 
 TION. 
 
 Suppose the fixed axis spoken of in the last arti- 
 cle to become free, so that the body may move in 
 any direction. Being pressed unequally in different 
 directions, by the centrifugal force, it will then im- 
 mediately alter its position, and the revolution will 
 begin to take place about some other imaginary axis 
 passing through the body ; this again, in its turn,, 
 will give place to some other, and so on, until out 
 of the infinity of axes, about which it may thus, in 
 
284? ILLUSTRATIONS OF MECHANICS 
 
 succession, be made to revolve, it falls upon one, 
 about which the centrifugal forces exactly balance 
 one another, and this axis, it will have no tendency 
 to change. In every solid body, there are three 
 such axes, called its principal axes. They intersect 
 in its centre of gravity, and are at right angles to 
 one another. 
 
 Although, when made to rotate accurately about 
 either of its principal axes, the body has no tend- 
 ency whatever to alter the axis of its rotation ; yet 
 its rotation may, or may not, when slightly deflected 
 from that axis, tend to return to it ; and it is of 
 importance to know whether this will, or will not be 
 the case ; for, practically, it is impossible by any 
 impulse, to cause the body, at the first instant of its 
 motion, to rotate accurately round either of its 
 principal axes, so that, when free, it cannot rotate 
 round either of those axes, unless of its own accord, 
 the rotation tend to pass into it. Now of the three 
 axes, there is only one into which the rotation thus 
 tends of its own accord to pass, and it is the shortest 
 of the three. If the body, being free to move, be 
 put in motion, not round this or any other principal 
 axis, its rotation will yet always tend to pass into 
 this shortest axis, and will eventually settle into a 
 rotation about it. 
 
 Although generally, any body, whatever may be 
 its form, has three principal axes of rotation, it yet 
 may have more. Any diameter of \ sphere, for 
 instance, is a principal axis of rotation. Of a cylin- 
 der, the axis or line joining the middle of one of 
 its circular ends to the middle of the other, is a 
 principal axis of rotation, being the longest it can 
 
THE AXES OF ROTATION OF THE PLANETS. 285 
 
 have, but any axis at right angles to this from its 
 middle point, is also a principal axis, than which it 
 can have none less. 
 
 So in a prolate spheroid, a solid, which may be 
 
 supposed to be generated by an ellipse revolving 
 
 fig. 70. round its greater diameter; this greater 
 
 srrrfo diameter is the longest principal axis of ro- 
 
 ^" tation, but any axis perpendicular to this 
 
 from its centre is also a principal axis of rotation. 
 
 These last axes are all of the same size, and are the 
 
 body's least principal axes of rotation. 
 
 In an oblate spheroid, which is generated by the 
 revolution of an ellipse about its shorter diameter, 
 fig. 71. this shorter diameter is a principal axis, 
 <<^gs and it is the shortest of the principal axes 
 of the spheroid; whilst any axis at right 
 angles to this, from its middle point, is a principal 
 axis, and these are its greatest principal axes. 
 
 245. THE PLANETS ROTATE ABOUT THEIR SHORT- 
 EST DIAMETERS. 
 
 The shortest diameter of an oblate spheroid, be- 
 ing its shortest principal axis, is that about which, if 
 any motion of rotation be communicated to it, it will 
 tend to rotate, and into a rotation about which its 
 motion, if left toit self, will ultimately settle (art. 24-4.). 
 We have a striking example of this fact in the system 
 of the universe. The planets are all oblate spheroids, 
 and it is about their least diameters that they all of 
 them rotate. Whether any cause have ever tended 
 to interfere with this rotation,, such as the shock of 
 some comet, or whether such a cause ever shall 
 operate, we know not ; but this we know, that what- 
 
286 ILLUSTRATIONS OF MECHANICS. 
 
 ever disturbance may be, for a time, produced, in 
 respect to the axis round which the rotation of any 
 planet takes place, if its form remain unaltered, it 
 will ultimately return to a rotation about its pre- 
 sent axis. There are, indeed, various minute natural 
 causes, always in operation, which might long ago 
 have changed the existing axis of the earth's ro- 
 tation *, had it not been that, into, a rotation about 
 which, it tends, from all other axes, to pass. This 
 change would involve a perpetual change in the 
 seasons of every place on the earth's surface. 
 
 Had its form been that of a prolate, instead of 
 an oblate spheroid, this case, of a perpetually chang- 
 ing axis of rotation would have occurred. The 
 least axes of the rotation of such a spheroid, are 
 any of those, at right angles to its greatest dia. 
 meter, from its centre ; about some of these it would 
 always tend, from all others, to rotate ; but it would 
 have no tendency to rotate about one of them rather 
 than the other ; and the slightest disturbance, aris- 
 ing from a change in the condition of the earth's 
 mass, the mere effect, indeed, of the tides of the air 
 and sea, would be sufficient to make its rotation 
 pass from one axis to another a change which, once 
 commenced, would never again cease. 
 
 246. EXPERIMENTAL ILLUSTRATION OF THE 
 TENDENCY OF A BODY'S ROTATION ABOUT ANY 
 OTHER AXIS, TO PASS INTO ONE ROUND ITS 
 SHORTEST PRINCIPAL AXIS. 
 Let a body be suspended, hanging by a string ; 
 
 freely from any point which is not the extremity of its 
 
 * Any such change, once commenced, would go on for ever, 
 even when the first cause of it had ceased. 
 
THE PRINCIPAL AXIS OF ROTATION. 287 
 
 shortest principal axis of rotation ; and let the string 
 then be rapidly turned round, which may be done by 
 twisting it, and allowing it to untwist ; the body will 
 thus be made to rotate about an axis which is not 
 its shortest principal axis of rotation ; its rotation 
 will therefore tend to leave this axis, and to pass 
 into a rotation about its shortest principal axis; and 
 it will do this with so great a force (if the motion 
 be sufficiently rapid) as to overcome the bodies 
 weight which tends to keep it in its first vertical 
 position, so that it will gradually lift itself up, bring- 
 ing its rotation continually nearer to its shortest 
 principal axis ; until, with a sufficiently rapid rota- 
 tion, it will (so far as the eye can perceive) find 
 that axis, and will rotate about it. 
 
 A very ingenious instrument is constructed by 
 Messrs. Watkins and Hill, of Charing Cross, by which 
 this rotation is made easy. It is 
 represented in the accompany- 
 ing figure. A very simple com- 
 bination of wheels, which will 
 be easily understood, from the 
 cut, communicates a rapid rota- 
 tion to the string, from which 
 bodies of various forms are sus- 
 pended, from any other axe; 
 than their shortest permanent 
 axes. With different increasing 
 velocities they alter their po- 
 sitions, continually approaching 
 to a rotation about their short- 
 est principal axes. In the 
 progress of this change, a remarkable optical phe- 
 
 fig. 72. 
 
288 ILLUSTRATIONS OF MECHANICS. 
 
 nomenon presents itself. The body beginning to re- 
 volve obliquely, the place to which each part of it 
 returns, after the interval of a revolution, is, in the 
 intermediate time, left vacant ; so that the sensation 
 of vision is from that place received, not continu- 
 ously, but impulsively. So rapid, however, are the 
 impulses, that one sensation remains until it is re- 
 placed by the next ; and the body appears at one 
 and the same time, to fill the whole space, whose 
 parts it in reality occupies in succession ; a phe- 
 nomenon analogous to that of the continuous circle 
 of flame shown by a fire-brand which is whirled 
 rapidly round. 
 
 24-7. THE FORCE WITH WHICH A BODY MOVES is 
 
 NEVER GENERATED INSTANTANEOUSLY. 
 
 The force with which a stone falls, in the very 
 first instant of its fall, would scarcely be perceptible ; 
 it continually accumulates as the stone descends, and 
 if it were allowed continually to descend without 
 resistance, would soon become irresistibly great. 
 
 The force of a cannon ball is not communicated 
 to it instantaneously, but by impulses of the air 
 liberated from the gunpowder, which impulses are 
 continually repeated until it finally leaves the barrel. 
 The longer the barrel is, the longer these impulses 
 are continued, and therefore the greater is the ac- 
 cumulated force. In some parts of the eastern 
 Archipelago, and in South America, the savages are 
 accustomed to propel small poisoned arrows, to the 
 shafts of which, tufts of feather are attached, through 
 long slender tubes, by blowing into them. The 
 
MOMENTUM NOT GENERATED INSTANTLY. 289 
 
 velocity is thus accumulated in the arrow by continued 
 impulses of the breath, until it leaves the tube, as in 
 the bullet by a continual expansion of liberated gas. 
 If a rope be attached to a ball fired from a can- 
 non, as in Captain Manby's apparatus for saving 
 shipwrecked mariners, the rope will almost always 
 be broken, for the rapid motion of the ball cannot 
 instantaneously be communicated to the parts of the 
 rope, nor so rapidly as the ball moves. The elasti- 
 city of the rope has a tendency to prevent this 
 rupture, because it allows of a certain motion of 
 one part whilst the rest does not move, and during 
 the time of this motion, it operates, to communicate 
 the motion of the first part to the second.* 
 
 The proverbial velocity of an arrow is due to the 
 continued action of the bow-string upon it, as the 
 bow expands ; and it is for this reason that the 
 distance of the flight is greatly dependant upon the 
 length of the bow ; the string remaining in contact 
 with the arrow, and impelling it longer, as the bow 
 by reason of its length, admits of being further 
 drawn back. 
 
 The balistce of the ancients, with which they threw 
 great stones and arrows, are similar instances of the 
 accumulation of velocity ; which was in these, pro- 
 duced by the elastic force with which ropes extended 
 and doubled, and then many times twisted, tended 
 to untwist themselves 
 
 Rams and goats, when they fight, recede before 
 they rush upon one another, that they may gradually 
 accumulate a great velocity of impact. 
 
 * The motion is thus propagated -through the rope like tha 
 undulation of an elastic medium. 
 U 
 
290 ILLUSTRATIONS OF MECHANICS. 
 
 The length of neck in some birds enables them to 
 accumulate velocity in their heads through a great 
 distance, and a blow from their beaks thus becomes 
 of irresistible force ; and to a like cause is to be at- 
 tributed the extraordinary violence with which a 
 serpent infixes its fangs.* 
 
 24-8. IF A GUINEA BE PLACED UPON A CARD AND 
 THE WHOLE BALANCED ON THE TlP OF THE 
 FINGER, A SHARP BLOW STRUCK UPON THE 
 EDGE OF THIS CARD WILL CAUSE IT TO SLIP 
 FROM UNDER THE GUINEA, AND TrfE LATTER 
 WILL BE LEFT ALONE ON THE FlNGER. 
 
 The force which tends to make the guinea move 
 with the card, is its friction upon it; this is but a 
 small force, a comparatively long continued action 
 of which would be necessary, to communicate to 
 the guinea a force of motion sufficient to cause it 
 to move as fast as the card is made to move by the 
 finger. This long continued action not being allowed, 
 the guinea is made to move but very little, and the 
 card passes from under it, leaving it nearly where it 
 was. 
 
 249. THE EFFECT OF SWINGING, RIDING, ETC. ON 
 THE CIRCULATION OF THE BLOOD. 
 
 Dr. Arnot has pointed out a remarkable effect, 
 which the principles which govern the communica- 
 tion of motion, probably have, upon the circulation 
 of the blood. It is well known that in all the veins, 
 
 * Serpents have been known, thus striking, to miss their prey, 
 and unable to controul the violence of the blow, to drive their 
 stings into their own bodies. 
 
A CANDLE FIRED THROUGH A BOARD. 291 
 
 there are valves, which open towards the heart ; 
 now in some of the great veins which ascend from 
 the lower part of the body to the heart, it cannot 
 but be, that when the body is made to descend sud- 
 denly, the blood should as it were be left behind it in 
 the vein, on the same principle that if a phial partly 
 filled with a liquid be made suddenly to descend, the 
 liquid will be made to strike against the top. This 
 tendency of the blood to remain behind, when the 
 vein, descends in swinging or riding, or even in walk- 
 ing, forces it through the valves of the vein, and thus 
 probably quickens the circulation. The peculiar 
 sensation felt in the descent of the body in swinging, 
 is probably to be attributed to this cause. 
 
 250. A CANDLE FIRED FROM A MUSKET WILL 
 
 PIERCE THROUGH A THICK BOARD. 
 
 When a body is struck, it is for the most part only 
 a few of the points on its surface which receive the 
 blow ; to communicate the effect of this blow, (the 
 motion of the parts immediately about the point of 
 impact) to all the rest of its mass, certain time is 
 required. Thus when a soft body is struck in one 
 place, a certain time is required, before the other parts 
 of it can be made so to feel the effects of the blow 
 as to admit of the surface yielding, greatly, even in 
 the place where it is struck ; and until this is the 
 case, the effect is the same as t'hough the body were 
 perfectly hard. It is thus that if the PALM OF THE 
 HAND be struck with force on the surface of water, 
 the blow will be resisted, at the first instant, almost 
 as though by a solid body. Nay a MUSKET BALL 
 u2 
 
292 ILLUSTRATIONS OF MECHANICS. 
 
 when fired against water is, it is said, repelled by it, 
 and even flattened; and a CANNON BALL fired over 
 the surface of a smooth sea, rebounds from it, as 
 from a hard plane. 
 
 These circumstances sufficiently explain the per- 
 foration of a board by a soft body, like a candle, when 
 fired from a musket. The parts of the candle can- 
 not yield until after a certain time ; until that time 
 has passed, they are like the parts of a solid, and 
 before it has passed, the candle has gone through the 
 door. 
 
 251. A MUSKET BALL PASSES THROUGH A PANE 
 
 OF GLASS WITHOUT CRACKING IT. 
 
 The explanation of this fact is the same with that 
 of the last, the indentation of the surface produced 
 by the first impact has not time to propagate itself, 
 so as to crack the pane before the ball has passed 
 through it. Thus, although if thrown by the hand 
 it would shatter the whole ; being projected with this 
 velocity, it carries away only so much, as will leave, 
 room for it to pass ; and were the pane suspended by 
 a thread,it would not break the thread, or even cause 
 it to oscillate. A sheet of paper placed edgeways 
 may, for a like reason, be perforated by a pistol ball, 
 without being knocked down ; and a door half open, 
 pierced by a cannon ball without being shut. 
 
 A cannon ball has been known to carry off the 
 extremity of a musket, without the soldier who carried 
 it feeling the stroke ; as the head of a thistle may, 
 by a rapid blow be struck off, without perceptibly 
 bending the stalk. It is for a like reason to that 
 explained above, that a cannon ball moving with 
 
MOMENTUM NOT DESTROYED INSTANTLY. 293 
 
 great velocity passes through the side of a ship, leav- 
 ing a clear aperture, whilst a spent ball splinters it. 
 
 252. THE FORCE WITH WHICH A BODY MOVES is 
 
 NEVER DESTROYED INSTANTANEOUSLY. 
 
 A cannon ball which impinges against a wall causes 
 a certain yielding of the substance of the wall, and 
 of its own substance before it is stopped ; this occu- 
 pies time. A bomb enters the ground some distance 
 before its descent is arrested, and that building only 
 is bomb proof, the covering of which is of a thick- 
 ness exceeding the distance to which the bomb will 
 thus of necessity sink in it. 
 
 Bales of cotton have sometimes been placed to 
 receive the impact of balls, and have stopped them, 
 because by their elasticity they continually resist 
 the progress of the ball as it enters them, and this 
 continual resistance more effectually takes away their 
 great force of motion (although in itself it is so 
 small) than a much greater, but momentary resist- 
 ance, would. 
 
 Dr. Arnot has given a very striking illustration of 
 this fact, drawn from the comparative strength of 
 iron and hempen cables, 
 
 A ROPE CABLE, being not far from the same in 
 weight with an equal bulk of water, is so buoyed up 
 by the water, as not to hang in any considerable 
 curve from the ship to the anchor, but to be distended 
 in a straight line, whilst an IRON CABLE being much 
 heavier, hangs in a deep curve. Thus the force of 
 the ship's motion, as when at anchor she is beat about 
 by the wind, can only be counteracted by the stretch- 
 u 3 
 
294- ILLUSTRATIONS OF MECHANICS. 
 
 ing of the substance of the rope cable, whilst it is 
 gradually counteracted merely by the tightening of 
 the curve of the chain cable. 
 
 253. ACCUMULATION AND DESTRUCTION OF THE 
 FORCE OF MOTION IN A MOVING BODY. DIS- 
 TINCTION BETWEEN FORCE OF MOTION AND 
 
 FORCE OF PRESSURE. 
 
 Let the force of gravity be imagined to become 
 for an instant extinct, and a ball with a string at- 
 tached to it, to be placed, in the void space, at some 
 distance above a table, in the top of which is a 
 small hole through which the string passes. Gra- 
 vity being extinct, the ball will rest, unsupported, in 
 the position in which it has been placed. Suppose 
 now that the string is pulled, through the hole in 
 the table, by a series of impulses, communicated to 
 the ball : the force of motion communicated to it 
 by each of these impulses the ball will retain, for 
 nothing opposes itself to its motion, and the force 
 of motion in a body is indestructible, except by the 
 action of some opposing force. Retaining thus 
 the force of motion communicated to it by each 
 impulse, the ball will at length, strike the table 
 with an aggregate force of motion, equal exactly 
 to the sum of all the separate impulses which it has 
 received. Moreover, this will manifestly be the 
 case whether it receives many or few impulses 
 before it reaches the table, whether each of them 
 be great or small, and whether they be commu- 
 nicated at longer or shorter intervals of time. 
 Thus it is true, if the impulses be infinitely near to 
 
FORCE OF MOTION AND PRESSURE. 295 
 
 one another, or if the string is pulled continuously. 
 Now, if the string be thus pulled continuously, the 
 number of impulses which the ball receives before 
 it reaches the table, must be infinite ; so that the 
 sum of all these must be infinite as compared with 
 any one of them, and therefore the force of motion 
 with which the ball strikes the table, infinite as com- 
 pared with the force with which the string is at any 
 instant pulled. Now, let the ball rest upon the 
 table, and let the string be pulled with precisely 
 the same force as before, each separate impulse of 
 the force with which the string is pulled, will now 
 be encountered by the resistance of the table, 
 whilst before, it was the accumulation of these 
 impulses which it had to encounter. Moreover, 
 each separate force is infinitely small as compared 
 with their sum. The table then, encounters in 
 the one case a force infinitely greater than in the 
 other. In the one case the force exerted upon the 
 table by the ball was one of motion, in the other, 
 one of pressure : this example points out the cha- 
 racteristic difference between them, and thus it is 
 seen that any force of motion is infinite as com- 
 pared with any force of pressure, every force of mo- 
 tion being the accumulation of an infinite number 
 of elements, of which accumulation, every force of 
 pressure is in the nature of one element. * It has, 
 in fact, been shown in the immediately preceding 
 articles, that force of motion in every case requires 
 a finite time, and the operation of a series of im- 
 
 * If it be not one element of that actual sum, it is at any rate 
 comparable to one of its elements, and bears a finite ratio to it. 
 U 4- 
 
296 ILLUSTRATIONS OF MECHANICS. 
 
 pulses to its production, and is never generated 
 instantaneously. It is in its nature an accumulation ; 
 these impulses are the elements of that accu- 
 mulation, and this time is necessary to their aggre- 
 gation. That force of pressure and force of motion 
 thus stand in the relation of parts to a whole, suf- 
 ficiently accounts for many remarkable analogies 
 between the phenomena of these two descriptions 
 of force, and therefore between the sciences of 
 Statics and Dynamics. 
 
 Since force of motion is the sum of a series of 
 impulses, it is evident that a series of such im- 
 pulses in an opposite direction, is required in any 
 case to destroy it; and thus it is sufficiently explained 
 why force of motion is never destroyed instan- 
 taneously. 
 
 254-. GRAVITATION. 
 
 Now, there pervades all material existences a 
 force, analogous to that which we have been de- 
 scribing by the illustration of a string continually 
 pulling a ball. Every portion of matter, impels 
 towards itself every other portion of matter, at 
 every instant of time, and under every variety of 
 circumstances in which these portions of matter 
 may be placed, whether of repose or motion. The 
 earth is a huge mass of matter, every particle of 
 which thus exerts a continual attraction upon every 
 other particle of it. This attraction produces in 
 all bodies on its surface, a tendency to descend 
 towards its centre. If this tendency be resisted by 
 any intervening obstacle, so that each impulse of 
 the earth's attraction is separately counteracted, 
 
MOMENTUM CANNOT BE MEASURED BY WEIGHT. 297 
 
 there results a pressure upon the obstacle, called 
 WEIGHT. If there intervene no obstacle, the body 
 moves towards the earth's centre, continually accu- 
 mulating the impulses of its attraction, increasing 
 the rapidity of its descent, and acquiring a greater 
 and greater FORCE OF MOTION ; which force of mo- 
 tion^ being the accumulation of an infinite number 
 of separate gravitating impulses, is infinite as com- 
 pared with the before-mentioned force of pressure 
 or weight, which is in reality but one of these 
 impulses. The wonderful force of an impact to 
 overcome the resistance of the parts of any solid 
 mass to rupture, is thus fully explained. Cohesive 
 force is in the nature of a force of pressure, and 
 therefore infinitely small as compared with any 
 force of motion, so that it of necessity yields to 
 any impact, however slight, at the moment of im- 
 pact. Thus a weight which, resting upon a table, 
 does not even indent its surface, being let fall upon 
 it, crushes it. 
 
 255. No FORCE OF MOTION OR IMPACT CAN BE 
 
 COMPARED WITH, OR MEASURED BY, A WEIGHT. 
 
 We cannot, for instance, say that the force with 
 which a body moves, is a force of so many pounds, 
 or so many times a given weight, for it is infi- 
 nitely greater than any given weight. How, then, 
 shall we compare the different quantities of this 
 hidden but mighty principle, in different but equal 
 portions of the same moving mass, or of different 
 moving masses ? Clearly by the quantities of mo- 
 tion which it communicates to them (see art. 206.) 
 
298 ILLUSTRATIONS OF MECHANICS. 
 
 256. UNIFORM, ACCELERATED, AND RETARDED, 
 MOTION. 
 
 Motion is change of place. The motion of a 
 body is said to be uniform when the distance, be- 
 tween the places occupied by it in any two successive 
 instants of time, is always the same. It is accelerated 
 when this distance, for any two successive instants, 
 is greater than for the preceding two. 
 
 It is retarded when it is less. 
 
 The motion of a body, if it be uniform, is mea- 
 sured by the space it actually describes in a given 
 time. 
 
 If it be accelerated or retarded, its motion at any 
 instant is measured by the space it would have de- 
 scribed in a given time, had the motion, which it had 
 at that instant, been continued uniformly through 
 that time. The time thus used as the standard of 
 comparison is one second. 
 
 257. VELOCITY. 
 
 The space which a body, moving uniformly, de- 
 scribes in one second, or the space which a body 
 whose motion is accelerated or retarded, would 
 move through in one second, if its motion had con- 
 tinued uniform during that second, is called its 
 velocity. 
 
 258. ACCELERATING FORCE. 
 A body acted upon by a series of different im- 
 pulses, or by a force which constantly impels it, 
 acquires continually more velocity. 
 
THE ACCELERATION OF GRAVITY. 299 
 
 If the force impelling the body be constant, the 
 additional velocities communicated to the body by 
 it in different equal intervals of time are equal; if the 
 force be variable, the additional velocities thus com- 
 municated to it are unequal. 
 
 The additional velocity which the body actually 
 acquires in each successive second of time, if it 
 be impelled by a constant force, is called the ac- 
 celerating force upon it ; or if it be impelled by 
 a variable force, the additional velocity which it 
 would acquire, if from any given instant that force 
 remained during one second of time a constant force, 
 is called the accelerating force upon the body at 
 that instant. 
 
 Since the body retains all its increments of force, 
 and therefore of velocity, its whole velocity after 
 any number of seconds of time, from the commence- 
 ment of its motion, will equal the velocity with 
 which it first began to move, added to the increments 
 of velocity which it has continually received. 
 
 259. THE LAW OF THE ACCELERATING FORCE 
 OF GRAVITY. 
 
 Bodies moving to one another by reason of that 
 attraction which pervades all matter, and is called 
 gravitation, receive continually greater accessions 
 of velocity in each second of time, as they approach 
 one another more nearly. The accession of velocity 
 or accelerating force at any one distance from the 
 centre of either body, being to that at any other, as 
 the square of the second distance is, to the square 
 of the first. This law is usually cited as that of the 
 
300 ILLUSTRATIONS OF MECHANICS. 
 
 inverse square of the distance. The accelerating 
 force of gravity being said to vary inversely, as the 
 square of the distance. Thus bodies falling at the 
 surface of the earth, receive continually greater in- 
 crements of velocity in each second as they ap- 
 proach its centre. Nevertheless the distance through 
 which a body can be made to fall at the earth's 
 surface being exceeding small, as compared with the 
 whole distance to its centre, this variation in the 
 accelerating force of falling bodies is exceedingly 
 small, and indeed imperceptible. 
 
 For all practical purposes we may therefore con- 
 sider the augmentations of velocity which a body 
 falling at or near the earth's surface receives, in 
 each successive instant of time, to be the same. 
 This constant accelerating force or increment of 
 velocity will subsequently be shown by experiment 
 to be 32 f e et. 
 
 260. GRAVITATION A FORCE INSEPARABLY AND 
 
 UNIVERSALLY ASSOCIATED WITH MATTER. 
 
 Gravitation is fixed in matter eternally and in- 
 separably. No lapse of time wears it away, no 
 modification of circumstances in which it can be 
 placed no appliance of artificial means or power 
 of other natural forces upon it, removes or can re- 
 move, the slightest conceivable portion of it. You 
 may crush the parts of a body into a powder, apply 
 to it the power of heat, and melt it into a liquid 
 or you may, by a yet intenser application of heat, 
 dilate it into a gas ; you may make of it a chemical 
 solution ; bring again to its original form of a 
 
GRAVITY IMMUTABLE AND UNIVERSAL. 301 
 
 solid analyse it again and again combine and 
 recombine it : through all these changes you will 
 not in the slightest conceivable degree have affected 
 the gravity or weight of any one of its particles. 
 
 Not only is the power of gravitating thus unalter- 
 ably infixed in matter, but it is infixed in it univer- 
 sally. 
 
 There is no place on the earth's surface where 
 there is matter and not weight there is no matter 
 known to exist in our system of the universe, which 
 does not gravitate ; and if we carry on our inquiries 
 beyond the limits of our system, into the fathomless 
 depths of space, we find there the STARS gravitating 
 towards one another. It is a recent discovery of 
 astronomy, that those multiple stars which, being 
 examined by powerful telescopes, are seen to revolve 
 round one another, and of which there are many, 
 are in their motions subject to certain laws, which 
 prove them to be attracted towards one another by 
 the force of gravity or rather by a force subject 
 to the same laws as that w r hich attracts all things on 
 the surface of our earth towards its centre, and our 
 earth itself towards the sun. 
 
 Such is the eternal, immutable nature of gravita- 
 tion, and such is its universality. 
 
 Although the force of gravity thus coexists uni- 
 versally with matter, yet does it not reside in the 
 same manner and degree in all matter ; there is not, 
 for instance, throughout all matter the same quantity 
 of the principle of gravitation (whatever it may be) 
 associated with the same volume. 
 
 Thus the component materials of the planets are 
 such, that were they all of the same volume or size, 
 
302 ILLUSTRATIONS OF MECHANICS 
 
 they would not, nevertheless,, all weigh the same. 
 And it is scarcely possible to take up any two por- 
 tions of the matter which composes the earth's sur- 
 face of which, equal volumes, would be found to 
 have the same weight or gravitating power. 
 
 261. THE GRAVITATION OF THE BODIES AROUND 
 US TO THE GREAT MASS OF THE EARTH, IS 
 
 A SENSIBLE FORCE ; THEIR GRAVITATION TO- 
 WARDS ONE ANOTHER, ALMOST INSENSIBLE. 
 
 Gravitation is the aggregate of the attractions of 
 the elements of which the earth is composed, each 
 such element attracting each other ; thus the pebble 
 under our feet is attracted, individually, by every 
 other of the pebbles, that are scattered around it; and 
 by all those that are strewed over the earth's wide 
 surface by every particle of fluid, air, or water, upon 
 the earth, and by every atom of its solid substance. 
 Whilst the aggregate of the attractions of the ele- 
 ments which compose the earth is a finite and sen- 
 sible force, thus known to us as gravitation, the 
 mutual attractions of any of these finite portions of 
 it, which come within the scope of our immediate 
 observation, are so small as to be insensible, except 
 to the most delicate admeasurements. All the sen- 
 sible objects around us, no doubt, gravitate towards 
 one another, although we do not perceive it, by rea- 
 son of the exceeding small amount of their gravitating 
 power, and the forces which, in every case, oppose 
 themselves to its taking effect. Thus, if I take up 
 two stones which lie side by side, I immediately 
 perceive the attraction of the great mass of the earth 
 
THE ATTRACTION OF MOUNTAINS. 303 
 
 upon them, but I remain wholly unconscious of their 
 attraction upon one another. 
 
 If two balls of lead were suspended by strings 
 from the ceiling of a room at the same height, they 
 would gravitate towards one another, and did no force 
 oppose itself to their motion, however small the force 
 of their gravitation, they would approach to con- 
 tact. But whilst they are thus attracted towards 
 one another, each is attracted by all the elements 
 which compose the great mass of the earth, and the 
 direction of this much greater attraction must in 
 each be disturbed, before they can, in obedience to 
 their mutual gravitation, approach one another; their 
 approach is thus rendered so small as to be imper- 
 ceptible. If they were placed on a horizontal plane, 
 their friction upon it would in like manner be suffi- 
 cient to retain them apart, and if they floated in a 
 fluid the resistance of the fluid. Nevertheless 
 there are cases in which attraction of gravitation 
 may be rendered sensible, in comparatively small 
 masses of matter. 
 
 262. THE ATTRACTION OF MOUNTAINS. 
 
 The greatest mountain on the earth's surface is 
 not the 59th millionth of its bulk ; the attraction of 
 such a mountain upon a ball of lead is, therefore, 
 as nothing, compared with the attraction of the whole 
 earth upon that ball of lead : yet would this attrac- 
 tion produce a deviation of the plumb line which 
 might be rendered sensible. Bouguer was the first 
 who traced a deviation of the plumb line from the 
 attraction of a mountain. On the sides of Chim- 
 the highest of the Andes, by observations 
 
304- ILLUSTRATIONS OF MECHANICS. 
 
 made under circumstances of extraordinary diffi- 
 culty, he ascertained that mountain to attract the 
 plumb line 7" or 8" from the perpendicular. Chitn- 
 bora90 is volcanic, and its attraction is diminished 
 by an immense cavity which it encloses. 
 
 In 1772 Maskelyne, by observations similarly 
 made at the foot of Mount Shehallian in Scotland, / 
 found a deviation of the plumb line of '5$% ~// 
 
 In 1824 M. Carlini found the attraction of Mount 
 Cenis to produce a variation in the oscillations of a 
 pendulum, to correct which it was necessary to 
 lengthen it by the -0082677th of an inch. These 
 experiments of Maskelyne and Carlini present a 
 means of comparing the attractions of the earth and 
 the mountain in each case, and therefore the masses 
 or quantities of matter in the earth and mountain, 
 which masses are proportional to their attractions ; 
 and the quantity of matter in the mountain being 
 estimated, by observing of what material it is com- 
 posed, and measuring its bulk ; we are enabled to 
 tell, from this comparison, what is the actual quan- 
 tity of the material, or the mass, of the earth. 
 Knowing thus the quantity of matter in the earth, 
 and knowing also, from astronomical admeasure- 
 ments, its bulk or volume, we can tell its mean den- 
 sity. The observation of Maskelyne gave 4'56 for 
 this mean density, and that of Carlini 4-39. From 
 the near agreement of these two observations, we 
 conclude, with great probability, that the earth's 
 mean density is about four times that of water. 
 
CAVENDISH S EXPERIMENT. 
 
 305 
 
 263. THE EXPERIMENTS OF CAVENDISH. 
 
 To the deviation of the plumb line the weight of 
 the plummet opposes itself. It is evident that a much 
 more delicate test of the existence of an attraction 
 would be obtained, if the plummet could be balanced. 
 No contrivance of this kind can, however, be made 
 to show the attraction of a mountain, because the 
 attraction of so great and comparatively distant a 
 mass, would affect equally the ball and the counter- 
 poise ; but such a contrivance may be applied to 
 show the attraction of a/ess and nearermass; and, with 
 this more delicate indication, the attraction of a mass 
 very greatly less, has been rendered even more sen- 
 sible than that of the mountain, upon the plumb 
 line. The following is the admirable experiment of 
 Cavendish. A and B are two balls of lead fixed to 
 fig. 73. 
 
 the extremities of a lever, and capable of being put 
 in motion round an axis which coincides with cd 
 produced, a and b are two smaller balls suspended, 
 by slender silver wires, from the extremities of a 
 rod ef. The wires which suspend these balls are 
 afterwards continued to meet in d, where the whole 
 is suspended by a third wire c d, about which, the 
 least conceivable force is sufficient, to communicate, 
 x 
 
306 ILLUSTRATIONS OF MECHANICS. 
 
 motion to the rod and its suspended balls. The 
 whole of this last-mentioned apparatus, of the rod 
 and smaller balls, is contained, and separated from 
 the rest, by a case, adapted to its form and to the 
 motion which is to be given it. The section of this 
 case is represented in the figure by the shaded line ; 
 it is intended to protect the motion of the balls 
 from any impulses of the air. That the oscillations 
 of the balls may be seen, small apertures are left in 
 the case at e andy, at the extremities of the rod. 
 Yet, more effectually to get rid of causes of dis- 
 turbance, and to obtain a uniform temperature, 
 Cavendish inclosed the whole of his apparatus in 
 a room, without door or window, and into which 
 was no other aperture than one for the admis- 
 sion of the reflected light and heat of a lamp, 
 and a second in which was fixed a telescope T, 
 through which the extremity of the rod might be 
 seen. The lever, or arm, which carried the balls A 
 and B, could be turned by a mechanical contrivance 
 adapted to that purpose, to which motion was given 
 outside of the chamber. When this arm was thus 
 turned, its position was of necessity made to cross 
 that of the light rod ef, carrying the lesser balls a 
 and b; and the greater balls were thus placed in 
 such a position that their attractions upon the lesser 
 balls should both conspire to turn the rod ef; to 
 which motion of the rod, no other force would op- 
 pose itself than the feeble resistance to torsion of 
 the wire c d. 
 
 In the experiments of Cavendish, the large balls 
 A and B were, in weight, somewhat more than 3 cwt. 
 each ; and their attraction upon the smaller balls, 
 
CAVENDISH'S EXPERIMENT. 307 
 
 when the arm carrying them was deflected, was 
 sufficient immediately to cause a deflection of the 
 rod ef, which, after a number of oscillations on 
 either side, at length took up a position nearly in 
 the line joining the centres of the greater balls, 
 deviating from that position only by the amount 
 due to the torsion of the wire c d. It appears from 
 theory, that the time of each oscillation, before the 
 rod eventually rests, is a measure of the attraction of 
 the balls, and sufficient to determine it, allowance 
 being made for the effects of the torsion. And it 
 was thus that Cavendish determined the attractions 
 of the greater balls upon the less ; this he compared 
 with the attraction of the earth upon these lesser 
 balls; and thus he was enabled to compare the 
 mass of the earth with the mass of the greater balls ; 
 and knowing the size of the earth and the size of 
 the balls, he thence obtained a comparison between 
 the densities of the two ; that is, between the density 
 of the earth and the density of lead. He thus 
 found the density of the earth to be 5*48, or about 
 5 \ times that of water. 
 
 The mass of the earth is a unity in terms of which, 
 the astronomer determines the masses of all the 
 bodies of our system of the universe. 
 
 The apparatus of Cavendish is therefore, in fact, 
 a scale in which the earth, sun, moon, and planets* 
 may be considered to have been weighed. 
 
308 ILLUSTRATIONS OF MECHANICS. 
 
 264. THE ATTRACTION OF THE EARTH WOULD 
 CAUSE ALL BODIES, WHETHER THEY WERE 
 LIGHT OR HEAVY, TO FALL TOWARDS ITS SuR- 
 FACE WITH THE SAME RAPIDITY, WERE IT NOT 
 FOR THE RESISTANCE OF THE AIR. 
 
 If a light body a piece of paper for instance 
 and a heavy, but less, body a piece of metal be let 
 fall from any height, at the same time, the heavy body 
 will soon be seen to have passed the other, and it will 
 reach the earth before it. That the air is the prin- 
 cipal cause of this difference may at once be shown, 
 by doubling up the paper till it is nearly of the 
 same size with the metal ; they will then fall nearly 
 in the same times. But the question may be sub- 
 mitted to the test of a conclusive experiment. It is 
 a common experiment with the air-pump to adapt 
 to the top of the interior of a high glass tube, a me- 
 chanical contrivance, on which a piece of money 
 and a feather being placed, they can both be let fall 
 at the same instant, by turning a screw on the out- 
 side of the tube. This tube is placed upon the 
 plate of an air-pump, and the air having been ex- 
 tracted from it, the screw is turned, and the piece 
 of money and the feather being let fall at the same 
 instant, reach also the bottom of the tube together. 
 If the experiment be repeated with the air only 
 partially extracted from the tube, the coin will a 
 little gain upon the feather; and if no air be ex- 
 tracted, the difference of the times of descent will 
 be considerable. 
 
THE VELOCITY DUE TO GRAVITY. 309 
 
 265. THE VELOCITY WHICH is COMMUNICATED TO 
 
 A BODY FALLING FREELY BY GRAVITY. 
 
 Bodies falling freely, near the earth's surface, 
 have communicated to them, equal additions of 
 velocity in equal times ; and since by the first law 
 of motion (art. 93.) none of these increments of the 
 velocity are lost *. but all accumulated in the falling 
 body ; it follows, that its actual amount at any time, 
 must be proportional to the time during which the 
 body has fallen. If, for instance, a body has fallen 
 through ten seconds, since in each second the attrac- 
 of the earth will have communicated to it the same 
 addition of velocity, and since all these additions of 
 velocity will be retained in it, its actual velocity must 
 be five times that which it would have had after 
 falling one second. 
 
 The velocity which gravity thus communicates to 
 a falling body in each second of time near the earth's 
 surface is 32^ feet ; so that after falling five seconds, 
 its velocity will be five times this amount, after ten 
 seconds ten times this amount, and so on. 
 
 This velocity is so great, that it would never have 
 been possible to ascertain its amount by direct ob- 
 servations on the fall of heavy bodies. 
 
 Could we, however, by any contrivance neutralise 
 the gravitating tendency of a body to any known 
 amount, reduce it, fo* instance, to one-half, or 
 one-tenth, or one-hundredth of what it was, since we 
 should diminish the velocity, communicated to it in 
 
 * The resistance of the air is here put out of the question, 
 X 3 
 
310 ILLUSTRATIONS OF MECHANICS. 
 
 each second, precisely to the same amount, we 
 might thus render its motions so slow, that the}' 
 might be observed and measured; we might thus 
 find the amount of the additional velocity actually 
 communicated to it in each second, and this multi- 
 plied by the known number of times by which we 
 had previously diminished the force of its gravity, 
 would give us the velocity which that force would 
 communicate in each second, when undiminished. 
 This is the object of Atwood's machine. 
 
 *266. ATWOOD'S MACHINE. 
 
 Let m and n be two equal weights, suspended at 
 the extremities of a string, which passes 
 over a pulley, as shown in the figure, and 
 imagine the pulley to be without friction, 
 and the string to be without weight and 
 perfectly flexible. It is clear that the 
 weights m and n being equal, the tend- 
 ency of each to descend, will be exactly 
 neutralised by that of the other, and they 
 will rest. Let now a small weight ^ 
 equal to any known fraction of m and 
 n, say the tenth of either or the twentieth 
 of their sum, be added to m. No force 
 whatever will act to counteract that with which ^ 
 tends to descend for all the force in m and n 
 is neutralised no portion of the force with 
 which [* tends to descend will therefore be destroyed. 
 It will nevertheless not take effect in such a way, as 
 to cause /x to descend as it would, if it descended 
 freely ; for p cannot move without communicating 
 
ATWOOD'S MACHINE. 31 1 
 
 an equal motion to m and n. Throughout the 
 bodies m n and /A, an equal force must therefore be 
 distributed, to produce this motion ; and that force 
 can only come from the gravitating force of /x ; this 
 force, being that with which p would actually de- 
 scend if left to itself, is therefore, by this contriv- 
 ance, made to be equally distributed through the 
 bodies m n and ^ and to operate um>n them in 
 common, with an energy less, in proportion, as the 
 mass through which it is thus diffused is greater. 
 In the case we have supposed, the mass through 
 which this force of p is thus diffused, is equal to 
 twenty-one times p ; the force actually existing in 
 each portion of it, is therefore the 2 1st part of what 
 it was in each portion of p, and //. will, in this corn- 
 bin ation, descend with -^-st part of the force that it 
 would, if it descended freely ; that is with -J-jSt part 
 of the ordinary force of gravity. This change being 
 made in the amount of the force effective on /x, leaves, 
 nevertheless, the law under which that force takes 
 effect the same, and reducing the velocity which it 
 produces in each second to the 21st part, enables 
 us to measure that velocity, and, taking it twenty- 
 one times, to estimate what it would be if the body 
 fell freely. 
 
 The conditions we have supposed of a perfect 
 absence of friction in the pulley, and of weight and 
 rigidity in the string, cannot be realized. They 
 are nevertheless approached to, in the machine shown 
 in the accompanying figure, which is called At- 
 wood's machine, and which serves, when accurately 
 constructed, to verify the law of the descent of heavy 
 bodies with great precision, 
 x 4? 
 
312 
 
 ILLUSTRATIONS OF MECHANICS. 
 
 The string is a slender thread of silk, and, to get 
 rid of friction, the axle of the pulley Q is made to 
 rest, at each extremity, upon 
 the circumferences of two 
 wheels, which turn with it, 
 and thus offer a greatly less 
 resistance to its motion than 
 a collar would. These wheels 
 being made with great care, 
 and accurately balanced, and 
 their axes being very small, 
 the various resistances to the 
 motion of the pulley, are in a 
 great degree got rid of. 
 
 A pendulum clock R, beat- 
 ing seconds, is affixed to the 
 ' machine, and there is a me- 
 chanical contrivance connect- 
 ed with it, by which the pulley 
 is set free, and the descent of 
 the weights made to commence, 
 at the commencement of a 
 particular second. The gra- 
 vitation of the weight p, con- 
 tinually adding to the velocity with which the bodies 
 move, in order to determine that velocity at any 
 particular instant, it becomes necessary to remove, 
 at that instant, the cause of acceleration, so that the 
 motion may continue, for a time, the same as it 
 was then. This is done by causing the descending 
 body to pass through a ring P, moveable along 
 a vertical scale. By trial, this ring is fixed in 
 such a position, that the descending weight shall 
 
ATWOOD'S MACHINE. 313 
 
 pass through it, precisely at the instant at which 
 the velocity is to be measured ; after one, two, 
 three, or any other number of beats of the pendu- 
 lum ; the weight /x, which is to produce the motion 
 is moreover made of the form of a small rod or 
 bar, of a length greater than the diameter of the 
 ring. Thus, whilst the two weights m and /*, are in 
 the act of passing through the ring, that is, at the 
 instant for which the velocity was to be measured 
 the weight //. will be removed ; and no force, thus, 
 remaining to accelerate the motion, it will become 
 uniform, and may be measured. In order to effect 
 this measurement, a flat piece of wood, moveable 
 along the scale, is placed, by trial, in such a posi- 
 tion, that the descending weight shall strike it pre- 
 cisely after one beat of the pendulum from the time 
 when it passed through the ring. The distance 
 marked upon the scale between the position of P, 
 and that of this second sliding piece, measures the 
 space, which the descending body describes, uni- 
 formly, in one second, with the velocity which it 
 had acquired at the instant of passing through P, 
 that is, after the given known number of seconds 
 of its motion. Now it is found by these experiments 
 that the velocity, thus acquired, in a descent of two 
 seconds, is double of that acquired in a descent of 
 one second ; that acquired in a descent of three 
 seconds, triple, that acquired in four seconds, quad- 
 ruple, &c. 
 
 Thus then, the body, thus falling acquires in each 
 second an equal amount of additional velocity, 
 which (if, as we have supposed, p is -^th of 
 m or n,) is -fa st part of the velocity which it 
 
314? ILLUSTRATIONS OF MECHANICS. 
 
 would have acquired, had it fallen freely : so that 
 a body falling freely, would acquire equal additions 
 to its velocity in each second. From experiments 
 thus made, it is found that the addition made to a 
 body's velocity in each second of its descent, when 
 it falls freely, near the earth's surface, is 32^ feet, 
 or more accurately 386 28 inches. 
 
 This is its increase of velocity in each second, 
 near the earth's surface. It would not be the same 
 at greater distances from it : at twice the distance 
 from the earth's centre that we are, it would only 
 be th what it is here ; at three times the distance 
 -Jth ; and four times the distance, -Jgth ; at five 
 times -^-tli. The law of this variation, which is 
 easily seen, is called that of the inverse square of 
 the distance. 
 
 267. DESCENT OF A BODY BY GRAVITY. 
 
 It has been shown (art. 266.), that a body, 
 whatever may be its weight, descending freely by 
 gravity, near the earth's surface, always increases 
 the velocity with which it descends, by 32 feet, 
 during every second, of its descent. 
 
 From this it may be calculated, that the space 
 through which it descends in a given number of 
 seconds, is equal to the square* of that number, 
 multiplied by one half of 32 feet, or by 16^ feet. 
 Thus, for instance, a body which descends freely 
 by gravity, during two seconds, will fall through a 
 space equal to the square of 2, that is 4, multiplied 
 
 * The square of a number is the product of that number 
 when multiplied by itself. 
 
DESCENT OF A BODY BY GRAVITY. 315 
 
 by 16-^2 feet; or it will fall, in that time, through 
 64 J feet. In 3 seconds, it will fall through 9 times 
 16^ feet, or 144f feet. In 4 seconds, through 16 
 times 16-j^ feet, or 257^ feet. In 18 seconds, 
 through 324- times 16^ feet, or 5211 feet that is, 
 a mile within 57 feet. 
 
 From this relation, of the space to the time, and 
 from the consideration that the velocity, after any 
 number of seconds, is equal to 32 feet multiplied 
 by that number of seconds, it is easily found, that 
 the velocity acquired in falling through a given 
 height, must equal the square root of the product, of 
 32 feet by twice that height. Thus, for instance, 
 the velocity acquired in falling through 144;f feet, 
 must equal the square root of 289J feet, multiplied 
 by 32 ; which multiplication being performed, and 
 the square root extracted, there will be obtained the 
 number 96J, for the number of feet per second 
 of the velocity of the body, after it has fallen 
 that height. By a similar calculation, the velocity 
 acquired in falling through 257^ feet will be 
 found to be 128f feet, and that acquired in falling 
 through 5211 feet, 579 feet. 
 
 The velocity acquired by a body in thus falling 
 through any given height, is called the velocity due 
 to that height. 
 
 268. A BODY PROJECTED DOWNWARDS OR 
 UPWARDS. 
 
 If the body have not acquired its whole velocity 
 in falling, but has been projected downwards, it will 
 retain the velocity of its projection, and acquire. 
 
316 ILLUSTRATIONS OF MECHANICS. 
 
 besides, an increment of velocity of 32 feet, in 
 each successive second. Thus the whole velocity 
 will equal that of projection, added to the product, of 
 32^ feet, by the number of seconds, during which 
 the body has descended. 
 
 If the body be projected upwards, instead of 
 downwards, its velocity upwards, will be diminished 
 by 32^- feet in every second, until it is wholly de- 
 stroyed. The body will then begin to fall, and its 
 velocity, will from that time, increase continually by 
 32 J- feet per second as before. 
 
 Since, in its descent, the body will acquire, in each 
 second, as much velocity as it lost, in its ascent ; and 
 that in the seconds which intervened, between any 
 period in the ascent and the period of its greatest 
 ascent, the body lost all the velocity it had at the first 
 mentioned period; also, since it will acquire just so 
 much velocity in descending, through that number 
 of seconds ; it follows that the body has, at any 
 number of seconds after the period of its greatest 
 height, just the same velocity which it had, at as 
 many seconds before it attained that greatest height ; 
 and thus that, the velocities of its ascent and descent 
 being in every successive instant (measuring the 
 time from the period of its greatest height) the 
 same, its motion in every respect, and the spaces it 
 describes, will be the same. 
 
 Thus, the times of its ascent and descent, will be 
 the same ; and it will return to the earth's surface 
 with the same velocity, with which it was projected 
 from it. 
 
DESCENT UPON AN INCLINED PLANE. 317 
 
 269. To FIND THE DEPTH OF A WELL BY LETTING 
 
 A STONE FALL INTO IT. 
 
 Let the number of seconds between the time when 
 the stone is let fall, and that when the sound of its 
 striking the bottom reaches the ear, be observed. 
 This will best be done by counting the beats of a 
 seconds pendulum. This time includes that of the 
 falling of the stone to the bottom and the return of 
 the sound to the ear. The velocity of sound being 
 assumed to be uniform, and at the rate of 1130 feet 
 per second * ; a very simple algebraical calculation, 
 gives us the following approximate rule : " Mul- 
 tiply the square of the observed number of seconds, 
 by 565, and divide the product, by the observed 
 number of seconds increased by 35, the quotient 
 will be the depth of the well, in feet." 
 
 Thus, let it be supposed, that 5 seconds intervene, 
 between the instant when the stone is let fall and 
 that when it is heard to strike the bottom. The 
 square of 5 is 25, and this multiplied 565, gives the 
 product 14,125, which is to be divided by 5 in- 
 creased by 35, that is, by 40. The quotient of this 
 division is 353^, which is nearly the depth of the 
 well, in feet. 
 
 270. VELOCITY OF THE DESCENT OF A BODY 
 
 UPON AN INCLINED PLANE. 
 
 If a body be supposed to slide down an inclined 
 plane, without any resistance, it will acquire, when 
 
 * The experiments of Flamstead and Halley give 1142 feet 
 for the velocity of sound. Recent experiments appear, however, 
 to show it to be yet less than the number we have assumed. 
 
318 ILLUSTRATIONS OF MECHANICS. 
 
 it reaches the bottom, a velocity, precisely equal to 
 that which it would acquire, by falling freely to the 
 same level, from a height equal to that of the plane. 
 Thus, if the point from which it fell be at a per- 
 pendicular height of 144f feet above the base, the 
 velocity acquired by falling down the plane will be 
 96^ feet per second ; that being the velocity which 
 it would have acquired by f ailing freely, or without 
 the plane, through 144- J feet (art. 267.). The velocity 
 is thus, entirely independant of the length of the plane, 
 and is the same, for instance, for a body falling down 
 a plane which is twice the length of another, provided 
 its height be the same. This may be easily under- 
 stood. The resistance of the plane, tends to 
 neutralise the gravitating force of the descending 
 body, in a degree dependant upon the smallness of 
 its inclination. If it be not inclined at all, or per- 
 fectly horizontal, it entirely neutralises the gravitating 
 force, so that the body does not descend at all ; if it 
 be slightly inclined it takes away some, but not all the 
 gravitating power, and the body descends slowly; if 
 it be greatly inclined it takes away but little, and 
 the body descends rapidly. Now if the inclined 
 plane be but little inclined, it must be of great length 
 to have a certain perpendicular height, and therefore 
 the body, descending slowly, must be long in de- 
 scending it ; if it be more inclined, the length cor- 
 responding to that height is less, and the time of 
 describing it, less ; so that as on the one hand, 
 by making the plane more and more inclined, 
 less and less of the body's gravity is taken away 
 from it ; on the other hand, the time through which 
 that gravity acts upon it in its descent, becomes, 
 
DESCENT UPON A CURVE. 319 
 
 with this increasing inclination, less and less ; and, 
 by a remarkable relation, it happens, that these causes 
 just compensate one another. The greater time of 
 descending the longer plane just compensates for the 
 less/orce with which the body descends it; so that the 
 whole velocity which that less force communicates, 
 acting through that longer time, is just equal to the 
 whole velocity which the greater force communicates, 
 acting through the less time ; and in both cases the 
 same amount of velocity is ultimately produced. 
 
 271. VELOCITY OF DESCENT UPON A CURVE. 
 
 When a body descends freely upon a curve, the 
 resistance of the curve neutralises, not as in the 
 inclined plane, the same portions of its gravitating 
 force at all points of its descent ; but different por- 
 tions of it at different points. Nevertheless the 
 velocity acquired in the descent is subject to the 
 same law as that acquired on the inclined plane ; 
 it is the velocity due to the height, the velocity 
 which the body would acquire in falling freely from 
 a height, equal to that of the point from which it 
 has descended, above that to which it has descended, 
 upon the curve. This remarkable property obtains, 
 whatever may be the form or the length of the curve 
 or rather it is a property which would obtain, if 
 there were no resistance of the air, and no friction 
 
 *272. TIME OF A BODY'S DESCENT UPON A CURVE. 
 
 The quantity of the descending body's gravitating 
 force, which at each point of its descent is neutral- 
 ised by the resistance of the curve, depends upon 
 
320 ILLUSTRATIONS OF MECHANICS. 
 
 the inclination of the element* of the curve, at that 
 point, to the horizon. Now by varying the form of 
 the curve, we can in any way vary this inclination ; 
 we can therefore in any way vary and modify the 
 ^neutralised or effective gravitation of the body, 
 so that from a force acting with the same energy at 
 all points (as in the case of free descent, or of de- 
 scent upon an inclined plane), we can convert it 
 into a force, varying, according to any law, from 
 point to point. Now if the effective force on the 
 descending body, could by any form of the curve 
 on which it descends, be thus so modified as to vary 
 directly as its distance along the curve, from the 
 point where its descent is to terminate, then would 
 the time of its descent to that point be the same, 
 from however great a distance along the curve it 
 had descended to reach it. 
 
 Thus if the form of a curve, A B, could be so 
 B contrived that, (its resistance 
 ip neutralising, at every point, 
 a certain portion of the force 
 by which a body descend- 
 ing upon it, would other- 
 wise be accelerated), that 
 which remained should, at 
 each point of its descent, be 
 proportional to the distance 
 of that point from the point 
 A, to which the body is to descend ; then the 
 time required by the body to descend from any 
 
 * The curve may be supposed to be made up of an infinite 
 number of exceedingly small straight lines, and the element 
 here spoken of to be one of these lines. 
 
DESCENT UPON A CURVE. 321 
 
 point P on the curve, to A, would be precisely the 
 same, as the time required to descend from Q to A> 
 or from B to A. 
 
 This may be understood without much difficulty. 
 Let us suppose that the distance measured along 
 the curve from A to P is twice that measured from 
 A to Q, then the force accelerating a body which 
 falls from P, is, by supposition, twice that accelerat- 
 ing a body which falls from Q; and in the first second 
 the body falling from P, will fall, along the curve, 
 twice as far as that from Q. Let P be the place 
 which the one body reaches at the end of the first 
 second, and Q l that which the other reaches. There- 
 fore P P A is equal to twice Q Q,, and hence it is 
 easily seen, that since A P is twice A Q, A P l must 
 be twice A Q x . Since then after the expiration of 
 the first second, one body is twice as far from A as 
 the other, the force urging the one down the curve 
 at the commencement of the second second, is twice 
 that urging the other, so that during the second 
 second the one will acquire, by the action of this 
 force, twice as much additional velocity as the other 
 will acquire. Also it begun that second with twice 
 as much velocity as the other. The one beginning 
 the second, then with twice the velocity of the other, 
 and acquiring twice as much additional velocity 
 during the second, must move during the second 
 through twice the space. Thus, if P 2 and Q 2 be, 
 respectively, the places of the bodies after the second 
 second) then P 1 P 2 equals twice Q { Q,,. And rea- 
 soning in the same way, in respect to the third 
 second and every succeeding second, we shall find 
 
 Y 
 
322 ILLUSTRATIONS OF MECHANICS. 
 
 that during each second, the one body describes 
 twice the space that the other does. After the 
 number of seconds which will bring then the body 
 which fell from Q to A, the body which fell from 
 P, (having described in each second twice as great a 
 distance as the other,) will on the whole, have de- 
 scribed a space equal to twice Q A ; that is it will 
 have described the whole space P A. Or, in other 
 words, it too will at that instant have arrived at A. 
 The same reasoning applies whatever proportion 
 the distances of the two bodies from A may have 
 borne at the beginning of their motion. They will, 
 on the supposition which has been made, arrive at 
 the same instant at A. A curve possessing the 
 property we have supposed, is called a tautochronous 
 or isochronous curve. 
 
 273. THE CYCLOID is AN ISOCHRONOUS CURVE. 
 
 If, exactly on the circumference of a circular 
 board A B, the point of a pencil P were fixed, and, 
 
 fig- 77. 
 
 the board being laid flat on a piece of paper, if it 
 were then made to roll along the straight edge of a 
 rule C D, the pencil would describe on the paper a 
 curve called a cycloid ; and this curve would pos- 
 sess the property of isochronism described in the 
 last article. If a piece of wood or metal were 
 
THE CYCLOIDAL PENDULUM. 323 
 
 bent exactly to the form of this curve, and if, 
 being placed upright, balls were allowed to roll from 
 different points in it; then, if there were no resist- 
 ance of friction or the air, these would be found 
 all to reach the lowest point of the curve in the 
 same time. 
 
 274-. To MAKE A PENDULUM OSCILLATE IN A 
 CYCLOID. 
 
 A body cannot roll on a curved surface, such as 
 that supposed in the last article, without friction; 
 and this friction cannot but materially interfere 
 with the equality of the times of its descent. The 
 most effectual way of getting rid of this friction is 
 to substitute, for the resistance of the curve, the ten- 
 sion of a string, to which the body is suspended ; 
 provided the tension of this string can be made to 
 act at every point precisely as the reaction of the 
 curve does. Thus, for instance, if the curve were 
 a circle, this would be easy. A body suspended 
 from a string and allowed to oscillate, would os- 
 cillate precisely under the same circumstances 
 that a body sliding without friction on the surface 
 of a circle would. The tension of the string, 
 and the reaction of the surface, being both of 
 them forces perpendicular to the circumference of 
 the circle, and acting so as to keep the body in the 
 circle. 
 
 To make a body, suspended from a string, de- 
 scend of itself in a curve of the form of a cycloid, 
 the direction of the string being always perpendicu- 
 lar to the direction of the descent, would, however, 
 
 Y 2 
 
324 ILLUSTRATIONS OF MECHANICS. 
 
 seem to be nearly an impossible task ; nevertheless, 
 by a remarkable property of the cycloid, it is easily 
 effected. That property is the following : 
 
 If there be shaped out two surfaces, accurately of 
 
 the form of half cycloids, as represented by AB and 
 
 4 AC, and if they be placed to- 
 
 jig, to, /i\ 
 
 gether so that their extremities 
 B join in A, and their bases are 
 ~7 i n the same horizontal line ; 
 
 and if a body P be suspended 
 "" 'D*^ between them from the point 
 
 A, by a string whose length is such that it will 
 just wind over either of the half cycloids from A 
 to B or from A to C ; then, this body being left to 
 itself, the string will, in the subsequent oscil- 
 lations, so wind itself on the cycloidal cheeks AB 
 and AC, and unwind itself from them, as to cause 
 the body to describe a curve BDC, which is ac- 
 curately a cycloid. Thus, then, from whatever 
 point it is made to fall, the body P will, under 
 these circumstances, fall in the same time to D, 
 and passing that point,, to whatever height in the 
 opposite curve DC it ascends, it will fall from that 
 point back again to D in the same time. So that 
 all its oscillations will be isochronous, or performed 
 in the equal times. That great desideratum, a 
 perfectly isochronous pendulum, would, by this 
 contrivance, be obtained, were it not that it is im- 
 possible to find any substance of which the string 
 AP can be formed, which shall be sufficiently 
 strong, and yet so flexible, that no force shall be 
 required to bend it on the two cycloidal cheeks, and 
 
THE SIMPLE PENDULUM. 325 
 
 such, that no adherence shall take place between 
 it and them. These causes of error, slight as they 
 appear, are yet sufficient so materially to affect the 
 oscillations of a pendulum thus formed, as to render 
 it greatly inferior to the simple pendulum, which 
 we are about to describe. 
 
 275. THE SIMPLE PENDULUM. 
 
 If a line were drawn from D to A (see Jig. 78. in 
 last article), then a circle described from the point 
 A with the radius AD, would accurately coincide 
 with the cycloid BDC at D and for some short 
 distance on either side of that point ; so that a 
 body, suspended by a string from the first-men- 
 tioned point, and oscillating freely in this circle, 
 would, in point of fact, for some distance on either 
 side of D, be oscillating in the cycloid BDC, and 
 its oscillations would therefore be isochronous, so 
 long as they were confined within that limited dis- 
 tance on either side of D; but if they exceeded that 
 distance, then the path of the body thus suspended, 
 deviating from the cycloid, the oscillations would 
 deviate from their isochronism. Thus then we 
 have a simple pendulum whose small oscillations 
 are isochronous. Moreover, the cycloids AB and 
 AC may be made of any size ; therefore AD may 
 be of any length,, so that the pendulum may be of 
 any length. From which it follows that the small 
 oscillations of a simple pendulum, of any length 
 whatever, are isochronous. 
 
 This law of the isochronism of the simple pen- 
 dulum, was one of the discoveries of Galileo. It 
 Y 3 
 
326 ILLUSTRATIONS OF MECHANICS. 
 
 was first observed by him, it is said, when very 
 young, in the oscillations of a lamp suspended 
 from the roof of the metropolitan church of Pisa. 
 He was struck by the equality of the times in 
 which the lamp returned from oscillation to oscil- 
 lation, as its motion gradually subsided ; and this 
 observation of a child became in the mind of the 
 man, a principle of philosophy, on which some 
 of the greatest discoveries of science have been 
 founded. 
 
 276. To DETERMINE THE TlME IN WHICH A 
 PENDULUM OF ANT GIVEN LENGTH WILL 
 PERFORM ITS OSCILLATIONS. 
 
 The oscillations of a simple pendulum, which are 
 made in a circle, coincide, if they be small, with 
 oscillations in a cycloid. From this consideration 
 it is shown by an easy process of the integral cal- 
 culus (see Pratt 1 s Mechanics, p. 370.), that the 
 number of seconds occupied by each oscillation of 
 a pendulum of a given length, in this country 
 (where the force of gravity is such as to accelerate 
 the descent of a falling body by 32 feet, or more 
 accurately by 32' 19084 feet in each second), may 
 be found by extracting the square root of the 
 length of the pendulum (measured in feet), and mul- 
 tiplying this square root by the decimal 0-55372.* 
 Thus, if it were required to find what would be the 
 
 * This rule is represented by the algebraical formula 
 # = 0*55372 V L, where t is the time of an oscillation in seconds, 
 and L the length of the pendulum in feet. 
 
THE SIMPLE PENDULUM. 327 
 
 time between each beat of a pendulum 9 feet long, 
 we must extract the square root of 9, which gives 
 us 3, and multiply *55372 by this square root; 
 whence we have 1*66116 for the number of seconds 
 
 277. TO DETERMINE WHAT MUST BE THE LENGTH 
 
 OF A SIMPLE PENDULUM, so AS TO BEAT ANY 
 GIVEN NUMBER OF SECONDS. 
 
 By a simple transformation of the rule in the 
 last article, which every one acquainted with al- 
 gebra will understand, we obtain a rule to determine 
 what must be the length of a pendulum, that its 
 oscillations may be of any duration that we may 
 require. " Square, or multiply by itself, the number 
 of seconds which the pendulum is to beat, and 
 multiply this result by the number 3*2616, the 
 product will the required length in feet." Let it 
 be required for instance, to find what must be the 
 length of a pendulum, so as to beat once in every 
 2 seconds. Squaring 2 we get the number 4, and 
 multiplying this number by 3*2616, we have 13*0464, 
 or a little more than 13, for the number of feet. 
 If it were required to find the length of a pendulum 
 which would beat single seconds, we must square 1, 
 which gives us 1, and this^ multiplied by 3*2616, 
 gives 3*2616, or somewhat more than 3^ for the 
 length in feet. 
 
 Y 4 
 
328 ILLUSTRATIONS OF MECHANICS. 
 
 278. TO MEASURE THE FORCE OF GRAVITY AT 
 ANY PLACE, BY OBSERVING THE BEATS OF A 
 PENDULUM. 
 
 The force which causes the motion of a pendulum 
 is gravity ; its motion at any place must therefore 
 be dependant upon the energy of the force of gra- 
 vity at that place. The following is the very simple 
 relation which connects them. " If the length of the 
 pendulum were divided, by the additional velocity 
 which gravity communicates to a falling body in 
 each second at the place of observation, and the 
 square root of this quotient being extracted, if the 
 result were multiplied by the number 3*1415, that 
 product, would equal the number of seconds in 
 each oscillation."* From this relation, an easy 
 process of algebra gives us this other. " If the 
 length of a pendulum be divided by the square of 
 the number of seconds which it requires to com- 
 plete each of its oscillations, and if this quotient 
 be multiplied by the number 9*8696, this last pro- 
 duct will exactly equal the number of feet by 
 which gravity will at that place, increase the ve- 
 locity of the descent of a falling body in each 
 second of time." This number of feet is what is 
 called the measure of the force of gravity at that 
 
 * This relation is expressed by the mathematical formula, 
 t = TT V ^> where t is the time of an oscillation in seconds, L the 
 length of the pendulum, g the acceleration of gravity at the 
 place of observation, and IT the number 3-1415, which is half 
 the circumference of the circle, whose radius is unity. 
 
THE SIMPLE PENDULUM. 329 
 
 place. Suppose, for instance, it were observed at 
 any place, that a pendulum whose length was 
 13'04?64 feet, beat once every two seconds; and it 
 were required to ascertain from this fact what was 
 the force of gravity at that place. Dividing the 
 length by the square of 2, or 4, we have 3*2616, 
 which, being multiplied by 9'8696, gives 32-1908, 
 which is very nearly the force of gravity in this 
 country. In making observations with the pen- 
 dulum, to determine the force of gravity at dif- 
 ferent places, it is usual, at each observation, to 
 alter its length, until it is such as to make it beat 
 single seconds ; the above rule then becomes greatly 
 more simple. " Let the length of the pendulum at 
 which it beats seconds, be accurately measured. 
 This length, multiplied by the number 9*8696, will 
 be the measure of the force of gravity at that 
 place. 
 
 279. THE FORCE OF GRAVITY DIMINISHES AS 
 
 WE APPROACH THE EQUATOR. 
 
 By observations such as these, it is found that 
 the force of gravity diminishes as we approach the 
 equator; a less length being required to make a 
 pendulum beat seconds there than here ; so that a 
 pendulum clock which went truly here, would, if 
 carried there, go too slow, and would require to 
 have its pendulum shortened. This striking phe- 
 nomenon is explained by the flattened shape of the 
 earth. Were it a perfect sphere, the force of gravity 
 would be the same every where upon its surface. 
 A table contained in the Appendix, and extracted 
 
330 ILLUSTRATIONS OF MECHANICS. 
 
 from the " Physique" of Pouillet, contains the result 
 of the various observations which have been made 
 with the pendulum, and a comparison of these 
 results with those which are given by theory, on 
 the supposition that the earth is accurately of the 
 form of a spheroid. 
 
 280. To FIND THE DEPTH OF A MINE BY OB- 
 SERVING THE BEATS OF THE PENDULUM. 
 
 The force of gravity as we descend into the earth, 
 does not vary by the law as it does when we de- 
 scend towards the earth's surface from the regions 
 above it. 
 
 A person descending from the top of a high moun- 
 tain, and making observations from time to time 
 with a pendulum, would find the force of gravity 
 increasing continually until he reached the level of 
 the sea ; if, then, he descended a deep mine, ob- 
 serving his pendulum, as before, from distance to 
 distance, he would find the force of gravity, instead 
 of increasing, to diminish continually. The reason 
 of this may be explained as follows : let the earth's 
 mass be supposed, when he has descended to any dis- 
 tance, to be divided into two parts. one a spherical 
 shell, extending over the whole of its surface, and 
 having for its thickness the depth to which he has de- 
 scended, and the other a solid sphere included in this 
 shell and filling it. Now it is a remarkable fact, that 
 the attractions of the different elements of a spherical 
 shell, of whatever thickness, upon a body, any where 
 situated in the interior or hollow of the shell, ex- 
 actly counterbalance one another ; so that the 
 
THE SIMPLE PENDULUM. 331 
 
 body, being drawn in every direction alike, has no 
 tendency to move in any one direction rather than 
 another ; and were the earth hollow, and its cavity 
 a sphere, could we descend into it, we might float 
 about in the void space, without, any effort every 
 muscular exertion would, indeed, be a source of 
 inconvenience and danger to us, and the principal 
 anxiety of our lives would be to guard ourselves 
 .against these continual collisions, upon the opposite 
 walls of our prison-house, which each effort would 
 tend to produce. 
 
 Since, then, this shell of the earth above him 
 exerts no attraction upon a person who descends 
 into it, the whole force by which he is attracted 
 must be that of the solid sphere which it encloses. 
 Now this sphere, beneath him, diminshes its dia- 
 meter perpetually as he descends; whilst his position 
 remains, in respect to this lesser sphere, precisely 
 the same as it was in respect to the greater, when 
 he was at the surface ; he may, in fact, be consi- 
 dered as standing continually, in his descent, on the 
 surface of a diminishing sphere; being then attracted 
 continually, under the same circumstances, but by a 
 less quantityof matter, it is clear that he must be less 
 attracted. 
 
 It is found that this diminution of the attraction, 
 is exactly proportional to the diminution of the 
 distance from the earth's centre ; and applying this 
 principle to determine the effect of the diminished 
 attraction on the motion of the pendulum, we have 
 the following rule to determine the depth of a 
 mine. 
 
 Observe the number of beats which the pen- 
 
332 ILLUSTRATIONS OF MECHANICS. 
 
 dulum loses in one day, by being carried into 
 the mine; ^ths, or nearly -^th of that number 
 of seconds, will give the depth of the mine in 
 miles. 
 
 281. THE CENTRE OF OSCILLATION. 
 
 A simple pendulum is supposed to be a material 
 point, suspended from a string without weight. Such 
 a pendulum can have no real existence. Every ma- 
 terial body which we can cause to oscillate is, in 
 reality, a combination of material points, and there- 
 fore a compound pendulum. If each of the material 
 points of which it were composed, were free to 
 oscillate alone, each would have (art. 276.) its own 
 different time of oscillation, dependant upon its 
 own distance from the point of suspension. By rea- 
 son of the connexion of these points into one mass, 
 they are all made to have the same time of oscil- 
 lation ; the times of oscillation of some being thus 
 made longer by their union with the rest, and those 
 of others shorter. Now, between those points 
 whose times of oscillation are made longer than 
 they would be if they were free, and those whose 
 times are made shorter, it is evident that there must 
 be some point, where one of these states passes into 
 the other, and where, therefore, the time of oscil- 
 lation is neither made longer nor shorter, but is the 
 same as it would be if the particle were free ; 
 this point is called the CENTRE OF OSCILLATION. 
 
 Its position may be determined for any body, 
 whose parts are of geometrical forms, by certain 
 rules of analysis. Having thus calculated the dis- 
 
THE CENTRE OF OSCILLATION. 333 
 
 tance of this point from the point of suspension, we 
 know at what distance a single point, suspended 
 freely, would oscillate in exactly the same time, that 
 the whole of the compound pendulum does. Now, 
 knowing this distance, we can tell what would be 
 the time of oscillation of this single point, since it 
 would be that of a simple pendulum (see art. 276.) : 
 we can therefore tell the time of oscillation of the 
 compound pendulum. Or, conversely, observing 
 the time of the oscillation of the compound pen- 
 dulum, we can calculate where its centre of oscil- 
 lation must be. Both these calculations require, 
 however, a knowledge of the force of gravity at the 
 place of observation. 
 
 82. PRACTICAL METHOD OF DETERMINING THE 
 CENTRES OF PERCUSSION AND GYRATION. 
 
 It is a remarkable fact, dependant upon some 
 dynamical relation, which has not,we believe, hitherto 
 been traced out that the centre of oscillation, in 
 respect to any given axis of suspension of a body, 
 is also its centre of percussion in respect to that 
 axis ; determining the one, therefore, we, in fact, 
 determine the other. Now it has been shown (art. 
 281.), that the centre of oscillation of a body 
 may be found in respect to any axis, simply by 
 observing the time which it requires to complete 
 each of its oscillations, when suspended from that 
 axis. This time being known, the length of a simple 
 pendulum which will oscillate in the same time, may 
 be found (art. 277.)> and this length is the dis- 
 tance of the centre of oscillation, that is of the 
 
334 ILLUSTRATIONS OF MECHANICS. 
 
 centre of percussion, from the axis of suspension.* 
 Thus, then, to determine by experiment the dis- 
 tance of the centre of percussion from the axis, 
 let the axis be placed in a horizontal position, 
 and the body suspended, freely, from it : let the 
 body then be slightly deflected from the position 
 in which it rests, and allowed to oscillate about it ; 
 observe the time, in seconds, of any one of its oscil- 
 lations (they will all be equal) ; square this num- 
 ber of seconds, and multiply it by 3*2616 ; the pro- 
 duct will be the distance required. 
 
 Having thus ascertained the position of the centre 
 of percussion, and knowing that of the centre of 
 gravity, we can readily determine from these the 
 centre of gyration. We have only to take the pro- 
 duct of the distances of these two points from the 
 axis of suspension, and extract the square root of 
 
 * There is, moreover, a very simple method of determining 
 the centre of percussion or oscillation, in respect to a particular 
 axis of suspension, from a knowledge of what is its position 
 when suspended from another axis. Thus wishing to determine 
 the centre of percussion of a forge hammer, we might suspend 
 it from any length of cord like a pendulum, and observe the 
 time of one of its oscillations. This fact, with a knowledge of 
 the position of its centre of gravity, would be sufficient to 
 enable us to determine the centre of percussion of the hammer, 
 about the axis round which it actually works. This fact is 
 mentioned here because there is a great practical inconvenience 
 in determining the oscillations of so large a mass, when sus- 
 pended thus from a string, rather than about the axis round 
 which it works. The calculation to which reference is made 
 above, would not probably be intelligible to those not versed in 
 analytical mechanics, and those who are will need no explana- 
 tion of it. 
 
THE PENDULUM OF BORDA. 
 
 835 
 
 that product : the number thus obtained will deter- 
 mine the distance of the centre of gyration from 
 the axis. The centre of percussion is the same with 
 the centre of spontaneous rotation (art. 230.). 
 
 ^ 283. THE PENDULUM OF BORDA. 
 
 The parts of this pendulum are represented in 
 the accompanying figure. C is a sphere of platinum, 
 a metal, less than any other subject to 
 alteration in its dimensions, from changes 
 in temperature. To this ball the piece 
 D is made to adhere, by turning its under 
 surface of a cup-like form, and accurately 
 of the same curvature as the surface of the 
 sphere, and then rubbing a little grease 
 upon it. E is a piece which screws upon 
 D, and in the centre of which is a small 
 hole, for the attachment of a copper wire, 
 which is to suspend the ball. A similar 
 contrivance attaches the opposite extre- 
 mity of the wire to the suspending piece 
 F ; which . is composed of a cylindrical 
 piece of steel, passing through, at right angles, and 
 fixed in, a triangular prism of steel, called a knife 
 edge. When the pendulum is used the two ends of 
 this, knife edge are made to rest upon two agate 
 planes, whose surfaces are accurately horizontal, and 
 upon the same level ; these are supported upon an 
 upright iron frame, with a contrivance for adjusting 
 their position, and the pendulumhangs suspended be- 
 
336 
 
 ILLUSTRATIONS OF MECHANICS. 
 
 tween them. The frame, with the pendulum supported 
 upon it, is represented in the next cut. Since, by reason 
 of variations in tempera- 
 ture, the wire is subject to 
 continual variations in its 
 length, it becomes neces- 
 sary at each observation 
 to measure its length, with 
 accuracy. Beneath the 
 pendulum a contrivance is 
 represented which is spe- 
 cially directed to this ob- 
 ject. It consists of an iron 
 pedestal solidly fixed, at 
 the top of which is a ver- 
 tical stand or column, 
 capable of being raised by 
 means of a screw. When 
 the length of the pendulum 
 is to be measured, the 
 screw is slowly turned 
 until, in its oscillations, the 
 ________ ball just grazes the top of 
 
 this column. The stand is 
 then fixed. The pendulum is removed from the agate 
 planes which support it, and is replaced by a rod, 
 carrying a knife-edge exactly as the pendulum does. 
 The opposite extremity of this rod is bored hollow, 
 and a cylindrical piece of brass fitted into this hollow 
 end, may be made by means of a screw to advance 
 any distance out of it so as to lengthen the rod. The 
 rod being suspended by its knife-edge on the agate 
 planes precisely as the pendulum was, is made to 
 
BORDA'S METHOD OF COINCIDENCES. 337 
 
 oscillate, and the screw is turned so as to lengthen 
 it, until at length, in its oscillations, it grazes the top 
 of the stand G, as the pendulum did. It is then 
 taken off and measured, and its length is the precise 
 length, between the point of suspension of the 
 pendulum and the bottom of the ball. 
 
 It will be observed, however, that this is a com- 
 pound pendulum, so that the length thus measured 
 is not the length of the simple pendulum which 
 would oscillate in the same time. To find that length, 
 we must find the centre of oscillation. In the case 
 of this pendulum, the parts of which are of very 
 simple geometrical forms, this is done by calculation 
 without much difficulty. And being thus found to 
 be at a particular point C, for any given length of 
 the wire, so that the length of the simple pendulum 
 is C F ; it is easily shown by the formulae, that when 
 the wire is made to vary slightly in length, the dis- 
 tance of C from F will vary |>y very nearly the same 
 quantity. Thus then to find the length of the simple 
 pendulum which will oscillate in the same time, we 
 have only to diminish the measurement, taken as 
 above, by the constant and known distance C A. 
 
 284. BORDA'S METHOD OF COINCIDENCES FOR 
 OBSERVING THE TlME OF OSCILLATION OF A 
 PENDULUM. 
 
 Let a pendulum clock be placed behind the 
 pendulum whose oscillations are to be observed, and 
 let its pendulum be made, nearly but not quite, of 
 z 
 
338 ILLUSTRATIONS OF MECHANICS. 
 
 the same length*, or of such a length as to oscil- 
 late nearly, but not quite, in the same time ; the 
 points of suspension of the two being imme- 
 diately behind one another. If now both pen- 
 dulums be set in motion at the same instant, 
 and looked at in front, after the first oscillation 
 they will be seen to move differently, one gaining 
 upon the other a little, at each oscillation, and 
 this crossing of the oscillations will continue, 
 until one has gained upon the other a complete 
 oscillation, when for an instant their motions will 
 coincide, again to deviate, in each succeeding oscil- 
 lation, until another complete oscillation is gained. 
 Neglecting then all the separate oscillations, let all 
 these coincidences be observed for a given time, say 
 three hours. The hand of the clock will show how 
 many oscillations its pendulum has made, and the 
 number of coincidences will show the number of 
 oscillations which the other pendulum has gained 
 or lost upon it. Adding or subtracting the number 
 of coincidences, from the number of oscillations 
 shown by the clock, we shall get the exact number 
 made by the pendulum we wish to observe, in the 
 three hours. Dividing the number of seconds in 
 the three hours by this number of oscillations, we 
 shall have the duration of each oscillation, in seconds. 
 
 * This change of the length ot its pendulum will alter the 
 going of the clock ; but that is immaterial ; the hand will still 
 register the number of the oscillations, which is all that is 
 required. 
 
CENTRES OF OSCILLATION AND SUSPENSION. 339 
 
 285. To DETERMINE EXPERIMENTALLY THE Po- 
 SITION OF THE CENTRE OF OSCILLATION OF 
 A BODY WITHOUT KNOWING THE FORCE Off 
 GRAVITY AT THE PLACE OF OBSERVATION. 
 
 It is a remarkable property of the centre ot 
 oscillation, which was first given by theory, that if 
 a body be suspended from any point, and the time 
 of its small oscillations about that point be ob- 
 served, and if it be then suspended from another 
 point, this second point being that which was its 
 centre of oscillation before; then its time of os- 
 cillation will now be found to be precisely the same 
 as it was when suspended from the first point, 
 that point having become its centre of oscillation 
 to this new point of suspension. This property is 
 usually described as that by which the centres of 
 suspension and oscillation are convertible. What is 
 meant by it, may perhaps be more clearly under- 
 stood as follows. If A be any point in a body 
 from which it is suspended, and B be its centre of 
 oscillation in respect to the point of suspension A, 
 and if the body had been suspended from B 
 instead of A, its centre of oscillation would have 
 been A instead of B. 
 
 Since in both cases the distance of the centre of 
 oscillation from the point of suspension would 
 have been the same, it is clear that the times of 
 oscillation would have been the same. 
 
 Thus then to determine by experiment, the centre 
 
 of oscillation B of a pendulum, about any point of 
 
 SUSPENSION, we have only to find by experiment, 
 
 a point B about which it will oscillate in the same 
 
 z 2 
 
340 ILLUSTRATIONS OF MECHANICS. 
 
 time as it does about A ; that is, we must suspend 
 it from different points, until at length we find one 
 in respect to which this equality obtains. 
 
 286. CAPTAIN RATER'S PENDULUM. 
 
 A most ingenious contrivance, introduced by 
 Capiain Kater, greatly facilitates the experimental 
 determination of the position of the centre of oscil- 
 lation described in the last article. On the rod of 
 his pendulum is placed a moveable or sliding weight. 
 By moving this weight, the form of the oscillating 
 body, and thus the position of its centre of oscilla- 
 tion, may be changed, so that when by trials as 
 described above, two points are found about which 
 the times of oscillation are nearly the same, by moving 
 this weight, they may, without any further change in 
 the position of the points, be made exactly the same. 
 By means of the slide the pendulum itself is in fact 
 altered, so as to have its centre of oscillation in the 
 point we wish it. 
 
 In Captain Kater's pendulum, the point B being 
 roughly determined to be the centre of oscillation to 
 the point of suspension A. triangular pieces of steel 
 called knife-edges are fixed through the middle of 
 the rod at those points. The projecting extremities 
 of the knife-edges at one of these points, say A, 
 oeing made to rest, by their angles, upon agate planes, 
 the pendulum is allowed to oscillate freely, and the 
 time of oscillation observed. Its position is then 
 reversed, and it is allowed to oscillate in the same 
 way upon the knife-edge at B. If the time of oscil- 
 lation is the same as before, then B is the centre of 
 
KATER'S PENDULUM. 341 
 
 oscillation, and ail that is required is known. If the 
 time of oscillation be not the same, the sliding 
 weight is moved until it becomes the same. When 
 this is the case the centre of oscillation is in B, and 
 A B is the length of a simple pendulum which 
 would oscillate in the same time. If then the time 
 of oscillation be observed, the force of gravity may 
 be calculated by the rule (art. 278.), in which A B 
 is to be taken for the length of the pendulum. 
 
 Sometimes two moveable weights are used, one of 
 which is moved by means of a micrometer screw, to 
 effect a more delicate adjustment.* It is a remark- 
 able fact, proved by analysis, that the result, in 
 experiments made with this pendulum, will not be 
 affected, if for the knife-edges cylindrical axes be 
 substituted. 
 
 Mr. Lubbock has shown in the " Philosophical 
 Transactions for 1830," that a slight deviation of 
 the knife-edges, from a position accurately transverse 
 or perpendicular to that in which the pendulum tends 
 to oscillate, is of no importance if it be a deviation 
 sideways or horizontally; but that a deviation of one 
 degree, vertically, would be sufficient to increase the 
 number of vibrations by 3 in 24 hours. An error 
 in placing the agate planes truly in a horizontal 
 position, has a yet greater effect. The sixth part of 
 a degree of deviation will in this case cause an 
 increase of 6 vibrations in 24 hours. 
 
 * For a detailed account of the experiments of Captain 
 Kater, see the Philosophical Transactions for 1818. 
 
 2 3 
 
342 ILLUSTRATIONS OF MECHANICS. 
 
 287. COMPENSATION PENDULUMS. 
 
 A pendulum, of whatever material it may be 
 formed, necessarily varies its dimensions with every 
 change of temperature. From this cause arises a 
 variation in the position of its centre of oscillation, 
 and in the time of each of its oscillations. In the 
 pendulums used for clocks, it becomes necessary to 
 introduce some contrivance for compensating this 
 variation, where great accuracy is required. The 
 method described in the last article, of varying the 
 position of the centre of oscillation, by means of a 
 sliding weight, could the weight be made to slide, of 
 itself, into a different position with each variation of 
 temperature, might evidently answer the purpose. 
 It is, in fact, somewhat on this principle, that the 
 compensation pendulum is formed. The general 
 tendency of the expansion of the material of the 
 pendulum is evidently to lengthen it, and to carry 
 its centre of oscillation lower; a compensation 
 would be made, if there were a part of this pen- 
 dulum whose mass, was, by its expansion, raised 
 higher. The one tending to raise and the other to 
 depress the centre of oscillation, by each additional 
 degree of temperature, it is clear that these ele- 
 ments, possibly, might be so combined as to keep 
 it exactly in the same place. 
 
 One of the simplest contrivances of this kind, 
 at once the earliest and practically the best, is 
 GRAHAM'S PENDULUM. The rod S B of this pen- 
 dulum is of steel. It carries a frame or etirrup 
 D B, on which is supported a glass cylinder G H con- 
 taining mercury. By every increase of temperature, 
 
GRAHAM'S PENDULUM. 34-3 
 
 the steel rod elongates, carrying the centre of os- 
 fig. 81. cillation of the whole, farther from the 
 point of suspension. But, by the same 
 change of temperature, the mercury rises 
 in the cylinder, thus carrying the centre 
 of oscillation upwards, and towards the 
 point of suspension. A right adjustment 
 of the quantity of mercury to the length 
 of the rod, will cause these two opposite 
 effects to neutralise one another, and. pre- 
 serve the centre of oscillation in its ori- 
 ginal position. To determine this quantity 
 of mercury, it js customary to assume, that 
 the surface of the mercury must be made 
 to remain always at the same distance 
 from the point of suspension. So that, 
 by whatever distance it may be depressed 
 by the elongation of the rod, it may be 
 raised the same distance, by its own expan- 
 sion. The computation on this principle 
 is easily made. We have only to know 
 the fraction of its length, by which a rod 
 of steel elongates for each additional degree 
 of temperature, and the similar fraction by 
 which a given quantity of mercury increases its 
 bulk. The former fraction is '00000636, and the 
 latter -00001066. From this last fraction, we could 
 readily ascertain by how much the column of mer- 
 cury in the glass cylinder, increased its height, and 
 elevated its centre of oscillation, were it not that 
 the same variation of temperature which causes an 
 expansion of the mercury, causes also an expansion 
 of the glass of the vessel which contains it, increasing 
 z 4 
 
344 ILLUSTRATIONS OF MECHANICS. 
 
 the capacity of the vessel. Nevertheless, this dis- 
 turbing cause may also be taken into the calculation, 
 or at any rate allowance may be made for it by 
 experiment; and thus the quantity of mercury in 
 the glass cistern may be so adjusted as to preserve a 
 position of the centre of oscillation, which ap- 
 proaches to uniformity. 
 
 There are, however, causes of variation in the 
 position of the centre of oscillation, and in the time 
 of oscillation, other than those which have been 
 spoken of, and which appear scarcely to admit of 
 compensation. The tirst is the difference of the 
 times requisite to communicate the variation of each 
 degree of temperature to the two metals, mercury 
 and steel. From a communication recently made 
 by Mr. Dent to the British Association of Science, 
 it appears that the mercury of this pendulum requires 
 nearly four times the interval to acquire a given 
 variation of temperature that the steel rod does. 
 During the whole of this interval the pendulum 
 cannot then be in a state of compensation, and there 
 must be a variation in its beats. Another cause of 
 variation, first noticed by Mr. Dent, and uncompen- 
 sated, is in the varying elasticity of the spring. 
 This elasticity diminishes as the temperature increases, 
 and to this cause Mr. Dent traced, in some of his 
 experiments, an error of nearly 2 seconds in 24 hours, 
 produced by an artificial elevation of the tempera- 
 ture of the spring to 95. He has recommended the 
 substitution of a cast iron cylinder for the reception 
 of the mercury, instead of one of glass. A much 
 more truly cylindrical form can be given to such a 
 vessel, by turning, than a glass cylinder can possibly 
 receive ; it can be more easily fixed to the rod; 
 
HARRISON'S PENDULUM. 345 
 
 moreover it possesses this great practical advantage, 
 that the mercury can be boiled in it, to expel the 
 bubbles of air which, when it is first filled, or after 
 it has been packed up and removed, are very liable to 
 adhere to its interior surface, displacing the mercury. 
 In his experiments to determine the qualities of a 
 pendulum, thus constructed, Mr. Dent's attention 
 was directed to the fact, hitherto unobserved, that 
 the rate of the pendulum was singularly affected by 
 radiant heat. He found that heat radiated from the 
 fire of the room, in which his experiments were made, 
 affected differently the pendulum having the glass 
 vessel, and that having the iron vessel ; the mercury 
 in the former preserving a temperature always 5 
 higher, than that in the latter. The heat radiated 
 from a lamp, was even sufficient to produce an in- 
 equality of 2, and it was only got rid of, completely 
 by screening both the fire and the lamp. 
 
 Directed by this fact, Mr. Dent recommends that 
 the cistern of the mercurial pendulum should always 
 be blackened with a composition of lamp black and 
 spirits of wine. 
 
 288. HARRISON'S COMPENSATION PENDULUM. 
 
 In this pendulum, known as the gridiron pen- 
 dulum, a system of bars, of steel and brass, are com- 
 bined in such a way, as that whilst the elongation 
 of the steel bars tends to depress the bob E of the 
 pendulum, that of the brass bars tends to elevate it: 
 and the lengths of these bars are so adjusted, that 
 the depression thus produced by the former, for each 
 degree of temperature, shall just be equalled by the 
 elevation produced by the latter. 
 
346 
 
 ILLUSTRATIONS OF MECHANICS. 
 
 The shaded lines in the accompanying figure 
 fig. 82. represent the steel bars, and the light 
 lines the brass bars. The pendulum 
 is suspended from the cross piece 
 A B, to the extremities of which are 
 fixed the steel bars A C and B D, 
 carrying the cross bar C D, through a 
 hole, in which the rod which carries 
 the bob passes. On this cross piece 
 C D, rest the two brass bars c a and 
 d b, supporting the cross piece a b. 
 Now, it is evident that the cross 
 piece a b is depressed by the elon- 
 gation of the rods A C and B D, 
 carrying with them the piece C D, 
 and that it is elevated by the elon- 
 gation of the brass bars c a and d b. 
 If then, the bars were of such lengths 
 that the elongation of the one pair 
 should just equal that of the other, 
 then the bar a b would exactly keep 
 its place ; or if they were of such 
 lengths that the elongation of the 
 brass bars exceeded that of the steel bars, then the 
 bar a b would be elevated, and by a proper adjust- 
 ment we might thus cause it to be elevated_, by 
 any quantity we chose. Reasoning in the same 
 manner, with regard to the next pair of the steel 
 bars of the system, and the next pair of brass bars, 
 it is apparent that, supposing the bar at a b to retain 
 its position, we can cause the bar e d to retain its 
 position, or to vary it in any way we like, by pro- 
 perly adjusting the lengths of the bars ; and that 
 
HARRISON'S PENDULUM. 34-7 
 
 this control over the position of the bar e d, is ren- 
 dered yet more perfect, by that which we possess, by 
 a similar adjustment, over the position of a b. Being 
 thus able to give to the position of e d any elevation 
 we like for each variation of a degree of temperature, 
 we can cause it to raise itself by just as much as 
 that variation of a degree of temperature, causes 
 the single steel bar which carries the bob E, to 
 elongate ; so that the bob itself shall accurately 
 remain at the same height. 
 
 The proper lengths of the bars are easily calcu- 
 lated, from the consideration that the elongation of 
 each pair of bars of brass, ultimately elevates the 
 bar e d, whilst that of each pair of steel bars, ulti- 
 mately depresses it. Now there are three pair of 
 bars of steel, and three of brass in the pendulum 
 shown in the cut ; moreover, each pair of steel bars 
 is longer than its corresponding pair of brass bars. 
 If then brass only expanded by the same quantity 
 as steel, for each degree of temperature, then the 
 bar e d would be less raised by each variation of a 
 degree, than it would be depressed, or, on the whole, 
 it would sink for each additional degree, instead of 
 rising as it is required to do. Brass elongates, how- 
 ever, more than steel, for each additional degree of 
 temperature, and it is for this reason that it is used : 
 the expansion of brass is, for every degree of tem- 
 perature, about -|ds that of steel. It is, however, 
 a mistake to suppose that an adjustment of the rods 
 which preserves the position of the bob E, pre- 
 serves a uniformity in the oscillations of the pen- 
 dulum. That uniformity can only be produced, by 
 a constant position of the centre of oscillation of the 
 
34-8 ILLUSTRATIONS OF MECHANICS. 
 
 whole. Now the variations of the lengths of the 
 bars inclosed in the figure A C B D, necessarily pro- 
 duce a variation in the position of the centre of 
 oscillation of that figure, and, therefore of the whole 
 pendulum. 
 
RETARDATION OF MOTION. 349 
 
 CHAP. VI. 
 
 THE RETARDATION OF MOTION THE PRINCIPLE OF 
 VIRTUAL VELOCITIES THE MEASURE OF THE DYNAM- 
 ICAL EFFECT OR THE ACTION OF AN AGENT THE 
 
 DYNAMICAL EFFECTS OF DIFFERENT AGENTS THE 
 MOVING AND WORKING} POWERS IN A MACHINE THE 
 
 MOVING AND WORKING POWERS IN ANY MACHINE ARE 
 EQUAL, ABSTRACTION BEING MADE OF THE RESISTANCES 
 WHICH OPPOSE THEMSELVES TO THE MOTIONS OF THE 
 PARTS OF THE MACHINE UPON ONE ANOTHER THE 
 
 MOVING POWER IN A STEAM-ENGINE THE WORKING 
 
 POWER IN A STEAM-ENGINE. 
 
 289. THE RETARDATION OF A BODY'S MOTION. 
 
 IF a body, having acquired a certain velocity by 
 the action of any accelerating force, be brought to 
 rest, and then projected back again with an equal 
 velocity, in such a way that it shall traverse in the 
 opposite direction the same path as it did before, 
 being acted upon at the same points of its path by 
 exactly the same forces ; but now in opposite direc- 
 tions to its motion, as before they acted in the same 
 directions, so as now to have become retarding in- 
 stead of accelerating forces; then will these take 
 away the force of the body's motion at the same 
 places, precisely by the same quantities that before 
 they increased it ; so that, in describing the same 
 length of path, it will now lose as much of its velo- 
 city as before it gained in that length of path. 
 Thus, then, in the same length of path in which 
 before it gained all its velocity it will now lose it 
 ally and will stop, of its own accord, precisely at the 
 
350 ILLUSTRATIONS OF MECHANICS. 
 
 same point from which before it began to move. A 
 stone, for instance, falling from any height to the 
 ground, and then being projected upwards with a 
 velocity equal to that which it acquired in falling 
 from that height, will ascend again (or, rather, 
 would ascend, if the air offered no resistance to its 
 motion) precisely to the same height from which it 
 fell. For a like reason, if the body P (fig. 75. 
 art. 272.) be allowed to descend freely on the curve 
 from P to A, and then projected back again from 
 A towards P, with a velocity equal to that which 
 it acquired in its descent, it will ascend (friction 
 and the resistance of the air not being considered) 
 precisely to P, and there of its own accord stop. It 
 is manifest that exactly the same result must follow, 
 if, instead of projecting the body thus backward,, up 
 the curve A P, we place another equal and similar 
 curve at A, similarly inclined, but turned the other 
 way, so that the two shall form similar branches of 
 the same curve, like those D B and D C of the 
 curve BCD (fig. 78. art. 274.): the body will then 
 project itself up one of these curves with the velocity 
 which it has acquired in descending down the 
 other, and will ascend upon the former to a height 
 precisely equal to that from which it has descended 
 on the latter ; so that, if it fall from B, then (fric- 
 tion, and the resistance of the air not being con- 
 sidered) it will ascend to C. This reasoning, which 
 is true of a body descending upon a curve, mani- 
 festly applies to a body suspended to a string, and 
 oscillating like a pendulum. This suspension is, 
 indeed, but another way of causing the body to 
 descend on a curve. 
 
VELOCITY OF PROJECTION UP A CURVE. 351 
 
 290. THE VELOCITY OF A BODY'S PROJECTION UP 
 A CURVE MAY BE FOUND BY OBSERVING THE 
 HEIGHT TO WHICH IT ASCENDS UPON IT. 
 
 If a body be projected up a curve, and we observe 
 the vertical height to which it ascends to lose all its 
 velocity of projection, we know the height from 
 which it must fail to acquire an equal velocity. 
 
 We can find, then, what the velocity of its pro 
 jection was, for we can tell what would be the 
 velocity acquired in falling down the curve from 
 the observed height ; that velocity being the same 
 as would be acquired in falling freely, or without 
 the curve, through that height. (See arts. 271. 
 and 267.) It is thus that, in the Ballistic Pendulum 
 (art. 215.), the velocity with which the pendulum 
 begins to move, and hence that with which the ball 
 first strikes upon it, is determined by observing the 
 height to which it first oscillates. The following 
 experiment, illustrative of the principle stated in 
 this article, was made by Desaguliers. He took 
 two hollow cylinders, each of them closed at one 
 extremity, and, having filled them with gunpowder, 
 he caused the open extremity of the one to fit into 
 the open extremity of the other. To similar points 
 in the sides of these cylinders were then attached 
 strings of the same length, fastened at their other 
 extremities to the same point in a horizontal axis ; 
 and, the whole hanging freely from these strings, 
 the gunpowder was exploded. 
 
 The force of motion communicated to each by 
 the explosion should, according to the principles 
 explained in article 211., be the same. Whence, 
 
352 ILLUSTRATIONS OF MECHANICS. 
 
 knowing their masses, might readily be calculated 
 the ratio of the velocities of the bodies immediately 
 after the explosion, and hence the relation of the 
 vertical heights to which they would afterwards 
 respectively ascend. This calculation being made, 
 and the heights being observed, the experiment and 
 calculation were found accurately to coincide.* 
 
 291. THE DEPTH TO WHICH A CANNON OR MUS- 
 KET BALL ENTERS INTO A BLOCK OF WOOD, OR 
 AMASS OF EARTH AGAINST WHICH IT is FIRED, 
 
 VARIES AS THE SQUARE OF THE VELOCITY 
 WITH WHICH IT IMPINGES UPON IT. 
 
 The resistance of such a mass is evidently the 
 same, or nearly so, at every point to which the ball 
 enters: it constitutes therefore a uniformly retarding 
 force. Now, if the ball be supposed to emerge again 
 from the mass into which it has been fired, com- 
 mencing its motion from the point to which it has 
 before been made to sink into it ; if, moreover, at 
 every point of its motion of emergence it be ima- 
 gined to be accelerated by a force precisely equal to 
 that by which it was> as it entered, retarded at that 
 point ; the resistance being, in fact, conceived to be 
 turned in the opposite direction, and converted into 
 
 * The object of Desaguliers, in making this experiment, 
 was to verify that " law of mechanics which is known as that 
 of the equality of action and re-action." The discussion of 
 this law is advisedly omitted in this work. To every person 
 acquainted with the elementary principles of algebra it will be 
 apparent that in the experiment of Desaguliers, the heights 
 to which the cylinders ascended should be inversely as the 
 squares of their weights. 
 
VIRTUAL VELOCITIES. 353 
 
 accelerating forces, then (art. 289.) it will acquire, 
 at the point where it actually emerges, a velocity 
 precisely equal to that with which it before entered 
 the mass there ; since, moreover, the force with 
 which it was resisted when it entered the mass was 
 a uniformly retarding force, the force with which it 
 will be accelerated, as, on this hypothesis, it leaves 
 it, will be a uniformly accelerating force, like that 
 of gravity, and subject to the same description of 
 law. The velocity which it acquires in thus leaving 
 it will then be equal to the square root of some 
 constant number multiplied by the depth (art. 267.), 
 or it will vary as the square root of the depth. 
 Thus, then, the velocity of the first impact varies as 
 the square root of the depth, and conversely the 
 depth varies as the square of the velocity of im- 
 pact. This fact was proved experimentally in a 
 great number of instances by Robins. ( See Robin s 
 Mathematical Tracts, by Wilson, vol. i. p. J 52.) 
 
 292. THE PRINCIPLE OF VIRTUAL VELOCITIES. 
 
 The principle known by this name arises out of 
 that relation between forces of motion and forces 
 of pressure, which has been pointed out in the pre- 
 ceding pages of this work (art. 253.). It embraces 
 every question of equilibrium, and may be con- 
 sidered as including the whole science of statics* ; 
 
 * The principles of the parallelogram of forces, and the 
 equality of moments upon either of which the whole science of 
 statics may be considered to be founded, may readily be 
 deduced from it. 
 
 A A 
 
354? ILLUSTRATIONS OF MECHANICS. 
 
 and it is especially important that it should be 
 known to practical men under its accurate and 
 most general form, because vague and exceedingly 
 erroneous notions of it are prevalent amongst 
 workmen, and conclusions false at once in practice 
 and in theory are deduced from it. To under- 
 stand what is meant by the virtual velocity of a force 
 (which is the only difficulty in the matter), lej; a 
 system of forces be supposed to be in equilibrium, 
 and let the points of application of two or more 
 of these forces be supposed to be capable of dis- 
 placement, the displacement of any one point 
 bringing about a displacement of the rest. Sup- 
 pose, moreover, a displacement of this kind to be 
 actually made in the system, but let it be an ex- 
 ceedingly small displacement, so that all the move- 
 able points of application afterwards occupy positions 
 different from those they occupied before, but ex- 
 ceedingly near to them, and all the forces applied 
 to them act in directions different from those in 
 which they acted before, but exceedingly near to 
 those directions. From the new point of applica- 
 tion of each force, drop a perpendicular upon the 
 previous direction of that force ; then the line 
 intercepted between the previous point of ap- 
 plication of that force and the foot of this per- 
 pendicular, will be what is called the VIRTUAL 
 VELOCITY of the force. This definition will be 
 more readily understood by a reference to the 
 accompanying diagram, where the arrows Pj p 2 , 
 P 2 P-* P 3 Ps P* ?4> P 5 P 5 > are supposed to represent 
 forces in equilibrium applied to the points p n p 2 , p 3 , 
 p 4 , which points are supposed moreover to be 
 
VIRTUAL VELOCITIES. 355 
 
 capable of displacement under certain limitations. 
 A small displacement is made in one of these points 
 
 of application, as, 
 for instance, p 15 
 r - /7 ' which is moved 
 to any other point 
 near to it, as q 1? 
 the force upon 
 that point now 
 acting in the di- 
 rection Q! q lB 
 This displacement 
 of the direction 
 and point of ap- 
 plication of one of the forces necessarily brings 
 about a corresponding displacement of all the rest ; 
 their new positions are supposed to be represented 
 by Q 2 q 2 > Q 3 q 3 Q* q*> Qs q 5 > and their new points 
 of application by q 2 q 3 q 4 q 3 . From these last men- 
 tioned points let perpendiculars qiV , q2v 2 , q 3 v 3 , q 4 v 4 , 
 qsV 5 , be supposed to be drawn upon the previous 
 directions of the forces, or these directions pro- 
 duced if necessary ; then the lines piv t , p 2 v 2 , p 3 v 3 , 
 p 4 v 4 , p 5 v 5 , intercepted, on the directions of the ori- 
 ginal directions of the forces, between their points 
 of application, and the feet of the perpendiculars, 
 are the VIRTUAL VELOCITIES of their respective 
 forces. This being thoroughly understood, the 
 enunciation of the principle of virtual velocities 
 becomes easy. It is this : 
 
 293. IF ANY NUMBER OF FORCES BE UNDER ANY 
 CIRCUMSTANCES IN EQUILIBRIUM, AND TO 
 A A 2 
 
356 ILLUSTRATIONS OF MECHANICS. 
 
 ANY OR ALL OF THEIR POINTS OF APPLICA- 
 TION THERE BE COMMUNICATED INDEFINITELY 
 SMALL MOTIONS IN ANY DIRECTIONS ; THEN 
 THESE FORCES, BEING EACH MULTIPLIED BY ITS 
 CORRESPONDING VIRTUAL VELOCITY, AND THE 
 SUM OF THESE PRODUCTS BEING TAKEN IN 
 RESPECT TO THOSE FORCES, THE DISPLACEMENTS 
 OF WHOSE POINTS OF APPLICATION ARE TO- 
 WARDS THE DIRECTIONS OF THEIR FORCES, AND 
 THE SUM IN RESPECT TO THOSE WHOSE DIS- 
 PLACEMENTS ARE FROM THE DIRECTIONS OF 
 THEIR FORCES ; THE ONE SUM SHALL EQUAL 
 
 THE OTHER. 
 
 Thus, referring to the diagram, let the forces 
 P i? P 2 , P 3 , be supposed to be respectively multi- 
 plied * by their virtual velocities, p^j, p 2 v 2 , p 3 v 3 , p 4 v 4j 
 p 5 v s ; then, it being observed that the displacements 
 of the points p 2 , p 3 , and p 5 , are towards the direc- 
 tions of the forces acting upon those points, whilst 
 the displacements of the points p x and p 4 are from 
 the directions of the forces acting at those points ; 
 by the principle of virtual velocities, the sum of 
 the above-mentioned products, in respect to the 
 first three, shall equal their sum in respect to the 
 two others. That is the sum of the products 
 P 2 by p 2 v 2 , P 3 by p 3 v 3 , P 5 by p 5 v 5 , shall equal the 
 sum of the products, P x by p^, and P 4 by p 4 v 4 .f 
 
 It is evident that if the displacement of any point 
 *ake place actually in the direction of the force 
 
 * It is here meant that the number of units in the force is 
 to be multiplied by the number of units of length in the vir- 
 tual velocity. 
 
 f This relation is expressed algebraically thus : 
 
VIRTUAL VELOCITIES. 357 
 
 applied at that point, then the perpendicular will 
 vanish, and the virtual velocity will be the actual 
 displacement of the point of application. 
 
 If, for instance, the point p t had been displaced 
 not to q ly but actually in the line of direction of the 
 force P! or along the line P t p x to any point r, then 
 P! r, the actual displacement of the point of applica- 
 tion of P! would have been also its virtual velocity. 
 
 If, moreover, the system to which the forces are 
 applied had been such that, the point of application 
 of any one being displaced actually in the line of 
 direction of that force, the points of application of 
 all the rest should have been displaced in the lines 
 of direction of their respective forces, then the 
 actual displacements of all would have been their 
 virtual velocities. 
 
 A particular case of the principle of virtual 
 velocities may then be enunciated under the fol- 
 lowing form : 
 
 " When the relation of the parts of a system acted 
 upon by any number of forces is such that the point 
 of application of any one force being displaced in 
 the line of the direction of that force, then the dis- 
 placements thereby produced in all the other points 
 of application shall be in the lines of direction of 
 their respective forces; then each force being multi- 
 plied by its actual displacement, the sum of these 
 products in respect to those whose displacements are 
 from the directions of the forces shall equal the sum 
 in respect to those whose displacements are towards 
 those directions" 
 
 The circumstances here supposed obtain in respect 
 to almost all the simple MECHANICAL POWERS, as 
 
 A A 3 
 
358 ILLUSTRATIONS OF MECHANICS. 
 
 they are usually applied, and in respect to a great 
 number of compounded machines, especially such 
 as act by animal power. 
 
 Take, for instance, the various applications of 
 the systems of LEVERS, shown in page 129. In 
 each, when the lever is first put into operation, the 
 points of application of the power and weight are 
 made to move in vertical directions ; that is, in the 
 lines of direction in which they severally act. The 
 virtual velocity of each is therefore its actual dis- 
 placement^ so that by the principle of virtual 
 velocities the displacement or motion of the point 
 of application of W, multiplied by W, is equal to 
 the displacement or motion of the point of applica- 
 tion of P, multiplied by P. 
 
 Now, it is evident in jig. 25., that since A is 
 eight times as far from the fulcrum as W, therefore 
 the displacement of A must equal eight times that 
 of W. Thus, then, it follows, that W, multiplied by 
 the displacement of W, equals P multiplied by 
 eight times the displacement of W, and therefore, 
 that W equals eight times P ; as it was shown to 
 be by the principle of the equality of moments. 
 
 Again, inj^. 26., since A is nine times as far from 
 the axis of motion as W is, it evidently moves nine 
 times as fast ; therefore, by the principle of virtual 
 velocities, W, multiplied by the displacement of W, 
 equals P multiplied by nine times the displacement 
 of W, so that W equals nine times P, as it ought. 
 
 Again, in the wheel and axle (art. \35.fig. 31.), 
 the displacements of the power and weight evi- 
 dently take place in the lines of the directions of 
 those forces ; these displacements are therefore the 
 
VIRTUAL VELOCITIES. 359 
 
 virtual velocities. So that by the principle of vir- 
 tual velocities, W, multiplied by the displacement of 
 W, is equal to P multiplied by the displacement of 
 P ; but it is evident that the displacement of W 
 (being the length of string wound on the lesser 
 cylinder) is to that of P (being the length of string 
 wound off the greater cylinder) as O A to OB: 
 hence it follows, by this principle, that W, mul- 
 tipled by O A, is equal to P multiplied by O B ; 
 which relation was also shown to result from the 
 principle of the equality of moments. 
 
 If the relation of the parts of the system be 
 such that after the first small displacement, causing 
 all the various points of application to take up new 
 positions near the first, the forces shall, under 
 these altered circumstances, be still in equilibrium ; 
 then a second small displacement, similarly produced 
 in each out of its second position into a third will, 
 like the first, be subject to the principle of virtual 
 velocities ; and if, in these third positions, they are 
 , in equilibrium, then a fourth displacement will be 
 subject to the same law, and so on. From this it 
 follows, that if the system be such that the forces 
 applied to it are continually in equilibrium, through- 
 out all the displacements to which they are sub- 
 jected, then if, after any number of such displace- 
 ments, each force be multiplied by the sum of all the 
 virtual velocities corresponding to these displace- 
 ments, the equality spoken of before shall obtain 
 between the sum of these products, in respect to 
 those displacements which take place towards the 
 direction of the force and the sum of those which 
 take place from it. 
 
 A A 4 
 
360 
 
 ILLUSTRATIONS OF MECHANICS. 
 
 Thus, for instance, if the balls A l and B,, and the 
 bar which connects them, be in equilibrium about 
 
 Fig. 84, 
 
 the point F, that point, supporting the centre of 
 gravity of the whole system, and the whole be 
 turned round into a series of new positions, differing 
 slightly from one another, and represented by the 
 dotted lines, then, since in each position the system 
 will be in equilibrium, it follows, by the principle 
 of virtual velocities, that the weight B 1? multiplied 
 by the sum of the virtual velocities BI Vi, B 2 V 2 , &c., 
 shall equal the weight A^ multiplied by the sum 
 of the virtual velocities A 1 V 1? A 2 V 2 , &c. Now 
 the former sum is evidently equal to the vertical 
 line B 4 N, and the latter to A 4 M ; thus, then, it 
 follows that the product of B 4 N by Bj equals the 
 product of A 4 M by A. 
 
 If the displacements of all the forces of the 
 system take place actually in the lines of the opera- 
 tion of the forces, and the equilibrium remain after 
 
VIRTUAL VELOCITIES. 361 
 
 every displacement, then the condition of an ex- 
 ceedingly small displacement disappears from the 
 enunciation of the general principle. In this par- 
 ticular case, each force being multiplied by its 
 actual displacement, however great it may be, the 
 sum of these products, in respect to those displace- 
 ments which take place towards the direction in 
 which the force acts, shall equal the sum in respect 
 to those which take place from that direction. 
 
 This is the principle known to workmen, as that 
 by which what is gained in power is lost in velocity. 
 Its application is limited to the particular case 
 last described ; applied beyond those limits, it leads 
 to serious errors. 
 
 All the systems of pulleys represented in fig. 65. 
 p. 216. offer illustrations of it. In the first, the 
 single fixed pulley, it is evident that the displace- 
 ment of the power is exactly equal to that of the 
 weight ; and, since the product of the former, by its 
 displacement must equal that of the latter by its 
 displacement, it is evident that to make up this 
 equality the power must equal the weight. 
 
 In the second -system, the string which carries the 
 power evidently lengthens by as much as the two 
 strings which carry the weight, together shorten ; 
 that is, by twice as much as either shortens sepa- 
 rately ; so that the displacement of the power is 
 equal to twice that of the weight. By the principle 
 of virtual velocities, then, the weight multiplied by 
 the weight's displacement equals the power multi- 
 plied by twice the weight's displacement; so that 
 the weight equals twice the power. 
 
 In the fourth system, the power evidently descends 
 
362 v ILLUSTRATIONS OF MECHANICS. 
 
 by twice as much as the first moveable pulley 
 ascends. 
 
 Again, this last ascends by twice as much as the 
 second moveable pulley ascends ; so that the power 
 ascends by four times as much as the second 
 moveable pulley. This second moveable pulley 
 ascends similarly, by twice as much as the third, and 
 by four times as much as the fourth ; so that, on the 
 whole, it is evident that the power is displaced by 
 eight times as much as the weight. By the principle 
 of virtual velocities, the weight, then, multiplied by 
 the weight's displacement, equals the power multi- 
 plied by eight times the weight's displacement. So 
 that the weight equals eight times the power t A 
 similar method of reasoning may be very easily 
 applied to all the other systems, except the third, 
 which offers some difficulty. 
 
 In this system the displacement of the power is 
 made up of the lengthening of the string to which 
 it is attached and the descent of the pulley over 
 which that string passes. Now, the lengthening of 
 the string which carries the power results partly 
 from the ascent of the weight, and is in this re- 
 spect the same as in the last case of the single 
 moveable pulley, equalling twice the ascent of 
 the weight ; and partly it results from the descent 
 of the pulley over which it passes, in this re- 
 spect equalling the descent of that pulley, and 
 therefore equalling the ascent of the weight ; so 
 that, upon the whole, the string which carries the 
 power lengthens by three times the ascent of the 
 weight ; again, the pulley over which this string 
 passes descends by as much as the weight ascends, 
 
OF MACHINES. 363 
 
 so that altogether the power P descends by four 
 times as much as the weight ascends. By the 
 principle of virtual velocities, therefore, the weight 
 equals four times the power. 
 
 294. OF MACHINES. 
 
 A machine is an assemblage of parts destined to 
 receive the operation of an agent, and to transmit it 
 to the point where it is to be applied, modifying it 
 in the transmission, according to the circumstances 
 under which it is to be applied. Thus, in a ma- 
 chine there are to be considered, 1st, the circum- 
 stances under which the operation of the moving 
 power is received ; 2dly, the circumstances by 
 which it is modified during its transmission ; 3dly, 
 the circumstances under which it is applied at its 
 working points. The power which operates directly 
 from the agent we shall here call the MOVING 
 POWER OR ACTION on the machine ; the power ac- 
 tually applied by the machine at its working points 
 in the performance of its work, we shall call the 
 WORKING POWER OR ACTION on the machine. It 
 is evident that the moving power produces the 
 working power, and also the motion of all the parts 
 if the machine, overcoming the resistances which 
 oppose themselves to the motions of those parts ; 
 so that the working power is essentially less than 
 the moving power in all cases, and in complex 
 machines greatly less, by reason of the great num- 
 ber of surfaces which in those machines are made 
 to move upon one another, and the great amount 
 of the resistances which for that reason oppose 
 themselves to their motion. 
 
364 ILLUSTRATIONS OF MECHANICS. 
 
 295. THE STATE OF THE MOTION OF A MA- 
 CHINE IS, AT FIRST, A STATE OF ACCELE- 
 RATED MOTION. 
 
 This is evident from the principles laid down in 
 art. 253. Each part of the machine must have, 
 before it can move, a FORCE OF MOTION or MO- 
 MENTUM communicated to it, and such momentum 
 being in its nature an accumulation of pressures, 
 requires, in every case, TIME, and a series of im- 
 pulses to its accumulation. 
 
 296. THE FORCES OPERATING IN A MACHINE 
 BEING IN EQUILIBRIUM IN EVERY RELATIVE 
 POSITION WHICH THE PARTS OF THAT MA- 
 CHINE CAN BE MADE TO ASSUME, ANY Mo- 
 
 MENTUM OR FORCE OF MOTION THROWN INTO 
 THE MACHINE WILL REMAIN IN IT CONTIN- 
 UALLY, UMIMPAIRED AND UNALTERED. . 
 
 In the statement of this principle, all consider- 
 ation of the resistance of the air is omitted, and the 
 friction of bodies in motion is supposed not to be 
 affected by the velocity of motion (see art. 172.). 
 The truth of it is immediately evident from the 
 consideration, that the forces operating upon the 
 machine including the friction of its parts, and 
 every other form of its resistances being supposed 
 in every position of its parts to be in equilibrium*, 
 it follows that there cannot at any period of its 
 
 * The equilibrium here spoken of, and every where else in 
 this work, is that of the state immediately bordering upon 
 motion. 
 
OF MACHINES. 36 
 
 motion be any force opposing itself to the force 
 of the motion of its parts ; this force, then, by 
 the principle of the permanence of the force of 
 motion (art. 193.), being once communicated, must 
 remain in the machine unimpaired. 
 
 If the forces operating upon a machine be not in 
 the state of equilibrium bordering upon motion 
 when motion is first communicated ; or if this 
 condition of equilibrium does not continue through- 
 out the motion of the parts of the machine ; then the 
 whole quantity of motion operating in the machine 
 will continually vary ; if the power be in excess it 
 will increase, if the resistance be in excess it will 
 diminish. In the former case the excess of the 
 power over that necessary to produce equilibrium 
 (remaining unopposed) continually generates addi- 
 tional momentum ; in the latter case the excess of 
 the resistance, over that portion of it which is over- 
 come by the power, operating in a direction opposite 
 to the motion, continually diminishes, and even- 
 tually destroys it. 
 
 Although in the first period of the motion of a 
 machine, the power operating in it may be greater 
 than that which would produce an equilibrium with 
 the resistance, yet practically, in every machine, 
 that relation of these forces, which is necessary 
 to their equilibrium (and which is accompanied by 
 a permanence of the force of motion), grows up 
 shortly after the motion has commenced. It is a 
 LAW imposed in the economy of the creation 
 around us, that no motion shall pass a certain 
 finite limit. 
 
 A few examples will render this sufficiently evi- 
 dent : 
 
366 ILLUSTRATIONS OF MECHANICS. 
 
 A ship, when at rest upon the water, and with her 
 anchor weighed, is in a state of equilibrium bor- 
 dering upon motion ; the pressures upon her bows 
 and stern are equal, and any force, however slight, 
 acting upon her horizontally in the direction of 
 her length, would be sufficient to move her. Her 
 sails are unfurled, and she receives the impulse of 
 the wind, a power which, if it continued, as at 
 first, unopposed, would continually accumulate ve- 
 locity in her, until she flew through the water as 
 fleet at least as the wind itself. That equilibrium, 
 however, of the forces upon her head and stern, 
 which obtained at first, does not remain ; the forces 
 upon the head, constituting the resistance, increase 
 with the motion*, and those upon the stern dimi- 
 nish ; and in a short time the impulse of the wind 
 upon the sails, and the pressure of the water upon 
 the stern, come to be together precisely equalled 
 by the increased resistance upon the bows. The 
 state of equilibrium is now, then, reproduced ; and 
 as long as it is kept up, the vessel moves on with 
 the quantity of force of motion which it had when 
 it passed into this state of equilibrium, unimpaired. 
 
 Again, let us suppose a pulley suspended at any 
 height, however great, above the earth's surface, 
 and a string of equal length to pass over it, carrying 
 at its extremities two unequal weights. Suppose 
 the greater weight to be drawn up, and the whole 
 machine then to be left to itself, the excess of the 
 greater over the lesser weight will evidently be an 
 unopposed power, and will communicate motion to 
 the system ; which motion, by the continual im- 
 * They increase as the square of the velocity. 
 
OF MACHINES. 367 
 
 pulses of this power, would be continually acce- 
 lerated, with no other limit than that of the height 
 through which the weight is allowed to descend ; 
 so that by increasing this height we could accumu- 
 late velocity and force of motion to any conceivable 
 extent, were it not for the resistance of the air ; 
 this would effectually limit any such accumulation. 
 It is a resistance which would be found rapidly to 
 increase with the velocity of the descent, and which 
 would soon become so great as entirely to baffle 
 any further effort of the power to increase the ra- 
 pidity of the motion ; in short this resistance would 
 soon pass into a state of equilibrium with the moving 
 power, and from that period the velocity of the 
 descent would be uniform, becoming what is tech- 
 nically called the terminal velocity. It is shown by 
 theory, and has been confirmed by numerous expe- 
 riments, that this terminal velocity of a descending 
 body is very soon acquired, and is by no means a 
 considerable velocity. Dr. Hutton has calculated 
 that a leaden ball one inch in diameter, could not, 
 by descending freely through the air (even if the air 
 were every where of the same density as at the 
 earth's surface) acquire a velocity of more than 
 260 feet per second. This velocity it would acquire 
 in falling through 2687 feet, or about half a mile.* 
 
 * Theoretical deductions on these subjects have been more 
 or less confirmed by numerous experiments in artillery prac- 
 tice. The method of the experiments was this : Bullets 
 fired vertically into the air, were received, on their descent, upon 
 planks of soft wood, and the velocity of the descent was judged 
 of from experimental data by the depths to which they sank 
 in the wood. 
 
368 ILLUSTRATIONS OF MECHANICS. 
 
 We shall take as our third and last example the 
 case of the LOCOMOTIVE CARRIAGE. 
 
 The pressure which opposes itself to the motion 
 of a carriage upon a railroad, where the road is ac- 
 curately level or horizontal, is about 8 Ibs. per ton 
 weight ; so that in a train weighing, carriages and 
 all, 10 tons, there would not be more than 1 80 Ibs. 
 ef resistance to be counterbalanced, that the whole 
 might be placed in a state bordering upon motion ; 
 and, as the engine of every locomotive carriage is 
 capable of producing upon its piston a far greater 
 pressure than this, it might be imagined that this 
 excess of power would produce a continually acce- 
 lerating motion, and that when this had attained its 
 greatest limit, consistently with the safety of transit, 
 the steam must be thrown off, and the pressure re- 
 duced to 1801b., to prevent any further accumu- 
 lation. In reality, however, instead of the velocity 
 of a locomotive being thus difficult to control and 
 keep down to limits consistent with safety, it has 
 been found impracticable to get it up even to those 
 limits which public expectation had fixed itself 
 upon, and which public convenience may be sup- 
 posed to demand. To a preservation of the con- 
 dition of the state bordering upon motion, it 
 is necessary that the cylinder should be contin- 
 ually filled and refilled with steam of the requisite 
 pressure. Thus to a rapid motion a rapid pro- 
 duction of steam becomes necessary ; and on this 
 the dimensions of the fire-place and boiler, and the 
 force of the draught of air, soon place a limit. 
 Again the resistance of the air increases with the 
 square of the velocity with which the carriage 
 moves ; so that when it moves with any considerable 
 
DYNAMICAL ACTION OR EFFECT. 369 
 
 degree of velocity, the motion of the carriage comes 
 to be opposed by this cause with a force adding 
 itself to the resistance of its friction, and soon 
 greatly exceeding it. The amount of this resist- 
 ance on the broad surfaces of the carriages will be 
 judged of when it is stated that it is equal to the 
 the pressure which a wind, moving with the velocity 
 of the carriages, would produce upon them at rest, 
 if that wind moved exactly in the line of the road ; 
 and, moreover, that, by the experiments of Smea- 
 ton, a wind moving with the velocity of from 30 to 
 35 miles an hour is a very high wind, almost 
 amounting to a gale. 
 
 297. THE DYNAMICAL EFFECT, OR THE AMOUNT 
 OF THE ACTION OR EFFICIENCY OF ANY AGENT, 
 
 IS MEASURED BY THE PRESSURE WHICH IT 
 EXERTS MULTIPLIED BY THE SPACE THROUGH 
 WHICH IT EXERTS IT. 
 
 For it is evident that the pressure exerted remain- 
 ing the same, the action or effect will vary as the 
 space through which it is exerted, and that the 
 space remaining the same it will vary as the pressure 
 exerted ; thus, by the rules of proportion, when both 
 vary, the action or effect will vary as their product. 
 
 Thus, for instance, a horse drawing a loaded car- 
 riage over six miles of road will exert a double 
 action and produce a double effect when his load 
 is doubled, and therefore his constant pressure 
 upon it doubled ; a triple effect when he draws 
 a triple load ; a quadruple effect when he draws 
 a quadruple load over this six miles of road, and so 
 
 B B 
 
370 ILLUSTRATIONS OF MECHANICS. 
 
 on ; so that the space he traverses remaining the 
 same, his effect will vary as the pressure which 
 he applies. Again, his load, and therefore the 
 pressure he applies, remaining the same, the effect 
 he produces will vary as the space he traverses. 
 Thus if he draw the same load twelve miles instead 
 of six, his effect will be doubled; if eighteen, 
 tripled, and so on. Since, then, his action or effect 
 varies as the pressure he applies when the space is 
 constant, and as the space when the pressure is con- 
 stant, it follows that when neither is constant it 
 varies as their product. 
 
 Thus the dynamical action or effect of a horse 
 which draws a load of 6 cwt. over two miles of 
 level road, is the same with that of a horse which 
 draws 4- cwt. over three miles ; since 6 X 2 is equal 
 to 4 x 3.* 
 
 The dynamical effect of a weight of 4 cwt. acting 
 to impel or to resist the motion of a machine 
 through 10 feet, is to that of a weight of 5 cwt. acting 
 through 12 feet as 2 to 3 ; since the product of 4 by 
 10 or 40, is to the product of 5 by 12 or 60 in that 
 ratio. 
 
 * This equality may perhaps be understood better by some 
 persons thus : the effect of 6 cwt. drawn over two miles is the 
 same as that of 1 2 cwt. over one mile ; for whether two horses 
 draw each 3 cwt. over the same mile or draw these over two 
 successive miles, the same dynamical effect is evidently pro- 
 duced. By exactly the same reasoning it is evident that the 
 effect of 4 cwt. drawn over three miles is the same as that of 
 1 2 cwt. over one mile : both of these dynamical effects being 
 therefore equal to 1 2 cwt. drawn over one mile, are equal to 
 one another. 
 
DYNAMICAL ACTION OR EFFECT. 371 
 
 298. THE DYNAMICAL EFFICIENCIES OF DIF- 
 FERENT AGENTS. 
 
 There are two ways of speaking of the dynamical 
 effect of an agent. We may speak of it as the 
 mean effect produced in a given period, as for in- 
 stance, one minute of the operation of that agent ; 
 or we may speak of it as the whole effect which 
 that agent is capable of producing, before its ope- 
 ration is withdrawn, or its powers become extinct. 
 In the former sense we speak of the mean effect 
 which a horse drawing a load is capable of pro- 
 ducing per minute, or of the effect which a given 
 quantity of fuel burning in the furnace of a steam 
 engine is capable of producing (by the interven- 
 tion of the water and steam,) upon the piston per 
 minute ; in the latter sense we speak of the whole 
 dynamical effect which a horse is capable of pro- 
 ducing during its life ; or a bushel of coals before 
 it is burned out. 
 
 299. THE DYNAMICAL EFFECT OF A HUMAN 
 AGENT. 
 
 The muscular power of a man is usually made to 
 operate either by his legs or his arms, rarely by 
 both together. It has been estimated that by the 
 action of his legs upon a treadwheel, he can raise 
 his own weight, about 150lbs., 10,000 feet per day ; 
 which gives a dynamical effect of 1,500.000 per 
 day, or 3125 per minute, supposing the work to 
 be continued eight hours a day. 
 
 A man who ascended a hill 10,000 feet high, 
 B B 2 
 
3?2 ILLUSTRATIONS OF MECHANICS. 
 
 would do a good day's work ; a result which cor- 
 roborates the preceding. 
 
 In respect to the dynamical effect of a man 
 working with his arms, we have the authority of 
 Smeaton, that a good labourer can thus raise 370 Ib. 
 10 feet high per minute ; so that his dynamical 
 effect is 3700 ; being somewhat greater with his 
 arms than his legs. Desaguiliers makes the dy- 
 namical effect of a man working with his arms, 
 5500 per minute : this is, however, considered too 
 high an estimate. 
 
 300. THE DYNAMICAL EFFECT OF A HORSE. 
 
 A horse drawing a weight out of a well over a 
 pulley can, according to Desaguiliers, raise 200 Ibs. 
 for eight hours together, at the rate of 2-J- miles 
 or 13,200 feet, per hour. This gives for the dy- 
 namical effect of a horse per minute 29,333. 
 
 The usual estimate of the dynamical effect per 
 minute of a horse, called by engineers a HORSE'S 
 POWER, is 33,000. 
 
 Mr. Smeaton states it to be 22,000. 
 
 301. THE POWER OF A LIVING AGENT TO 
 PRODUCE A GIVEN DYNAMICAL EFFECT* 
 
 A distinction must be made between the dy- 
 namical effect produced by a living agent, and its 
 power of producing that effect as affected by the cir- 
 cmstances under which it is produced. Thus the dyna- 
 mical effect of a load of 200 Ibs. raised by a horse for 
 8 hours a day, at the rate of 2- miles an hour, 
 is the same with that of 20 Ibs. raised for the same 
 
DYNAMICAL ACTION OR EFFECT. 373 
 
 period at 25 miles an hour ; but the power of pro- 
 ducing this effect, considered as residing in t!u* 
 horse, is not the same ; in fact, the action exerted by 
 the horse to produce these two effects is different ; 
 he has to carry the weight of his body, lifting it 
 a certain height at every step, much farther in the 
 one case than the other. The distinction between 
 the two, is that between the moving and the working 
 power in a machine. The moving action or effect 
 includes the motion communicated to the machinery 
 of the horse's body, the working action or effect 
 only that applied to the load. 
 
 An animal is best capable of exerting its mus- 
 cular power against any resisting force, when it is 
 at rest. When it is in motion, a portion of its 
 muscular force is consumed in its motion. If the 
 rate at which a horse is travelling per hour in miles 
 be subtracted from 12, and the remainder squared, 
 a number will be obtained, which will, it is said, 
 represent the number of pounds of traction which 
 the horse is capable of exerting, when it moves with 
 this velocity. 
 
 Thus, if the horse be moving at the rate of 4? 
 miles per hour, this number being subtracted from 
 12, gives 8, which squared is 64. So that the 
 horse could, according to this rule, walking at 
 4 miles per hour, be able to draw with a force of 64 
 Ibs. Now 4 miles per hour is 35 C Z feet per minute. 
 The dynamical effect per minute of a horse, thus 
 drawing, would then be 22,528. 
 
 A waggon loaded with 86 tons, and therefore 
 requiring a traction of^ of this weight, or If tons, 
 may be drawn by 8 horses, at 2-Jr miles an hour, 
 B B 3 
 
374? ILLUSTRATIONS OF MECHANICS. 
 
 for 8 hours daily. This gives a dynamical ^effect 
 per minute of 41,066 for each horse. 
 
 A mail coach, of 2 tons weight, and travelling 
 at the rate of 10 miles per hour, may be worked 
 on a turnpike road both ways, by as many horses 
 as there are miles of road. The dynamical eifect 
 per minute may In this case be calculated as be- 
 fore: it will be found to be 8215, being scarcely 
 I of the effect which the horses would have been 
 capable of producing at the slower rate of the 
 waggon. 
 
 302. THE DYNAMICAL EFFECT OF ONE POUND 
 OF COALS. 
 
 The power of heat, which slumbers among the 
 particles of a mass of coal, is best called into 
 operation as a dynamical agent by combining it 
 with water under the form of steam. According 
 to Mr. Watt, a bushel of coals (84 Ibs.) will con- 
 vert into steam 10 cubic feet of water, so that 
 8*4 Ibs. is sufficient to vaporise 1 cubic foot. Now. 
 1 cubic foot of water, according to Tredgold 
 (p. 153.), will expand itself into 1711 cubic feet of 
 steam at temperature 212, and retaining an elas- 
 ticity equal to the pressure of one atmosphere. 
 These 1711 cubic feet of steam are therefore capa- 
 ble of propelling a piston of 1 foot square, under 
 the pressure of one atmosphere, through a distance 
 of 1 7 1 1 feet. Now, the pressure of the atmosphere on 
 a surface 1 foot square, is 21 20 Ibs. These 8*4 Ibs. 
 of coals, thus converting into steam a cubic foot of 
 water, are capable therefore, through this inter- 
 vention of the steam, of producing a dynamical 
 
DYNAMICAL EFFECT IN A MACHINE. 375 
 
 effect represented by the product 1711x2120, or 
 by 3,627,320. 
 
 This effect being produced by 8*4 Ibs,, the effect 
 of ] Ib. is obtained by dividing it by 8*4; by which 
 division we find 431,824 for the dynamical effect 
 which 1 Ib. of coals is capable of producing. 
 
 303. THE DYNAMICAL EFFECT OF ANY AGENT 
 OPERATING THROUGH A MACHINE WHICH 
 MOVES WITH A UNIFORM MOTION, IS THE 
 SAME WHATEVER THAT MACHINE MAY BE, 
 PROVIDED ONLY THE RESISTANCES OPPOSED 
 TO THE MOTIONS OF THE PARTS OF THE 
 MACHINE BY FRICTION AND OTHER OPPOSING 
 CAUSES BE THE SAME. 
 
 For to the state of the uniform motion of a 
 machine there is necessary that state of the equi- 
 librium of the pressures acting upon it which 
 borders upon motion (see art. 294.). And this 
 state of the equilibrium of the pressures acting 
 upon the machine supposes, by the principle of 
 virtual velocities, that the product of the power by 
 the space it describes should equal the sum of the 
 products of the resistances * by the spaces they 
 severally describe. Now, the product of any pres- 
 sure by the space through which it is made to act 
 
 is its DYNAMICAL EFFECT. 
 
 * The resistance upon any point of a machine implies a 
 force acting in a direction opposite to that in which the motion 
 of the point takes place. The power and the resistances in 
 the machine here spoken of, are all supposed to operate actually 
 in the lines of direction in which the points to which they are 
 applied move. 
 
S76 ILLUSTRATIONS OF MECHANICS. 
 
 Including then, among these resistances, together 
 with those upon the working points of the machine, 
 those offered by the frictions of its various inter- 
 mediate moving parts upon one another, the un- 
 countorbalanced weights of certain of them which 
 are raised as the motion goes on, and the resistance 
 of the air upon the motion of all ; it follows that 
 the dynamical effect of the power is equal to the 
 sum of the dynamical effects of the resistances ; 
 and that separating the resistances upon the work- 
 ing points of a machine from the rest of the 
 resistances upon it, and supposing these last to be 
 in every respect the same in different machines ; 
 then the same agent operating equally (that is, with 
 the same dynamical effect upon the receiving 
 organ) through these different machines, will pro- 
 duce the same aggregate dynamical effect upon the 
 working points of all. 
 
 That the state of the uniform motion of the 
 machine should have been attained is necessary to 
 the application of this principle, as is expressly 
 stated in the enunciation of it ; for in that state of 
 accelerating motion which precedes the uniform 
 motion of the machine, the distribution of pressure 
 and motion will vary not only with the frictions 
 and uncounterbalanced weights of the parts of diffe- 
 rent machines, but with their actual weights and 
 dimensions, and the distribution of their dimensions 
 in respect to their axes of motion (arts. 221. and 
 225.). 
 
 Since neither in these respects, nor in respect to 
 the frictions of their various surfaces of motion 
 upon one another, or their uncounterbalanced 
 
THE MOVING POWER IN A STEAM ENGINE. 377 
 
 weights, can there be a positive equality between 
 any two; and since in respect to machines gene- 
 rally there is in all these respects a great inequality, 
 it follows that generally the dynamical effects pro- 
 duced upon the working points of different machines 
 by equal operations of the same agent ARE NOT 
 THE SAME ; and, therefore, that to estimate the 
 actually working effects of the same agent on 
 different machines, it is necessary to know what 
 portion of the dynamical effect, made to operate in 
 each machine, is consumed in the resistances opposed 
 to the machine, elsewhere than at its working points, 
 and with this view to distinguish between the 
 moving and working powers, or the dynamical 
 effects produced at the moving and at the working 
 points ; between THE EFFECTS PRODUCED AT THE 
 POINT WHICH RECEIVES THE OPERATION OF THE 
 AGENT AND AT THE POINTS WHICH APPLY IT. 
 
 304-. THE DYNAMICAL EFFECT UPON THE MOVING 
 POINT, OR THE MOVING POWER, IN A STEAM 
 ENGINE. 
 
 In a steam engine the operation of the agent 
 (the steam) is received upon the piston. To esti- 
 mate the dynamical effect of this agent upon the 
 moving point, we have then to determine the 
 pressure of the steam upon the piston and the ve- 
 locity in feet with which the piston moves per 
 minute ; the product of these will give the dyna- 
 mical effect upon the piston per minute. This is 
 termed THE POWER of the engine. Compared with 
 the dynamical effect of a horse per minute, which 
 
378 ILLUSTRATIONS OF MECHANICS. 
 
 we have seen to be 33,000,, it determines what is 
 called the HORSE'S POWER of the engine. There 
 is very great difficulty, however, in determining 
 the elasticity of the steam in the cylinder and its 
 actual pressure upon the piston. The steam gauge 
 determines it under all circumstances with sufficient 
 accuracy in the boiler; but the elasticities of the 
 steam in the cylinder and in the boiler are not the 
 same : the former is influenced by the rapid state 
 of the motion of the steam through the narrow 
 passage of the steam pipes and its expansion into 
 the body of the cylinder, and especially it is in- 
 fluenced by the greater or less opposition which 
 the piston offers to this expansion. The deter- 
 mination of all these conditions is a problem of 
 great difficulty, and as yet it is an unsolved problem 
 of practical mechanics. 
 
 An instrument has indeed been contrived for mea- 
 suring the elastic force of the steam in the cylinder, 
 called the Steam Indicator. See Tredgold on the 
 Steam Engine (art. 560.). This instrument, at 
 best but an imperfect one, although many years 
 ago used by Watt, has only, we believe, of late 
 come to be employed to any extent by steam engine 
 manufacturers for estimating the powers of their 
 engines. It appears to admit of improvement, and 
 will probably before long be taken for the constant 
 guide of the practical engineer. 
 
 We are not aware of any published experiments 
 with the Indicator of sufficient precision and au- 
 thority to warrant their mention here. Whenever 
 such experiments shall be made, valuable theoretical 
 results cannot fail to be deducible from them. 
 
THE MOVING POWER IN A STEAM ENGINE. 379 
 
 It is customary for the engine maker to assume 
 that his engine is made to work with a certain 
 velocity of the piston and with a certain pressure 
 upon it ; and different makers have been accustomed 
 to assign different values to these quantities. The 
 engines of Watt were made to work with a pressure 
 of 7 Ibs. on the square inch, and the piston to travel 
 at 220 feet per minute. Tredgold gives, as the 
 best velocity of the piston, 120 times the square root 
 of the length of the stroke, in feet. It is very ques- 
 tionable whether any of these conclusions, consi- 
 dered as theoretical conclusions, are founded on 
 sufficient data. 
 
 As an example of the calculation of the dynamical 
 effect upon the piston of a steam engine, let us take 
 the following : 
 
 The cylinder of an engine has a diameter of 
 36 inches, and its piston a stroke of 7 feet, making 
 16 double strokes a minute ; the pressure upon the 
 piston of this engine was shown by the steam indicator 
 to average 10 Ibs. to the square inch. From these 
 data it may be calculated that the area of the piston 
 was 1017*8 square inches, and the whole pressure 
 upon it 10,178 Ibs.; moreover, that it moved at the 
 rate of 224 feet per minute ; so that the dynamical 
 effect per minute produced upon it was represented 
 by the product of these numbers or by the number 
 2,279,872 ; which, taking the dynamical effect of a 
 horse per minute to be 33,000, makes the horse- 
 power, as it is called, of the engine or the effect 
 produced upon its piston (not its working power) 
 equal to that of 69 horses. The actual pressure of 
 10 Ibs. per square inch upon the piston of this en- 
 
380 ILLUSTRATIONS OF MECHANICS. 
 
 gine was determined by Mr. Glyn with the steam in- 
 dicator. The engine was probably made to work 
 with 7 Ibs. or 8 Ibs. pressure, and would have been 
 called by the maker an engine of 55-horse power. 
 Had this engine worked without friction of its 
 machinery, this moving dynamical effect or moving 
 power of 69 horses, would have been propagated 
 through it without diminution, and distributed 
 among its working points, would have constituted 
 its working or useful effect. 
 
 305. THE DYNAMICAL EFFECT UPON THE WORK- 
 ING POINTS OR THE WORKING POWER OF A 
 
 STEAM ENGINE. 
 
 The dynamical effect produced at the working 
 points in a steam engine, is equal to the sum of 
 the pressures exerted there and performing the 
 work, each being multiplied by the space over 
 which it is made to operate. 
 
 The following example is from the monthly re- 
 ports of the working of the Cornish engines ; it 
 will sufficiently illustrate the method according to 
 which this calculation is usually made.* 
 
 * In the year 1811, the principal raining proprietors in 
 Cornwall determined, with a view to the encouragement of the 
 skilful manufacture and working of engines, to ascertain from 
 monthly reports, made by competent persons and with the 
 requisite precautions, and to make public, the useful effect of 
 their respective engines during that month, together with the 
 consumption of coals and the steam pressure in the cylinder. 
 For this purpose a mechanical contrivance, called the counter^ 
 was annexed to each engine, and accurately registered its 
 number of strokes ; and this registration, with the measured 
 dimensions of its pump and stroke, are sufficient data for 
 determining its useful effect, as shown in the above example. 
 
THE WORKING POWER OF A STEAM ENGINE. 381 
 
 The engine at the mine called the Wheal Hope, 
 works three pumps, and the length of the stroke of 
 each is 8 feet : their pistons support and lift, at 
 every stroke, columns of water, whose joint weights 
 are 27,766 Ibs., and in the month of December 
 1826, they are stated to have made 261,890 strokes. 
 Hence it may be calculated that the velocity of 
 the pistons was 46*9 feet per minute, and 27,766 Ibs. 
 of water being moved with this velocity, that the 
 working effect per minute, was the product of these 
 two numbers, or 13,022,254. 
 
 If the whole distance travelled by the pistons in 
 the month had been multiplied by the mean pressure 
 upon them, so as to obtain the whole working ef- 
 fect in the month, and this product had been divided 
 by the number of bushels of coals consumed in 
 the month, which was 1242, the quotient would 
 have been the working effect of each bushel of 
 coals in that engine, and it would have been found 
 to be 46,838,246. This number is called the DUTY 
 of the engine. It includes in its amount not only 
 the qualities of the engine, but of the fuel, and the 
 economy of the stokers in the use of it ; and espe- 
 cially, it would seem to depend upon the greater 
 or less escape of the heat, by radiation from the 
 surface of the boiler. 
 
 306. PRACTICAL METHOD OF DETERMINING THE 
 DYNAMICAL EFFECT AT ANY WORKING POINT 
 IN A MACHINE, OR THE WORKING POWER 
 
 OPERATING AT THAT PoiNT. 
 
 Let the work be thrown off from the shaft which 
 
382 ILLUSTRATIONS OF MECHANICS 
 
 conveys the power to that working point*, whose 
 dynamical effect is to be estimated. Let then a 
 friction-strap or break-wheel, such as that shown in 
 the accompanying figure, to which is connected 
 
 the rod or bar AB, be placed upon the shaft, and 
 its revolution with the shaft being prevented by 
 the stop D, let the strap be tightened upon the 
 shaft by means of the screw B, until the motion of 
 the machine is again brought back by the friction 
 of the strap, exactly to what it was before the work 
 was thrown off, a fact which will be indicated by 
 the shaft making now precisely as many revolutions 
 per minute as it did then. This being accomplished, 
 it is certain that the friction of the strap is pre- 
 cisely equal to the resistance of the work ; and that 
 the power before expended in performing the work, 
 is precisely equal to the power now expended in 
 overcoming the friction of the strap. It only re- 
 mains, therefore, to determine this last. For this 
 purpose let a weight be suspended from the ex- 
 tremity of the rod, and gradually increased, until 
 
 * The power will, in the majority of cases, be found to be 
 conveyed to each working point of the machine by such a 
 shaft, which may be considered as the channel along which it 
 flows. In any case where it is not, a shaft may be introduced 
 and made the medium of communication, for the express pur- 
 pose of this admeasurement. 
 
THE THEORY OF THE STEAM ENGINE, 383 
 
 the rod at length descends from the stop D, (against 
 which it has hitherto been pressed, and by the re- 
 sistance of which the friction of the strap has hi- 
 therto been overcome,) and assumes the horizontal 
 position shown in the figure. An equilibrium then 
 manifestly exists between the weight E, acting on 
 the arm of the lever CF, and the friction, acting on 
 the circumference of the shaft. From this relation, 
 the friction upon the shaft may at once be calcu- 
 lated ; and this friction in pounds, multiplied by 
 the distance in feet, traversed by the circumference 
 of the shaft per minute, gives the dynamical effect of 
 the ffiction at the shaft, and therefore the power upon 
 the working point, which was to be determined. 
 
 307. THE THEORY OF THE STEAM ENGINE. 
 
 Could we determine from a knowledge of the di- 
 mensions, and the combination of the parts of a 
 steam engine its cranks, axles, levers, pistons, 
 c. and the frictions of their surfaces of 
 contact, the conditions of the equilibrium of the 
 pressures acting in the machine, when in its 
 state bordering upon motion ; could we, in fact, 
 determine accurately, under the form of an ana- 
 lytical expression, that precise relation which ex- 
 ists between a power operating upon the piston 
 of a steam engine, and the resistances opposed to 
 the motion of the machine at its working points, 
 when motion is about to ensue by the power over- 
 coming the resistances at those points friction 
 being of course rigidly, included in the compu- 
 tation ; and did this analytical formula or com- 
 putation apply itself to ail the various positions ol 
 
384 ILLUSTRATIONS OF MECHANICS. 
 
 the piston,, and therefore of the beam, crank, levers, 
 &c. ; then we should know accurately under what 
 steam pressure upon the piston the engine would 
 perform any given work, and one of the most im- 
 portant elements of the theory of the steam engine 
 would be determined. 
 
 The next step in the investigation would be to 
 find, if it were possible, from given dimensions 
 of the furnace and boiler, the quantity of steam 
 which the engine would produce, and throw per 
 minute into the cylinder, of such a density as that 
 its elasticity should be sufficient to produce the re- 
 quired pressure per square inch upon the piston. 
 Every time the cylinder was filled with steam of 
 this density, the piston would be driven along it ; 
 and the number of times per minute that it would 
 be so filled, would be known by a comparison of 
 its capacity with the quantity of steam of the same 
 density, generated per minute in the boiler. The 
 pressure upon the piston being thus known, and its 
 velocity, the whole moving and working effect of 
 the engine, would seem to be known, and its theory 
 completely determined. 
 
 Three important elements in the computation 
 have, however, been here omitted: 
 
 1st. The temperature under which the steam 
 fills the cylinder influences greatly its elasticity, 
 and therefore its pressure upon the piston. 
 
 2dly. The velocity under which the steam passes 
 through the steam-pipe, from the boiler to the 
 cylinder^ controlling as it does the supply of steam 
 to the piston, and depending for its amount upon 
 the relative densities of the steam in the boiler and 
 
THE THEORY OF THE STEAM-ENGINE. 385 
 
 cylinder, of necessity influences the result; and must 
 be supposed to do so appreciably, until the contrary 
 is proved, or at least rendered probable. 
 
 3dly. The elasticity of the steam in the cylinder is 
 undoubtedly, in some degree, and probably to a great 
 extent, affected by the state of motion produced in 
 it by the influent jet of steam from the steam- 
 pipe ; and, like the last, this disturbing cause must 
 be supposed to have an appreciable amount, until 
 the contrary is proved. 
 
 These conditions, thrown into the problem, greatly 
 add to its difficulties, and appear to place it far 
 beyond the limits of any solution which has yet 
 been offered. 
 
 Of the various discussions of the theory of the 
 steam-engine which have been propounded for the 
 guidance of practical men, there are two which 
 may here be noticed; those of Mr. Tredgold and 
 M. de Pambour. 
 
 The theory of Mr. Tredgold appears to assume, 
 that the steam pressure upon the piston is wholly 
 controlled and governed by the pressure in the 
 boiler, and entirely independent of the resistance 
 upon the piston. It is scarcely possible to extract 
 any other meaning from the calculation given by 
 that author of the working or useful pressure on 
 the piston of a non-condensing engine*, (see 
 Tredgold, art. 367.) unless, indeed, the whole effect 
 
 * The following is the calculation given by him of the 
 effective or working pressure upon the piston (that is, the pres- 
 sure upon the piston, deducting the friction of the parts of the 
 engine and the resistances opposed to its motion by all other 
 causes acting to transmit it). 
 
 C C 
 
386 ILLUSTRATIONS OF MECHANICS. 
 
 of the resistance on the piston, upon the steam pres- 
 sure, be supposed to be included in his determin 
 ation of the first small element, -0069 of the calcu- 
 lation. 
 
 According to this calculation, the actual pressure 
 of the steam upon the piston, neglecting the effect 
 
 The effective pressure upon the piston is less than that in the 
 boiler, considered as unity. 
 
 By the force producing motion of the steam 
 
 into the cylinder ... -0069 
 
 By the cooling in the cylinder and pipes - -0160 
 By the friction of the piston and waste - '2000 
 By the force required to expel the steam 
 
 into the atmosphere - - - *0069 
 
 By the force expended in opening valves, 
 
 and friction of the parts of the engine - *0622 
 By the steam being cut off before the ter- 
 mination of the stroke - *1000 
 
 3920 
 
 To the expression in the text of the opinion he has formed 
 of the principles on which this calculation of the power of 
 a steam-engine and others of the same class are founded, the 
 author begs here to add, that it is by no means his wish to 
 be considered as extending this opinion to the general cha- 
 racter of Mr. Tredgold's work. That work contains a vast 
 mass of practical information, which will be sought for in vain 
 elsewhere ; and the many admirable plates and valuable papers 
 which have been added to the last edition of it, published by 
 Mr. Weale, will no doubt obtain for it a place in the library 
 of every man interested in the progress of practical science. 
 Nevertheless, in justice to the real interests of science, the 
 author is compelled to express an opinion that every single 
 question connected with the theory of the steam-engine ought, 
 n the existing state of our knowledge, to be received with 
 distrust and caution. 
 
THE THEORY OF THE STEAM-ENGINE. 387 
 
 of cutting off the steam before the termination of 
 the stroke, is only less than that in the boiler by the 
 small fraction -0229. 
 
 Now the pressure in the boiler can be measured, 
 and thence that upon the piston calculated, allowing 
 the loss of this small fraction of its amount in pass- 
 ing from the boiler to the cylinder; so as to deter- 
 mine the moving dynamical effect or moving power 
 of the engine according to Mr. Tredgold's rule ; 
 and it would be found, by comparing it with the 
 working dynamical effect, or working power, to 
 amount only to from one third to two thirds of it. 
 To reconcile the two, then, enormous allowance 
 must be made by those who adopt this rule for 
 friction and other causes opposed to the motion of 
 the engine. 
 
 Mr. Tredgold accordingly assigns to the piston 
 alone a friction amounting to no less than th 
 of the whole pressure upon it, and to the friction 
 of the machinery by which the motion of the piston 
 is transmitted T ^^ths. .Whence it may be calcu- 
 lated, that if an engine had a working power of 
 100 horses, 40 would be necessary to draw its piston 
 alone, and 12 to move the remaining portion of its 
 machinery. (See De Pambour's Theory of the 
 Steam-Engine, p. 7.) 
 
 It is due to the interests of science to state that 
 these calculations appear to be grounded in no 
 sound or recognised principles : they are deduced 
 from formulae which are to be considered as scarcely 
 more than empirical, and which do not appear to be 
 borne out by the practice of the steam-engine. 
 
 The theory of M. de Pambour makes the elas- 
 c c 2 
 
388 ILLUSTRATIONS OF MECHANICS. 
 
 ticity of the steam in the cylinder to depend en- 
 tirely upon the resistance which the piston opposes 
 to it, and the motion of the piston to be governed 
 entirely by the quantity of steam generated by the 
 engine per minute, at a given temperature, which 
 he calls its vaporising power. The elasticity of the 
 steam in the cylinder is, however, dependent upon 
 its temperature as well as its density ; and to com- 
 pare it, and therefore the pressure upon the piston, 
 with the vaporising power of the engine, it is neces- 
 sary to establish a relation between the two. 
 
 M. de Pambour states, from numerous experi- 
 ments made simultaneously with the thermometer 
 and manometer, applied both to the boiler of a 
 steam-engine and also to the tube, through which 
 the steam, after having terminated its effect, escaped 
 into the atmosphere, that during all its action in 
 the engine the steam remains in the state technically 
 denoted by the name of saturated steam; that is, it 
 remains at the maximum density for its temperature. 
 This fact, on the discussion of which it is impossible 
 here to enter, establishes the required relation of 
 density and temperature, and leads to a solution of 
 the problem under the conditions supposed. 
 
 If confirmed by subsequent observations, it can- 
 not but be considered a very valuable addition to 
 the theory of the steam-engine. 
 
APPENDIX. 
 
 ccS 
 
391 
 
 APPENDIX. 
 
 TABLE I. 
 
 COMPRESSIONS PRODUCED IN DIFFERENT SUBSTANCES BY EACH 
 ADDITIONAL PRESSURE OF ONE ATMOSPHERE, MEASURED IN 
 MlLLIONTHS OF THE WHOLE VOLUME OR BULK. 
 
 OERSTED. 
 
 COLL AD ON AND STURM. 
 
 Substances experi- 
 mented on. 
 
 Million ths. 
 
 Substances experi- 
 mented on 
 
 Millionths. 
 
 Mercury 
 Alcohol 
 Sulphuret of carbon 
 Water 
 
 1 
 
 20 
 30 
 46-1 
 
 Mercury 
 Sulphuric acid 
 Nitric acid 
 Ammonia 
 
 5-3 
 
 32-0 
 32-2 
 34-7 
 
 Sulphuric ether 
 
 60 
 
 Acetic acid 
 
 42-2 
 
 
 
 Water containing air 
 Water freed from air 
 
 49-5 
 51-3 
 
 
 
 Nritric ether 
 
 7V5 
 
 
 
 Essence of terebin-7 
 thum - - 3 
 
 73-0 
 
 
 
 Acetic ether 
 
 71-5 
 
 
 
 Hydrochloric ether^ 
 under the 1st / 
 
 85-9 
 
 
 
 atmosphere -3 
 
 
 
 
 Ditto, under the 9th 7 
 
 82' 5 
 
 
 
 atmosphere - J 
 
 
 
 
 Alcohol under the 7 
 
 _ 
 
 
 
 1st atmosphere -3 
 
 yb o 
 
 
 
 Ditto, under the 9th 7 
 atmosphere -3 
 
 93' 
 
 
 
 Sulphuric ether un-} 
 der the 1st atmo- ( 
 sphere, at temp. (" 
 0cent. -3 
 
 133-0 
 
 
 
 Ditto,ditto, at temp. ) 
 11 cent. -3 
 
 141-0 
 
 
 
 Ditto, under 24th } 
 atmosphere, at v 
 temp. cent. - j 
 
 150-0 
 
 c c 4 
 
392 
 
 ILLUSTRATIONS OF MECHANICS. 
 
 TABLE II. 
 
 LIQUEFACTION OF THE GASES. 
 
 Names of the Gases liquefied. 
 
 Temperature in 
 Degrees of the 
 Centig. Ther. 
 
 Pressure at which 
 Liquefaction is 
 produced in At- 
 mospheres. 
 
 Sulphurous acid 
 Cyanogen 
 Chlorine - - 
 
 7 
 7 
 15-5 
 
 2 
 4 
 
 Ammonia 
 
 
 
 5 
 
 The same - 
 
 10 
 
 6*5 
 
 Muriatic acid 
 
 -16 
 
 20 
 
 The same 
 
 - 4 
 
 25 
 
 The same 
 
 10 
 
 40 
 
 Carbonic acid 
 
 -11 
 
 20 
 
 The same - 
 
 
 
 36 
 
 Nitrous oxide 
 
 
 
 44 
 
 The same 
 
 7 
 
 51 
 
 TABLE III. 
 EXTENSIBILITY. 
 
 EXPERIMENTS ON THE DIRECT EXTENSIBILITY OF WOOD AND 
 IRON. 
 
 
 Ji 
 
 a 
 
 "r+ ta 
 
 
 Substance extended. 
 
 S H 
 
 If 
 
 Name of 
 
 
 '~ 
 
 C O 
 
 Experimenter. 
 
 
 Jl 
 
 II 
 
 
 Bars of oak 
 
 1 
 
 1176 
 
 Minard and Desormes. 
 
 Iron wire, No. 18 (in cables) 
 
 1 
 
 91 
 
 Vicat. 
 
 17 
 
 1 
 
 85 
 
 
 
 Bar iron - 
 
 1 
 
 15 
 
 82 
 2500 
 
 f Engineers of the Pont 
 I des Invalides. 
 Minard and Desormes. 
 
 . 
 
 18 
 
 10000 
 
 
 
 . 
 
 20 
 
 20000 
 
 
 
 ... 
 
 23 
 
 50000 
 
 
 
 _ 
 
 25 
 
 rupture 
 
 
 
 i i 
 
 
APPENDIX. 
 
 393 
 
 TABLE IV. 
 
 EXPERIMENTS BY MR. BARLOW ON THE DIRECT EXTENSIBILITY 
 OF WROUGHT IRON.* 
 
 
 Parts of the Bar extended by each additional Ton, in 
 
 
 MILLIONTHS of the whole Length. 
 
 Extending 
 Weight in 
 
 BARS ONE INCH SQUARE. 
 
 Tons. 
 
 Bar No. I. 
 
 Bar No. II. 
 
 Bar No. III. 
 
 Bar No. IV. 
 
 1 
 
 
 
 
 
 
 
 
 
 2 
 
 20 
 
 
 
 160 
 
 150 
 
 3 
 
 62 
 
 73 
 
 150 
 
 130 
 
 4 
 
 93 
 
 80 
 
 130 
 
 140 
 
 5 
 
 109 
 
 90 
 
 120 
 
 140 
 
 6 
 
 110 
 
 110 
 
 110 
 
 130 
 
 7 
 
 
 
 90 
 
 120 
 
 100 
 
 8 
 
 93 
 
 80 
 
 120 
 
 80 
 
 9 
 
 
 
 100 
 
 120 
 
 elasticity 
 
 
 
 
 
 destroyed. 
 
 BARS TWO INCHES SQUARE. 
 
 
 Bar No. V. 
 
 Bar No. VI. 
 
 Bar No. VII. 
 
 
 8 
 
 180 
 
 150 
 
 125 
 
 
 10 
 
 140 
 
 120 
 
 110 
 
 
 12 
 
 110 
 
 100 
 
 50 
 
 
 14 
 
 110 
 
 80 
 
 50 
 
 
 16 
 
 110 
 
 85 
 
 50 
 
 
 18 
 
 110 
 
 80 
 
 105 
 
 
 20 
 
 100 
 
 75 
 
 100 
 
 
 22 
 
 100 
 
 70 
 
 95 
 
 
 24 
 
 100 
 
 75 
 
 95 
 
 
 26 
 
 100 
 
 80 
 
 95 
 
 
 28 
 
 95 
 
 80 
 
 95 
 
 
 30 
 
 90 
 
 95 
 
 95 
 
 
 32 
 
 95 
 
 95 
 
 90 
 
 
 34 
 
 85 
 
 110 
 
 85 
 
 
 36 
 
 75 
 
 full elas- 
 
 90 
 
 
 38 
 
 95 
 
 ticity. 
 
 95 
 
 
 40 
 
 145 
 
 
 95 
 
 
 
 elasticity 
 
 
 elasticity 
 
 
 
 exceeded. 
 
 
 perfect. 
 
 
 
 
 
 i 
 
 * Compiled from a Report addressed to the Directors of the London 
 and Birmingham Railway. Fellowes, 1835. 
 
394 ILLUSTRATIONS OF MECHANICS. 
 
 Mean Extension per T.on, per Square Inch. 
 
 
 
 Millionths. 
 
 Millionths. 
 
 Bar 
 
 No. I. 
 
 - 98 
 
 Bar No. V. 
 
 - 108 
 
 
 II. - 
 
 - 90 
 
 VI. - 
 
 - 95 
 
 
 III. - 
 
 - 101 
 
 VII. - 
 
 - 84 
 
 
 IV. - 
 
 - 97 
 
 
 
 
 
 
 
 
 Mean 
 
 - 94 
 
 
 Mean 
 
 - 96 
 
 
 
 On these extremely valuable experiments, Mr. Barlow has 
 made the following remarks : " Collecting the results of 
 these seven experiments, and reducing them all to square 
 inches, we find that the strain which was just sufficient to 
 balance the elasticity of the iron, was in 
 
 Bar No. I. (re-manufactured iron) 10 tons. 
 
 IL ditto 11 tons. 
 
 III. new bolt 11 tons. 
 
 IV. ditto 10 tons. 
 V. (re-manufactured) 9*5 tons. 
 
 VI. ditto, from old furnace bars 8^5 tons. 
 VII. new bar, by Messrs. Gordon 10 tons. 
 
 We may consider, therefore, that the elastic power of good 
 iron is equal to about ten tons per inch, and that this force 
 varies from ten to eight tons in indifferent and bad iron. It 
 appears also, (considering -000096 as representing in round 
 numbers fi^th) that a. bar of iron is extended fooffith part of 
 its length by every ton of direct strain per square inch of its 
 section ; and, consequently, that its elastic limit will be fully 
 excited when it is stretched to the amount of yjftfth part of its 
 length." 
 
 " We have seen, that with about ten tons per square inch, a 
 bar is stretched -j^gth part of its length, and its elasticity 
 wholly excited or surpassed. Again, admitting 76 to be 
 the extreme range of the thermometer, in this country, be- 
 tween summer and winter, it appears from the very accurate 
 experiments of Professor Daniel, that a bar of malleable 
 iron will contract or expand with this change of temperature, 
 by sgggth part of its whole length." Now, by the preceding 
 experiments it appears that a bar extending by this fraction 
 of its length would exert a strain of five tons per square 
 inch on its abutments. Such, then, is the strain which a bar, 
 fixed between two immoveable obstacles in winter, would 
 exert against them in summer. 
 
APPENDIX. 
 
 395 
 
 TABLE V. 
 
 THE TENACITIES OF DIFFERENT SUBSTANCES, AND THE RESIST- 
 ANCES WHICH THEY OPPOSE TO DIRECT COMPRESSION. 
 
 Substances experimented on. 
 
 Tenacity in 
 Tons per 
 Square Inch. 
 
 Name of Ex- 
 perimenter. 
 
 Crushing 
 Force in Tons 
 per Square In. 
 
 Name of Ex- 
 perimenter. 
 
 Wrought iron, in wirefrom ) 
 l-20th to l-30th of an 
 inch in diameter - J 
 in wire l-10th of an inch 
 
 60 to 91 
 36 to 43 
 
 Lame 
 Telford 
 
 
 
 in bars, Russian (mean) 
 
 27 
 
 Lame 
 
 
 
 English (mean) 
 
 25JL 
 
 __ 
 
 
 
 hammered 
 
 30* 
 
 Brunei 
 
 
 
 rolled in sheets, and cut 7 
 lengthwise j 
 
 14 
 
 Mitis 
 
 
 
 ditto, cut crosswise 
 
 18 
 
 _ 
 
 
 
 in chains, oval linksfiin. 7 
 clear, iron 1| in. dia. 3 
 
 21* 
 
 Brown 
 
 
 
 ditto, Brunton's, iwithl 
 stay across link - 3 
 Cast iron, quality No. 1. 
 
 25 
 6to7f 
 
 Barlow 
 Hodgkinson 
 
 38 to 41 
 
 Hodgkinson 
 
 2. - 
 
 6 to 8 
 
 mm 
 
 37 to 48 
 
 _ 
 
 3.* . 
 
 6to9f 
 
 mm 
 
 51 to 65 
 
 _ 
 
 Steel, cast ... 
 
 44 
 
 Mitis 
 
 
 
 cast and tilted 
 
 60 
 
 Rennie 
 
 
 
 blistered and hammered 
 
 59| 
 
 
 
 
 
 shear ... 
 
 57 
 
 __ 
 
 
 
 raw ... 
 
 50 
 
 Mitis 
 
 
 
 Damascus 
 
 31 
 
 mm, 
 
 
 
 ditto, once refined 
 
 36 
 
 _ 
 
 
 
 ditto, twice refined - 
 
 44 
 
 
 
 
 
 Copper, cast ... 
 
 
 Rennie 
 
 52 
 
 Rennie 
 
 hammered 
 
 15 
 
 _ 
 
 46 
 
 __ 
 
 sheet ... 
 
 21 
 
 Kingston 
 
 
 
 wire .... 
 
 271 
 
 
 
 
 PJatinum wire 
 
 17 
 
 Guyton 
 
 
 
 Silver, cast ... 
 
 18 
 
 mm. 
 
 
 
 wire - 
 
 17 
 
 
 
 
 
 Gold, cast ... 
 
 9 
 
 
 
 
 
 wire 
 
 14 
 
 __ 
 
 
 
 Brass, yellow (fine) 
 
 8 
 
 Rennie 
 
 73 
 
 Rennie 
 
 Gun metal (hard) 
 
 16 
 
 
 
 
 Tin, cast - 
 
 2 
 
 mm 
 
 7 
 
 _ 
 
 wire ... 
 
 3 
 
 __ 
 
 
 
 Lead, cast ... 
 
 4-5ths 
 
 _ 
 
 31 
 
 _ 
 
 milled sheet 
 
 1* 
 
 Tredgold 
 
 
 
 * The strongest quality of cast iron is a Scotch iron known as the Devon 
 Hot Blast No. 3. : its tenacity is 9| tons per square inch, and its resistance 
 to compression 65 tons. 
 
396 ILLUSTRATIONS OF MECHANICS. 
 
 TABLE V. continued. 
 
 Substances experimented on. 
 1 
 
 Tenacity in 
 Tons per 
 Square Inch. 
 
 Name of Ex- 
 perimenter. 
 
 Crushing 
 Force in Tons 
 per Square In. 
 
 Name of Ex- 
 perimenter. 
 
 Lead wiife - . 
 
 1 1 
 
 Guyton 
 
 
 
 Stone, slate (Welsh) 
 Marble (white) 
 
 57 
 4 
 
 
 1-4 
 
 Rennie 
 
 Givry ... 
 
 1 
 
 
 
 
 Portland ... 
 
 
 m 
 
 1-6 
 
 
 
 Craigleith freestone 
 Bramley fall sandstone 
 
 - - 
 
 
 
 2-4 
 27 
 
 
 
 Cornish granite 
 Peterhead ditto 
 
 - - 
 
 - 
 
 2-8 
 
 
 
 Limestone(compactblk.) 
 
 - - 
 
 - 
 
 37 
 4 
 
 - 
 
 Aberdeen granite 
 
 _ 
 
 . 
 
 4 
 5 
 
 ~ 
 
 Brick, pale red - 
 
 13 
 
 - 
 
 56 
 
 .0 
 
 
 
 Hammersmith (pavior's) 
 ditto (burnt) 
 Chalk - - - - 
 
 - - 
 
 I 
 
 O 
 
 1-4 
 
 22 
 
 
 
 Plaster of Paris 
 
 03 
 
 
 
 ~" 
 
 Glass, plate - 
 
 4 
 
 
 
 
 Bone (ox) ... 
 
 2-2 
 
 
 
 
 Hemp fibres glued together 
 Strips of paper glued together 
 
 41 
 13 
 
 
 
 
 Wood, Box, spec, gravity '862 
 
 9 
 
 Barlow 
 
 
 
 Ash . -6 
 
 8 
 
 __ 
 
 
 
 Teak '9 
 
 7 
 
 
 
 
 
 Beech - 7 
 
 5 
 
 . 
 
 
 
 Oak - . -92 
 
 5 
 
 _ 
 
 17 
 
 _ 
 
 Ditto - *77 
 
 4 
 
 
 
 
 
 Fir - -6 
 
 5 
 
 
 
 
 
 Pear - -646 
 
 4? 
 
 _ 
 
 
 
 Mahogany -637 
 
 3} 
 
 . 
 
 
 
 Elm . 
 
 6 
 
 _ 
 
 57 
 
 ^ 
 
 Pine, American 
 
 6 
 
 , 
 
 73 
 
 
 
 Deal, white 
 
 6 
 
 
 
 86 
 
 
 
 TORSION. 
 
 M. Savart has shown, in a series of experiments, detailed 
 in the Annales de Chimie, August, 1829, on the torsion of 
 bars of different sections and dimensions 
 
 1st. That the ANGLES of torsion are in every case propor- 
 tional to the FORCES of torsion, so long as the torsion of the 
 bar remains within the elastic limits. 
 
 2dly. That in bars of the same section, subjected to the 
 same forces of torsion, the angles of torsion are directly pro- 
 portional to the LENGTHS of the bars. 
 
APPENDIX 
 
 397 
 
 TABLE VI. 
 
 EXPERIMENTS BY M. DULEAU UPON THE ANGLE or TOR- 
 SION IN BARS OF IRON. 
 
 
 
 
 Angle of Tor- 
 
 Nalnre of the Specimen. 
 
 Length of the 
 Part twisted. 
 
 Side of Square, 
 or Diameter 
 of Cylinder. 
 
 sion produced 
 by a Pressure 
 of 22 IDS. act 
 at a Leverage 
 
 
 
 
 of l-jfij Feet. 
 
 
 Feet. 
 
 Inches. 
 
 Degrees. 
 
 Round iron, English, > 
 marked Dowlais 3 
 
 7'9 
 
 78 
 
 4 
 
 Round iron, Perigord 
 
 9'5 
 
 91 
 
 3 
 
 Square iron, English, ) 
 marked C 2 - 3 
 
 13-5 
 
 79 
 
 *i 
 
 Square iron, Perigord 
 
 8-3 
 
 8 
 
 3-8 
 
 Flat iron, English 
 
 9'6 
 
 1 -32 x -337 
 
 11*4 
 
 TABLE VII. 
 
 EXPERIMENTS BY MR. G. RENNIE ON THE RUPTURE OF SQUARE 
 BARS OF DIFFERENT METALS BY TORSION : THE FORCE 
 BEING MADE TO ACT AT THE EXTREMITY OF A LEVER Two 
 FEET IN LENGTH. 
 
 Description of Material. 
 
 Length of 
 Piece. 
 
 Side of 
 Square 
 Section. 
 
 Mean Weight 
 producing Rup- 
 ture. 
 
 i Inches. 
 
 Inches. 
 
 Lbs. Oz. 
 
 Iron cast, horizontally 
 
 1 
 
 9 15 
 
 vertically 
 
 
 
 J 
 
 10 10 
 
 horizontally 
 
 i 
 
 ^ 
 
 7 3 
 
 . 
 
 1 
 
 A 
 
 8 1 
 
 . 
 
 i 
 
 ^. 
 
 8 8 
 
 vertically 
 
 | 
 
 | j 10 1 
 
 . 
 
 ^- 
 
 ^ 
 
 8 9 
 
 - 
 
 i 
 
 ^ 
 
 8 5 
 
 . 
 
 6 
 
 i 
 
 9 12 
 
 horizontally 
 
 
 
 ^ 
 
 93 12 
 
 - 
 
 o 
 
 1 
 
 74 
 
 i 10 
 
 1 
 
 52 
 
 Steel 
 
 
 
 i 
 
 17 1 
 
 Wrought iron, English 
 
 
 
 % 
 
 10 2 
 
 Swedish i 
 
 $ 
 
 9 8 
 
 Gun metal, hard 
 
 * 
 
 5 
 
 Yellow brass, fine - 
 
 
 
 4 11 
 
 Copper, cast 
 
 
 
 4 5 
 
 Tin - 
 
 f 
 
 1 
 
398 
 
 ILLUSTRATIONS OF MECHANICS. 
 
 TABLE VIII. 
 
 EXPERIMENTS BY MR. BRAMAH ON THE RUPTURE BY TOR- 
 SION OF SQUARE BARS BY WEIGHTS ACTING AT A LEVER- 
 AGE OF THREE FEET. 
 
 Description of Material. 
 
 Length of 
 Piece. 
 
 Side of 
 Square 
 Section. 
 
 Weight \ 
 producing | 
 Rupture. 
 
 
 Inches. 
 
 Inches. 
 
 Lbs. 
 
 Cast iron, alloyed with ^th 1 
 
 12 
 
 IT'S 
 
 215 
 
 of copper - - 3 
 
 
 16 
 
 
 - 
 
 24 
 
 IT'S 
 
 213 
 
 Mixture of equal parts of ^ 
 old Adelphi and Alfre- > 
 
 12 
 
 ITS 33 
 
 ton - j 
 
 
 ' 
 
 . 
 
 12 
 
 *i 
 
 310 
 
 _ 
 
 24 
 
 IT'O 
 
 280 
 
 Cast iron 
 
 12 
 
 1 
 
 238 
 
 - 
 
 24 
 
 1 
 
 218 
 
 TABLE IX. 
 
 EXPERIMENTS BY MR. DUNLOP ON THE RUPTURE BY TOR- 
 SION OF CYLINDRICAL BARS OF CAST IRON, WITH WEIGHTS 
 ACTING AT A LEVERAGE OF FOURTEEN FEET Two INCHES. 
 
 Length of the 
 Bar. 
 
 Diameter. 
 
 Weights produc- 
 ing Rupture. 
 
 Inches. 
 
 Inches. 
 
 Lbs. 
 
 2| 
 
 2 
 
 250 
 
 *t 
 
 & 
 
 384 
 
 3 
 
 2 i 
 
 408 
 
 3 
 
 2J 
 
 700 
 
 4 
 
 
 
 1170 
 
 5 
 
 3i 
 
 1240 
 
 5 
 
 SJ 
 
 1662 
 
 5 
 
 4 
 
 1938 
 
 6 
 
 4| 
 
 2158 
 
APPENDIX. 399 
 
 MB. HODGKINSON'S EXPERIMENTS ON THE MECHANICAL 
 PROPERTIES OF CAST IRON. 
 
 The experiments of Mr. Hodgkinson and Mr. Fairbairn 
 have been published, in the Seventh Report of the British 
 Association of Science, since our chapter on the strength of 
 materials went to press. Their great practical importance 
 will sufficiently account for their introduction here, as an ap- 
 pendix to that chapter. They have reference 
 
 1 st. To the resistance of cast iron to rupture by extension. 
 
 2d. To the resistance of cast iron to rupture by compres- 
 sion. 
 
 3d. To the resistance of cast iron to rupture by transverse 
 strain. 
 
 4th. To the destruction of the elastic properties of the 
 material as the body advances to rupture. 
 
 5th. To the influence of time upon the conditions of rup^ 
 ture. 
 
 6th. To certain relations of the internal structure of metals 
 to their conditions of rupture. 
 
 7th. To the relative properties in all these respects of HOT 
 
 AND COLD BLAST IRON. 
 
 The experiments on tension and compression were made by 
 means of a lever constructed for the purpose by Mr. Fair- 
 bairn, and admirably adapted to its use. A table given at 
 the end of this paper contains their principal results. 
 
 From this table it appears that the resistance of cast iron to 
 rupture by extension varies from 6 to 9 tons upon the square 
 inch ; and that to rupture by compression from 36 to 65 tons. 
 
 A series of experiments was directed to the verification of 
 the commonly assumed principle, that the forces resisting rup- 
 ture by extension, are the material being the same as the 
 areas of the sections of rupture ; and they appear fully to have 
 
4-00 ILLUSTRATIONS OF MECHANICS. 
 
 established this principle, not only in respect to iron but to 
 wood. 
 
 The experiments of Mr. Hodgkinson on transverse strain 
 present less of novelty and importance ; they fully, however, 
 confirm the views previously taken on this subject by him, 
 and detailed in articles 66. 68, &c. A series of them, directed 
 to the verification of the commonly assumed principle, " that 
 the strengths of rectangular beams of the same width, to resist 
 rupture by transverse strain, are as the squares of their depths," 
 fully established that law. 
 
 With regard to the destruction of the elastic properties of 
 the material, as it approaches to rupture, the experiments of 
 Mr. Hodgkinson possess great interest and importance. 
 
 It has been asserted by Mr. Tredgold, and commonly as- 
 sumed, that this destruction of elastic power, or displacement 
 beyond the elastic limit, does not manifest itself until the 
 load exceeds one third the breaking weight. 
 
 Mr. Hodgkinson found that, in some instances, this effect was 
 produced, and manifested in a permanent set of the material, 
 when the load did not exceed one sixteenth of the breaking 
 weight. Thus, a bar one inch square, supported between 
 props 4^ feet apart, which broke when loaded with 496 lb., 
 showed a permanent deflection, or set, when loaded with 1 6 lb. 
 In other cases, permanent sets were given by loads of 7 lb. 
 and 14 lb., the breaking weights being respectively 364 1 lb. 
 and 1120lb. These sets were therefore given by ^d and 
 ^gth the breaking weights respectively. Thus, then, there 
 would seem to be no such limits, in respect to transverse strain, 
 as those known by the name of elastic limits ; and it follows 
 from these experiments that the principle of loading a beam 
 within the elastic limit has no foundation in practice, 
 
 It was ascertained by a very ingenious experiment, that a 
 bar, subjected, under precisely the same circumstances, to ex- 
 tension and compression by transverse strain, gave, for equal 
 loads, equal deflections, in the two cases. 
 
 The most remarkable results on the subject of transverse 
 strain were, however, those of Mr. Fairbairn, having reference 
 
APPENDIX. 401 
 
 to the influence of TIME upon the deflection produced by a 
 given load. 
 
 A bar one inch square, supported between props 4 feet 
 apart, and loaded with 280 Ibs., being about |ths its breaking 
 weight, had its deflection accurately measured, from month 
 to month, for fifteen months, and it was found that, through- 
 out that period, the deflection was CONTINUALLY INCREASING ; 
 the whole increase in that period amounting to the fraction 
 043 of an inch. A bar of the same dimensions, similarly sup- 
 ported, and loaded with 336 Ibs., being about fths of its break- 
 ing weight, increased its deflection similarly, and in the same 
 period, by the fraction *077 of an inch. Another similar bar, 
 loaded with about Jths the breaking weight, similarly in- 
 creased its deflection by the '088th of an inch. The de- 
 flection of these bars still daily advances under the same loads, 
 and, a sufficient period having elapsed, will no doubt proceed 
 to rupture. A fourth bar of the same size was loaded with 
 448 Ibs., being very nearly its breaking load. It bore it for 
 thirty-seven days, increasing its deflection during the first few 
 days by the fraction -282 of an inch ; thence retaining the 
 same deflection until it broke. 
 
 The fact thus established, that a beam loaded beyond a 
 certain limit continually yields to the load, but with an exceed- 
 ingly slow progression, unless the load very nearly approach 
 the breaking load, is one of vast practical importance ; it opens 
 an entirely new field of speculation and inquiry. The ques- 
 tions, what are the limits of loading (if any) beyond which 
 this continual progression to rupture begins ? what are the va- 
 rious rates of progression corresponding to different loads beyond 
 that limit ? and what are the effects of temperature on these 
 circumstances ? remain, as yet, almost unanswered. 
 
 Another interesting feature of Mr. Hodgkinson's experi- 
 ments has, however, reference to certain relations of the in- 
 ternal structure of cast iron to the conditions of its rupture. 
 
 In the compression of short columns of different heights, 
 and of the same diameter, he found that where the height 01 
 the column exceeded a certain limit, the crushing force be- 
 D D 
 
4?02 ILLUSTRATIONS OF MECHANICS. 
 
 came constant, not varying as the height of the column was 
 increased, until it reached another limit; at which second 
 limit the column began to yield, not strictly by the crushing, 
 but by the bending of its material. 
 
 The first limit was a height of little less than three times 
 the radius of the column; the second limit was about six 
 times the radius of the column. For columns of different 
 heights, between these limits, and having equal diameters, 
 the force producing rupture by compression, was nearly the 
 same. When the column was less than the lower limit, the 
 crushing force became greater ; and when it was greater than 
 the higher limit, the crushing force became less. 
 
 These facts were at once explained by an examination of 
 the fragments of the ruptured columns. In all cases where 
 the height of the column exceeded a certain limit, the section 
 of rupture was found to be a plane inclined at nearly the 
 same angle to the axis of the column. The mean value of 
 this angle was 55, and in no case did the inclination of the 
 section vary from that angle more than 5. 
 
 Now the limiting height of the column at which this 
 oblique section first began to be distinctly and completely 
 made, was precisely that (equal to three times the radius) at 
 which the force producing rupture became independent of 
 height. One of these facts, indeed, completely explains the 
 other. For every height of the column above that limit, the 
 section of rupture being a plane inclined at the same angle to the 
 axis of the column, was a plane of the same size ; so that in 
 each case the cohesion of the same number of particles was to be 
 overcome, that the rupture might be produced; and the cohesion 
 of the same number of particles being to be overcome under 
 the same circumstances for each different height, the same 
 force would be required to overcome that cohesion ; until at 
 length that height (six times the radius) was attained at 
 which the column began to bend. This height once reached, 
 a pressure continually less as the column was longer became, 
 of course, sufficient to break it. 
 
 This property is not, however, limited to cast iron ; similar 
 experiments were made by Mr. Hodgkinson, and by Rondelet, 
 
APPENDIX. 
 
 403 
 
 fig. 86. 
 
 with columns of wrought iron, wood, bone, marble, and other 
 stones, and with the same result. 
 
 Although the angle with the axis of the direction of rupture 
 was always the same, yet the particular position round the 
 axis in which the section was made was not the same. 
 There may evidently be an infinite number of such planes 
 round a given point in the axis of the column, all in- 
 clined at the same angle of 55 to its axis ; and there is no 
 reason, in the nature of the material itself, provided it be 
 homogeneous, why it should affect one of these planes of 
 section rather than another. In the majority of cases one of 
 these planes of section will, however, be determined in preference 
 to the rest, by some want of homogeniety in the material, OF 
 by some inequality in the distribution of the compressing force 
 upon the top of the column. Still such a particular determina- 
 tion of the section of rupture may not possibly present itself 
 In that case, the rupture, having no tendency to take place in 
 one direction rather than another, will take place in all direc- 
 tions at once ; and thus the surface 
 of rupture will assume the form of 
 the surface of a double cone, of 
 which the two component cones have 
 a common apex, and from which the 
 sides of the column will break away. 
 In the accompanying figures are 
 represented the fragments of a column 
 which broke under these circum- 
 stances in the experiments of Mr. Hodgkinson. 
 
 In the case of rectangular columns, the section of rupture 
 will manifestly be the narrowest section which can be made 
 at the given inclination, or that sloping towards the nar- 
 rower face of the rectangle; because a section inclined at 
 the same angle, but sloping towards the wider face, would 
 oppose to rupture the cohesion of a much greater number of 
 particles than the other or narrower section. In the majority 
 of cases, this section will be made from one end of the top 
 of the column rather than the other ; but it may take place 
 from both ends at once. This case occurred, too, in the ex- 
 periments of Mr. Hodgkinson, and is represented in the cut, 
 DD 2 
 
ILLUSTRATIONS OF MECHANICS. 
 
 where the two planes of 
 rupture from the opposite 
 ends of the top of the column 
 are seen crossing one an- 
 other at the centre, and 
 dividing the column into 
 four wedge-like masses. 
 The last cut represents 
 fragments obtained in a 
 similar experiment with a shorter rectangular column, where 
 the height was not sufficient to allow of the one part sliding on 
 
 the other along its plane 
 
 Jiff. 88. _^^\^ of fracture. Thus it became 
 
 apparent that the material 
 had its first and easiest 
 direction of fracture at a 
 given angle of inclination 
 to the direction of the pres- 
 sure ; so that its first and 
 
 easiest fracture would take place, if allowed to do so, by the 
 sliding of one portion of it on the surface of another, at a 
 gi ven angle of inclination to the axis of the pressure. And thus 
 was completely explained the great increase of the strength of 
 the column when it was so short (less than three times the 
 radius) that one portion could not thus slide upon the other, 
 the height of the column being less than the perpendicular 
 height of the true plane of section, the upper portion was, in 
 this case, manifestly prevented from sliding upon the lower, by the 
 resting of its base upon the mass which supported the column. 
 
 Now not the least interesting feature of these experiments is, 
 that their results had long ago been anticipated by theory. 
 
 It is evident 'that when a column sustains a pressure in the 
 direction of its length, the tendency of the column to yield, by 
 the sliding of one portion upon the other along an oblique 
 section, will be influenced by two causes: first, it will be 
 greater as the inclination of the section to the direction of 
 the pressure is less ; on the principle, that the tendency of a 
 heavy mass to slide upon an inclined plane is greater, as the 
 inclination of the plane to the vertical is less : secondly, it will 
 be less as the inclination of the section to the direction of the 
 
APPENDIX. 405 
 
 pressure is less ; inasmuch as the number of the cohering par- 
 ticles in such a section, and therefore the actual coherence of 
 the whole section, is greater as the section is more oblique. 
 Thus, then, by the operation of one of these causes, as the 
 section is more oblique, the tendency to slide along it is 
 greater, and by the operation of the other it is less. There 
 must then be a particular obliquity of section for which these 
 causes most effectually neutralise one another, and the tendency 
 to rupture is the least. The position of this section was dis- 
 cussed by Coulomb, as early as the year 1773, (Memoires des 
 savans Etrangers, 1773,) and was found, neglecting the weight 
 of the material of the column, and the friction of the surfaces 
 which slip upon one another, and considering only the coherence 
 of these surfaces, to be inclined at 45 to the axis of the column. 
 Allowing for the effect of friction, and supposing, as is very 
 probable, that under these circumstances of intimate contact 
 it gives a limiting angle of resistance of 20, this theoretical 
 result of Coulomb is brought precisely to the practical result 
 of Mr. Hodgkinson, giving 55 for the obliquity of the section 
 of fracture. 
 
 A yet further confirmation of this fact is found in some ex- 
 periments of Professor Daniel, detailed in the first volume of 
 the Journal of the Royal Institution. Having immersed some 
 rectangular bars of hammered lead for a considerable time in 
 mercury, the solid metal became saturated with the fluid. 
 Any friction which the two surfaces of any section, slipping 
 upon one another, might have had, was thus taken away 
 by the intervention of the mercury, and the cohesion of the 
 particles of the bar was so destroyed, that it could not 
 sustain its own weight. Under these circumstances the 
 theory of Coulomb evidently points to an angle of 45, as that 
 at which the surfaces should slip. This is precisely the angle 
 at which they were found to slip. 
 
 From these facts it is apparent, that if columns be taken of 
 different diameters, and of heights so great as not to allow of 
 their bending, but yet sufficient to allow of a perfect separation 
 of the plane of fracture ; that is, if they be taken of heights 
 lying between three times and six times the radius of each ; 
 then their strengths being as the numbers of particles in their 
 D D 3 
 
408 
 
 ILLUSTRATIONS OF MECHANICS. 
 
 planes of fracture respectively, will be as the areas of those 
 planes ; moreover, the planes of fracture being inclined at equal 
 angles to the axes of the cylinders, their areas will be as the 
 transverse sections of the cylinders ; so that, in fact, the strengths 
 of the columns will be as the areas of their transverse sections. 
 This law Mr. Hodgkinson verified. Thus, for instance, the 
 mean of three experiments upon a column of an inch in 
 diameter, gave for the crushing force 6,426 Ibs., whilst the mean 
 of four on a column | of an inch in diameter, gave 14,542 Ibs. 
 The diameters of these columns were as 2 to 3; these sections 
 were, therefore, as 4 to 9 ; and this is near the ratio of the 
 crushing weights. 
 
 A series of experiments was directed by Mr. Hodgkinson 
 to the verification of this law, usually assumed in respect 
 to the transverse strength of rectangular beams, that, when 
 their lengths and breadths are the same, their strengths are as 
 the squares of their depths. 
 
 His experiments fully established this law. Thus he placed 
 between props, 4 feet 6 inches apart, castings of Carron iron 
 No. 2., which were all 1 inch broad, and respectively 1, 3, and 
 5 inches deep ; these broke respectively with weights of 452 Ibs., 
 3,843 Ibs., and 100,50 Ibs.; which are very nearly as the numbers 
 1 , 9, 25 ; that is, as the squares of the depths. 
 
 The following table contains a general summary of the results 
 obtained by Mr. Hodgkinson, in respect to the direct strengths 
 of hot and cold blast iron to resist compression and extension. 
 
 TABLE X. 
 
 1 
 ) 
 
 Description of Metal 
 
 Compressive 
 Force per 
 Square Inch. 
 
 Tensile 
 Force per 
 Square Inch. 
 
 Ratio. 
 
 Devon Iron, No. 3, Hot blast 
 Buffery Iron, No. 1. Hot blast 
 Ditto, No. 1. Cold blast 
 Coed-Talon Iron, No. 2. Hot blast 
 Ditto Cold blast 
 Carron Iron, No. 2. Hot blast 
 Ditto Cold blast 
 Ditto, No. 3. Hot blast 
 Ditto Cold blast 
 
 145,435 
 
 86,397 
 9 i,385 
 82,734 
 81,770 
 108,540 
 106,375 
 133,440 
 115,442 
 
 21,907 
 13,434 
 17,466 
 16,676 
 18,855 
 13,505 
 16,683 
 17,755 
 14,200 
 
 6-638:1 
 6-431: 
 5-34R: 
 4-961: 
 4-337: 
 8-037: 
 6-376: 
 7-515:1 
 8-129:1 1 
 
APPENDIX 
 
 407 
 
 TABLE XI. 
 
 GENERAL SUMMARY OF RESULTS, AS DERIVED FROM THE 
 EXPERIMENTS ON TRANSVERSE STRENGTH OF HOT AND 
 COLD BLAST IRONS. 
 
 
 Ratio of the 
 
 Ratio of the 
 
 Description of Metal. 
 
 strengths, that of 
 the cold blast being 
 
 powers to sustain 
 impact (cold blast 
 
 
 represented by 1000. 
 
 being 1000). 
 
 Carron iron, No. 2. 
 
 1000 990-9 
 
 1000 1005-1 
 
 Devon, No. 3. 
 
 1000 1416-9 
 
 1000 2785-6 
 
 Buffery, No. 1. 
 
 1000 930-7 
 
 1000 962-1 
 
 Coed- Talon, No. 2. 
 
 1000 1007 
 
 1000 1234 
 
 Ditto, No. 3. 
 
 1000 927 
 
 1000 925 
 
 Elsicar and Milton 
 
 1000 818 
 
 1000 875 
 
 Carron, No. 3. 
 
 1000 1181 
 
 1000 1201 
 
 Muirkirk, No. 1. 
 
 1000 927 
 
 1000 823 
 
 Mean 
 
 1000 1024-8 
 
 1000 1226-3 
 
 On the whole, then, it appears that the strength of hot blast 
 iron to resist transverse strain is greater than that of cold- 
 blast iron, in the ratio of 1024-8 : 1000; and that its strength 
 to resist impact is greater, in the proportion of 1226*3 : 1000. 
 
 ON THE CHEMICAL COMPOSITION OF HOT AND COLD BLAST 
 IRONS, AS ANALYSED BY DR. THOMPSON.* 
 
 The following differences of the two descriptions of metal 
 resulted from the investigations of Dr. Thompson : 
 
 1. The specific gravity of hot blast iron is greater than that 
 of cold blast iron, by about the 22d part. 
 
 2. It was found that manganese, silicon, and aluminum 
 were united with carbon in the composition of all cast iron ; 
 but that, of these foreign ingredients, carbon, silicon, and 
 aluminum entered into the composition of the hot blast 
 iron in a much less proportion than into the cold blast iron ; 
 
 * Report of Brit Ass. Sci. vol. vi. 
 D D 4- 
 
408 ILLUSTRATIONS OF MECHANICS. 
 
 in short, that the hot blast was greatly purer than the cold 
 blast iron. 
 
 The mean result of five analyses of different irons gave for 
 the hot blast iron No. 1 . the proportion of 6 J atoms of iron * 
 to 1 of carbon, silicon, aluminum ; and for the cold blast iroi? 
 No. 1 . the proportion 3 J : 1 . 
 
 The proportions in which carbon, silicon, and aluminum 
 entered into the cold blast iron were 4, 1, 1 ; and those in 
 which they entered into the hot blast iron, 1 2, 5, 2. 
 
 No trace of the ingredients silicon and aluminum was 
 found by Dr. Thompson in the best steel; but only the iron, 
 manganese, and carbon ; and he gives it as his opinion, that 
 the union of these two ingredients, silicon and aluminum, in 
 all English iron, is the reason why good steel can never be 
 made from it. 
 
 Dr. Thompson gives the following explanation of the 
 economy of the hot blast : 
 
 " The whole of the oxygen of the air of the hot blast com- 
 bines with the fuel as soon as it enters into the furnace ; whilst 
 the oxygen of the air of the cold blast is not all consumed im- 
 mediately, but makes its way upwards, and is gradually con- 
 sumed in its ascent, producing a scattered heat, which is of no 
 use in smelting the iron, but serves only to consume the fuel. 
 Whenthehot blast is used the combustion is concentrated towards 
 the bottom of the furnace ; with the cold blast it is much more 
 diffused. Hence the reason of the saving of the coals in the 
 former case, which constitutes the great advantage attending 
 the new method. This greater concentration of the combus- 
 tion must subject the iron to a greater heat than when the 
 combustion is more scattered. Hence the greater rapidity of 
 the process, and consequently the additional quantity of the 
 cast iron obtained from the furnace in a given time." 
 
 * With the iron is here included the small fractional proportion of mar., 
 ganese. 
 
APPENDIX. 
 
 409 
 
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APPENDIX. 
 
 413 
 
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 417 
 
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 ILLUSTRATIONS OF MECHANICS. 
 
 TABLE XIII. 
 
 THE HORIZONTAL THRUST OF A SEMICIRCULAR ARCH WHOSE 
 EXTRADOS IS A HORIZONTAL STRAIGHT LlNE. 
 
 
 D 
 
 
 HORIZONTAL THRUST. 
 
 ^\ 
 
 Values 
 of 
 AB 
 
 C 
 
 BD 
 
 
 
 BD IBD 
 
 O'l n-Q 
 
 BD -.BD__ ftvt 
 
 BD 
 
 BD 
 
 AC 
 
 AC 
 
 AC 
 
 AC 
 
 AC 
 
 AC 
 
 AC 
 
 AC 
 
 0-05 
 
 o-io 
 
 0-08174 
 0-10279 
 
 0-14797 
 0-16370 
 
 0-21762 
 0-22588 
 
 0-28877 
 0-28862 
 
 0-36060 
 0-35164 
 
 0-43277 
 0-41481 
 
 0-79541 
 0-73161 
 
 015 
 
 0-11894 
 
 0-17480 
 
 0-23111 
 
 0-28764 
 
 0-34429 
 
 0-40100 
 
 0-68504 
 
 0-20 
 
 0-13073 
 
 0-18191 
 
 0-23322 
 
 0-28460 
 
 ^33603 
 
 0-38747 
 
 0-64488 
 
 0-25 
 
 0-13871 
 
 0-18553 
 
 0-23237 
 
 0-27922 
 
 0-32607 
 
 0-37293 
 
 0-60727 
 
 0-30 
 
 0-14333 
 
 0-18604 
 
 0-29874 
 
 0-27145 
 
 0-31416 
 
 0-35687 
 
 0-57041 
 
 0-35 
 
 0-14054 
 
 0-18379 
 
 0-22258 
 
 0-26140 
 
 0-30023 
 
 0-33907 
 
 0-53335 
 
 0-40 
 
 0-14422 
 
 0-17913 
 
 0-21415 
 
 0-24924 
 
 0-28437 
 
 0-31953 
 
 0-49560 
 
 0-45 
 
 014124 
 
 0-17240 
 
 0-20374 
 
 0-23520 
 
 0-26674 
 
 0-29835 
 
 0-45693 
 
 0-50 
 
 0-13649 
 
 0-16396 
 
 0-19168 
 
 0-21957 
 
 0-24760 
 
 0-27573 
 
 0-41728 
 
 Note. This and the following table are extracted from the work of M. 
 Garidel, entitled Tables de la Poussee des Voutes. Paris, 1837. 
 
 TABLE XIV. 
 
 THE ANGLE OF RUPTURE IN A SEMICIRCULAR ARCH, THE 
 EXTRADOS BEING A HORIZONTAL STRAIGHT LINE. 
 
 
 ANGLES OF RUPTURE. 
 
 23 
 
 Values 
 
 C 
 
 of 
 AB 
 AC 
 
 BD 
 AC= 
 
 BD 
 
 BD 
 
 BD 
 AC=' 3 
 
 BD 
 
 =0-4 
 AC 
 
 BD 
 
 AC 1 
 
 0-05 
 
 68-0 
 
 59-19 
 
 5404 
 
 51-15 
 
 49-35 
 
 48-20 
 
 45-74 
 
 o-io 
 
 65-4 
 
 60-48 
 
 57-70 
 
 56-01 
 
 54-93 
 
 54-17 
 
 52-34 
 
 0-15 
 
 64-0 
 
 61-3 
 
 59-7 
 
 58'69 
 
 58-0 
 
 57-49 
 
 56-21 
 
 0-20 
 
 63-1 
 
 617 
 
 60-88 
 
 60-30 
 
 59-90 
 
 59-60 
 
 58-80 
 
 0-25 
 
 62-24 
 
 6176 
 
 61-44 
 
 61-22 
 
 61-05 
 
 60-94 
 
 60-59 
 
 0-30 
 
 61-3 
 
 61-42 
 
 61-54 
 
 61-60 
 
 6166 
 
 61-67 
 
 61-81 
 
 0-35 
 
 6017 
 
 60-80 
 
 61-21 
 
 61-54 
 
 61-78 
 
 61-98 
 
 62-56 
 
 0-40 
 
 58-8 
 
 59-8 
 
 60-52 
 
 61-05 
 
 61-48 
 
 61-67 
 
 62-9 
 
 0-45 
 
 57-32 
 
 58-53 
 
 5945 
 
 60-19 
 
 60-80 
 
 61-28 
 
 62-85 
 
 0-50 
 
 55-63 
 
 56-97 
 
 58-09 
 
 58-98 
 
 59-72 
 
 60-34 
 
 62-40 
 
APPENDIX. 
 
 419 
 
 . 
 
 > CO 
 
 3 * 
 
 H 
 
 S 
 
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 - 
 
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 1 
 1 
 
 c^ 
 
 E E 2 
 
4-20 
 
 ILLUSTRATIONS OF MECHANICS. 
 
 fcjj 
 
 |S 
 
 J 
 JZlQ 
 
 ^"c v '~ & < 
 
 tl B 9 ** fT 2 J3 S 
 
 ISv'SliaSl 
 
 Q jjJMv: 
 
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 1 
 
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 r< 
 
 W 
 
 H^ 
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 1 1 1 1 1 1 1 1 1 1 1 1 1 
 
 CiCiO^O^OiOiO^O^O^O^OiCiO^ 
 
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 thi 1 
 
 11 
 
APPENDIX. 
 
 EXPERIMENTS ON FRICTION, MADE AT METZ IN THE YEARS 
 1831, 1832, 1833. M. MORIN. 
 
 These experiments, into the mechanical details of which more 
 precautions were introduced, and in which greater mechanical 
 accuracy was probably attained, than in any which have pre- 
 ceded them ; and in the measurement of the results of which, 
 and the separation of the friction of the moving body from the 
 various other elements which complicated those results, admir- 
 able theoretical skill and ingenuity were exhibited *, have 
 placed the question of friction entirely in a new, and a far 
 more satisfactory position than it has before occupied. They 
 were made at the expense of the French government, under 
 the most favourable circumstances, by methods which have 
 been fully and clearly detailed ; and however opposed they may 
 be in their results to all former experiments, and especially to 
 those of Coulomb, it is impossible not to yield to them the 
 greatest confidence. 
 
 The principal conclusions drawn from these experiments 
 may be stated as follows : 
 
 They show the friction of two surfaces which have been for 
 a considerable time in contact to be not only different in its 
 amount^ but in its nature, from the friction of surfaces in con- 
 tinuous motion, especially in this, that this friction of qui- 
 escence is subject to causes of variation and uncertainty, from 
 
 * The contrivance, first suggested by M. Poucelet, by which the motion 
 of the moving surface was made to record itself through all the variations 
 of its velocity, as the weight which communicated motion to it accelerated 
 or retarded its descent, is one of the most remarkable and the most valua- 
 ble contributions which theory has ever made to practical mechanics : for 
 the details of it the reader is referred to the work of M. Moriu, entitled 
 " Nouvelles Experiences sur le Frottement." Paris, 1833. Bachelier. 
 This instrument admits of being applied under a modified form to deter- 
 mine the action or working dynamical effect of any part of a machine in 
 motion ; its determinations may be extended to every period and circum- 
 stance of the motion. Applied by a very simple contrivance to the cylin- 
 der of a steam engine, it would serve admirably the purpose of a steam 
 indicator, recording with precision every varying effort of the moving 
 power, and indicating the exact period of the motion when each such effort 
 was made. Results thus obtained from an extensive series of experiments 
 would constitute a body of facts invaluable as facts of reference to the 
 civil engineer. 
 
 E E 3 
 
422 ILLUSTRATIONS OF MECHANICS. 
 
 which the friction of motion is exempt. This variation does 
 not appear to depend upon the extent of the surfaces of con- 
 tact ; for, with different pressures, the ratio of the friction to the 
 pressure, or the co-efficient of friction, as it is called, varied 
 greatly, although the surfaces of contact were the same. * The 
 uncertainty which would have been introduced into every 
 question of practical mechanics, and especially of construction, 
 by this consideration, is, "however, removed by a second very 
 important fact developed accidentally in the course of the ex- 
 periments. It is this, that by the slightest Jar or shock, the most 
 imperceptible movement of the surfaces of contact, their friction 
 is made to pass from this state accompanying quiescence into 
 that entirely different state of friction which accompanies 
 motion ; and as every machine or structure of whatever kind 
 may be considered to be subject to such shocks or impercep- 
 tible motions of its surfaces of contact, it is evident that the 
 state of friction to be made the basis on which all questions of 
 statics are to be determined, should be that last mentioned, 
 which accompanies continuous motion. Now the LAWS of this 
 friction, thus accompanying motion, are shown by the ex- 
 periments of M. Morin to be of remarkable uniformity and 
 precision, and that, under an extensive range of variation, as 
 well in the pressures by which the surfaces are held in contact, 
 as in the dimensions of those surfaces. They are these, 
 
 1. The friction accompanying the motion of two surfaces 
 between which no unguent is interposed, bears the same pro- 
 portion to the force by which those surfaces are pressed to- 
 gether, whatever may be the amount of that force. 
 
 2. This friction is independent of the extent of the surfaces 
 of contact. 
 
 3. Where unguents are interposed, a distinction is to be 
 made between the case in which the surfaces are simply 
 unctuous, and in intimate contact with one another, and the case 
 in which the surfaces are wholly separated from one another 
 
 * Thus, for instance, in the case of oak upon oak with parallel fibres, 
 the co-efficient of friction of quiescence varied under different pressures, 
 but upon the same surface, from '55 to '76. 
 
Al'PENDIX. 
 
 423 
 
 by an interposed stratum of the unguent. If the pressure upon 
 a surface of contact of given dimensions be increased beyond a 
 certain limit, the latter of these cases passes into the first ; the 
 stratum of unguent being pressed out, and the unctuous sur- 
 faces which it separated from one another being brought into 
 intimate contact. As long as either of these two states remains, 
 the laws of its friction are not affected by the presence of the 
 unguent ; but in the transition from the one state to the other, 
 an exception is made to the independence of the friction upon 
 the extent of the surface of contact ; for supposing the extent 
 of two surfaces of contact, between which a stratum of unguent 
 is interposed, and which sustain a given pressure, to be con- 
 tinually diminished, it is evident that the portions of this pres- 
 sure which take effect upon each element of the surfaces of 
 contact will be continually increased, and that they may thus 
 be so increased as to press out the interposed stratum of unguent, 
 and cause the state of the surfaces to pass into that which we 
 have designated as unctuous, thereby changing the co-efficient 
 of friction. That law of friction, then, which is known as the 
 law of " the independence of the surface," is to be received, in 
 the case where a stratum of unguents is interposed, only within 
 certain limits. 
 
 It will be understood, from what has above been said, that 
 there are three states, in respect to friction, into which the 
 surfaces of bodies in contact may be made successively to pass : 
 one, a state in which no unguent is present ; the second, a 
 state in which the surfaces are unctuous, but intimately in con- 
 tact ; the third, a state in which the surfaces are separated by 
 an entire stratum of the interposed unguent. Throughout 
 each of these states the co-efficient of friction is the same ; but 
 it is essentially different in the different states, as will be seen 
 from the following tables. 
 
 4. It is a law common to the friction of all the states of 
 contact of two surfaces, that their friction, when in motion, is 
 altogether independent of the velocity of the motion. M. 
 Morin has verified this law, as well in various states of contact 
 without interposed fluids, as in cases where water, oils, grease, 
 8*4 
 
4-24 ILLUSTRATIONS OF MECHANICS. 
 
 glutinous liquids, syrups, pitch, &c., were interposed in a con- 
 tinuous stratum. 
 
 The variety of the circumstances under which these laws 
 obtain in respect to the friction of motion, and the accuracy 
 with which the phenomena of motion accord with them, may 
 be judged of from one example taken from the first set of ex- 
 periments of M. Morin upon the friction of surfaces of oak 
 whose fibres were parallel to the direction of their motion upon 
 one another. He caused the surfaces of contact to vary their 
 dimensions in the ratio of 1 to 84, from less than 5 square 
 inches to nearly 3 square feet ; the forces which pressed them 
 together, he varied from 88lbs. to 2205 Ibs., and the velocities 
 of their motion, from the slowest perceptible to 9*8 feet per 
 second causing them to be at one period accelerated motions, 
 at another uniform, at a third retarded ; yet throughout all this 
 wide range of variation, he in no instance found the co-efficient 
 of friction to deviate from the same fraction of 0*478 by more 
 than J|th of the amount of that fraction. 
 
APPENDIX. 
 
 425 
 
 I 
 
 CO 
 CO 
 
 3 
 
 P 
 
 2 
 
 "5 q 
 
 -t-T ' 
 
 4! i 
 
 i n 
 
 &< * ^ 
 
 w I : 
 
 
 
 I ~f 
 
 Limi 
 of R 
 
 si 
 
 fs, 
 
 5 
 
 g 
 
426 
 
 ILLUSTRATIONS OF MECHANICS. 
 
 
 if 
 
 
 
 
 
 
 
 
 
 
 
 
 w 
 
 
 
 
 111 
 
 
 
 
 
 
 1 
 
 L 
 
 
 
 
 
 
 
 
 
 
 S 
 
 a 
 
 
 
 
 i 111 
 
 
 
 
 
 
 1 
 
 U c 
 
 
 
 
 J;S TS ^^c a 
 
 
 
 
 
 
 <4- 
 
 -g 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 pjj eBs2 toff 
 
 
 
 
 
 
 [> 
 
 * 
 
 
 
 
 ^SjjjsS 
 
 
 
 
 
 
 5 
 
 ^ 
 
 
 
 
 n$ 1 
 
 
 
 
 
 
 | 
 
 c "rt 
 
 
 
 
 ISili 
 
 
 
 
 
 
 c 
 
 j'o 
 
 -V^ta/ 
 
 
 
 j 
 
 Pii 
 
 ti . 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 
 c c 
 
 < 
 
 
 
 b 
 
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 8 
 
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 US 
 
 
 
 g 
 
 9 
 
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 ! 
 
 
 
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 6 
 
 
 6 
 
 
 6 
 
 1 
 
 2s 
 fc 
 
 
 
 
 
 
 
 
 
 
 
 
 11 
 
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 I 
 
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 5 
 
 
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 If 
 
 S3 
 
 ft 
 
 
 
 
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 CO 
 
 
 S 
 
 J 
 
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 6 
 
 
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 o 
 
 
 
 
 
 
 
 
 
 
 
 SURFACES OF CONTACT. 
 
 ik upon oak, the fibres of both surO 
 faces being perpendicular to the direc- > 
 tion of the motion - - -3 
 
 ik upon oak, the fibres of the moving"] 
 surface being perpendicular to the 
 surface of contact, and those of the > 
 surface at rest parallel to the direc- 
 tion of the motion -J 
 ak upon oak, the fibres of both sur-} 
 faces being perpendicular to the sur- f 
 face of contact, or the pieces end to f 
 
 1m upon oak, the direction of the fibres 7 
 being parallel to the motion - -3 
 
 ak upon elm, ditto ... 
 
 1m upon oak, the fibres of the movingl 
 surface (the elm) being perpendicular | 
 to those of the quiescent surface (the V 
 oak) and to the direction of the mo- 
 tion 
 
 
 1 
 
 
 
 C 
 
 
 
 O 
 
 w 
 
 
 
 > 
 
 X 
 
 w 
 *1 
 
 P3 
 
 < 
 
 H 
 
APPENDIX. 
 
 427 
 
 ABcsc/rfl5%c 
 
 BE5 < s4)Ki** o)<u 
 
 f <" S..S SO <U T3 G. -S 
 
 1 .5 e^o ^ SH ' 
 
 .SrtfJ-^^aJo 
 
 O t- C< O 
 
 r~ c5 co ?* i^ 
 
 If) p 1C) -^ tp 
 
 6 6 666 
 
 S S i S 
 
 66 66 
 
 CO Op C3 OS 
 CO rf< T-H Tj< 
 
 rHOO Ir- 00 
 
 f*o 
 
 
 6 6 
 
 660 
 
 660 
 
 ^i 00 
 
 Oi CO 
 
 6 6 
 
 6 66 6 6 
 
 r**A^r^> 
 
 ' Is 
 
 w fl ,es 
 
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 r^ i^-* 
 tc ' *-"5 
 
 > T3 C * 
 
 S 
 
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 S c 8L|.j3 
 
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 23*||2| 
 
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 0X5 13 
 
 3 ' 
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 =^3 
 
 z2 
 
 ^5 ^S 
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 ll. 
 
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 53* S ^-g 
 
 gSglill'll 2 ! 
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 liilll!sis!l 
 
 373 3 S--- rfS* 3 *" cT<u 
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428 ILLUSTRATIONS OF MECHANICS. 
 
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 lift 
 
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 3 Ml*. 
 
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 I 111 l! i Jill H* rfl 
 
 g^ S c g o2 5 c c o ^^^ rt >, 
 
 l|llIS.il|g^||l ^lo|5 
 
APPENDIX. 
 
 429 
 
 
 *i T3 fi> s >-,- 
 M 0) JS O >- 
 
 
 
 X ^* C r; C '- 
 
 o^ J 1 S C > t3 
 
 1 
 
 11*1111 
 
 
 "All the above experir 
 with curried black 
 the phenomenon o 
 polish of the surfac 
 a state of theirs 
 to, and dependent i 
 
 c 
 
 I 
 
 '5 
 
 
 
 _j 
 
 8 S 
 
 8 S 
 
 0) 
 
 
 
 co o o >b ^-< 1> c^^ 
 
 tO if) (O CC CO O* CO CO G-J g 
 
 ? 9 
 
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 1 
 
 ^l O IC5 WJ O CO O5t^'f 1 O< 
 
 6 6 666 66666 
 
 S 3 
 
 
 
 OO Oi C* O O< C3 I^OOOiO 
 CO T O * CO 1C) iji CO 
 
 S & 
 
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 O^ O CO CO CO O Tf O O CO 
 CO 0< CO CO CO CN WCOO^^ 
 
 1 ! 
 
 fe 
 
 6 
 
 * co n r^ to co oioco^ 
 tc co 'pcpco cotccoeooi 
 6 6 6666 6666 
 
 srr 
 
 SCD > 
 ^g 5 
 
 = o> o e > ' 
 
 **llg 
 
 2 
 'S 
 tf 
 
 rt 
 
 2 i?J '"o^i^'o'o's'oo 1 g 
 I'Sd -53 * " ^ Sg | 
 fig-l S^ - S" 5* '21 
 S 3 2-S-go;! o g 0^ 
 
 lir"lll|' | |i i||i 
 
 "3 o. o 9 
 
 S 
 
 gSl ^1,11 -2 ^'f --5? 
 
 ^s" -> ' 
 
 A 
 
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 13 
 
 O 
 
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 S^jfi'S-fjSHl^lS^ll 
 
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 -g.S S S 53 .S S^5 5 | 1-ailJ l-g 
 H H O O ^OO 
 
ILLUSTRATIONS OF MECHANICS. 
 
 TABLE XVII. 
 
 EXPERIMENTS ON THE FRICTION OF UNCTUOUS SURFACES BT 
 M. MORIN. 
 
 In these experiments the surfaces, after having been smeared 
 with an unguent, were wiped, so that no interposing layer of 
 the unguent prevented their intimate contact. 
 
 
 FRICTION OF 
 
 FRICTION OF 
 
 
 MOTION. 
 
 QUIESCENCE. 
 
 SURFACES OF CONTACT. 
 
 || 
 
 d 
 
 !' 
 
 * i 
 
 
 11 
 
 |2| 
 
 11 
 
 l|j 
 
 
 5* 
 
 < 
 
 It 
 
 .*S 
 
 *<x 
 
 Oak upon oak, the fibres being pa- 7 
 rallel to the motion - - 3 
 
 0-108 
 
 6 10' 
 
 0-390 
 
 21 19' 
 
 Ditto, the fibres of the moving } 
 body being perpendicular to the > 
 motion - - -3 
 
 0-143 
 
 8 9 
 
 0-314 
 
 17 26 
 
 Oak upon elm, fibres parallel 
 
 0-136 
 
 7 45 
 
 
 
 Elm upon oak, ditto 
 
 0-119 
 
 6 48 
 
 0-420 
 
 22 47 
 
 Beech upon oak, ditto 
 
 0-330 
 
 18 16 
 
 
 
 Elm upon elm, ditto 
 
 0-140 
 
 7 59 
 
 
 
 Wrought iron upon elm, ditto 
 
 0-138 
 
 7 52 
 
 
 
 Ditto upon wrought iron, ditto 
 
 0-177 
 
 10 3 
 
 O1 18 
 
 6 44 
 
 Cast iron upon wrought iron, ditto - 
 
 0-143 
 
 8 9 
 
 U llo 
 
 
 Wrought iron upon brass, ditto 
 Brass upon wrought iron 
 
 0-ldO 
 0-166 
 
 9 6 
 9 5:6 
 
 
 
 Cast iron upon oak, ditto 
 
 0-107 
 
 6 7 
 
 0-100 
 
 5 43 
 
 Ditto upon elm, ditto, the un- 7 
 guent being tallow - -3 
 Ditto, ditto, the unguent being > 
 hog's lard and black lead - 3 
 
 0-125 
 0-137 
 
 7 8 
 7 49 
 
 
 
 Elm upon cast iron, fibres parallel - 
 
 0-135 
 
 7 42 
 
 0-098 
 
 5 36 
 
 Cast iron upon cast iron - - 
 
 0-144 
 
 8 12 
 
 
 
 Ditto upon brass - 
 
 0-132 
 
 7 32 
 
 
 
 Brass upon cast iron - 
 
 0-107 
 
 6 7 
 
 
 
 Ditto upon brass - 
 
 0-134 
 
 7 38 
 
 0-164 
 
 9 19 
 
 Copper upon oak 
 Yellow copper upon cast iron 
 
 0-100 
 0-115 
 
 5 43 
 6 34 
 
 
 
 Leather (ox hide) well tanned upon 7 
 cast iron, wetted . - -3 
 
 0-229 
 
 12 54 
 
 0-267 
 
 14 57 
 
 Ditto upon brass, wetted 
 
 0-244 
 
 13 43 
 
 
 
 i 
 
 
 
 I 
 
 The distinction between the friction of surfaces to which no 
 unguent is present, those which are merely unctuous, and those 
 
APPENDIX. 431 
 
 between which a uniform stratum of the unguent is inter- 
 posed, appears first to have been remarked by M. Morin ; it 
 has suggested to him what appears to be the true explanation 
 of the difference between his results and those of Coulomb. 
 He conceives, that in the experiments of this celebrated en- 
 gineer the requisite precautions had not been taken to exclude 
 unguents from the surfaces of contact. The slightest unc- 
 tuosity, such as might present itself accidentally, unless ex- 
 pressly guarded against such, for instance, as might have been 
 left by the hands of the workman who had given the last 
 polish to the surfaces of contact is sufficient materially to 
 affect the co-efficient of friction. 
 
 Thus, for instance, surfaces of oak having been rubbed with 
 hard dry soap, and then thoroughly wiped, so as to show no 
 traces whatever of the unguent, were found by its presence to 
 have lost ds of their friction, the co-efficient having passed 
 from 0-478 to 0-164. 
 
 This effect of the unguent upon the friction of the surfaces 
 may be traced to the fact, that their motion upon one another 
 without unguents was always found to be attended by a 
 wearing of both the surfaces ; small particles of a dark colour 
 continually separated from them, which it was found from 
 time to time necessary to remove, and which manifestly in- 
 fluenced the friction : now with the presence of an unguent 
 the formation of these particles, and the consequent wear of 
 the surfaces, completely ceased. Instead of a new surface of 
 contact being continually presented by the wear, the same 
 surface remained, receiving by the motion continually a more 
 perfect polish. 
 
432 
 
 ILLUSTRATIONS OF MECHANICS. 
 
 TABLE XVIII. 
 
 EXPERIMENTS ON FRICTION WITH UNGUENTS INTERPOSED, BY 
 M. MORIN. 
 
 The extent of the surfaces in these experiments bore such a 
 relation to the pressure, as to cause them to be separated from 
 one another throughout by an interposed stratum of the 
 unguent. 
 
 
 FRICTION 
 
 FRICTION 
 
 
 
 OF 
 
 OF 
 
 
 
 MOTION. 
 
 QUIESCENCE. 
 
 
 SURFACES OF CONTACT. 
 
 c 
 
 s 
 
 UNGUENTS. 
 
 
 
 3 
 
 
 
 
 
 
 
 5 "S tj 
 
 56 o'S 
 
 
 
 1 
 
 3 
 
 
 
 
 
 
 
 
 Oak upon oak, fibres parallel 
 
 0164 
 
 0-440 
 
 Dry soap. 
 
 Ditto ditto 
 
 0-075 
 
 0-164 
 
 Tallow. 
 
 Ditto ditto 
 
 0-067 
 
 . 
 
 Hogs' lard. 
 
 Ditto, fibres perpendicular 
 
 0-083 
 
 0-254 
 
 Tallow. 
 
 Ditto ditto 
 
 0-072 
 
 . 
 
 Hogs' lard. i 
 
 Ditto ditto 
 
 0-250 
 
 . 
 
 Water. 
 
 Ditto upon elm, fibres pa- 7 
 rallel j 
 
 0-136 
 
 - - 
 
 Dry soap. 
 
 Ditto ditto 
 
 0-073 
 
 0-178 
 
 Tallow. 
 
 Ditto ditto 
 
 0-066 
 
 
 Hogs' lard. 
 
 Ditto upon cast iron, ditto - 
 
 0-080 
 
 . 
 
 Tallow. j 
 
 Ditto upon wrought 7 
 iron, ditto j 
 
 0-C98 
 
 - - 
 
 Tallow. 
 
 Beech upon oak, ditto 
 Elm upon oak, ditto 
 
 0-055 
 0-137 
 
 "0-411 
 
 Tallow. 
 Dry soap. 
 
 Ditto ditto 
 
 0-070 
 
 0142 
 
 Tallow. 
 
 Ditto ditto 
 
 0-060 
 
 _ 
 
 Hogs' lard. 
 
 Ditto upon elm, ditto 
 Ditto upon cast iron, ditto 
 
 0-139 
 0'066 
 
 0-217 
 
 Dry soap. 
 Tallow. 
 
 
 
 
 f Greased and 
 
 Wrought iron upon oak, ditto 
 
 0-256 
 
 0-649 
 
 < saturated 
 
 
 
 
 C with water. 
 
 Ditto ditto ditto 
 
 0-214 
 
 . 
 
 Dry soap. 
 
 Ditto ditto ditto 
 
 0-085 
 
 0-108 
 
 Tallow. 
 
 Ditto upon elm, ditto 
 Ditto ditto ditto 
 
 0-078 
 0-076 
 
 - 
 
 Tallow. 
 Hogs' lard. 
 
 Ditto ditto ditto 
 
 0-055 
 
 _ 
 
 Olive oil. 
 
 Ditto upon cast iron, ditto 
 
 0-103 
 
 _ 
 
 Tallow. 
 
 Ditto ditto ditto 
 
 0-076 
 
 _ 
 
 Hogs' lard. 
 
 Ditto ditto ditto 
 
 0-066 
 
 o-ioo 
 
 Olive oil. 
 
 Ditto upon wrought > 
 iron, ditto J 
 
 0-082 
 
 . 
 
 Tallow. 
 
 Ditto ditto ditto 
 
 0-081 
 
 
 
 Hogs' lard. 
 
 Ditto ditto ditto 
 
 0-070 
 
 0115 
 
 Olive oil 
 
APPENDIX. 
 TABLE XVIII. (continued.) 
 
 433 
 
 
 FRICTION 
 
 FRICTION 
 
 
 
 OP 
 
 OP 
 
 
 
 MOTION. 
 
 QUIESCENCE 
 
 
 SURFACES OF CONTACT 
 
 1 1 
 
 g B - 
 *5 o 
 
 UNGUENTS. 
 
 
 "C 
 
 
 
 Wrought iron upon brass, ) 
 fibres parallel - 3 
 Ditto ditto ditto 
 
 0-103 
 0-075 
 
 - - 
 
 Tallow. 
 Hogs' lard. 
 
 Ditto ditto ditto 
 
 0-078 
 
 
 Olive oil 
 
 Cast iron upon oak, ditto 
 
 0-189 
 
 m 
 
 Dry soap. 
 
 
 
 
 r Greased, and 
 
 Ditto ditto ditto - 
 
 0-218 
 
 0-646 
 
 -J saturated 
 
 
 
 
 6 with water. 
 
 Ditto ditto ditto 
 
 0-078 
 
 O'lOO 
 
 Tallow. 
 
 Ditto ditto ditto 
 
 0-075 
 
 
 Hogs' lard. 
 
 Ditto ditto ditto 
 
 0-075 
 
 o-ioo 
 
 Olive oil. 
 
 Ditto upon elm, ditto 
 Ditto ditto ditto 
 
 0-077 
 0061 
 
 
 Tallow, 
 Olive oil. 
 
 Ditto ditto ditto 
 
 0-091 
 
 . 
 
 (Hogs' lard and 
 v. plumbago. 
 
 Ditto, ditto upon wrought 7 
 iron - - - J 
 
 - 
 
 o-ioo 
 
 Tallow. 
 
 Ditto upon Cast iron 
 
 0-314 
 
 m m 
 
 Water. 
 
 Ditto ditto 
 
 0-197 
 
 
 Soap. 
 
 Ditto ditto 
 
 0100 
 
 o-ioo 
 
 Tallow. 
 
 Ditto ditto 
 
 0-070 
 
 o-ioo 
 
 Hogs' lard. 
 
 Ditto ditto 
 
 0064 
 
 
 Olive oil. 
 
 Ditto ditto 
 
 0-055 
 
 
 f Lard <and 
 
 Ditto upon brass 
 
 0-103 
 
 
 I plumbago. 
 Tallow. 
 
 Ditto ditto 
 
 0-075 
 
 
 
 Hogs' lard. 
 
 Ditto ditto 
 
 0-078 
 
 
 Olive oil. 
 
 Copper upon oak, fibres pa- 7 
 rallel # - J 
 
 0-069 
 
 o-ioo 
 
 Tallow. 
 
 Yellow copper upon Cast iron 
 Ditto ditto 
 
 0-072 
 0-068 
 
 0-103 
 
 Tallow. 
 Hogs' lard. 
 
 Ditto ditto 
 
 0-066 
 
 _ 
 
 Olive oil. 
 
 Brass upon cast iron 
 
 0-086 
 
 0-10d 
 
 Tallow. 
 
 Ditto ditto 
 
 0-077 
 
 
 Olive oil. 
 
 Ditto upon wrought iron - 
 
 0-081 
 
 . 
 
 Tallow. 
 
 Ditto ditto 
 Ditto ditto 
 
 0-089 
 
 0-072 
 
 - - 
 
 f Lard and 
 [ plumbago. 
 Olive oil. 
 
 Ditto upon brass 
 
 0-058 
 
 
 
 Olive oil. 
 
 Steel upon cast iron 
 Ditto ditto 
 
 0-105 
 0-081 
 
 0-108 
 
 Tallow. 
 Hogs' lard. 
 
 Ditto ditto 
 
 0-079 
 
 
 
 Olive oil. 
 
 Ditto upon wrought iron - 
 Ditto ditto 
 
 0-093 
 0-076 
 
 - 
 
 Tallow. 
 Hogs' lard. 
 
 Ditto upon brass 
 
 0056 
 
 _ 
 
 Tallow. 
 
 Ditto ditto 
 
 0-053 
 
 _ 
 
 Olive oil. 
 
 Ditto ditto 
 
 0-067 
 
 . 
 
 [" Lard and 
 [ plumbago. 
 
 Tanned oxhide upon cast 7 
 
 0-365 
 
 
 "" Greased, and 
 saturated 
 
 iron - - 3 
 
 
 
 .with water. 
 
434 
 
 ILLUSTRATIONS OF MECHANICS. 
 
 TABLE XVIII. (continued.} 
 
 
 FRICTION 
 
 FRICTION 
 
 
 
 OP 
 
 OF 
 
 
 
 MOTION. 
 
 QUIESCENCE. 
 
 
 SURFACES OF CONTACT. 
 
 U 
 
 .SJ c 
 
 UNGUENTS. 
 
 
 
 |o| 
 
 
 
 o fe 
 
 i 'C 
 
 u * 
 
 
 Tanned ox hide upon cast ? 
 iron - - -3 
 
 0-159 
 
 - 
 
 Tallow. 
 
 Ditto ditto 
 
 0-133 
 
 0-122 
 
 Olive oil, 
 
 Ditto upon brass 
 Ditto ditto 
 
 0-241 
 0-191 
 
 
 Tallow. 
 Olive oil. 
 
 Ditto upon oak 
 Hempen fibres not twisted," 
 
 0-29 
 
 0-79 
 
 Water. 
 
 moving upon oak,the fibres 
 
 
 
 
 of the hemp being placed 
 in a direction perpendicu- > 
 
 0-332 
 
 0-869 
 
 f Greased, and 
 ^ saturated 
 
 lar to the direction of the 
 
 
 
 C with water. 
 
 motion, and those of the 
 
 
 
 
 oak parallel to it 
 The same as above, moving ) 
 upon cast iron - - 3 
 Ditto ditto 
 
 0-194 
 0-153 
 
 - - 
 
 Tallow. 
 Olive oil. 
 
 Soft calcareous stone of Jau-~) 
 
 
 
 
 , mont upon the same, with 
 
 
 
 
 a layer of mortar, of sand, 1 
 
 
 7 
 
 
 and lime, interposed after f 
 
 
 U 74 
 
 
 from JO to 15 minutes' con- 1 
 
 
 
 
 tact - - - J 
 
 
 
 
 A comparison of the results enumerated in the above table 
 leads to the following remarkable conclusion, easily fixing 
 itself in the memory, that with the unguents hogs' lard and olive 
 oil interposed in a continuous stratum between them, surfaces of 
 wood on metal, wood on wood, metal on wood, and metal on metal, 
 when in motion, have all of them very nearly the same co-efficient 
 of friction, the value of that co-efficient being in all cases included 
 between 0'07 and -08, and the limiting angle of resistance there- 
 fore between 4 and 4 35'. 
 
 For the unguent tallow the co-efficient is the same as the above 
 in every case, except in that of metah upon metals ; this un- 
 guent seems less suited to metallic surfaces than the others, and 
 gives for the mean value of its co-efficient O'lO, and for its limiting 
 angle of resistance 5 43'. 
 
 The experiments of which the above are results were all 
 made under considerable pressures, such as those under which 
 
APPENDIX. 
 
 t 
 
 436 
 
 the parts of the larger machines are accustomed to move 
 upon one another : under such pressures the adhesion of the 
 unguent to the surfaces of contact, and the opposition pre- 
 sented to their motion by its viscidity, are causes whose in- 
 fluence may be altogether neglected as compared with the 
 friction. In the case of lighter machinery, as, for instance, that 
 of clocks and watches, these considerations rise, however, into 
 importance. 
 
 TABLE XIX. 
 COMPARISON OF FRENCH AND ENGLISH MEASURES. 
 
 MEASURES OF LENGTH. 
 
 French. 
 
 English. 
 
 Millimetre 
 Centimetre 
 Decimetre 
 
 MStre 
 Myriametre 
 
 0*03937 inch. 
 0-393708 inch. 
 3-937079 inches 
 I 39-37079 inches. 
 < 3-2808992 feet. 
 C 1-093633 vard. 
 62138 miles. 
 
 MEASURES OF SUPERFICIES. 
 
 MStre Carre 
 Are 
 Hectare 
 
 1 196033 square yard. 
 0-098815 rood. 
 2 -47 1145 acres. 
 
 MEASURES OF CAPACITY. 
 
 Litre 
 
 Decalitre 
 Hectolitre 
 
 f 1-760773 pint. 
 i 0-2^00967 gallon. 
 2-2009668 gallons. 
 22009668 gallons. 
 
436 ILLUSTRATIONS OF MECHANICS. 
 
 TABLE XIX. (continued.) 
 
 MEASURES OF WEIGHT. 
 
 French. 
 
 English. 
 
 Gramme 
 Kilogramme 
 
 r 15-438 grains, troy. 
 < 0*643 penny weights, troy. 
 C G'03216 ounces, troy. 
 C 2-68027 pounds, troy. 
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INDEX. 
 
 CTON'S Modern Cookery 
 
 LCOCK'S Residence in Japan 
 
 LLIES on Formation of Christendom 
 
 Ipine Guide (The) 
 
 LVENSLEBEN'S Maximilian in Mexico .... 
 
 PJOHN'S Manual of the Metalloids 
 
 RNOLD'S Manual of English Literature. . . . 
 
 NOTT'S Elements of Physics 
 
 rundines Cami 
 
 utumn holidays of a Country Parson . . 
 YRE'S Treasury of Bible Knowledge 
 
 AGON'S Essays, by WHATELY 
 
 Life and Letters, by SPEDDINO 
 
 Works 
 
 "[ AIN on the Emotions and Will 
 
 ' on the Senses and Intellect 
 
 on the Study of Character 
 
 ALL'S Alpine Guide 
 
 ARNARD s Drawing from Nature 
 
 AYLDON'S Rents and Tillages 
 
 eaten Tracks 
 
 ECKER'S Charicles and Gallus 
 
 EETHOVEN'S Letters 
 
 SNFEY'S Sanskrit Dictionary 
 
 ERRY'S Journals and Correspondence .... 
 
 ;lliard Book (The) 
 
 LACK'S Treatise on Brewing 
 
 LACKLEY and FRIEDLANDER'S German and 
 
 English Dictionary 
 
 LAINE'S Rural Sports 
 
 -Veterinary Art, 
 
 HT'S Week at the Land's End 
 
 H'S Epigrams 
 
 _-.INE on Screw Propeller 
 
 OURNE'S Catechism of the Steam Engine.. 
 Handbook of Steam Engine 
 
 -- Treatise on the Steam Engine ... 
 OWDLER'S Family SHAKSPEARK ............ 
 
 OYD'S Manual for Naval Cadets ............ 
 
 RAMLEY-MOORE'S Six Sisters of the Valleys 
 RANDE'S Dictionary of Science, Literature, 
 and Art .................................... 
 
 RAY'S (C.) Education of the Feelings ..... 
 
 Philosophy of Necessity ........ 
 
 RINTON on Food and Digestion 
 
 RISTOW'S Glossary of Mineralogy 
 
 RODIE'S (Sir C. B.) Works 
 
 Constitutional History 
 
 ROWNE'S Exposition 39 Articles 
 
 xposi 
 
 UCKLE'S History of Civilization 
 ULL'S Hints to Mothers .................... 
 
 - Maternal Management of Children. 
 UNSEN'S (Baron) Ancient Egypt .......... 
 
 BUNSEN'S (Baron) God in History 
 
 Memoirs 
 
 3 
 4 
 
 BUNSEN (E.Dfi) on Apocrypha 20 
 
 's Keys of St. Peter 20 
 
 BDRKE'S Vicissitudes of Families 5 
 
 BURTON'S Christian Church 3 
 
 Cabinet Lawyer 
 
 CALVERT'S Wife's Manual 
 
 CATES'S Biographical Dictionary 
 
 CATS' and FARLIE'S Moral Emblems 
 
 Chorale Book for England 
 
 Christian Schools and Scholars 
 
 CLOUGH'S Lives from Plutarch 
 
 COLENSO (Bishop) on Pentateuch and Book 
 
 of Joshua 
 
 COLLINS'S Horse- Trainer's Guide 
 
 Commonplace Philosopher in Town and 
 
 Country 
 
 CONINGTON'S Chemical Analysis 
 
 Translation of VIRGIL'S sEneid 
 
 CONTANSEAU'S Pocket French and English 
 
 Dictionary 
 
 Practical ditto 
 
 id HOWSON'S Life and Epistles 
 
 of St. Paul. 
 
 COOK on the Acts 
 
 COPLAND'S Dictionary of Practical Medicine 
 
 COCJLTHART'S Decimal Interest Tables 
 
 Cox's Manual of Mythology 
 
 Tales of the Great Persian War 
 
 Tales from Greek Mythology 
 
 Tales of the Gods and Heroes 
 
 Tales of Thebes and Argos 
 
 CRESY'S Encyclopaedia of Civil Engineering 
 
 Critical Essays of a Country Parson 
 
 CROWE'S History of France 
 
 CRUMP on Banking, Currency, & Exchanges 
 
 DART'S Iliad of Homer 25 
 
 D' AUBIGNE'S History of the Reformation in 
 the time of CALVIN., 
 
 DAVIDSON'S Introduction to New Testament 
 
 DAYMAN'S Dante's Divina Commedia 
 
 Dead Shot (The), by MARK 
 
 DE BURGH'S Maritime International Law. . 27 
 
 DE LA RIVE'S Treatise on Electricity 
 
 DE MORGAN on Matter and Spirit 
 
 DE TOCQDEVILLE'S Democracy in America. . 
 DISRAELI'S Speeches on Parliamentary Re- 
 form . 
 
 DOBSON ontheOx 27 
 
 DovEonStorms 10 
 
 DYER'S City of Rome 2 
 
30 
 
 NEW WORKS PUBLISHED BY LONGMANS AKD CO. 
 
 E ASTLAKE'S Hints on Household Taste .... 17 
 
 EDWARDS' Shipmaster's Guide 27 
 
 Elements of Botany 13 
 
 ELLICOTT'S Commentary on Ephesians .... 19 
 
 Lectures on Life of Christ 19 
 
 Commentary on Galatians 19 
 
 -Pastoral Epist... 19 
 
 -Philippians, &c. . 19 
 
 -Thessalonian a... 19 
 
 ENGBL'S Introduction to National Music .. 
 
 Essays and Reviews 
 
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