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 Properties of matter. 
 
 3 1924 002 949 141 
 
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 tine Cornell University Library. 
 
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 the United States on the use of the text. 
 
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PK0PEETIE8 OF MATTEK 
 
 BY 
 
 P. a. TAIT, M.A., Sec. E.S.E. 
 
 FORMERLY FELLOW OF ST. PETEE'S COLLEGE, CAMBRIDGE ; 
 
 PROFEaaOR OF NATURAL PHILOBOPHT IN THE 
 
 UNIVERSITY OF EDINBURGH 
 
 EDINBUKGH 
 
 ADAM AND CHAELES BLACK 
 
 1885 
 
PREFACE. 
 
 The subject of this elementary work still forms — in 
 accordance with tradition from the days of Eobison, 
 Playfair, Leslie, and Forbes^ the introduction to 
 the course of Natural Philosophy in Ediiiburgh 
 University. 
 
 The work is (with the exception of a few isolated 
 sections) intended for the average student; who is 
 supposed to have a sound knowledge of ordinary 
 Geometry, and a moderate acquaintance with the 
 elements of Algebra and of Trigonometry. 
 
 But he is also supposed to have — what he can 
 easily obtain from the simpler parts of the two first 
 chapters of Thomson and Tait's Elements of Natural 
 Philosophy, or from Clerk-MaxweU's excellent little 
 treatise on Matter and Motion — a general acquaint- 
 ance with the fundamental principles of Kinematics 
 of a Point and of Kinetics of a Particle. To have 
 treated these subjects at greater length than 
 has here been attempted would have rendered it 
 imperative to omit much of the development of 
 
vi PEEFACE. 
 
 important parts of preliminary Physics, of which, so 
 far as I know, there is no modern British text-book. 
 The work was peremptorily limited to a small 
 volume ; so that the parts of these auxiliary subjects 
 which have been Admitted are. mainly of two kinds : 
 — those which are really introdv£tory to the books 
 just mentioned, because treating of matters usually 
 deemed too simple for special notice; and a few 
 which are in a sense mipplementary, because giving 
 valuable results not usually included in elementary 
 books. 
 
 It is my present intention to complete my series 
 of text-books by similar volumes on Dyrmmics, Sound, 
 and Electricity. Should I succeed in bringing out 
 such works, I shall thenceforth be enabled to in- 
 troduce references to one or other, instead of the 
 digressions which are absolutely necessary in every 
 self-contained elementary treatise devoted to one 
 special branch of Physics only. 
 
 P. G. TAIT. 
 
 OOLLEGE, EdINBUROH, 
 
 Mmch 5, 1885. 
 
CONTENTS. 
 
 CHAPTER I. 
 
 PAGE 
 
 Inteoductort ..... 1 
 
 CHAPTER II. 
 
 Some Htpoihesbs as to the Ultimate Steuctdre 
 
 OF Mattbe ..... 16 
 
 CHAPTER III. 
 
 Examples of Terms in Common Use as applied 
 
 TO Matter ..... 23 
 
 CHAPTER IV. 
 Time and Space .... 46 
 
 CHAPTER V. 
 Impenetrability, Pobositt, Divisibility . . 80 
 
 CHAPTER VI. 
 Inertia, Mobility, Centrifugal Foece . . 91 
 
 CHAPTER VII. 
 Gravitation . . . . .110 
 
 CHAPTER VIII. 
 Preliminary to Deformability and Elasticity . 142 
 
viii CONTENTS. 
 
 CHAPTER IX. • 
 
 FA.aE 
 
 Compressibility of Gases and Vapoues . . ,156 
 
 CHAPTER X. 
 Compression op Liquids . . . .180 
 
 CHAPTER XI. 
 Compressibility and Rigidity op Solids . . 192 
 
 CHAPTER XII. 
 Cohesion and Capillarity . . .225 
 
 CHAPTER XIII. 
 DippusioN, Osmose, Transpiration, Viscosity, etc. 254 
 
 CHAPTER XIV. 
 
 AGGEEaATION OP PARTICLES . . .274 
 
 APPENDIX. 
 
 I. Hypotheses as to the Constitution op 
 
 Matter. By Propessor Flint . . 287 
 
 II. Extracts prom Clerk -Maxwell's Article 
 
 "Atom" ..... 291 
 
 III. ViTRunus on Archimedes' Experiment . 307 
 
 IV. Singular Passage op the " Principia " . 307 ■ 
 
 INDEX ...... 317 
 
CHAPTEE I. 
 
 INTRODUCTORY. 
 
 1. We start with certain assumptions or Axioms, which 
 are not of an a priori character, but which the observa- 
 tions and experiments of many generations have forced 
 upon us •.^- 
 
 (1) That the physical universe has an objective ex- 
 
 istence. 
 
 (2) That we become cognisant of it solely by the aid 
 
 of our Senses. 
 
 (3) That the indications of the Senses are always 
 
 imperfect,, and often misleading ; but 
 
 (4) That the patient exercise of Eeason enable.? us to 
 
 control these indications, and gradually, but 
 surely, to sift truth from falsehood. 
 2. If, for a moment, we use the word Thing to 
 denote, generally, whatever we are constrained to allow 
 has objective existence — i.e. to exist altogether inde- 
 pendently of our senses and of our reason — we arrive at 
 the following conclusions : — 
 
2 PEOPERTIES OF MATTER. 
 
 A. In the physical universe there are but two classes 
 of things, Matter and Energy. 
 
 B. Time and Space, though well known to all (in 
 Newton's words, omnibm notissma), are not things. ^ 
 
 C. Number, Magnitude, Position, Velocity, etc., 
 are likewise not things. 
 
 D. Consciousness, Volition, etc., are not physical. 
 
 3. So says modern physical science, and to its gener- 
 ally received statements we cannot but adhere. Meta- 
 physicians, of course, who trust entirely to so -called 
 "light of nature," have their own views on this, as on 
 all other subjects ; but the number and variety of these 
 views, some of which are entirely incompatible with 
 others, form a striking contrast to the general consensus 
 of opinion on the part of those who have at least tried to 
 deserve to know. In the words of v. Helmholtz,^ one of the 
 chief living authorities, in science properly so-called. :--- 
 
 The genuine metaphysician, in view of a presumed 
 necessity of thought, looks down with an air of superiority 
 
 ' "Space is . . . regarded as a condition of tlie possibility of 
 phenomena, not as a determination produced by them ; it is a 
 representation d, priori which necessarily precedes all external 
 phenomena : " — 
 
 " Time is not an empirical concept deduced from any experience, 
 for neither co-existence nor succession would enter into oui; per- 
 ception, if the representation of time were not given d, priori.' — 
 Kant, CrUigm of Pure Reason ; Max MiiUer's Translation. 
 
 ^ "Hierhabeu wir denachteuMetaphysiker. Einer augeblichen 
 Denknothwendigkeit gegeniiber bliokt er hochmuthig auf die, 
 welche sich um Erforschung der Thatsachen bemiihen, herab. 1st 
 es sohon vergessen, wie viel TJnheil dieses Verfahren in den 
 fruheren Entwicklungsperioden der' Naturwissenschaften ange- 
 richtet hat ? " — Preface to the German Translation of the second 
 part of Thomson and TaiCs Natural Philosophy. 
 
INTRODUCTOKY. 3 
 
 on those who labour to investigate the facts. Has it 
 already been forgotten how much mischief this procedure 
 wrought in the earlier stages of the development of the 
 sciences 1 
 
 4. A stone, a piece of lead or brass, water, air, the 
 ether or luminiferous medium, etc., are portions of 
 matter; wound-up springs, water-power, wind, waves, 
 compressed air, heat, electric currents, as well as the 
 objective phenomena corresponding to our sensations of 
 sound and light, are examples of energy associated 
 with matter. 
 
 5. All trustworthy experiments, without exception, have 
 been found to lead to the conviction that matter is unalter- 
 able in quantity by any process at the command of man. 
 
 This is one of the strongest arguments in favour of 
 the objective existence of matter. It was used, at the 
 very end of last century, by Kumford in his memor- 
 able Inquiry concerning the Source of the Heat excited hy 
 Friction} 
 
 It forms also the indispensable foundation of modem 
 chemistry, whose main instrument is the balance, used 
 to determine quantity of matter with great exactness. 
 
 We may speak of this property, for the sake of future 
 reference, as the Conservation of Matter. It justifies one- 
 half of the statement in § 2, A. 
 
 6. So far the reader (if he resemble at all the average 
 student of our acquaintance) is not likely to feel much 
 difficulty. His every -day experience must have long 
 ago impressed on him the conviction of the objectivity 
 of matter, though perhaps he may not have learned to 
 express it in such a form of words. 
 
 1 Phil. Trans., 1798. 
 
< PROPERTIES OF MATTER. 
 
 But it is usually otherwise when he is told that 
 energy has an objective existence quite as certainly as 
 has matter. He has been accustomed to the working 
 of watermills, let us say, and he cannot but allow that 
 a "head" of water is something other than the water; 
 it is something associated with the water in virtue of its 
 elevation. He sees and (if he be of an economic turn) '. 
 he deplores the terrible waste of water-power which is 
 stupidly permitted to go on all over the world. He 
 allows that water-power does exist, but the waste which 
 he laments he looks upon as its omiiMlation. Till within 
 the last forty years or so the vast majority even of 
 scientific men held precisely the same opinion. 
 
 7. The modem doctrine of the Conservation of Energy, 
 securely based upon the splendid investigations of Joule 
 and others, completes the justification of our preliminary 
 statement. Energy, like matter, has been experimentally 
 proved to be indestructible and uncreatable by man. 
 It exists, therefore, altogether independently of human 
 senses and human reason, though it is known to man 
 solely by their aid. 
 
 The objectivity of energy is virtually admitted in a 
 curious way, by its being advertised for sale. Thus in 
 manufacturing centres, where a mill-owner has a steam- 
 engine too powerful for his requirements, he issues a 
 notice to the effect, "Spare Power to let." But,- of 
 course, the common phrase, " price of labour," at once 
 acknowledges the objectivity of work. 
 
 8. There is, however, a most important point to be 
 noticed. Energy is never found except in association 
 with matter. Hence we might define matter as the Vehicle 
 or Receptacle of Energy ; and it is already more than prob-/ 
 
INTRODUCTORY. 5 
 
 able that energy will ultimately be found) in all its! 
 varied forms, to depend upon motion of matter. 
 
 This is- advanced, for the moment, as a mere intro- 
 ductory statement, instances of which will be discussed 
 even in the present work; but its complete treatment 
 would require the introduction of branches of physics 
 with which we have here nothing to do. One great 
 argument in its favour is, that matter is found to consist 
 of parts which preserve their identity, while energy is 
 manifested to us only in the act of transformation, and 
 (though measurable) cannot be identified. 
 
 9. Besides their common characteristic, conservation, 
 and in strange contrast to it, we have their characteristic 
 difference. Matter is simply passive (inert is the , scien- 
 tific word) ; energy is perpetually undergoing transforma- 
 tion. The one is, as it were, the body of the physical 
 universe ; the other its life and activity. All terrestrial 
 phenomena, from winds and waves to lightning and 
 thunder, eruptions and earthquakes, are transformations 
 of energy. So are alike the brief flash of a falling star, 
 and the fiery glow from th6 mighty solar outbursts of 
 incandescent hydrogen. 
 
 10. From the strictly scientific point of view, the greater 
 part of the present work would be said to deal with 
 energy rather than with matter. Thus, when we speak 
 of weight as a property of matter, in the sense that a stone 
 of itself has weight, or even in the sense that the earth 
 attracts the stone, we go directly in the teeth of Newton's 
 distinct assertion, which will be quoted in its proper 
 place. 
 
 For such a statement (because confined to the attract- 
 ing bodies alone) implies the existence of Action at a 
 
6 PROPERTIES OF MATTER. 
 
 DistoMX, a very old but most pernicious heresy, of which 
 much more than traces still exist among certain schools, 
 even of physicists. 
 
 Gravitation, like all other mutual actions between 
 particles of matter, such as give rise to cohesion, resist- 
 ance to compression, elasticity, etc., must, so far as our 
 present kuowledge extends, be set down to the energy 
 which particles of matter are found to possess when 
 separated. The intervening mechanism by which this 
 is to be accounted for has, as yet, only been guessed at, 
 and none of the guesses have been successful. Clerk- 
 Maxwell's success in explaining electric and magnetic 
 attractions by stresses and rotations in the luminiferous 
 ether shows, however, that we need not despair of being 
 able to explain the tdtimate mechanism of gravitation. 
 
 But there is great convenience in separating, as far 
 as possible, the treatment of Mass, Weight, Cohesion, 
 Elasticity, etc., which we range under the general title. 
 Properties of Matter, from that of Heat, Light, Electric 
 Energy, etc., which can all in great measure be studied 
 without express reference to any one special kind of 
 matter — though, of course, as forms of energy, they 
 exist only (§ 8 above) in association with matter. Along 
 with these forms of energy must of course be treated 
 the allied properties of matter, such as specific heat, 
 refractive index, conductivity, etc. Such, therefore, are 
 foreign to the present work. And, even in popular 
 language, we invariably speak of the hardness of a body, 
 its rigidity, its elasticity, as belonging to it in much the 
 same sense as does its density or its atomic weight — 
 though in a very different sense than does its tempera- 
 ture or its electric potential. 
 
INTBODUCTORY. 7 
 
 It is, therefore, on the two grounds of custom and 
 
 convenience that we use the term Properties of Matter 
 
 ' as the title of this work. The error involved is not by 
 
 any means so monstrous as that which all agree to 
 
 perpetuate by the use of the term Cenfrifugcd Force. 
 
 11. The word Force must often, were it only for 
 brevity's sake, be used in the present work. As it does 
 not denote either matter or energy, it is not a term for 
 anything objective (§ 2, A). The idea it is meant to 
 express is suggested tons by the "muscular sense," just 
 as the ideas of brightness, noise, smell, or pain, are sug- 
 gested by other senses :t— though they do not correspond 
 directly to anything which exists outside us. 
 
 It is exceedingly difficult to realise the fact that 
 noise is a mere subjective impression, even when reason 
 has convinced us that outside the drum of the ear there 
 is nothing to correspond to it except a periodic com- 
 pression and dilatation of the air. 
 
 We need not, therefore, be surprised at the tenacity 
 with which almost all, even of scientific men, stUl chng 
 to the notion of force as something objective. 
 
 But if it were objective, what an absolutely astound- 
 ing fact would have to be faced by one who tries to 
 explain the nature of hydrostatic pressure; and who 
 finds that by the touch of a finger on a little piston he 
 can produce a pressure of a pound weight on every square 
 inch of the surface of a vessel, however large, if full of 
 water, and the same amount on every square inch of 
 every object immersed in it, even if that object consisted 
 of hundreds of square miles of sheets of tinfoil far enough 
 apart to let the water penetrate. between them. 
 
 When we communicate energy to a body originally at 
 
8 PROPERTIES OF MATTER. 
 
 rest, as in pushing or drawing a carriage, the impression 
 on our muscular sense does not correspond to the energy 
 communicated per second, but to the energy communi- 
 cated per inch of the motion. For experiment has proved 
 that what appears to our muscular sense as a definite 
 tension (in a cord, let us say) is associated with the com- 
 munication of energy, to any mass of matter whatever, in 
 direct proportion to the space through which it is exerted, 
 altogether independently of the speed with which the 
 mass may be already moving in the direction of the 
 tension ; so that in equal times energy is communicated 
 in direct proportion to that speed. When there is no 
 motion, no energy is communicated; and this would 
 certainly not be the case if communication of energy 
 corresponded to the time during which the tension was 
 said to act. 
 
 12. The muscular sense is far more deceptive than 
 any other, except, perhaps, that of touch. Conjurors, 
 ventriloquists, perfumers, and cooks make their liveli- 
 hood by the imperfections of our senses of sight, hear- 
 ing, smell, and taste respectively ; but he who has tried 
 the simple experiment of rolling a pea on the table 
 between his first and second fingers, after crossing one 
 over the other, will at once assign to the sense of touch 
 a place of honour in the list of these deceivers. And 
 the muscular sense well deserves a place beside it. 
 
 13. Many of the terms which are now used in a 
 strictly scientific sense had a humbler origin, having 
 been devised entirely for the popular expression of 
 common ideas. The term Work is a specially illustrative 
 one. Thus, in a draw-well, the work done in bringing 
 water to the surface would be reckoned at first in terms 
 
INTRODUCTORY. 9 
 
 of the quantity of water raiJsed ; — two raisings of a full 
 bucket lifting twice as much water as one. But then it 
 was found that, for the same quantity of water raised, 
 the work depended on the depth of the well : — doubled 
 depth corresponding to doubled work. But if the bucket 
 were filled with sand instead of water, more work was- 
 required, in proportion as sand is heavier than water. 
 All these statements were soon found to be compre- 
 hended in the simple form : — the work done is directly 
 proportional to the weight raised and also to the height 
 through which it is raised. Here the indications of the 
 muscular sense stepped in, and work came to have a 
 general meaning, viz. the product of the so-called 
 force exerted, into the distance through which it is 
 exerted. 
 
 Had they not possessed the muscular sense, men 
 might perhaps have been longer, than they have been in 
 recognising the important thing poteniial energy; but 
 when they had come to recognise it they would have 
 stated that when water is raised it gains potential energy 
 in proportion as it is raised, and perhaps they might 
 have found it convenient to use a single term for the 
 rate at which such energy is gained per foot of ascent. 
 This would probably not have been called " Force," but 
 it would have expressed precisely what the word force 
 now expresses. 
 
 Then they would have recognised that when energy 
 of any kind is transmitted by a driving-belt, the amount 
 transmitted is (ceteris pa/ribus) directly proportional to the 
 space through which the belt has run. They might 
 have invented a name for the rate of transmission per 
 foot-run of the belt ; they might even have called it the 
 
10 PROPERTIES OF MATTER. 
 
 tension of the belt ; but, anyhow, it would be precisely 
 what is now called force. 
 
 Let us look at the matter from another point of 
 view. 
 
 14. A stone, if let fall, gradually gains Unetic energy, 
 or energy of motion, and experiment shows that the 
 energy gained is directly proportional to the vertical 
 space fallen through. Hence we have come to say that 
 the stone is acted upon by a force (its weight, as we call 
 it) whose amount is practically the same at all moderate, 
 distances from the earth's surface. 
 
 Butj so far as we know the question scientifically, we 
 can only say that the stone has potential energy (just as 
 water in a mill-pond has head) in proportion to its eleva- 
 tion above the earth's surface ; and consequently, by the 
 conservation of energy, it ' acquires energy of motion in 
 proportion to the space through which it descends. 
 Why it has potential energy when it is raised, and why 
 that potential energy takes the first opportunity of 
 transforming itself into kinetic energy : — thus requiring • 
 that the stone shall fall unless it be supported-: — are 
 questions to be approached later. 
 
 15. That the statement above is complete, without 
 the introduction of the notion of force, is seen from the 
 fact that a knowledge of the kinetic eQefgy acquired, 
 after a given amount of descent, enables us to determine 
 fully the nature of the resulting motion even when the 
 stone is projeded, obliquely or vertically, not merely 
 allowed to fall. The question is reduced to one of 
 mathematics, or rather of kmematics, and as such the 
 noU'-mathematical student must, for the present, simply 
 accept the above statement as true. 
 
INTRODUCTORY. 11 
 
 And thus we have another of the many distinct and 
 independent proofs that Force is a mere phantom sugges- 
 tion of our muscular sense ; , though there can be no 
 doubt that, in the present stage of development of 
 science, the use of the term enables us greatly to con- 
 dense our descriptions. 
 
 But it is a matter for serious consideration whether 
 we do not connive at a species of mystification by. thus 
 employing a term for a mere sensation, corresponding 
 to nothing objective, even if it be employed solely to 
 shorten our statements or our demonstrations. 
 
 Everyone knows that matter («.gr. com, gold, diamonds) 
 has its price ; so (as we saw in § 7) has energy. We are 
 not aware of any case in which force has been ofi'ered for 
 sale. To "have its price" is not conclusive of objec- 
 tivity, for we know that Titles, Family Secrets, and 
 even Degrees, are occasionally sold ; but "not to have 
 its price " is conclusive against objectivity. 
 
 16. These introductory remarks have been brought 
 in with the view of warning the reader that we are 
 dealing with a subject so imperfectly known that at 
 almost any part of it we may pass, by a single step sis it 
 were, from what is acquired certainty to what is still 
 subject for mere conjecture. 
 
 An exact or adequate conception of matter itself, 
 could we obtain it, would almost certainly be something 
 extremely unlike any conception of it which our senses 
 and our reason will ever enable us to form. Our object, 
 therefore, in what follows, is mainly to state experi- 
 mental facts, and to draw from them such conclusions 
 as seem to be least unwarrantable. 
 
 17. But, for the classification of the properties of 
 
12 PROPERTIES OF MATTER. 
 
 matter, whether our classification be a good one or 
 not, it is necessary that we should have a definition of 
 matter. 
 
 From what was said in last section it is obvious that 
 no definition we can give is likely to be adequate. All 
 that we can attempt, then, is to select a definition which 
 (while not obviously erroneous) shall serve as at least a 
 temporary basis for the classification we adopt. 
 
 18. Numberless definitions of matter have been pro- 
 posed.^ Here are a few of the more important : — 
 (a) That which possesses Inertia (§ 9). 
 (/3) The Beceptade or Vehicle of Energy (§ 8). 
 (7) Whatever exerts, or can be acted on by, Force. 
 (S) Whatever can be perceived by our senses, especi- 
 ally the sense of Touch. This is closely akin 
 to the well-known definition of matter as a 
 Permanent Possibility of Sensation. 
 (e) Whatever can occupy space. 
 (^ Whatever, in virtue of its motion, possesses 
 
 Energy. 
 (r)) Whatever, to set it in motion, requires the ex- 
 penditure of Work. 
 (ff) [ToniceUi, LezioniAccademiche, 1715, 'p. 25.] La 
 materia altro non h, che un vaso di Circe incan- 
 tato, il quale serve per ricettacolo della forza, e 
 de' momenti dell' impeto. La forza poi, e gl' 
 impeti, sono astratti tanto sottili, son quintes- 
 senze tanto spiritose, che in altre ampoUe non 
 si posson racchiudere, fuor che nell' intima cor- 
 pulenza de' solidi naturali. 
 
 ^ A remarkable collection of such (now historical) speculations, 
 due to Frofesspr Flint, is given in Appendia: I. 
 
INTRODUCTORY. 13 
 
 (t) [The Vortex Hypothesis of W. Thomsoa] The 
 rotating parts of an inert perfect fluid ; which 
 fills all space, but which is, when not rotating, 
 absolutely unperceived by our senses. 
 
 19. The mutual incompatibility of certain pairs of 
 these definitions shows that some of them, at least, must 
 be of the so-called metaphysical species (§ 3). 
 
 (a), (/3), (f), (j?), above, have much in common, and, 
 with further knowledge, may perhaps be found to differ 
 in expression merely. At present, from want of in- 
 formation, we cannot be certain that any two of them 
 are precisely equivalent. 
 
 Berkeley virtually asserted that all motion is produced 
 by the direct action of spirits on matter. Even then, 
 the statement (/3) that matter is the receptacle or vehicle 
 of energy holds good (but how then does energy exist in 
 the spirit 1). 
 
 But the statement that matter is whatever can exert 
 force (7) is to be rejected ; though it was virtually intro- 
 duced by Cotes in his Preface to the Prmdpia. 
 
 (S) must be rejected, if only because there is another 
 thing besides matter (in the physical universe) which we 
 know of, and of course only through our senses (§ 1). 
 But this is not all the error ; for we get the notion of force 
 through our muscular sense (§11), and force is not matter, 
 not even a thiTig. 
 
 Torricelli's language is poetical, and therefore his 
 statement (0) must not be taken too literally. In his 
 time, as in all subsequent time till within the last 
 quarter of a century, energy and force were very rarely 
 distinguished from one another. Even now they are too 
 often confounded. 
 
14 PROPERTIES OF MATTER. 
 
 (i), the most recent of these speculations, has the 
 curious peculiarity of making matter, as we can perceive 
 it, depfend upon the existence of a particular kind of 
 motion of a medium which, under many of the defini- 
 tions above, would be entitled to claim the name of 
 matter, even when it is not set in rotation. 
 
 20. But as we do not know, and are probably incapable 
 of discovering, what matter is, what we want at present 
 is merely a definition which, while not at least ebnously 
 incorrect, shall for the time serve as a working hypothesis. 
 
 We therefore choose (e) above, i.e. we define, for the 
 moment, as follows : 
 
 Matter is whatever can ocmgy space. 
 
 Experience has proved that it is from this side that 
 the average student can most easily approach the subject, 
 i.e. here, as it were, the contour lines of the ascent (§ 80) 
 are most widely separated. 
 
 21. But this definition involves three distinct pro- 
 perties : — (1) the Volume, (2) the Form or Figure, of 
 the space occupied; and (3) the nature or quality of the 
 Occupation. 
 
 Hence the older classical works almost invariably 
 speak of matter as possessing— (1) Extension, (2) Form, 
 and (3) Impenetrability. It is mainly for the sake of the 
 first of these, and the preliminary discussions which it 
 necessarily introduces, that we have chosen the above 
 definition as our starting-point. 
 
 .22. Before we take these up in detail, however, it may 
 be useful to devote a short chapter to a digression on 
 some of the more notable of the hypotheses which have 
 been propounded as to the ultimate structure of matter. 
 We advisedly use the word struciv/re instead of nature, for 
 
INTRODUCTORY. 15 
 
 it must be repeated, till it is fully accepted, that the dis- 
 covery of the ultimate nature of matter is probably 
 beyond the range of human intelligence. 
 
 Another chapter, of a very miscellaneous character, 
 will follow, devoted to the examination of some of the 
 terms popularly applied to pieces of matter, and a rapid 
 glance at the physical truths which underlie them. This 
 is introduced to give the reader, at the very outset of 
 his work, a general idea of its nature and extent. 
 
CHAPTEE II. 
 
 SOME HYPOTHESES AS TO THE ULTIMATE STRUCTURE 
 OF MATTER. 
 
 23. The hard Atom, glorified in the grand poem of 
 Lucretius, but originally conceived of, some 2400 years 
 ago, by the Greek philosophers Democritus andLeucippus, 
 survives (as at least an unrefuted, though a very improb- 
 able, hypothesis) to this day. Newton made use of 
 the hypothesis of finite, hard, atoms to explain why the 
 speed of sound in air was found to be considerably 
 greater than that given by his calculations ; which were 
 accurate in themselves, but founded on erroneous or, 
 rather, incomplete data. But in this problem Laplace 
 found the vera causa, and in consequence Newton's 
 apparent support of the hypothesis of hard atoms is no 
 longer available. 
 
 Many of the postulates of this theory are with diffi- 
 culty reconciled with our present knowledge; some 
 have been contemptuously dismissed as " inconceivable." 
 But any one who argues on these lines becomes, ipso 
 facto, one of the so-called metaphysicians. 
 
 Let us briefly consider the main statements of this 
 theory, but without regard to the order in which 
 Lucretius gives them. 
 
ULTIMATE STRUCTURE OF MATTER. 17 
 
 24. Nature works by invisible things ; thus paving- 
 stones and ploughshares a^e gradually worn down 
 without the loss of any visible particles. 
 
 Reproduction [i.e. agglomeration of scattered particles 
 so ae to produce visible bodies] is slower than decay 
 [i.e. the breaking up of bodies into invisible particles], 
 and therefore there must be a limit to breakage, else the 
 breaking of infinite past ages would have prevented any 
 reproduction within finite time. Hence there exists a 
 least in things [i.e. unbreakable parts or Atoms, " strong 
 in solid singleness "]. 
 
 But there is also void in things, else they would be 
 jammed together, and unable to move. Here Lucretius 
 takes the case of a fish moving in water, showing that 
 void is necessary in order that it may be able to move. 
 [Our modern knowledge of circulation, i.e. the motion of 
 fluids in re-entrant paths, shows that this reasoning is 
 baseless.] 
 
 There can be no third thing besides body and void. 
 For nothing biit body can touch and be touched ; and 
 what cannot be touched is void. [Here we have the 
 germ of the erroneous definition of matter (S) in § 18 
 above.] 
 
 The atoms are infinite in number, and the void in 
 which they move [space] is unlimited. 
 
 They have dilferent shapes; but the number of 
 shapes is finite, and there is an infinite number of 
 atoms of each shape. ■ 
 
 Nothing whose nature is apparent to sense consists 
 of one kind of atoms only. 
 
 The atoms move through void at a greater speed 
 than does sunlight. 
 
 
 
18 PROPERTIES OF MATTER. 
 
 Besides this, there is a great deal of curious specula- 
 tion as to how a vertical downpour of atoms [supposed 
 to be a result of their weight] is, in some arbitrary way, 
 made consistent with their meeting one another and 
 agglomerating into visible masses of matter. 
 
 The basis of the whole of Lucretius' reasoning in 
 favour of the existence of atoms lies in the gratuitous 
 assumption that reproduction is slower than decay. 
 This is by no means consistent with our modern know- 
 ledge, for potential energy of different masses [whether 
 gravitational or chemical] is constantly tending to the 
 agglomeration of parts, and on a far grander scale than 
 that in which any known cause tends to decay or 
 breaking up. 
 
 But if there be hard atoms, they must (in all known 
 bodies) have intervals between them; for compressi- 
 bility : — i.e. capability of having the component atoms 
 brought more closely together : — is a characteristic of all 
 known bodies. [Contrast this mode of arriving at the 
 conclusion that " there must be void in things," with the 
 erroneous mode employed by Lucretius.] 
 
 25. A refinement of this theory, mainly due to 
 Boscovich, gets rid of the material atom altogether, 
 substituting for it a mere mathematical point, towards 
 or from which certain forces tend. It is justified by the 
 assertion that we know matter only by the effects which 
 it produces (or seems jto produce), and therefore that, if 
 these effects can be otherwise explained, we need not 
 assume the existence of substance or body at all. This 
 theory was, at least in part, accepted by so great an 
 experimenter and reasoner as Faraday. But the fatal 
 objection to which it is exposed is that .it does not seem 
 
ULTIMATE STRUCTURE OF MATTER. 19 
 
 capable of explaining inertia, which is certainly a distinct- 
 ive (perhaps the most distinctive) property of matter. 
 
 This theory must be regarded as a mere mathe- 
 matical fiction, very similar to that which (in the 
 hands of Poisson and Gauss) contributed so much to 
 the theory of statical Electricity ; though, of course, it 
 could in no way aid inquiry as to what electricity is. 
 
 26. A inuch more plausible theory is that matter is 
 continuous {i.e. not made up of particles situated at a 
 distance from one another) and compressible, but in- 
 tensely heterogeneous ; hke a plum-pudding, for instance, 
 or a mass of brick-work. The finite heterogeneousness 
 of the most homogeneous bodies, such as water, mercury, 
 or lead, is proved by many quite independent trains of 
 argument based on experimental facts. If such a con- 
 stitution of matter be assumed, it has been shown ^ that 
 gravitation alone would sufiice to explain at least the 
 greater part of the phenomena which (for want of know- 
 ledge) we at present ascribe to the so-called Molecular 
 Forces. 
 
 27. The most recent attempt at a theory of the 
 structure of matter, the hypothesis of Vortex Atoms, is of 
 a perfectly unique, self-contained character. Its postu- 
 lates are few and simple, but the working out of anything 
 beyond their immediate consequences is a task to tax to 
 the utmost the powers of the greatest mathematicians 
 for generations to come. A vortex filament, in a perfect 
 fluid, is a true " atom ;'' but it is not hard like those of 
 Lucretius ; it cannot be cut, but that is because it 
 necessarily wriggles away from the knife. 
 
 The idea that motion is, in some sort, the basis of 
 ' W. Thomson, Proc. B.S.E., 1862. 
 
20 PROPERTIES OF MATTER. 
 
 what we call matter is an old one; but no distinct 
 conceptions on the subject were possible until v. Helm- 
 holtz, a quarter of a century ago, made a grand contribu- 
 tion to hydrokinetics in the shape of his theory of vortex 
 motion.'' He proved, among other entirely novel pro- 
 positions, that the rotating portions of a perfect incom- 
 pressible fluid, in which there is no finite slipping, 
 mamtain their identity: — being thus for ever definitely 
 differentiated from the non- rotating parts. He also 
 showed that these rotating portions are necessarily 
 arranged in continuous, endless filaments: — forming 
 closed curves, which may be knotted or linked in any 
 way : — unless they extend to the bounding surface of the 
 fluid, in which alone they can have ends. Thus, to give 
 ends to a closed vortex filament {i.e. to cut it), we must 
 separate the fluid mass itself, of which it is a portion : — 
 so that on Thomson's theory we must (virtually) sever 
 space itself. 
 
 Such vortex filaments (though necessarily of an im- 
 perfect character) are produced when air is forced to 
 escape from a box, through a circular hole in one side, 
 by sharply pushing in the opposite side. If the air in 
 the box be filled with smoke, or with sal-ammoniac 
 crjrstals, the escaping vortex ring is visible to the eye; 
 and the collisions of two vortex rings, which rebound 
 from one another, and vibrate in consequence of the 
 shock, as if they had been made of solid india/-rubber, 
 are easily exhibited. Experimental results of this kind 
 led Sir W. Thomson^ to propound the theory that matter, . 
 such as we perceive it, is merely the rotating parts of a 
 
 ' Crelle, 1858. Translated in FMl. Mag., 1867. 
 » Proe. B.8.E., 1867. 
 
ULTIMATE STRUCTUEE OP MATTER. 21 
 
 fluid which fills all space. This fluid, whatever it be, must 
 have inertia : — that is one of the indispensable postulates 
 of V. Helmholtz's investigation ; and the great primary 
 objection to Thomson's theory is, that it explains matter 
 only by the help of something else which, though it is 
 not what we call matter, must possess what we consider 
 to be one of the most distinctive properties of matter. 
 
 28. But this theory is stUl in its infancy, and we 
 cannot as yet teU whether it will pass with credit the 
 severe ordeal which lies before it, when the properties 
 of vortices (which must be discovered by mathematical 
 investigation) shall be compared, one by one, with the 
 experimentally ascertained properties of matter. As we 
 have already said, this theory is self-contained ; no new 
 hypotheses can be introduced into it; so that it possesses, 
 as it were, no adaptability, or capability of being modi- 
 fied, but must fall before the very first demonstrated 
 insufficiency, or contradiction, if such should ever be 
 discovered. 
 
 29. But the really extraordinary fact, already known 
 in this part of our subject, is the apparently perfect 
 similarity and equality of any two particles of the same 
 kind of gas, probably of each individual species of 
 matter when it is reduced to the state of vapour. Of 
 such parts, therefore, whether they be further divisible 
 or not, each species of soHd or liquid must be looked 
 on as built up. This similarity of parts, very. small 
 indeed but still of essentially finite magnitude, has 
 been so well treated by Clerk-Maxwell that, instead of 
 insisting upon it here, we give a considerable extract 
 from one of his remarkable articles in Appendix II. 
 below. 
 
22 PROPERTIES OF MATTER. 
 
 30. The further treatment of the subject of structure, 
 involving the question of how the component parts (be 
 they atoms or not) of bodies are put together, must be 
 deferred foi: a little. What has been said above must 
 be looked on as a mere preliminary sketch, not intended 
 even to be fully understood until the experimental data, 
 on which all our reasoning must be based, are brought 
 before the reader as completely as our limits permit. 
 
CHAPTER III. 
 
 EXAMPLES OF TEEMS IN COMMON USE AS APPLIED TO 
 MATTER. 
 
 31. Before we proceed to a more rigorous treatment 
 of our subject, it may be well to consider what physical 
 truth underlies each of some of the many adjectives in 
 common use as applied to portions of matter, such as 
 Massive, Heavy, Plastic, Ductile, Viscous, Elastic, Rigid, 
 0]pague, Blue, Coherent, etc. 
 
 This course secures a twofold gain, so far as the 
 beginner is concerned, for, first, he is thus introduced, in 
 a familiar way, to some of the more important terms 
 which are indispensable in scientiiic description; and 
 second, he obtains a glance here and there through the 
 whole subject of Natural Philosophy, because the pro- 
 gramme before us is so vague as to leave room for 
 innumerable digressions, each introducing some novel 
 but important fact or property. But we must endeavour 
 to be brief, for whole volumes would have to be written 
 before this subject could be nearly exhausted. 
 
 32. Every one who has used his senses to any purpose 
 knows, before he comes to the study of our science, a 
 great many of its phenomena, among them some of the 
 yet unexplained. But he knows, as it were, each by 
 
24 PKOPEETIES OF MATTER. 
 
 itself, and only in its more prominent features; the 
 analysis of the appearances or impressions- which he has 
 seen and experienced,' and the explanation of the physi- 
 cal fact or process which underlies each of them, are 
 absolutely necessary before he can understand the mode 
 in which they must be grouped, and the reasons for such 
 grouping. 
 
 33. Thus he knows that the moon keeps company 
 with the earth, never receding nor approaching by more 
 than a small fraction of the average distance. He. also 
 knows that the earth keeps, within narrow limits, at a 
 definite distance from the sun. He has a general notion, 
 at least, that the state of matters on the earth would 
 become serious, as regards both animal and vegetable 
 life, if we were to approach to even half our present 
 distance from the sun, or recede to double that distance. 
 But he would require to be a Newton if, without instruc- 
 tion, he could divine that these results are due to the 
 very cause which keeps the bob of a conical pendulum 
 moving in a horizontal circle. 
 
 He sees ripples running along on the surface of a 
 pool, but requires to be told that their motion depends 
 upon the cause which rounds the drops of water on a 
 cabbage-blade, or in a shower, and which renders it 
 almost impossible to keep a water-surface clean. 
 
 He sees what he calls a flash of lightning, but he 
 requires to be told that what he sees is merely particles 
 of air heated so as to be self-luminous. 
 
 He looks at the stars and thinks he sees them as they 
 are, but he requires to be informed that he sees even 
 the nearest of them only as it was three years ago, and 
 that it may have changed entirely in the interval, 
 
TEKMS APPLIED TO MATTER. 25 
 
 And he will certainly require to be informed, even 
 with patient iteration) that air is made up of separate and 
 independent particles : — the number of which in a single 
 cubic inch is expressed by twenty-one places of figures, a 
 multitude altogether beyond human conception : — a busy 
 jostling crowd, each member of which darts about in all 
 directions, impinging on its neighbours some eight thou- 
 sand million times per second. 
 
 But when he has got so far, and has been told that 
 this astounding information is as nothing to what we 
 feel convinced that science can yet reveal, he cannot 
 help marvelling alike at the arcana'of physics, and at the 
 patient efforts of genius which have abeady penetrated 
 so far into the darkness shrouding its mysteries. 
 
 34. Take the terms Massive and Heavy as applied to a 
 piece of matter, or the coirespondiug substantives, the 
 Mass and the Weight of a body. 
 
 The terms are usually regarded as synonymous, but 
 in their origin they are completely distinct. The one is 
 a property of the body itself, and is retained by it with- 
 out increase or diminution wherever in the universe the 
 body may be situated. The other depends for its very 
 existence on the presence of a second body, and diminishes 
 more rapidly than the distance between the two increases. 
 The destructive eifects of a cannon-ball are due entirely 
 to its mass and to the relative speed with which it im- 
 pinges on the target, and would be exactly the same (for 
 the same relative speed) in regions so far from the earth, 
 or other attracting body, that the ball had practically no 
 weight at all. 
 
 When an engine starts a train on a level railway, or 
 when a man projects a curling-stone along smooth ice. 
 
26 PROPERTIES OF MATTER. 
 
 I the resistance which either prime mover has to overcome 
 is due to the mass of the t»ody to be moved. Its weight, 
 except indirectly through friction, has nothing to do 
 with it. So when we open a large iron gate properly 
 hinged, it is the mass with which we have to deal ; if it 
 were lying on the ground and we tried to lift it, we 
 should have to deal simultaneously with its weight and 
 with its mass. 
 
 The exact proportionality of the weights of bodies to 
 their masses, at any one place on the earth's surface, 
 was proved experimmtally by Newton, and is thus no 
 mere truism, but an essential part of the great law of 
 gravitation. 
 
 Thus a pound of matter is a definite amount, or mass, 
 of matter, unchangeable whithersoever that matter may 
 be carried. But the weight of a pound of matter, or a 
 " pound-weight," as it is commonly called, is a variable 
 quantity, depending upon. the position of the body with 
 respect to the earth; and changes (to an easily measurable 
 amount) as we carry the body to different latitudes, even 
 without leaving the earth's surface. 
 
 35. The common use of the balance as a means of 
 measuring out equal' quantities of matter is justified by 
 ' Newton's result ; but the process is essentially an in- 
 I direct one, for the balance tells only of equality of weight. 
 If the earth were hollow at the core, the balance would 
 cease to act in the cavity. Bodies would preserve their 
 masses there, but be deprived of weight. 
 
 To sum up for the present, the mass of a body is 
 estimated by its inertia, and is taken as the measure of 
 the amount of matter in the body ; while the weight is 
 an accidental property, connected with the presence of 
 
TERMS APPLIED TO MATTER. 27 
 
 another mass of matter. But it is a most remarkable 1 
 fact that under the same given external conditions the 
 weight depends upon the quantity only, and not on the/ 
 quality, of the matter in a body. 
 
 If a body, A, becomes heavy in consequence of the 
 presence of another body, B, so in like wise does B become 
 heavy in consequence of A's presence. And the weights 
 x)f the two, each as produced by the attraction of the 
 other, are exactly equal. Hence, if they be free to 
 move, the qumtities of motion (i.e. the momenta) produced 
 in a given time are equal and opposite. [Newton's Lex iii. 
 § 128.] But as the momentum is the product of the mass 
 and the velocity, the parts of the velocities .of the two 
 bodies, due to their mutual gravitation alone, will be in 
 amount inversely as their masses. Thus, though the weight 
 of the whole earth, produced by the attraction of a stone, 
 is exactly equal to that of the stone produced by the 
 attraction of the earth, the consequent rate of fall of the 
 earth towards the stone is less than that of the stone 
 towards the earth in the same proportion as the mass of 
 the stone bears to that of the earth, and is therefore 
 usually so small as to escape observation. The moon, 
 however, is a stone whose mass is not excessively smaller 
 than that of the earth, and the consequences of the earth's / 
 fall towards the moon have to be taken account of in 1 
 astronomy. 
 
 36. To properties such as mass, which depends on 
 the size as well as on the material of a body, and 
 weight which, in addition, depends' on a second body, 
 there correspond what are called sipedfic properties, 
 characteristic of the substance and independent of the 
 dimensions of the particular specimen examined. 
 
28 PROPERTIES OF MATTER. 
 
 I Thus the mass of a cubic foot of any kind of matter 
 
 j may be called its specific mass. But this quantity is 
 
 ! usually expressed by the term Density. 
 
 \ The weight of a cubic foot of each particular kind of 
 
 I matter in any locality may be called the specific weight. 
 
 I But as this varies, though in the same proportion for all 
 
 \ bodies, from place to place, we use instead of it the ratio 
 
 lof the weight of a cubic foot of the substance to that of- 
 
 ia cubic foot of some standard substance. This is called 
 
 the Specific Gravity. Pure water, at the temperature 
 
 called 4° C, is usually taken as the standard substance. 
 
 Newton's experimental result shows that the density 
 
 and the specific gravity are proportional to one another, 
 
 so that if the density of water at 4° 0. be taken as unit- 
 
 I density; a table of specific gravities is identical with a 
 
 ;' table of densities. But we must repeat, the coincidence 
 
 I is an experimental fact, not in any sense a tndsm. 
 
 \ Specific gravity is, in general, much more easily 
 
 i measured with accuracy than is density, so that it is 
 
 usually the property to be directly determined, the other 
 
 \ simply following from it in consequence of Newton's 
 
 j discovery. 
 
 37. To vary the subject widely, let us now consider 
 the term Viscous as appUed to fluids. The contrasted 
 adjective is usually taken as Mobile. 
 
 When a liquid partially fiUs a vessel, and has come 
 to rest, it assumes a horizontal upper surface. If the 
 vessel be tilted, and held for a time in its new position, 
 the liquid will again ultimately settle into a definite 
 position, with its surface again horizontal. Practically it 
 occupies the same bulk in each of these positions. Hence 
 the only change it has suffered is a change oiform. 
 
TEEMS APPLIED TO MATTER, 29 
 
 But this change of form is much more rapidly attainedN 
 in some liquids than in others, even when they are of 
 nearly the same density. Some (such as sulphuric ether) 
 attain their equilibrium position so quickly that they 
 retain energy enough to oscillate about it for some time 
 before coming to rest ; others (such as treacle) assume 
 it only after a long time and, unless in great masses and 
 when violently disturbed, do not oscillate but gradually 
 creep to their final shape. Hence we call treacle viscous. 
 To analyse this result, let us consider (in a very ele- 
 mentary case, for the general analysis of the process 
 requires higher mathematical methods than we can | 
 employ in a work like this) what is involved in Shear : — V 
 i.e. change of form of a body without change of bulk 
 
 38. When water flows, without eddies, slowly in a ! 
 rectangular channel of uniform width, we know, by obser- i 
 vation of particles suspended in it, that the upper parts | 
 flow faster than the lower, and (practically) in such a[ 
 way that a column of the water, originally straight and 
 vertical, inclines, as a whole, forwards more a,nd more in/ 
 the direction of its motion. Hence in a vertical section,^ 
 along the middle of the channel, the particles originally ) 
 forming the line ab in the figure will, after the lapse of j 
 a certain time, be found approximately in the line a'b'.l 
 Similarly those which were originally in cd parallel to\ 
 ab, will.be found in c'd', parallel to a'b', and so situated \ 
 that a'c'='ac, and of course also b'd' = bd. The figures ! 
 ad, a'd', are thus parallelograms on equal bases and 
 between the same parallels, and therefore equal in area. 
 This shows that the water enclosed between vertical 
 cross sections through ab and cd has the same volume as 
 that between inclined sections (perpendicular to the sides) 
 
30 
 
 PROPERTIES OF MATTER. 
 
 .passing through a'V and dd'. There has thus been 
 /change of form only in this mass of water, and we see 
 
 a' 
 
 Fig. 1. 
 
 Ithat it has been produced by the sliding of every hori- 
 zontal layer of the water over that -immediately beneath 
 I it. [The same result follows even if a'b' be not straight, 
 i for c'd' will necessarily be equal and similar to it.] A 
 good illustration of the nature of this kind of distortion 
 will be seen in the leaves of an opened book, especially 
 a thick one, such as the London Diiredory.. It is often 
 well ejdiibited by piles of copies of a pamphlet, or of 
 quires of note-paper curiously arranged in a shop-window. 
 Now when there is resistance to sliding of one solid on 
 another we call it friction. Thus the viscosity of a fluid 
 is due to its internal friction, just as the slower motion 
 at the bottom than at the top of the channel is to be 
 ascribed to the friction of the fluid against the solid. 
 
 39. We now see vih/y it is that disturbances of liquids 
 gradually die away : — why the waves on a lake, or even 
 on an ocean, last so short a time after the storm which 
 produced them has ceased. Also why winds (for there 
 is friction in gaseous fluids as well as in liquids, though 
 the mechanical explanation of its origki may not be quite 
 the same) gradually die out. In either case the energy 
 . apparently lost is, as in the case of friction of solids, 
 
TERMS APPLIED TO MATTEE. 31 
 
 merely transformed into heat. We also see why it is 
 that winds have the power of raising water-waves. 
 
 The stirring of water, or oil, and the measurement / 
 of the rise of temperature when the whole had come to \ 
 rest, the work done in stirring being also determined, 
 was one of the processes by which Joule found, with 
 great accuracy, the dynamical equivalent of heat. 
 
 40, It is very instructive to watch the ascent of an 
 air-bubble in glycerine, and to compare it with that of 
 an equal bubble in water. The experiment is easily 
 tried with long cylindrical bottles, nearly full of different 
 liquids, but having a small quantity of air under the 
 stopper. When the bottle is inverted the bubble has 
 to traverse the whole column. 
 
 The (apparent) suspension in water of mud, and ex- 
 ceedingly fine sand (to whose presence the exquisite 
 colours of the sea and of Alpine lakes are mainly due) 
 ■ is merely another example of viscosity. So is that of 
 fine dust, and of cloud particles, in the air. Stokes 
 calculates that a droplet of water, a thousandth of an 
 inch in diameter, cannot fall in still air at a much greater 
 rate than a inch and a half per second. If it be of one- 
 tenth of that size it will fall a hundred times slower, i.e. 
 not more than one inch per minute ! This result, that 11 
 the resistance in these cases varies as the diameter, and II 
 not as the section of the drop, is very remarkable. I' 
 
 41. Bodies are called Elastic or Nbrtrelasfic. Compare, I 
 for instance, the properties of a wire of steel with those of I 
 a lead wire ; or of a piece of india-rubber and a piece of 
 clay or putty. But the popular use of these terms is 
 usually, very inaccurate. The blame rests mainly with 
 the ordinary text-books of science, which are (as a rule) 
 
32 PROPERTIES OF MATTER. 
 
 singularly at fault as regards the whole of this special 
 subject, including even its most elementary parts. 
 
 Elasticity, in. the correct use of the term, implies that 
 property of a body in virtue of which it recovers, or 
 
 / tends to recover, from a deformation. 
 
 The phrase " tends to recover " is scarcely scientific ; 
 we should preferably say "requires the continued appli-. 
 cation of deforming stress to prevent recovery, entire or 
 partial, from deformation." 
 
 Kinematics show us that any deformation, however 
 complex, is made up of mere changes of hulk and of form. 
 (See Chap. VIII.) 
 
 A distortion may therefore be wholly Compression, 
 or wholly Shear (§ 37), or made up of these in any way. 
 Hence there are two distinct kinds of Elasticity, viz. 
 Elasticity, of Bulk and Elasticity of Form. The former is 
 possessed in perfection by all fluids, while the second is 
 wholly absent. In solids both are present, but neither 
 in perfection. 
 
 Thus we see that, as a necessary preliminary to in- 
 vestigations on elasticity of bodies, we must study their 
 capabilities of being distorted : — a whole series of pro- 
 perties, such as compressibility, extensibility, rigidity, etc. 
 
 T 42. In popular language, bodies are said to be WTdte, 
 Black, Blue, Red, etc. The investigation of the underlying 
 scientific facts, on which these depend, is partly physical 
 (and therefore within our scope), but also partly physio- 
 logical. The subject is thus a somewhat complex one. 
 
 What do we mean by White Light ? This is a question 
 much more physiological than physical Probably the 
 
 1 true answer to it depends upon circumstances, or con- 
 ditions, which may be varied indefinitely, and with them 
 
TERMS APPLIED TO MATTER. 83 
 
 will, of course, vary what is described in terms of 
 them. 
 
 Thus, in a room lit by gas, a piece of ordinary writing- 
 paper, or of chalk, appears white : — at least if we have 
 been in the room for some little time. But if,-beside it, 
 there be another piece of the same paper or chalk on 
 which, through a chink, a ray of sunlight is allowed to 
 fall (weakened, if necessary, so as to make the two 
 appear of nearly the same brightness), we at once call the 
 first piece of paper or chalk yellow, allowing the second 
 to be white. Here we enter on a purely physiological 
 question. In fact, if we accustom ourselves, for a suffi- 
 ciently long time, to the observation of bodies in a room 
 lit up only by burning sodium (which gives almost 
 homogeneous orange light), we may ultimately come to 
 regard bright bodies, such as chalk, etc., as being white : — 
 others, of course, being merely of different shades, or 
 degrees, of blackness. This, therefore, is foreign to our 
 present subject. But, for all that, it furnishes us with 
 the means of answering an important question somewhat 
 different from that proposed above : — viz. What do we 
 mean by a white body I 
 
 43. Suppose two sources to exist in the room, giving 
 different kinds of homogeneous light ; one being incan- 
 descent sodium as before, the other incandescent lithium, 
 which (at moderate temperatures) gives a homogeneous 
 red light. Chalk and ordinary writing-paper wiU still 
 appear as white bodies to an eye which has become 
 accustomed to the light in the room ; other bodies appear 
 darker, but some are reddish, some of an orange tint. 
 
 And thus we obtain the idea that a white body is one 
 which sends to the eye, in nearly the same proportion 
 D 
 
84 PROPERTIES OF MATTER. 
 
 as it receives them, the various constituents of the light 
 which falls upon it ; while a black body sends none ; and 
 coloured bodies send back light which, while (in general) 
 necessarily made up of the same constituents as the 
 incident light, contains them in different prffportions to 
 those in which they fell upon it. [It would only con- 
 fuse the student were we here to refer to Fhwreseence.] 
 
 44. Thus white light would seem to be a mere relative 
 term. It is conceivable that the inhabitants of worlds 
 whose sun is a blue star, or a red star (and there are 
 many notable examples of such stars), may have their 
 I peculiar ideas of white light, formed from their own cir- 
 ■ cumstances; as ours is formed from the light of our 
 own sun, which is what, in contrast with these, we 
 must call a yellow star. 
 
 However this may be, the discussion above has shown 
 
 what is meant by a white body. A blue body is, by 
 
 similar reasoning, one which returns blue rays in greater 
 
 proportion than it does those of other visible light. It 
 
 is therefore said to absorb the other rays in greaterpro- 
 
 portion than it absorbs the blue rays. 
 
 I Now we are in a position to understand why blue and 
 
 I yellow pigments, mixed together, give green : — while a 
 
 ' disc, painted with alternate segments of the same blue 
 
 j and yellow, appears of a purplish colour when made to 
 
 rotate rapidly. For the light given out by the rotating 
 
 1 disc is a mkiwre, in the proportion of the angles of the 
 
 sectors, of the kinds of bght returned by the blue and 
 
 I yellow separately. But that which the mixed powders 
 
 I send back has in great part penetrated far enough into 
 
 the mass to run, as it were, the gauntlet of absorption 
 
 |by each of the separate components in turn, and there- 
 
TERMS APPLIED TO MATTER. 35 
 
 f 
 
 fore is finally made up of those rays only which are not 
 freely absorbed by either. 
 
 To this discussion we need only add, in illustration 
 of the conservation of energy, that a body is always 
 found to be heated in proportion to the amount of light 
 energy which it absorbs. 
 
 45. Shifting our ground again, we next take the 
 words Malleable, Ductile, Plastic, and Friable, as applied 
 to solid bodies. 
 
 All of these refer specially to the behaviour of solids 
 under the action of forces which, tend to change their 
 form; for the change of volvme of solids, even under 
 very great pressures, is usually very small. The first 
 three indicate that the body preserves its continuity 
 while yielding to^ such forces, the fourth that it breaks 
 into smaller parts rather than change its form. And, 
 in popular use at least, the terms imply in addition that 
 the body is not elastic. 
 
 46. The most perfect example of a malleable body is 
 metallic gold. The gold leaf employed for " gilding," as 
 it is called, is prepared by a somewhat tedious process, 
 which requires a high degree of skill in the workman. 
 The gold is first rolled into sheets thinner than the 
 thinnest writing-paper (thus already showing a high 
 amount of plasticity) j next it is beaten out between 
 leaves of vellum, till its surface is increased, and there- 
 fore its thickness diminished, some twenty-fold. A small 
 portion of this fine leaf is then placed between two pieces 
 of gold-beater's skin ; and a more skilful workman, with 
 a lighter hammer, again extends its surface twenty-fold. 
 This operation can be repeated without tearing the thin 
 film of metal, so great is its tenacity. 
 
36 PROPERTIES OF MATTER. 
 
 Here we have one dimension (thickness) diminished 
 in a marked manner, but the product of the other two 
 dimensions (the surface of the leaf) is' of course pro- 
 \ portionally increased. 
 
 47. The action of the hammer may be practically 
 viewed as equivalent to that of an intense pressure 
 exerted through a very small volume, thus at every stroke 
 applying a finite .amount of energy. One portion of this 
 is changed into heat in the hammer, the anvil, and the 
 gold leaf ; the rest is employed in doing work against the 
 molecular forces of the gold, and thus altering its form. 
 
 To show that this is the true explanation of the 
 observed effect, we may vary the experiment by 
 subjecting a leaden bullet to the action of a hydrostatic 
 press. A few strokes of the pump suffice to bruise the 
 bullet into a mere cake. The process is essentially the 
 same as that of gold-beating, but lead is by no means so 
 malleable as gold. 
 
 48. This leads us, in our present discursive treatment 
 of parts of our subject, to inquire how it is, that by 
 means of such a machine as the Bramah press, a man 
 can apply pressure sufficient to mould a piece of lead, 
 whose shape he could scarcely alter to a perceptible 
 amount by the direct pressure of the hand. 
 
 Here we have a first inkling of the Function of a 
 Mackine. A machine is n.erely a contrivance by which 
 we can apply work in the way most suitable for the 
 purpose we have in hand. Work (as a form of energy) 
 is a real thing, whose amount is conserved. But we 
 have seen that it can be measured as the product of 
 two factors — the (so-called} force exerted, and the space 
 through which it is exerted. Hence, because even when 
 
TEEMS APPLIED TO MATTER. 87 
 
 a machine is perfect it can give out only the energy 
 communicated to it, if there be but one movable part 
 to which energy is supplied and another by which it is 
 given off, the simultaneous linear motions of these 
 two parts must be in the inverse ratio of the forces 
 applied to them, or exerted by them, in the direction of 
 these motions respectively. Thus we are not concerned 
 with the interior structure, or mode of action, of a 
 perfect machine ; all we need to know is the necessary ; 
 relation of the speeds of the two parts or places at which ;' 
 energy is taken in and given out. This is a matter of;' 
 kinematics, and can be made the subject of direct! 
 measurement when the machine is caused to move., ■. 
 whether it be transmitting work or not. ; 
 
 The statement just made is embodied in the ver/ 
 nacular phrase — i 
 
 i 
 What is gained in power is lost in speed. i 
 
 Objections may freely be taken to this form of words,! 
 but it is meant to imply precisely what was said above 
 as to the action of a perfect machine. 
 
 If the machine be imperfect, as, for instance, if there 
 be frictional heating during its working, the heat sol 
 produced represents some of the energy given to the] 
 machine, and the remainder of it is alone efficient. 
 
 49. A substance is said to be ductile when it can be 
 drawn into very fine wires — i.e. when it admits of great 
 exaggeration of one of its three dimensions (length) at 
 the expense of the product of the other two (cross 
 section). Wire-drawing is, essentially, a very coarse 
 operation, for it has to be effected by finite stages, the 
 wire being drawn in succession through a number of 
 
38 PROPERTIES OF MATTER. 
 
 holes in a hard steel plate, in which each hole is a little 
 smaller in diameter than the preceding one. The more 
 nearly continuous the operation is made, the more 
 tedious and therefore the more costly it becomes. 
 f The associated tenacity and plasticity of silver render 
 I it one of the most ductile of metals. And an ingenious 
 idea of WoUaston's enables us, as it were, to impart to 
 other metals much of the ductility of silver. His idea 
 may be briefly explained by analogy as follows. Suppose 
 a glass rod, whose core is coloured, be drawn out while 
 softened by heating, the diameter of the core is found 
 to be reduced in the «ame proportion as is that of the 
 rod. Thus, to obtain platinum wires much finer than- 
 could be procured by direct drawing, WoUaston suggested 
 the boring of a hole in the axis of a cylindrical rod of 
 silver, plugging the hole with a platinum wire which 
 just fitted it, and then drawing into fine wire the. com- 
 pound cylinder. When this operation has been carried 
 to its limit, practically determined by the ductility of 
 the silver, the diameter of the platinum has been 
 reduced nearly in the same proportion as that of the 
 silver ; and the silver may be at once removed from the 
 fine platinum core by plunging the whole in an acid 
 which freely attacks silver but has no efi'ect on plat- 
 inum. 
 
 50. Plasticity is shown, on the large scale, by many 
 substances which, in hand specimens, appear fragile in 
 the extreme. Glacier -ice is one of these, but its 
 behaviour is so closely connected with its thermal 
 properties that we can only msjntion it here. 
 
 The whole earth, though its rock -structure appears 
 so rigid, has been found to be more plastic (under the 
 
TERMS APPLIED TO MATTER. 39 
 
 tidal attraction of the moon) than a globe of glass of the 
 same size would have been. 
 
 But it is specially under the action of small but 
 persistent forces that bodies, which are usually regarded 
 as brittle or friable, show themselves to be really plastic. 
 A good example of this is given by an experiment due 
 to Sir W. Thomson. Cobbler's wax is usually regarded 
 as a very brittle body ; yet if a thick cake of it be 
 •laid upon a few corks, and have a few bullets laid on its 
 upper surface, the whole being kept in a great mass of 
 water to prevent any but small changes of temperature, 
 after a few months the corks will be found to have 
 forced their way upwards to the top of the cake, while 
 the bullets will have penetrated to the bottom. 
 
 51. For variety, let us next take the terms Trans- 
 paremt, Translucent, and Opague. 
 
 These refer, of course, to the behaviour of a substance 
 with regard to the passage of light through it. In 
 common speech, a pane of ordinary window-glass is called 
 transparent, while a piece of corrugated or of ground 
 glass is translucent: — the latter transmits rays, no doubt, 
 but with their courses so altered that they are no loiiger 
 capable of producing distinct vision of the source from 
 which they come. Consistency would require that the 
 term translucent should also be applied to irregularly- 
 heated air, or to a mixture of water and strong brine 
 before diffusion has rendered it uniform throughout. 
 
 Translucent is hardly a scientific word, unless we 
 choose to limit its application to heterogeneous bodies. 
 In science we speak of the degree of transparency of a 
 homogeneous substance ; as, for instance, water more or 
 less coloured, and employed in greater or less thickness. 
 
40 PKOPEETIES OJ MATTER. 
 
 In such cases, besides the inevitable surface-reflection, 
 there is more or less absorption ; and the percentage of 
 any definite kind of incident Hght which unit thickness 
 of the substance transmits is called its transparency for 
 that kind of light. 
 
 Opacity may arise from either of the.two causes just 
 mentioned. Light may enter a body, and be unable to 
 proceed farther, as is the case with lamp black. Or 
 it may fall on a highly polished surface, such as thinly 
 silvered glass, and be in great part reflected without 
 entering. 
 
 In the former case it is said to be absorbed; and, 
 when this happens, the absorbing body is raised in 
 
 ' temperature. The incident energy is converted from 
 
 I the light form into that of heat. 
 
 In the latter case part only can enter the body ; and, 
 if it meet in succession other reflecting surfaces in suffi- 
 cient number, practically the whole of it may be reflected. 
 This is the case with a heap of pounded glass, a cloud, 
 
 I a mass of snow, or of froth or foam. All of these 
 materials are transparent, but they reflect some of the 
 incident light ; and, in consequence of the multiplicity 
 of surfaces which the light has to encounter, the greater 
 
 j part of it is reflected before it has penetrated deeply into 
 
 1 the mass. Hence the whiteness and brightness of snow 
 
 I and clouds in full sunshine. 
 
 — 52. We have here an excellent opportunity of calling 
 the student's attention to the distinction : — a very pro- 
 found one : — between Eeat and Temperatwe. 
 
 For we have seen that energy, in the form of light, 
 when absorbed, becomes heat in the absorbing body, and 
 
 I thus raises its temperature. But if the same quantity of 
 
TERMS APPLIED TO MATTER. 41 
 
 heat had been given to a body, of the same nature but 
 of twice the mass, the rise of temperature ■would have 
 been only half as great. The very form of words here 
 used shows at once how different are the meahings of 
 the words temperature and heat. . For the quantity of 
 hea,t (so much energy, a real thing) is perfectly definite, 
 but the effect it produces on the temperature (a mere 
 state) depends on the quantity and quality of the mass 
 to which it is communicated. 
 
 Heat is therefore a thing, something objective ; 
 temperature is a mere condition of the body, with which 
 the heat is temporarily associated; a condition which 
 in certain cases determines the physical state of the body 
 itself, and in all cases determines its readiness to part 
 with heat to surrounding bodies or to receive it from 
 them. 
 
 Heat may, in this connection, but only for illustration, 
 be compared with the air compressed into the receiver 
 of an air-gun ; temperature would then be analogous to 
 the pressure of that air. Neither of two receivers would 
 (except by diffusion, with which we are not at present 
 concerned) give air to the other, when a pipe is opened 
 between them, if the pressure were the same in both ; 
 but air would certainly flow from the receiver in which 
 the pressure is greater to the other ; and this, altogether 
 independently of the relative capacities of the two re- 
 ceivers, or the consequent amounts of their contents. 
 — 53. As another example, take the terms Cohesive, In- 
 I coherent, Bepulsive. 
 
 \ A lump of sandstone has considerable tenacity, which, 
 { of course, is to be ascribed to those molecular forces of 
 ' which we spoke in § 26. But when, in virtue of its 
 
42 PROPERTIES OF MATTER. 
 
 friability, it has been pounded down into sand, it becomes 
 an incoherent powder. And we know that it must at 
 some tinie previously have been in this form, for it often 
 contains fossil plants or fish, and it may even have pre- 
 served (perhaps for a million or more of years) records of 
 surface -disturbance in the form of dents made by rain 
 or hail, or by the feet of birds or reptiles. 
 
 The graphite, or plumbago, which forms the material 
 for the finest drawing- pencils, is a somewhat rare and 
 valuable mineral. In cutting it up into " leads," how- 
 ever carefully, a considerable portion is reduced to powder 
 — i.e. sawdust. But if this powder be exposed, in mass, 
 to pressure sufficient to bring its particles once more 
 within the extremely short mutual distance at which the 
 molecular forces are sensible, these forces again come 
 into play, and the powder becomes a solid mass, which 
 can in turn be sawn into " leads " for a somewhat in- 
 ferior class of pencils. 
 
 The whole of this part of the subject, especially as 
 regards liquids, will be fully treated later, so that we 
 need not further consider it here. 
 
 54. But let us contrast, with the behaviour of the 
 particles of a solid or a liquid, that of the particles of a 
 gas or vapour. Such substances require to be subjected 
 to external pressure in order to prevent their particles 
 from being widely scattered. When a small quantity 
 of air is allowed to enter an exhausted receiver it dilates 
 so as to occupy with practical uniformity the whole 
 I interior of the receiver, however large that may be. 
 
 This result was, naturally enough, at first ascribed to 
 a species of repulsion between the various particles ; but 
 \ the notion was found to be an erroneous one. For the 
 \ 
 
TEEMS APPLIED TO MATTER. 43 
 
 eflfects of a true repulsion, capable of producing the 
 practically infinite dilatation already spoken of, could not 
 all be consistent with the corresponding observed results. 
 The mode of departure from them depends upon the law 
 according to which the repulsion may be supposed to 
 vary with the distance between two particles. Some 
 assumed laws would give as a consequence that the 
 particles would all be driven to the sides of the vessel, 
 leaving the interior void. Others would require that 
 the pressure should change in value if we were to take 
 half the gas and confine it in a vessel of half the content. 
 Others would make it dififerent at different parts of the 
 surface unless the vessel were truly spherical, etc. etc. 
 
 The true explanation of the phenomenon becomes 
 obvious to us when we apply heat to the gas. For it 
 then appears that the pressure requisite to maintain the 
 whole at a constant volume increases as the temperature 
 is raised ; and thus that heat is, in some way, the ccmse 
 of the pressure. 
 
 55. Hence we are led to what is called the Kinetic 
 Theory of gases, whose fundamental assumption is that 
 the particles dart about in all directions (with an average 
 speed which is greater the higher the temperature), 
 impinging on one another, and also upon the sides of the 
 containing vessel. This continued series of very small 
 but very numerous impacts (each, by itself, absolutely 
 escaping observation) is perceived by our senses as the 
 so-called " pressure " exerted by the gas. Experiment 
 showed that, when a gas is confined in a vessel of definite 
 size, the changes of its pressure are nearly proportioned 
 to the changes of temperature, as measured by a mercury 
 thermometer, whether these changes be in the direction 
 
44 PROPERTIES OF MATTER. 
 
 of a rise or a faU. If we assume, for a moment, that 
 this statement is true for all ranges of temperature, even 
 beyond those attainable in experiment, it leads us to the 
 question : — At what temperatwe does the pressure of a gas 
 vanish ? 
 
 Calculations carried out in the above way showed 
 that, under the assumption just mentioned, all gases 
 cease to exert pressure at one common temperature 
 (about - 273° C.) Thermodynamical theory comes to our 
 assistance, and shows that the above guess is not far 
 from the truth : — th^t a body, cooled to - 274° C, cannot 
 be cooled any farther ; that it then is, in fact, totally 
 deprived of heat. 
 
 We might, therefore, fancy that a gas, if it could be 
 brought to this temperature, would be reduced to a mere 
 layer of incoherent dust or powder, deposited by gravity 
 on the lower surface of . the containing vessel. But 
 experiment has shown that gaseous particles, even while 
 in motion, if only close enough together, exert mutual 
 molecular forces, so that the result (on the gas) of the 
 conditions above specified would probably be its assum- 
 ing a liquid or even a solid form. 
 
 56. We speak of bodies as Hard and Soft. These are 
 barely scientific terms ; because, unless they are strictly 
 defined, they may bear a great variety of meanings. 
 
 Thus, for instance, we have the mineralogist's Scale 
 of Ha/rdness, which is often of great practical value in 
 , field-work.. For there are numerous instances in whifch 
 1 two quite different minerals (sometimes a very valuable 
 1 and a very common one) are almost undistinguishable 
 Ifrom one another so far as colour, density, and crystal- 
 line form are concerned. Chemical tests (even the com- 
 
TERMS APPLIED TO MATTER. 45 
 
 paratively coarse blowpipe tests), though they would 
 settle a question of this kind at once, are not readily 
 applied in the field. Hence the use of the scale of 
 hardness, in which minerals are arranged, so that every 
 one can scratch the surface of any other which is lower 
 in the scale. By carrying a set of twelve small specimens 
 of selected minerals only, the finder of a doubtful crystal 
 can readUy determine its rank among them as regards 
 scratching ; and can thus often settle in a moment what 
 would otherwise require some time, even with the 
 facilities of a laboratory. 
 
 In such a scale diamond, of course, stands at the top, 
 while native copper, one of the toughest of substances, is 
 far below it. 
 
 But if we were to test relative hardness by some 
 other method, say by blows of a hammer, we should be 
 led to arrange our specimens in a very difierent order. 
 The scale above spoken of is, therefore, by no means a 
 scientific one, though, as we have seen, it may often 
 ' give easily some useful information. 
 
CHAPTEE IV. 
 
 TIME AND SPACE. 
 
 57. We begin with an extract from Kant, ,who, as mathe- • 
 matician and physicist, has a claim on the attention of 
 the physical student of a different order from that pos- 
 sessed by the mere metaphysicians. 
 
 " Time and space are two sources of knowledge, from 
 which various h priori synthetical cognitions can be 
 derived. Of this pure mathematics give a splendid 
 example in the case of our cognitions of space and its 
 various relations. As they are both pure forms of 
 sensuous intuition, they render synthetical propositions h 
 priori possible. But these sources of knowledge b, priori 
 (being conditions of our sensibility only) fix their own 
 limits, in that they can refer to objects only in so far as 
 they are considered as phenomena, but cannot represent 
 things as they are by themselves. This is the only field 
 in which they are valid; beyond it they admit of no 
 objective application. This peculiar reality of space and 
 time, however, leaves the truthfulness of our experience 
 quite untouched, because we are equally sure of it, 
 whether these forms are inherent in things by themselves, 
 or by necessity in our intuition of them only. Those, 
 
TIME AND SPACE. 47 
 
 on the contrary, who maintain the absolute reality of 
 space and time, whether as subsisting or only as inherent, 
 must come into conflict with the principles of experience 
 itself. For if they admit space and time as subsisting 
 (which is generally the view of mathematical students of 
 nature), they have to admit two eternal infinite and self- 
 subsisting nonentities (space and time), which exist with- 
 out their being anything real only in order to comprehend 
 all that is real. If they take the second view (held by 
 some metaphysical students of nature), and look upon 
 space and time as relations of phenomena, simultaneous 
 or successive, abstracted from experience, though repre- 
 sented confusedly in their abstracted form, they are 
 obliged to deny to mathematical propositions h priori 
 their validity with regard to real things (for instance in 
 space), or at all events their apodictic certainty, which 
 cannot take place h ^posteriori, while the h priori concep- 
 tions of space and time are, according to their opinions, 
 creatures of our imagination only."^ 
 
 On matters like these it is vain to attempt to dogma- 
 tise. Every reader must endeavour to use his reason, 
 as he best can, for the separation of the truth from the 
 metaphysics in the above characteristic passage. 
 
 58. We must now take up, as indicated in § 21, the 
 property Extension, which is one of those expressly in- 
 cluded in our provisional definition of matter. 
 
 It implies that all matter has volume, or bulk. The 
 thinnest gold leaf has finite thickness, the finest wire 
 has a finite cross section. 
 
 In popular language this is recognised by the use of 
 the associated terms length, breadth, and thickness. 
 ^ Critique of Pure Reason. Max Miiller's Translation. 
 
48 PROPEETIES OF MATTER. 
 
 r In other words, the term extension recognises the 
 essentially Tridimensional character. of space. 
 
 Why space should have three dimensions, and not 
 more nor less, is a question altogether beyond the range 
 of human reason. Only those who fancy that they know 
 what space is, would venture (at least after well con- 
 sidering the meaning of their words) to frame such a 
 question. 
 
 59. The proof that our space has essentially three 
 dimensions is given in its most conclusive form by the 
 statement, based entirely upon experience, that 
 
 To assign the relative position of two points in space, three 
 numbers {of which one at least must be a multiple of the vmt 
 of length) are necessary, and are sufficient. 
 
 It is an easy matter for us, accustomed to tridimen- 
 sional space, to imagine one or more of its dimensions 
 to be suppressed. In fact so-called Plane Geometry is the 
 geometry of one particular kind of two-dimension space ; 
 Spherical Trigonometry that of another. We cannot well 
 speak of the gmnetry of space of one, or of no dimensions ; 
 but the idea, meant to be expressed, is a correct one, 
 though the term is inappropriate. 
 
 When, however, we try to conceive space of four or 
 more dimensions, we are attempting to deal with some- 
 thing of which we have not had experience ; and thus, 
 though we may by analogy extend our analytical and 
 other processes to an imagined space, in which the 
 relative position of two points depends on more than 
 three numerical data, we can form no precise idea of how 
 the additional dimensions would present themselves to us. 
 
 A few remarks on this subject will be made at the 
 end of the chapter. 
 
TIME AND SPACE. 49 
 
 60. Space of no dimensions is a geometrical point, of 
 which nothing further can be said. 
 
 61. Space of one dimension : — let us call that dimen- 
 sion Length : — ^is a mere geometrical line which may be 
 curved or straight. But to be sure of the existence of 
 this characteristic, and to understand its true nature, we 
 must have cognisance of space of two dimensions if it be 
 a plane curve, of three if it be tortuous. The study of 
 all the properties of space of one dimension, though 
 an excessively simple affair, is of very great intrinsic 
 importance, besides being a necessary step towards that 
 of the higher orders. We will, therefore, treat it so 
 fully that a far less amount of detail will be necessary 
 when we come to two and to three dimensions. 
 
 62. Every one, whether he be aware of the fact or 
 not, is acquainted by experience with at least the elements 
 of this subject. Suppose, for instance, we take as our 
 one-dimensioned space any one of the roads or railways 
 leading from Edinburgh to London; which we will, 
 for the moment, suppose to be straight, and to run due 
 south. The milestones, set up at equal distances along 
 the road, mark the positions of various points in terms 
 of the one dimension, length, which is alone involved, or, 
 rather, to which for the present we restrict our consider- 
 ation. And a Gazetteer or a Bailway Guide gives us the 
 positions of the towns or stations along the road or line : 
 the position of each being fully described by a single 
 number, understood as a multiplier of a mile or of some 
 other specified unit of length, and with a qualification 
 which will presently be introduced. 
 
 But these numbers refer to the distance from some 
 assumed starting-point, or Origin as it is technically 
 
50 PROPERTIES OF MATTER. 
 
 called ; say, in this case, London. Thus we find in an 
 old Eoad Guide, for the particular one-dimensioned space 
 called the East Coast Route, a column of data from which 
 
 we extract the following : — 
 
 Miles. 
 London . . 
 
 York 196 
 
 Berwick . . ... 337 
 Edinburgh 395, 
 
 Fractions of miles are omitted to avoid mere arithmetical 
 complication. 
 
 From this table, by ordinary subtraction, we form a 
 list, as below, of the lengths of what we may call the 
 various stages of the route. Thus — 
 
 Miles. 
 
 London to York 196 
 
 York to Berwick . . . .141 
 
 Berwick to Edinburgh . . . .58 
 
 It will be seen that, in this list, the origin from which 
 each number is measured is the first named of the two 
 corresponding places, and the number itself is found by 
 subtracting, in the first list, the number corresponding 
 to the first of the two places from that corresponding to 
 the second. 
 
 63. Now let us at once take the only step which 
 presents any difficulty. Choose York as our origin, and 
 boldly apply the rule just given, no matter what the 
 
 consequences may be. The result is — 
 
 Miles. 
 
 London -196 
 
 York 
 
 I Berwick ...... 141 
 
 Edinburgh . . ... 199 
 
 Here there is no difficulty whatever in understanding 
 
TIME AND SPACE, 51 
 
 the numbers for Berwick and for Edinburgh. They are, 
 as before, the numbers of miles by which Berwick and 
 Edinburgh are separated from York. Also the number 
 for London, when York is the origin, differs from that 
 for York, when London is the origin, only hy change of 
 sign. 
 
 So that we at once recognise the meaning of the 
 negative sign as applied to a length in our one-dimen- 
 sional space : — it measures the length in the opposite 
 direction to that in which a positive length is measured. 
 
 The necessity for this convention, and its extreme 
 usefulness, were early recognised in Cartesian geometry, 
 but they had long before been applied in common 
 arithmetic as well as in algebra. 
 
 Perhaps the simplest view we can take of the subject 
 is that afforded by a man's " balance " at the bank. So 
 long as this is on the right side {i.e. positive) he can draw 
 • any less amount and still be on the credit side ; if he - 
 overdraws (i.e. takes more out of the bank than his 
 balance), the difference is negative, and he is to that 
 amount indebted to the bank. 
 
 64. In the first of the three little tables above, all 
 the places involved lay to the north of the origin (London), 
 and were all therefore affected by the same sign (which . 
 we happened to take as +). When York was taken as 
 origin, Berwick and Edinburgh were to the north, and 
 their numerical quantities were still +. But London, 
 being to the south, had a - number. 
 
 It would be easy to give multipUed examples of this, 
 but they are unnecessary. The only additional com- 
 ments which we need make are these : — 
 
 (1.) When the northward direction along a line was 
 
52 PROPERTIES OF MATTER. 
 
 [ called +, the southward necessarily became - . Simi- 
 larly, had we chosen southward as +, northward would 
 have become - . 
 
 (2.) We chose for our special example a northward- 
 running line, but we might equally well have chosen an 
 eastward one, etc. Hence pairs, such as N. and S., E. 
 and W., up and down, etc., must be regarded as having 
 their members contrasted exactly as are the + and - 
 of Algebra or of Analytical Geometry. 
 
 And, just as a displacement in one direction along a 
 line may be regarded as +, while a displacement in 
 the opposite direction must then be regarded as - , so 
 it is with rates of motion, i.e. Speeds, in space of one 
 dimension. 
 
 Thus the relative speed of two trains running 
 northward, A at 60 miles an hour, B at 40, is 20 miles 
 an hour northward as regards A seen from B, and 20 
 southward as regards B seen from A; so if A be moving 
 southward at -60 miles an hour, and B northward at 
 40, the speed of A with regard to B is 100 miles per 
 hour southward, and of B with regard to A 100 miles 
 northward. 
 
 65. A precisely similar distinction is observed when 
 our one -dimensional space is a curved line;^take for 
 example the orbit of a planet. To describe fuUy the 
 position of the planet, when the orbit is given, one number 
 alone (say the angle^edor, the angle which the radius- 
 vector, or line joining the centres of planet and sun, 
 makes with some fixed line in the plane of the ofbit) is 
 required. This, however, must again be qualified as -f- 
 or - . (In the case of angles, we agree to call them + 
 when they are measured in the opposite direction to that 
 
r 
 
 TIME AND SPACE. 53 
 
 of the motion of the hands of a watch ; that is, when 
 they are described in the same sense as that in which 
 the northern regions of the earth' turn about the polar 
 axis). Angular velocities in one plane are similarly- 
 characterised. 
 
 In all cases where motion is restricted to one line the 
 same thing holds. Thus the position of a pendulum is 
 at every instant- completely assigned by the angle the 
 rod makes with the vertical, provided we are also told 
 on which side the displacement is. 
 
 The record kept by a self-acting tide-gauge gives at 
 any instant the elevation or depression (again + and - ) 
 of the water from the mean level. Similarly with regis- 
 tering barometers, thermometers, etc. But, for the full 
 appreciation of the indications of these records, they are 
 usually made in two dimensions by an important principle 
 which will presently be explained. 
 
 66. In what precedes we have been dealing with a 
 kind of space in which the only displacements are 
 forward or backward ; nothing is possible (nor even 
 conceivable) sideways or upwards. 
 
 This characteristic applies to Time, as well as to space 
 of one dimension, and therefore we should expect to 
 find, as we do find, that (with the necessary change of 
 a word or two) all that has just been said -with reference 
 to relative position is true of events in time, as well as of 
 points in one-dimensional space. There is no such thing 
 as motion or displacement in time, so that this part of 
 the analogy is wanting. Every event has its definite 
 epoch, for ever unalterable. And of course there is no 
 going sideways or upwards, as it were, out of the one- 
 dimensional course of time. 
 
64 PROPERTIES OF MATTER. 
 
 r 
 
 Thus we find that to assign definitely the position of 
 an event in time, provided our origin is assigned, all 
 we need know is a single number (a multiplier of the 
 time-unit) with its sign, + or - , signifying time after or 
 time before that origin. 
 
 Our usually adopted origin is the Christian era, and 
 - we speak of 1885 a.d. as the present year, while the date 
 of the battle of Marathon is recorded as 490 B.C. The 
 difiference between the characteristics A.D. and B.C. is 
 of precisely the same nature as that between north and 
 south, or + and - . 
 
 Hence, if we wish to find the interval between the 
 present time and the battle of Marathon, we have to 
 subtract +1885 (the position of the new origin) from 
 - 490. The result is - 2375, i.e. Marathon was fought 
 2375 years ago. Thus to change the origin, or epoch, 
 we must perform precisely the same operation as that 
 which gave us the table in § 63, from the first table in § 62, 
 Thus to change our chronology to the year of the world 
 (designated by A.M.) or to the old Eoman (marked 
 Aitr.c), all we need do is to subtract from each date 
 (A.D. or B.C., regarded as + and - respectively) the 
 assumed date of the creation of the world (4004 B.C.) 
 or of the foundation of the city of Eome (753 B.C.) 
 
 We need say no more on such a matter. Every intel- 
 ligent reader can make new and varied examples for 
 himself. 
 
 67. Passing next to space of two dimensions, whether 
 plane or spherical, we see at once from a map, or a globe, 
 that the position of a place is given by two numbers, its 
 Latitude and Loihgitude. But each of these has to be quali- 
 fied for definiteness by the -f or - sign, or something 
 
TIME AND SPACE. 
 
 55 
 
 equivalent. Thus we have N. or S. latitude, and E. or 
 W. longitude. 
 
 But there are two methods, specially applicable to the 
 plane, which deserve closer attention in view not only 
 of their intrinsic usefulness, but also of their bearing 
 on the general question of tridimensional space. These 
 are known in geometry as Bedcmgular and Fola/r co- 
 ordinates. 
 
 68. In the first we assume two reference lines at 
 right angles to one another, both passing through the 
 origin, and assign the position of a point by giving its 
 distances from these two lines. These distances are 
 looked on as drawn to the point from either line, and 
 each therefore changes sign when the point is taken on 
 the other side of the corresponding reference line. This 
 
 -X 
 
 X 
 
 is symbolised in the cut. Oa;, Oy are the two reference 
 lines, the origin. The perpendiculars PM, PN, let fall 
 from P on these lines, completely, and without ambiguity, 
 define its position. For if we know OM or NP, the oc 
 
56 PROPERTIES OF MATTET?. 
 
 I of P, ie. its distance from Oy, that condition alone 
 limits our choice for P to points lying in PM, a line 
 drawn parallel . to Oy and everywhere at the assigned 
 distance, x, from it. Similarly, y being given as ON or 
 MP, the choice of points is limited to those on the line 
 
 i NP, all of which have this property. 
 
 i But two lines at right angles to each other must 
 intersect, and in one point only. Thus the point P is 
 determined by the conditions without ambiguity. 
 
 If P lie to the left of Oy, its x is negative ; if below 
 Ox, its y is negative. The lettering above, at the ends 
 of the lines bounding the four quadrants, shows at a 
 glance the signs of x and y when P is situated in any 
 one of them. 
 
 In general, any given relation, between the x and y of 
 a point, limits its position to a definite Curve in the plane 
 of the reference lines. It is often very convenient to 
 represent such a relation by a curve ; and, in fact, most 
 self-registering instruments actually trace such a curve 
 for us. Thus, if intervals of time measured from a 
 definite instant (represented by 0) be laid off along 
 Ox, with the corresponding heights of the thermometer, 
 barometer, tide, etc., erected as perpendiculars at their 
 extremities, we have a curve showing the mode in which 
 temperature, pressure of the, atmosphere, etc., change as 
 time goes on. But stich curves can be traced by a pencil 
 attached to the instrument, or by photographic processes, 
 on a long band of paper which is drawn horizontally past 
 it, at a uniform rate, by clockwork. 
 
 69. In the second method the data are the length of 
 OP (the radius -vector), and the magnitude of <MOP 
 (the angle-vector), § 65. These are usually denoted by 
 
TIME AND SPACE. 57 
 
 r and 6. Here r is always taken as a positive (or rather 
 a signless) quantity, while 6 is positive if it be measured 
 round from Ox cmmter-dochmse. 
 
 This is the method adopted by a surveyor when, 
 with a chain and a theodolite, he measures a field. His 
 reference line, Oa;, is usually given by a magnetic needle 
 attached to the theodolite. He measures the angle 
 a;OP and the distance OP, P being a corner (let us 
 say) of the field. These two data, determined for each 
 prominent part of the boundary, enable him to plot the 
 field ; and therefore contain all the necessary numerical 
 data for calculating its area, etc. 
 
 It is also the method usually employed in dealing 
 with orbital motion of any kind in one plane. 
 
 Comparing the two methods, we see that the 
 directed line OP may be resolved (as it is called) into 
 OM and MP, lines in directions perpendicular to one 
 another. Also that this resolution, in any direction, is 
 effected by means of the cosine of the angle involved. 
 
 < 
 
 For a; = OM = OP cos xOV = r cos 9, 
 < 
 ,t/ = MP = OP cos 2/OP = r sin 0. 
 
 It is clear that, though we have hitherto spoken of 
 and P as the simultaneous positions of two points, we 
 may look on them as successive positions of OTie (moving) 
 point. If we look on the displacement as having been 
 produced uniformly, and in one second, it represents in 
 magnitude and direction the Velocity of the moving point ; 
 and OM, MP represent, on the same scale, its resolved 
 parts or components, parallel to Ox and Oy. 
 
 70. As examples, we give one or two results which 
 wUl be specially useful to us in later chapters. 
 
S8 PROPERTIES OF MATTER. 
 
 If a point be moving, in any manner whatever, we 
 may consider its velocity alone, independent altogether 
 of the actual path piursued. Take any fixed point, O, 
 
 and let OP represent, in 
 magnitude and direction, 
 the velocity of the mov- 
 ing point. Aiter one unit 
 
 FlQ. 3. 
 
 of time let the velocity he 
 represented by OPj ; after 
 two units, by OPg ; and so 
 on. It is clear that all 
 the points P, P^, Pg, etc., lie on some definite curve, 
 which will be more accurately traced the greater the 
 number of points we obtain ; i.e. the smaller we assume 
 our unit of time. If the motion whose properties are 
 thus studied be that of a particle of matter, this curve 
 (which is called the Eodograph) is necessarily continuous, 
 for the velocity cannot alter by starts either in magnitude 
 or in direction. And, as OP passes to a proximate 
 position, OQ, by having a velocity PQ compounded with 
 it, the acceleration of OP is the velocity with which P 
 moves in its curve. If the path be a plane curve, the 
 hodograph is .also plane. 
 
 This construction enables us at- once to solve a 
 number of elementary problems in kinematics, which 
 will be of great use to us in the sequel. 
 
 71. Thus, suppose a point to move uniformly, with 
 speed V, in a circle of radius E. OP in the hodograph 
 (Fig. 3) has constant length V, and its direction rotates 
 uniformly. Hence the hodograph is another circle, also 
 uniformly described in the same sense {i.e. clockwise or 
 counter-clockwise), and in the same period of time. 
 
TIME AND SPACE. 
 
 Hence the speed of P must be such that it describes a 
 circle of radius V, in the time that a point whose speed 
 is V takes to go round a circle of radius E. It must, 
 therefore, be V^/R. Also the direction of this speed is 
 perpendicular to OP, and therefore along the radius of 
 the first circle. And, because both circles are described 
 clockwise, or the opposite, its direction is towards the 
 centre. 
 
 Let, now, the figure repre- 
 sent the circle of radius E, 
 and draw any diameter, AOA'. 
 Then N moves round with 
 speed V, and the acceleration of 
 its motion is V^E along NO. 
 Eemembering that accelerations 
 and velocities are resolved like 
 lines, we see that if NM be 
 
 drawn perpendicular to AA', the speed of the point M 
 wUl be 
 
 V|^ along MC, 
 
 and its acceleration 
 
 r=^ ;=== also alone MC, and towards C. 
 R CN ° 
 
 The motion of M, thus defined, is called Simple Har- 
 monic. It obviously consists in a vibration back and 
 forward along the line AA', the speed being greatest at 
 0, and vanishing at A and A'. The special character- 
 istic is that the acceleration is always directed towards 
 0, and is proportional to the displacement of M from that 
 point. 
 
 72. If we use Newton's Fluxional Notation, in which 
 
60 PROPERTIES OP MATTER. 
 
 the rate at which a quantity increases per unit of time 
 
 is expressed by putting a dot over the symbol for that 
 
 quantity, a second dot placed over it will signify the 
 
 rate at which that rate increases, and so on: 
 
 Thus, if CM above be denoted by x, the speed of M 
 
 is X, and its acceleration is x. And we see at once from 
 
 the result of last section that 
 
 .._ V2 
 !c - - p K, 
 
 the negative sign being prefixed because x is directed 
 to the right in the figure, and the acceleration to the 
 left. Whenever, in future chapters, we meet with a 
 relation of this kind, we will, therefore, interpret it as 
 expressing simple harmonic motion. The multiplier 
 of the right-hand side depends only on the ratio of V to 
 E : — what is called the angular velocity. If we denote 
 this by ft), the equation may be written 
 
 x= - oi^x ; 
 and this belongs to all simple harmonic motions, what- 
 ever be their range of vibration, provided the angular 
 velocity in the corresponding uniform circular motion be 
 £0, or the period 277/0). 
 
 73. The result above was obtained by projecting 
 uniform circular motion on a diameter of the circle, or, 
 what comes to the same thing, on a plane perpendicular 
 to the plane of the circle. 
 
 But an exceedingly interesting result is obtained by 
 projecting the circular motion on any other plane. In 
 orthogonal projection equal areas are projected into 
 equal areas, and a circle is projected into an ellipse, 
 whose centre is the projection of the centre of the circle. 
 
 Hence the projection gives motion in an ellipse, the 
 
TIME AND SPACE. 
 
 61 
 
 radius-vector drawn from the centre of the ellipse tracing 
 out equal areas in equal times, and the acceleration 
 being still directed inwards along the radius-vector, and 
 proportional to it. 
 
 74. Another extremely useful result may be obtained 
 by supposing the uniform angular velocity in the circle 
 to be maintained, but with a continual shrinMng of the 
 radius at a rate measured by « times its length at each 
 instant. 
 
 The velocity of the moving point is thus made up of 
 two components, one along the circle, the other along 
 the radius, each proportional to the radius. Hence the 
 path is a spiral, which makes a constant angle with the 
 radius, what is called the Equiangular, or Logarithmic, 
 Spiral. The radius-vector still revolves uniformly.^ 
 
 Let PE be the spiral, SP any radius. Then, if PT 
 
 Pio. 6. 
 
 be the velocity of P, and a the angle between its direc- 
 tion and that of PS, we see at once that 
 
 PTsina = wSP, PTcosa = kSP, 
 whence k = w cot a. 
 
 If SQ be equal and parallel to PT, Q is a point in the 
 
 hodograph. But as PT, and therefore SQ, is propor- 
 
 1 Proe. B.S.E., December 19, 1867. 
 
62 PEOPERTIES OF MATTER. 
 
 iional to SP, and the angle QSP is the supplement of 
 a, the hodograph is the same spiral rotated through a 
 given angle, and altered in its linear dimensions by the 
 
 factor (^jjT^). Thus the hodograph of the hodograph is. 
 another similar spiral, again turned through the same 
 angle, and with its radii altered in the ratio (-^Y. If 
 
 ° ' \sma/ 
 
 PU be drawn, making an angle a with TP produced 
 backwards, and meeting QS in U, it will therefore be 
 the direction of acceleration at P. 
 
 But PU may be resolved into PS, SU, the first of 
 which is along the radius-vector, the second parallel to 
 the tangent at P. The parts of the acceleration in these 
 directions are, respectively, 
 
 (^^YpS, and 2(^^y SP cos a. 
 V sin a/ Vsin a/ 
 
 The latter of these, by the first equation above, may be 
 
 written as 
 
 \ sm a/ 
 
 T^™ sin a cos a ^ , _,„ „ .„„ 
 PT = 2wcota.PT = 2kPT. 
 
 Hence the motion of P is due to an acceleration along, 
 and proportional to the length of, PS, and another 
 along, and proportional to the length of, TP. 
 
 Thus the resolved part of the motion along any line 
 in the plane possesses the same characteristics. If x 
 represent the distance between the projections of S and 
 of P, on such a line, we see at once that we have 
 
 2m cot a.x ^-rr X, 
 
 or, introducing the value of k above, 
 
 a = - 2kX - {(O^ + K^X. 
 
 This diff'ers from the equation for simple harmonic 
 
TIME AND SPACE. 63 
 
 motion by the term involving i. But the preceding 
 investigation shows us that an equation of this form 
 represents the resolved part (in some definite direction) 
 of uniform circular motion with angular velocity <», the 
 radius of the circle shrinking by the fraction k of its 
 amount in each second. (This is the same thing as 
 saying that its logarithm diminishes by k in unit of 
 time.) Or we may call it simple harmonic motion 
 whose Bade is constantly diminishing at a definite rate. 
 
 75. The step to three-dimensional space is now easy. 
 WewUl take it from a somewhat altered point of view. 
 Our reference system is now three planes at right angles 
 to one another; say the floor, the north wall, and the 
 west wall of a room, the corner in which these three 
 meet being for the time our origin. 
 
 And the position of a point is determined without 
 ambiguity if we know its distances from these planes, 
 with the proper sign of each. 
 
 For, knowing only its distance from the floor, we 
 limit it to the horizontal plane which is everywhere at 
 that distance from the floor. Similarly the second con- 
 dition limits it to a definite plane parallel to the north 
 wall. These two conditions together limit it to a certain 
 horizontal line lying east and west. The third condition 
 limits it to a certain plane parallel to the west wall ; and 
 this is intersected in one point, and one only, by the 
 east and west line just mentioned. That one point is 
 the sole which satisfies all three conditions. 
 
 Thus, let represent the origin, yOa the north wall, 
 %Qx the west wall, and xOy the floor. The figure is 
 drawn as seen by an eye equidistant from these three 
 planes, and in the room, i.e. on the positive side of each 
 
64 
 
 PROPERTIES OF MATTER. 
 
 of them. And it will be noticed that the lettering 
 X, y, as of the ends of the edges, which meet in O, is so 
 applied that rotation from Ox to Oy, from Oy to Oz, and 
 from Oz to Ox again, will all alike appear to be cownter- 
 clockmse. 
 
 -y--.- 
 
 -X 
 
 The position of ^ny point, P, is then found thus i — 
 Draw PN perpendicular to the iloor, meeting it in N ; 
 thence NM perpendicular to Oa;. Then OM = a; is the 
 distance of P from the north wall ; MN = y is its dis- 
 tance from the west wall ; and NP = a is its distance 
 from the floor. 
 
 If P assume a new position which requires it to pass 
 through, any one of these planes, the corresponding co- 
 ordinate changes sign ; if it pass through an- edge (i.e. 
 the intersection of two of these planes) two co-ordinates 
 change sign ; and if it pass through (where the three 
 
TIME AND SPACE. 65 
 
 planes meet) all three co-ordinates become negative. 
 This is illustrated by the negative lettering at the 
 (dotted) prolongation of each edge through 0. 
 
 76. But, in analogy with the second method of § 69, 
 we see that the position of P will be fully specified if 
 we know the vertical plane through in which it lies 
 (i.e. the plane «0N), the angle NOP in that plane, and 
 the length of OP. The first is determined if we know 
 the angle kON. Hence we determine P by its distance 
 from 0, and two angles which (together) enable us to 
 assign the direction of OP. The angle jcON is called the 
 Admuth of the plane »0N ; let us denote it by 0. The 
 angle NOP is called the Altitude of P, as seen from ; 
 let us denote it by cj). Also let the length of OP be, as 
 before, called r. 
 
 Comparing, as before, the results of the two methods, 
 
 we see that ON = r cos ^, and therefore 
 
 K = OM = ON cos fl = r cos <j> cos 8, 
 y = MN = ON sia $ = r cos (p sin 8, 
 % = NP = r sin 0. 
 
 The elements of spherical trigonometry show that the 
 multipliers of r, in the values of «, y, z respectively, are 
 the cosines of the angles between the line OP and the 
 lines Oa;, Oy, Qz. Hence the more symmetrical method, 
 in which these cosines are represented by I, m, n respect- 
 ively, gives 
 
 x = rl, y = rm, z = rn, 
 
 with the condition 
 
 It is easy to see that the remark in § 69 as to resolu- 
 tion of a velocity in two dimensions holds with respect 
 to three. 
 
 F 
 
66 PROPERTIES OF MATTER. 
 
 Then Newton's Second Law of Motion (Chap. VI.) at 
 once extends these conclusions to Forces. 
 
 77. A remark of great importance must be made 
 here. We saw in § 68 that a point was determined, in 
 X, y co-ordinates {i.e. plane space of two dimensions), as 
 the intersection of two straight lines, to one of which 
 it was confined by its x being given in value, to the 
 other by the value of its y. But any two independent 
 conditions connecting x and y wiU also determine their 
 values. Such a condition connecting x and y is known 
 as the equation of a curve, and, when given, limits the 
 position of P to that curve. Two such conditions, 
 therefore, give P by the intersection of two curves, on 
 each of which it must lie. Such a condition applied to 
 a physical particle is called a Degree of Constraint. In 
 two-dimensional space a free particle has but two Degrees 
 of Freedom, one of which is removed by each degree 
 of constraint to which it is subjected. 
 
 78. Similarly we saw that, in three dimensions, the 
 point given by x, y, z is determined as the intersection 
 of three planes, on each of which it must lie. But any 
 one condition connecting the values of x, y, and z is 
 the equation of a surface, and, when it is given, a particle 
 at the point is subjected to one degree of constraint. 
 When free, it has but three degrees of freedom; and 
 three degrees of constraint, by completely determining 
 its X, y, and «, fix its position. 
 
 We should arrive at the same result by considering 
 relations among the r, 6, <j> co-ordinates. But it sufiS.ces 
 to consider merely what constraint each of these imposes 
 when its value is given. All points for which r has a 
 given value, lie on a sphere whose centre is at 0, When 
 
TIME AND SPACE. 67 
 
 is given, the point must lie somewhere in the vertical 
 plane «0N. When <f> is given, it must lie somew^here on 
 a right cone of which is the vertex and Oa the 
 axis. 
 
 [The two latter statements are easily illustrated by 
 means of a telescope, mounted (in the common way) on 
 a stand which allows it to rotate about a horizontal, or 
 about a vertical, axis. Place it in any azimuth, and 
 vary its altitude, it turns in a vertical plane about the 
 horizontal axis. Place it at any altitude, and vary its 
 azimuth, it rotates conically about the vertical axis. 
 Hence, by means of these co-ordinates, or conditions, 
 each definite point in its axis is constrained to lie on a 
 sphere, a plane, and a cone, simultaneously.] 
 
 79. Two devices are in common use for enabling us 
 to represent, on a plane (or other space of two dimen- 
 sions) the third dimension. 
 
 Thus, in an Admiralty chart, we find the sea-area 
 marked over with figures denoting Soundings :—i.e. the 
 average depth of the water at certain places is written 
 in in fathoms. These soundings are of course more 
 numerous at places where there are shoals or intricate 
 channels. But it is obvious that, if they were numerous 
 enough, they would enable us to construct a model of 
 the sea-bottom. The soundings, therefore, supply, as it 
 were, the necessary third dimension. But this process, 
 though usually sufficient for purposes of navigation, is 
 at best a rude and incomplete one. 
 
 The other method, however, rises to a very high 
 order of scientific importance, not merely from the point 
 of view for which it was originally devised, but on account 
 of the extent to which its essential principles are now 
 
68 PROPERTIES OF MATTER. 
 
 applied throughout the whole range of physics. We 
 therefore devote some space to its full explanation. 
 
 80. This is called the method of Contour Lines, 
 employed with great effect in Ordnance Survey Maps. 
 
 A contow line passes through all places which a/re at the 
 same height above the sea-level. 
 
 Thus the sea-margin is the contour line of no elevation. 
 Suppose the water to rise one foot (vertically). There 
 would be a new sea-margin, encroaching more on the 
 land than the former ; encroaching most at places where 
 the beach has the gentlest slope, not encroachiag at all 
 on a perpendicular cliff, and thrust out (seawards) from 
 an overhanging cliff. This is the contour line of one foot 
 elevation. It is clear that by supposing a gradual rise 
 of the sea, or subsidence of the land, foot by foot, we 
 could obtain a series of curves (each a sea-margin) 
 gradually circumscribing the uncovered portion of the 
 land, and finally closing in over its highest peak. We 
 require no such natural convulsion as that just imagined. 
 Cloud strata, or fog-banks, with definite horizontal sur- 
 faces, constantly show us these appearances in hilly 
 countries. But it is a simple matter of Levelling to trace 
 out contour lines, and to draw them on a map of the 
 district. For practical purposes it is usually sufficient 
 to draw them for every 50 or 100 feet of additional 
 elevation above the searlevel. 
 
 The celebrated Parallel Roads of Glen Boy are merely 
 contour lines, etched on the sides of the valley by long- 
 continued but slight agitation of the margin of the water 
 which filled the glen to various depths in succession, as 
 the barriers which dammed it up were, at intervals, 
 broken down. 
 
TIME AND SPACE. 
 
 69 
 
 Eeferring to § 78, we see that a surface can be 
 expressed in terms of one relation between x, y, and z. 
 Let the plane of Ox, Oy, be that of the sea-level, and let 
 the relation expressing the surface of the land be 
 
 »=/fe y)- 
 Then the contour lines, as traced on the (two-dimension) 
 map are the curves 
 
 / (a !/.) = 0, / (b, !/) = 100, / (x, y) = 200, etc. 
 81. To familiarise the student with the general 
 
 appearance of contour lines, and their relation to the 
 form of the corresponding surface, we give those of a 
 
70 PEOPERTIES OF MATTER. 
 
 right cone whose axis is vertical, of a hemisphere, and 
 of a fusiform or spindle-shaped body. 
 
 The fusiform body, whose contour lines are drawn, is 
 formed by the rotation of a quadrant about a vertical 
 tangent, the point of contact being the apex. And the 
 contour lines are drawn, in each case, at successive 
 heights increasing by one -fifth of the whole height 
 of the figure. Thus the distances between successive 
 contours, in the two last figures, form the same series 
 but in opposite order. 
 
 The equality of distance between the successive contour 
 lines of the cone indicates uniform steepness throughout. 
 In the hemisphere the lines are closer together near the 
 boundary of the figure, in the spindle they close in on 
 one another towards the centre ; the hemisphere being 
 steepest at its edges, and the spindle surface steeper 
 towards the point. 
 
 82. In fact, the Gradient of a surface in any direction 
 (i.e. the amount of rise per horizontal foot) is obviously, 
 at any point, inversely as the distance in that direction 
 between successive contour lines, for they are traced at 
 successive equal difierences of level ; and thus the dis- 
 tance between them, along any line drawn on the map, 
 is the space by which we must advance horizontally 
 along that line while ascending or descending through 
 100 feet. 
 
 83. The line of steepest slope at any point of a surface 
 is represented on the map by the shortest line which 
 can be drawn to the nearest contour line. Thus it cuts 
 the contour lines at right angles, and is the path along 
 which a drop of water would trickle down. It is there- 
 fore called a Strearrirline. 
 
TIME AND SPACE. 
 
 71 
 
 84. If the surface be like that of a saddle, concave 
 upwards along the horse's back, convex upwards across 
 it, we have at the middle of the saddle what is called, 
 in geography, a Col or mountain-pass — the lowest point 
 of the ridge between two neighbouring summits. The 
 characteristic of the col is that, at such a point, a 
 contour line intersects itself. The following figure shows 
 the general form of the contours near such a point. 
 
 In the shaded regions depicted to the right and left of 
 the col the ground rises, in the unshaded regions 
 depicted above and below it falls. [The figures on the 
 contour lines show their order of altitude above the sea- 
 level.] 
 
 Other very special peculiarities might be mentioned, 
 but they are not necessary for the beginner; and the 
 more mathematical reader can easily work them out for 
 himself.^ 
 
 ' See Cayley, Phil. Mag., XVIII. 264; Clerk-Maxwell, aid., 
 December 1870. 
 
72 PROPERTIES OF MATTER. 
 
 85. If we draw, by the help of the contour lines, the 
 stream-lines (which, § 83, cut them at right angles), we 
 find that they have the following property. In regions 
 al)0ve the level of a col, they fall away on both sides 
 from that particular one of their number which passes 
 from a mountain Summit down to the col, and thence up 
 to the neighbouring summit. This particular line, then, 
 is the Watershed, separating two valleys or drainage areas. 
 If we follow their course into regions below the col, we 
 find that they usually approach to the special stream- 
 lines drawn downwards from the col on opposite sides. 
 These will therefore be fed by all the httle riUs in 
 succession, and thus they become the Watereowses. A 
 watercourse is thus the stream-line drawn from a col so 
 as to pass through an Imit, or lowest point of the surface. 
 
 86. So far, we have been dealing with contour lines 
 in the ordinary sense of the word. But essentially the 
 same sort of thing is presented by the meteorological 
 curves called Isobars, and by Isothermak, Lines of Equal 
 Magnetic Variation, of Equal Dip, etc. etc. In each case 
 the lines are drawn, on a two-dimension map, so as to 
 pass through all places where the barometer, or the 
 thermometer, stands at a given reading or level, where 
 the compass deviates a given amount from true north, 
 etc. etc. Thus they have a characteristic similar to 
 that of contour lines, viz. that all points on any one 
 line possess some definite property to exactly the same 
 amount. These applications of the principle are of great 
 importance, but they do not belong so immediately to 
 our subject as do others, of which we will now give an 
 example or two. 
 
 87. Just as water trickles from places at higher, to 
 
TIME AND SPACE. 73 
 
 ' others at lower, level, and as heat flows, in a conducting 
 body, from places of higher, to others of lower, tempera- 
 ture, so electricity is said to flow from places of higher, 
 to places of lower, potential. Hence, to study the flow 
 of electricity in a sheet of metal, we require to know the 
 lines of equal potential. 
 
 The first investigation of this subject, by Kirchhoff,^ 
 supplies an exceedingly beautiful example. 
 
 Putting the wires attached to the ends of a galvanic 
 battery into contact with a very large sheet of uniform 
 tinfoil, at points A and B, we estabhsh and maintain a 
 definite difference of potential between those points of 
 the sheet. Hence there is a steady flow of electricity 
 from the one to the other ; and it must pass, at every 
 point, in a direction perpendicular to the equipoterdial 
 line passing through that point. Thus, to find the lines 
 of flow of electricity, we must have a means of, as it were, 
 contouring the plate electrically, and finding its lines of 
 equal potential. This is furnished by a galvanometer, 
 for that instrument indicates at once any current passing 
 through its coil of wire. But, if the ends of its coil be 
 kept at equal potentials, no current will pass. Hence, if 
 we put one end of the galvanometer coil in contact with 
 the tinfoil at any point, P, and move the other end about 
 on the foil until no current passes, the point, Q, with 
 which it is then in contact, is at the same potential as 
 P. By fishing about, therefore, we can, point by point, 
 trace out the equipotential line PQ passing through P. 
 And the same may be done for other points, till we have 
 
 -^vered the tinfoil with as many lines of this kind as we 
 desire. 
 
 ' Fogg. Ann., 1845, Ixir. 
 
Fio. 9. 
 
TIME AND SPACE. 76 
 
 In the special case which we have taken, it was found 
 that when the plate of tinfoil is very large in comparison 
 with the length AB, these lines are circles, whose centres 
 lie in the line AB, and in each of which the ratio BQ/AQ 
 is the same throughout ; though of course its values are 
 diiferent for different circles of the series. A few of these 
 circles are given (in full lines) in the figure. [To ensure 
 proper contact with the battery, little circular discs of 
 copper (indicated in white) are attached to the tinfoil at 
 A and B. The edges of these (on account of the superior 
 conductivity of the copper) are equipotential lines. The 
 points A and B are not exactly at the centres of these 
 discs.] 
 
 Now geometry tells us that the lines, which cut at 
 right angles all circles drawn according to the above 
 law, belong to another series of circles : — viz. those 
 which are determined by the condition that each passes 
 through the two points A and B. These circles (some 
 of which are represented by the dotted lines in the 
 figure) are therefore the current lines along which the 
 electricity passes in the tinfoil. 
 
 The full circles are drawn for successive equal 
 changes of potential ; and the dotted circles which are 
 drawn are so selected that the amount of electricity 
 which flows in a given time through the space bounded 
 by portions of each contiguous pair is the same. If the 
 full lines be regarded as contour lines of a surface, the 
 right hand side of the figure represents a hill, and the 
 left hand an exactly equal and similar hollow ; so that the 
 halves, as separated by the straight contour line, would 
 exactly _;?< into one another if the whole could be folded 
 along that line. 
 
76 PROPERTIES OF MATTER. 
 
 If both A and B be connected with the positive pole 
 of the battery, and its negative pole be connected with 
 a massive ring of copper, or other good conductor, which 
 borders the sheet of tinfoil all round at a very great dis- 
 tance from A and B, the equipotential lines are what 
 mathematicians call Cassini's Ovals. One of them is the 
 Lemniscate of Bernoulli, and its double point corresponds 
 to a col. The figure resembles in general form that of 
 § 84, and the current lines are a series of rectangular 
 hyperbolas. 
 
 88. As a final example, somewhat more pertinent to 
 our present work, take the relation between the pressure, 
 volume, and absolute temperature, of a given mass of air. 
 
 Experiment has proved that when any two of these 
 three quantities are given, the third is determined. 
 Calling themp, v, and i respectively, the relation between 
 them is (nearly enough for our present purpose) found 
 to be represented by the expression 
 
 pv = Bt (1) 
 
 where E is a known constant quantity. [In a later 
 chapter we wiU study the exact relation. What we seek 
 at present is an illustration of method, not a specially 
 exact representation of fad.'] 
 
 Now we may treat p, v, and t just as we treated x, y, z 
 in § 80 above. In this statement lies the essence of the 
 value of the contour-line idea as applied to questions of 
 general physics. 
 
 Thus the experimental relation among j?, v, t, (1) above, 
 may be looked on as the equation of a surface. Let us 
 draw its contour lines on the plane in which p and v are 
 measured. 
 
TIME AND SPACE. 
 
 77 
 
 Equation (1) shows that these lines are all rectangular 
 hyperbolas, of which the asymptotes are the axes of 
 volume and pressure, Ov and Op. Any line of equal 
 pressure ^WiVg is divided so that Av^, Av^ Av^, etc., are 
 
 proportional to the absolute temperatures. So with a line 
 of equal volume Bpj^aPs- ^^^ o'l^ special advantage of 
 this mode of representation is, that the worh required 
 to compress the gas at any constant temperature, as fj, 
 from volume OB' to volume OB, is given by the area 
 B'^'j^jB, which is contained between the curve t-^, the 
 axis of volume, and the lines of equal volume By^ 
 Bpj. This follows at once from the fact that the work 
 done during an elementary change of volume dv, under 
 pressure ^, is represented by pdv; a little element of 
 
78 PEOPEETIES OP MATTER. 
 
 area bounded by the curve, the axis of v, and two con- 
 tiguous ordinate^. 
 
 Draw a tangent .PT to one of these curves at a point 
 P, and draw PQ parallel to Ov. The compressibility of 
 a gas, at constant temperature, is the percentage change 
 of volume per unit increase of pressure. It is therefore 
 represented by 
 
 QP J_ J_ 
 QT ■ OM' QT' 
 
 or (by a property of the hyperbola) p^^, 
 
 i.e. it is inversely as the pressure. 
 
 The expansibility, at constant pressure, is found simi- 
 larly by producing QP to cut the proximate curve t^ in 
 R ; for it is expressed by the percentage change of 
 volume per unit rise of temperature, that is 
 
 PR J_ (^-yOM 1 _1 
 0M'<2-<i' «iOM ' h-h h 
 
 This is a mere portion of what is called, in Thermo- 
 dynamics, Watt's Diagram of Energy, the whole of which 
 is an application of the contour idea. 
 ] 89. We must now, as promised in § 59, say a few 
 1 words as to a (possible 1) fourth dimension of space. 
 I Let us treat this from the point of view of what we 
 ' may imagine would be the experience of beings, endowed 
 with something corresponding to human reason and 
 human senses, but inhabiting space of one or of two 
 dimensions. 
 
 In one -dimensional space the inhabitants can have 
 length only, and have absolutely no hint from experience 
 of what another dimension could be. Yet we might 
 imagine them, if they were not mere points, to experi- 
 
TIME AND SPACE. 79 
 
 ence some perfectly novel, and to them unaccountable, 
 sensation in passing from part to part of their space 
 if its cwvatwe were not everywhere the §ame. 
 
 Similarly, an inhabitant of two-dimensional space (a 
 bookworm, as Sylvester once called him) might, even if 
 his dimensions were both finite, pass from place to place 
 in a plane or a spherical space without feeling any new 
 sensation. But, if a part of his space were creased or 
 folded, he might be imagined to feel some strange sen- 
 sation while he passed through such a part. This, 
 again, is a question of cwvatwe, which would be totally 
 unintelligible to a being whose experience had not pre- 
 pared him for it. 
 
 So, if there should be a fourth dimension, our three- 
 dimensioned space may appear to a four-dimensional 
 observer to have something analogous to curvature or 
 creasing; and if, in the course of our system's rapid 
 progress through space, we. should come to a region of 
 that kind, we may fancy that some absolutely novel 
 form of experience would be the result. 
 
 90. Speculations of this nature, however, though 
 
 I based to a certain extent on scientific facts, necessarily 
 
 (involve the question of sensation or perception ; and, in 
 
 so far as they do so, they pass from the domain of 
 
 f)hysical science into the realms of Physiology. 
 
CHAPTER V. 
 
 IMPENETRABILITY, POROSITY, DIVISIBILITY. 
 
 91. Our working definition of matter (§21) involves 
 another property besides those discussed in last chapter — 
 viz. IrnpenetraMIMy. The sense in Trhich we are to 
 understand this term depends upon the use of the word 
 occv/py as applied to space. 
 
 On the theory of ultimate atoms, whether the old 
 hard atom (§ 23) or the vortex atom (§ 27), the occupa- 
 tion of space is absolute so far as each atom is concerned. 
 Where one atom is, it fills space to the absolute exclusion 
 of every other. But space is not cordmuously filled by 
 the atoms of any visible portion of matter (§ 24) ; hence 
 there may be mixtures of atoms of different kinds, which 
 will be the more perfect and intimate the smaller we 
 suppose the individual atoms to be. But there is no use 
 in discussing questions of this kind, at least until we 
 prove the existence of atoms. Thus the strictly scientific 
 use of the term impenetrability need not occupy us. 
 
 92. There is, however, a semi -scientific use of the 
 word which is of some importance. For, whether matter 
 be impenetrable in the strict sense or not, we may use- 
 fully discuss the consequences of its not hemg pemtrated. 
 Thus the hollow of a mould, and only the hollow, is 
 
IMPENETRABILITY— DIVISIBILITY. 81 
 
 accessible to the liquid metal poured into it. Otherwise 
 " casting " would be impossible. 
 
 One of the most important of these consequences 
 was long ago given by Archimedes, — viz. a mode of 
 easily comparing the volumes of bodies of such irregular 
 or complex shapes as to defy the powers of calculators 
 working from mere linear measurements. All that is 
 required is to immerse them successively in a vessel 
 partly filled with water, and note the amount by which 
 the level of the water is disturbed, i.e. (in the usual 
 phrase) the amount of water displaced. Bodies which, 
 when thus tried, displace equal amounts of water have 
 equal volumes, however different may be their figures. 
 
 93. Hence, to measure the volume of an irregularly- 
 shaped body : — a lump of stone or coal, for instance :— 
 grease it or varnish it all over, to prevent water from entering 
 its pores, i.e. to secure non-penetration; and immerse it 
 completely in a vessel partly filled with water. Mark 
 the height to which the water-level rises. Withdraw the 
 stone, and pour in mercury until the same disturbance 
 of water-level is again produced. The volume of the 
 mercury is the same as that of the stone. The mercury 
 has the advantage of taking at once the form of any 
 vessel in which it may be placed, so that its volume 
 may be at once determined by pouring it into a properly 
 graduated beaker. 
 
 This simple consideration forms one of the bases of 
 the common method of measuring specific gravity (§ 36) 
 by weighing a body, first in air, and then when it is 
 suspended in water. 
 
 94. But it is not solids (such as the stone above) or 
 other liquids (such as mercury) alone which can thus 
 
82 PROPERTIES OF MATTER. 
 
 displace their own bulk of water. Air will do equally- 
 well. Thus a diving beU is merely an open-mouthed 
 vessel, inverted and let down into water. The air it 
 contains is not penetrated by the water, and thus dis- 
 places (just as a solid or a liquid would have done) its 
 own volume of water. Its volume, no doubt, becomes 
 less as the bell descends under the water, but this is 
 due to the increase of hydrostatic pressure to which it 
 is subjected. Still, however it be compressed, as it is 
 not penetrated it displaces at every instant its bulk of 
 water. 
 
 95. When one liquid mixes with, or when it combines 
 with another, it does not usually displace its own bulk 
 of the other. In such cases there is interpenetration. 
 
 Thus, when twenty-seven parts (by weight) of water 
 are mixed with twenty-three of alcohol, the volume of 
 the mixture is less by 3 '6 per cent than the sum of the 
 volumes of the constituents. 
 
 When an alloy of tin and copper, such as used to be 
 employed for the specula of large reflecting telescopes, 
 is formed, the joint bulk may be as much as 7 or 8 per 
 cent less than the sum of the bulks of the constituents. 
 
 And Faraday showed that, when potassium is oxidised, 
 the resulting potash has a less volume than either of the 
 constituents. 
 
 96. But, as a rule, in these cases of contraction, other 
 physical phenomena present themselves. Thus the 
 mixture of alcohol and water above described becomes 
 more than 8° C. warmer than the components, if both 
 were taken at the same ordinary temperature. A rod 
 of tin dipped into melted copper (at a very dull red 
 heat) produces vivid incandescence as it melts and is 
 
IMPEITETRABILITY— DIVISIBILITY. 83 
 
 alloyed. And the combination of oxygen and potassium 
 develops kinetic energy to an almost explosive amount. 
 
 There are other cases, which vre need not treat of 
 here (especially as they belong properly to Heat and to 
 Chemistry), in which the volume of a mixture is greater 
 than the sum of the volumes of its constituents. 
 
 97. These examples show that Archimedes' notable 
 discovery might altogether have failed in its application. 
 For he is said to have been asked to find whether the 
 votive crown made for Hiero of Syracuse really consisted 
 of the amoiint of gold furnished for its manufacture, or 
 whether a part of the gold had been abstracted, and its 
 place supplied by an equal weight of silver. 
 
 He procured lumps of silver and of gold, each equal 
 in weight to the crown. These he immersed successively 
 in a vessel filled to the brim with water, measuring in 
 each case the amount of overflow, which he found to be 
 greater for silver than for gold. The vessel being once 
 more filled, the crown itself was immersed, and was 
 found to displace more water than did the gold. Hence, 
 by calculation, Archimedes found how much silver had 
 been substituted for an equal weight of gold.^ This 
 calculation, of course, must have proceeded on the sup- 
 position that the bulk of the alloy was equal to the sum 
 of the bulks of the component metals. 
 
 But interpenetration, of which he had no knowledge, 
 might have completely baffled the great mathematician. 
 
 If a similar question were raised now, it would of 
 course be decided at once by the processes of the chemist, 
 not by those of the physicist. 
 
 98. We have seen (§ 24) that, on the hypothesis of 
 ' The original passage is gi¥en as Appendix III. 
 
84 PROPERTIES OF MATTER. 
 
 hard atoms, there must necessarily be interstices between 
 them, else bodies could not be compressible. 
 
 But it is an experimental fact, independent of all 
 hypotheses, that bodies in general are Porous. By the 
 term pores we do not refer to mdble channels, such as 
 those which run in all directions through a piece of 
 sponge, but to microscopic channels, which pervade 
 even the most apparently homogeneous substances, such 
 as solid lead, silver, gold, etc. 
 
 The proof that such channels exist was given experi- 
 mentally by Bacon, who tried to compress water by 
 squeezing or hammering a leaden shell filled with water 
 and closed. The water exuded like perspiration through 
 the pores of the lead. The Florentine Academicians 
 tried the same experiment with a silver shell, but 
 obtained the same result. They then tried to prevent 
 the escape of the water by thickly gilding the shell, but 
 again in vain. 
 
 99. When a comer of a piece of blotting-paper, or of 
 a lump of loaf-sugar, is dipped in water, we see (especially 
 if the water be coloured) the rapid entrance it effects 
 into the pores. Why it enters, under these conditions, 
 is another question. 
 
 The porosity of wood, necessary for the circulation of 
 the sap, is beautifully shown by the fact that, from 
 microscopic examination of a thin sKce of a fossil tree, a 
 botanist can tell at once the species to which it belonged. 
 The greater part of the material of the wood has dis- 
 appeared for it may be millions of years, but its micro- 
 scopic structwre has been preserved by the infiltration of 
 silicious or calcareous materials which, hardening in the 
 pores, have thus preserved a perfect copy of the original. 
 
IMPENETEABILITY— DIVISIBILITY. 86 
 
 The rapid passage of gases through unglazed pottery, 
 sheet indiarubber, iron and (hot) steel, etc., shows the 
 porosity of these bodies in a very remarkable manner. 
 So does the strange absorption of hydrogen by a mass 
 of palladium. (See Chapter XIII.) 
 
 Another beautiful instance is afforded by the silicious 
 concretion, Tahasheer, found in the joints of sugar-canes. 
 It is opaque when dry, but when immersed in water for 
 a short time becomes transparent. Certain agates, called 
 Hydrophones, exhibit the same property. 
 
 100. No decisive proof of the porosity of vitreous 
 bodies, such as glass, seems yet to have been obtained. 
 That they form almost a solitary class of exceptions to an 
 otherwise general rule seems highly improbable. And 
 instances, such as those given below, seem to indicate 
 that these vitreous bodies have not yet been proved to 
 be porous only because we have not discovered the proper 
 mode of testing them. 
 
 When polished marble is wetted with water, very 
 little enters the pores; while oil, on the contrary, is 
 rapidly absorbed. 
 
 A bag of cambric or gauze, whose pores are visible 
 to the eye, holds mercury securely, until sufficient pres- 
 sure is applied to force out the liquid. 
 
 Glazed pottery-ware, which is practically impervious 
 to hydrogen and to pure water, is rapidly penetrated by 
 a strong aqueous solution of bichromate of potash. This 
 solution, crystallising in the pores, disintegrates the 
 whole, just as water, freezing in the pores of a rock, 
 gradually breaks its surface-layers into small fragments, 
 to be afterwards washed down to the plain as alluvial 
 soil. 
 
86 PROPERTIES OF MATTER. 
 
 The question of the porosity of colloidal bodies, such as 
 gelatine, albumen, and, from some points of view, india- 
 rubber, is somewhat puzzling. We will refer to it in 
 Chapter XIII. 
 
 101. The Bivisihility of matter, in the strict scientific 
 sense, at once raises the question of the existence of 
 finite atoms. For, if there be such atoms, division has 
 them for its limit, whatever processes may be employed.-' 
 We are not prepared to face this aspect of the question, 
 and must, therefore, confine ourselves to examples of 
 extremely minute Bivision. 
 
 An " impalpable " powder is one which gives no gritty 
 sensation when we rub it between the thumb and fingers. 
 The process of Lengatim, depending on fluid friction, is 
 employed for the assortment of solid particles into packets 
 of different degrees of fineness. Thus, if ground emery 
 be thrown into a tall vessel full of water, we may remove 
 from the bottom of the vessel successive crops, as it were, 
 of gradually increasing fineness. Yet even the finest 
 of these powders can be used for grinding metallic or 
 glass surfaces, showing that its particles still possess 
 the same properties as do the coarser. 
 
 Silica may be thrown down, by chemical processes, 
 in such an extreme state of division that when it is dried 
 and poured into a trough it behaves almost like a liquid. 
 Especially when it is heated, we observe that, like a 
 liquid, it is capable of propagating waves. Calcined 
 magnesia and other very fine powders show similar 
 properties. 
 
 102. Even the rough process of scratching the 
 polished surface of glass with a diamond point can be 
 
 ^ See again Appendix II. 
 
IMPENETRABILITY— DIVISIBILITY. 87 
 
 carried out by machinery to such an extent of delicacy 
 that groups of equidistant parallel lines may be traced, 
 some of which can only be " resolved " into their com- 
 ponents by the very best microscopes ; others, which we 
 have every reason to believe capable of resolution, have 
 not yet been resolved. These pieces of ruled glass are 
 known to microscopists as Ndbert's Test. 
 
 Ordinary gold-leaf, though prepared by a very rough 
 process, has a thickness of about l/300,000th of an 
 inch only. But, as Faraday showed, it can be rendered 
 very much thinner by immersion in a solvent such as 
 cyanide of potassium. And, by a species of inversion 
 of WoUaston's process (§ 49), i.e. drawing into very fine 
 wire a silver rod thickly gilt, we obtain a continuous 
 film of gold, whose thickness is estimated at less than 
 l/4,000,000th of an inch. 
 
 103. The average size of the particles of water in a 
 light cloud is easily estimated from the diameter of the 
 coloured rings, or Coronce, which it produces when it 
 covers the sun or moon.^ If the radius of the inner- 
 most ring be 15°, the diameter of the particles must be 
 about l/13,000th of an inch. Such must be the average 
 size of the dust particles which have produced the recent 
 remarkable sunsets, as well as the corona still seen about 
 the sun when no cloud is visible. The length of time 
 these particles have remained in suspension is accounted 
 for in § 40 above. 
 
 104. Leslie, in his Natwal Philosophy (1823), says : 
 " A single grain of musk has been known to perfume a 
 large room for the space of twenty years. Consider how 
 often, during that time, the air of the apartment must 
 
 1 Tait's Light, §§ 172, 238. 
 
88 PROPERTIES OF MATTER. 
 
 have been renewed and have become charged with fresh 
 odour ! At the lowest computation the musk had been 
 subdivided into 320 quadrillions of particles, each of 
 them capable of affecting the olfactory organs." He 
 does not tell us how the computation was made, nor 
 even what we are to understand by quadrillions. 
 
 [The usual British reckoning gives a quadrillion as a 
 billion billions, each billion being a million millions, 
 while the French reckoning makes it only a thousand 
 million millions. This confusion is entirely removed 
 by the modern mode of writing large numbers, which 
 we know only in rough approximation. We write the 
 first two or three significant figures, and indicate the 
 number of the remaining ones by the corresponding 
 power of 10.] 
 
 Thus Leslie may have meant either 320 x 10^*, or 
 320 X lO'^^ If, as is most probable, he meant the former 
 of these numbers, the result of his computation has been 
 singularly verified by recent discoveries, some of them 
 apparently altogether unconnected with the question 
 before us. 
 
 105. One of the most striking instances of division is 
 that furnished by holding, in an otherwise slightly 
 luminous flame, a particle of common salt or of some 
 other metallic chloride. Fox Talbot, in 1826, wrote : 
 " A particle of muriate of lime on the wick of a spirit- 
 lamp will produce a quantity of red and green rays for a 
 whole evening without being itself sensibly diminished." 
 Swan traced the source of the peculiar orange ray which 
 appears in the light of almost every flame to the wide 
 diffusion of exceedingly small quantities of common salt. 
 These phenomena are nowadays known to all in connec- 
 
IMPENETEABILITY— DIVISIBILITY. 89 
 
 tion with Spectrum Analysis. The quantity of salt which, 
 for a considerable time, will continue to give the orange 
 tinge to the flame of a large Bunsen lamp is minute in 
 the extreme. The effect is now proved to be due to 
 vapowr of sodium. 
 
 106. A conviction of the practically infinite divisi- 
 bility of matter must be held by all who believe in the 
 "dilutions" which are at least popularly supposed to 
 be one of the main characteristics of homoeopathic 
 medicines. When a single grain of atropine is dissolved 
 in a gallon of water, and one drop of this is added to 
 another gallon of water, we have what is called the first 
 dilution. Add a drop of this to a third gallon of water, 
 and we have the second dilution. And so on. Tenth 
 dilutions are said to be sometimes administered. If we 
 take the diameter of a drop as about l/8th of an inch, 
 we find, by an easy calculation, that (as there are 277 
 cubic inches in a gallon) the tenth dilution should eon- 
 tain about 2/10*" of a grain of atropine per drop ! If 
 that drop were magnified to the size of the whole earth, 
 the atropine in it (magnified, of course) would correspond 
 to a particle of somewhere about one three-biUionth of 
 an inch in diameter ! ! 
 
 107. The kinetic theory of gases informs us that, in 
 a cubic inch of any gas at atmospheric pressure, and at 
 ordinary temperatures, there are about 3 x 10^" detached 
 particles absolutely similar and equal to one another. 
 These cannot be Lucretian atoms, for they have each 
 many difierent modes of vibration, even when they 
 belong to a simple and not to a compound gas. Here 
 we reach the limit of our present knowledge as to division 
 of matter. What is the structure of these gaseous par- 
 
90 PROPERTIES OP MATTER. 
 
 tides on which their vibrations depend (§ 29), and ho-w 
 far further divisible each particle must be supposed in 
 consequence, are matters beyond our knowledge. [These 
 results of the kinetic gas theory are confirmed by alto- 
 gether independent lines of physical reasoning with 
 which we are not concerned here.] 
 
 We may take, as a rough approximation, that the 
 grained structure (§ 26) of the most homogeneous bodies 
 is of the order of 5 x 10^ to the inch linear. To give a 
 notion of the amount of division which this indicates, 
 suppose we magnify a cubic inch of such a substance to 
 a cube whose side is the diameter of the earth. The 
 earth's diameter is 5 x 10^ inches, very approximately. 
 Thus, in the enormously magnified cube, there is one 
 particle in every cubic inch or so. We say nothing, 
 it is to be noticed, as to the size of the particle or 
 granulation itself. [The estimates hitherto made of 
 this quantity can hardly be called even rough approxi- 
 mations. But probably the particle does not occupy so 
 much as 5 per cent of its share of the whole content.] 
 
 All that can be said of the estimates above is that 
 they are, at least nearly, of the proper order of magni- 
 tude. And it is curious to find that the result^ of 
 Leslie's old "computation" (§ 104) agrees fairly enough 
 with our present knowledge. 
 
CHAPTEE VI. 
 
 INERTIA, MOBILITY, CENTRIFUGAL FORCE. 
 
 108. We commence with Newton's 
 
 First Law of Motion. 
 
 Every body perseveres in its state, of rest or of uniform 
 motion in a straight line, except in so far as it is compelled 
 by forces to change that state. 
 
 The property, thus enunciated as belonging to all 
 bodies, is usually called Inertia. And it is clear from 
 the statement above that it may be described as passivity, 
 or dogged perseverance, but in no sense whatever as 
 revolutionary activity. This consideration will be found 
 presently to be of great importance. 
 
 Matter is, as it were, the plaything of force ; sub- 
 mitting to any change of .state that may be imposed on 
 it, but rigorously persevering in the state in which it is 
 left, until force again acts upon it. 
 
 109. The state referred to is one of rest, or of uniform 
 motion in a straight line (of which rest is a mere particular 
 case). Here we meet with a serious difficulty. 
 
 All motion (including rest, or null motion) is, from 
 the very nature of space, essentially relative. 
 
 Eelatively to what, then, are we to say that a body 
 
82 PROPERTIES OF MATTER. 
 
 not acted on by force moves uniformly in a straight line t 
 The answer, so far as we can give it, is — 
 
 Relatweh) to amy set of lines drawn in a rigid body of 
 finite dimensions, which is not acted on by force, cmd which 
 has no rotation. 
 
 As will be seen later, Newton has pointed out a 
 physical test, by which it can be ascertained whether a 
 body has rotation or not. 
 
 But questions of this kind can only be alluded to, 
 certainly not fully discussed, in an elementary work 
 
 110. The grand proof of the truth of this as well as 
 of the other Laws of Motion is furnished by the celestial 
 motions. So irregular is the motion of the moon, when 
 considered carefully, found to be, that no amount of the 
 most exact observation alone (i.e. unaided by physical 
 investigations) could enable us to predict its place, even 
 twenty -four hours beforehand, with anything like the 
 accuracy with which it is predicted four years before- 
 hand in the Nautical Almanac. So convinced have 
 astronomers become of the truth of the laws of motion, 
 which are necessarily involved in all their lunar and 
 planetary calculations, that when a discrepancy between 
 prediction and observation is found to occur no one now 
 questions the bases of the calculation. The discrepancy 
 is used to correct our previous estimates of the elements 
 of the lunar or planetary orbit; or, as in the notable 
 case of Uranus, it is employed as an indication of where 
 to seek for some undiscovered body whose influence has 
 not been taken into account. 
 
 111. Familiar instances of Inertia present themselves 
 in all directions. When a railway carriage is running 
 uniformly on a straight piece of road, we become uncon- 
 
INERTIA— CENTRIFUGAL FORCE. 93 
 
 scious of the motion except by looking out at external 
 bodies; but we detect at once any sudden change of 
 speed. If the motion be checked by suddenly applying 
 the brake, their inertia (which really maintains their 
 motion) appears to urge the passengers forwards. A 
 sudden starting of the train produces the opposite eifect. 
 While the steady motion continues a conjuror can keep 
 a number of balls in the air just as easily as if the car- 
 riage were at rest. But these things need not surprise 
 us. Our rooms are always like railway carriages in 
 respect of their perfectly smooth, but very rapid, motion 
 round the earth's axis. The whole earth itself is flying 
 in its orbit at the rate of a million and a half of miles 
 per day; yet we should have known nothing of this 
 motion had our globe been perpetually clouded over 
 like that of Jupiter. The whole solar system is travel- 
 ling with great speed among the fixed stars, but we 
 know of the fact only from the minutely accurate obser- 
 vations of astronomers, aided by all the resources of the 
 Theory of ProbaMUties. 
 
 112. When a bullet is dropped in a uniformly running 
 carriage, it strikes the same point of the floor whatever 
 be the speed of the motion ; for, by its inertia, it pre- 
 serves while falling the forward motion of the carriage 
 which it obviously had while it was held in the hand. 
 But, if the bullet be dropped from the yard of a vessel 
 to the deck, it will not fall always on the same spot, 
 however uniform be the ship's progress, if there be any 
 pitching. For, when the vessel pitches, the yard moves 
 forward alternately faster and slower than does the deck. 
 Now the top of a tower (unless it be at the pole) is 
 farther from the earth's axis than is the foot, or the 
 
94 PROPERTIES OF MATTER. 
 
 ground on which the tower is built ; and, therefore, as 
 both complete their revolution in twenty-four hours, 
 the top of the tower moves permanently faster than the 
 base. Hence even a truly spherical bullet, dropped 
 from the top, does not fall vertically. It deviates to the 
 east of the vertical, because it preserves while falling its 
 superior eastward speed. Thus we have one physical 
 proof of the earth's rotation. 
 
 113. The upsetting of buildings by an earthquake 
 furnishes a striking instance of inertia. So does the 
 almost perfect immunity we experience from the millions 
 of meteoric stones which are constantly encountering the 
 earth with planetary velocities. This is due to the 
 inertia of the air, which, in its turn, is one indispensable 
 cause of the destructive action of a tornado ; just as, on 
 a smaller scale, a cannon-ball would be harmless without 
 inertia, while an earthwork, without inertia, would afford 
 no defence. But we need give no more instances — the 
 reader will easily supply others from his own experience. 
 
 114. So far, we have been speaking of inertia as mani- 
 fested by the tendency of a body to persevere in its 
 motion with wmltered speed. But we must carefully note 
 that this is only one part of Newton's Law. The state 
 in which he tells us that bodies persevere by inertia is 
 not one of imiform motion merely, but of motion in a 
 straight line. The preservation of the rectilinear path is 
 quite as essential a part of the functions of inertia as is 
 the preservation of the uniform speed. Hence, just as 
 we attribute any change in the speed of a body to the 
 action of force, so, if its line of motion be not straight, 
 whether the speed be unaltered or no, this also must be 
 due to the action of force. 
 
INERTIA-CENTKIFUGAL FORCE. 95 
 
 115. How the force must be applied whicli causes a 
 body, in spite of its inertia, to move in a curve is easily 
 understood from some common instances, though it is 
 pretty obvious that it must be in a direction per^pevdicula/r 
 to tliat of the motion ; and, of course, in the plane in 
 which the curvature takes place. For any force in the 
 direction of motion must tend only to increase or to 
 diminish the speed. 
 
 It is found that, at a curve on a railway line, it is the 
 outer of the two rails which is most worn (i.e. that one 
 which forms the convex side of the track). And, when 
 a sharp curve has to be taken rapidly, the outer rail has 
 generally to be laid a little higher than the other. But 
 (except when the brake is on) the pressure is mainly 
 perpendicular to the rails. 
 
 When we whirl a sling with a stone in it, we feel the 
 tension of the cord (which is const&ntly pulling the stone 
 from its natural straight path in towards the hand) 
 increased as we cause the sling to rotate faster. 
 
 A bullet, suspended by a string, forms what we call 
 a simple pendulum. It can, by proper initial projection, 
 be made to rotate uniformly in a horizontal circle. Here 
 the tension of the string may be resolved into two parts ; 
 one vertical, which supports the weight of the bullet, 
 the other horizontal, which continually deflects the bullet 
 from its natural rectilinear path. If the string could be 
 made long enough, the time of rotation might be made 
 twenty -four hours; and if the pendulum were then set up 
 at the north pole, and made to describe its circle in the 
 positive direction (§ 65), it would appear to remain sus- 
 pended at rest in the air, the supporting string not being 
 vertical ! If it were made to rotate in the negative 
 
96 PROPERTIES OF MATTER. 
 
 direction, it would appear to complete a rotation in 
 twelve hours. 
 
 The moon is caused to move in an (approximately) 
 circular path about the earth by the same attraction 
 which causes stones to fall vertically downwards. 
 
 116. One of Newton's remarks on the First Law of 
 Motion runs thus : — 
 
 " A hoop, whose parts by their cohesion perpetually 
 draw one another aside from rectilinear motions, does 
 not cease to rotate, except in so far as it is retarded by 
 the air." 
 
 Thus the uniformity of the earth's rotation about its 
 axis, which is the basis of our measurement of time, is 
 .merely an example of the First Law of Motion. 
 
 But when a fly-wheel, or a grindstone, is made to 
 rotate so fast that the cohesion of its parts is no longer 
 capable of supplying the forces requisite to keep them 
 moving in their circular paths, it bursts (this is the tech- 
 nical word), and the fragments fly off in paths which are 
 tangential and rectilinear, except in so far as gravity 
 modifies them. 
 
 If the rotating body be plastic, as must have been 
 the case long ago with the earth as a whole, its form 
 will be modified by the tendency of every particle to 
 preserve its rectilinear path. Thus it swells out in aU 
 directions perpendicular to the axis of rotation. Jupiter 
 and Saturn, being much larger than the earth, and also 
 ■ rotating more rapidly, show this effect in a much greater 
 degree. 
 
 A beautiful example is furnished by suspending an 
 endless chain by a cord,, and (by very rapidly twisting 
 the cord by means of multiplying gear) throwing it into 
 
INERTIA— CENTRIFUGAL FORCE. 97 
 
 rotation. When the rotation of the whole is sufficiently 
 rapid it assumes almost exactly the form of a horizontal 
 circle, all its links being equidistant from the (vertical) 
 axis of rotation. 
 
 117. The old notion, probably suggested by such 
 instances as the pull which the stone in a sling seems to 
 exert on the hand, was that bodies have a tendency to 
 fly outwards from the centre about which they are revolv- 
 ing. Hence they were said to exert Oentrifugal Force, 
 and a Centripetal Force was of course required to balance 
 this. The term Centrifugal Force has become rooted 
 in our scientific language. It is a convenient enough 
 expression, provided we do not split it up, thus taking 
 it to imply force, and flying from a centre ; but interpret 
 it merely as indicating that, to keep a body moving in 
 a curve instead of in its natural straight line, a force 
 directed towards the centre of curvature is always required. 
 But, as the third law of motion (§ 128) tells us, a force is 
 only one-half of a stress, so that when force is exerted 
 to pull the body inwards from the tangent, an equal force 
 must be exerted at the centre tending outwards from it. 
 
 We might quite as justly speak of the Onward Fwce 
 of a cannon-ball, which requires a resistance to check it ; 
 as of the centrifugal force (understood not as a single 
 term but as two words, each with its ordinary meaning) 
 which must exist because it requires centripetal force 
 to balance it. 
 
 118. Calculating (as in § 71) from the earth's mean 
 equatorial radius, 3962 miles, and the number of seconds 
 in a sidereal day, 86164, we find that the acceleration of 
 a point on the equator is about 0'1116 feet per second. 
 Thus about ^i^^th of its weight is required merely to 
 
 H 
 
98 PROPERTIES OF MATTER. 
 
 a body on the earth's surface at the equator. By 
 this amount its weight (as indicated by a Bprmg-halcmce) 
 would be diminished. 
 
 If the earth had been revolving seventeen times faster 
 than it does, this apparent diminution of weight would 
 have been 17^ (or 289) times greater than it is, i.e. bodies 
 at the equator would have shoTvn no apparent weight, pro- 
 vided they moved along with the ground below them. 
 
 119. As is pointed out in the preceding extract from 
 Newton (§ 116), a wheel or other body, rotating about 
 an axis and not acted on by forces, perseveres by inertia 
 in its uniform rate of rotation. But it does more; it 
 preserves (even when acted on by small forces) the 
 direction of its axis of rotation, provided at least that it 
 be rotating about its axis of greatest or of least moment 
 of inertia (§ 132). It is for this reason that rifling of 
 the bore of a gun has been introduced ; and also that a 
 skilled player, when throwing a quoit, gives it rotation 
 in its own plane. 
 
 The rotation of the earth about its axis is a more 
 complex phenomenon, because it takes place under the 
 action of considerable forces which tend to make the earth 
 revolve about axes lying in its equator. Yet, because 
 the moments of inertia (§ 132) about aU such axes are 
 approximately equal, the period of the daily rotation is not 
 altered, though the direction of the polar axis is affected 
 by Precession and by Nutation. We cannot, however, 
 do more than allude to matters of this order of diflBculty. 
 They are all beautifully illustrated by means of Gyro- 
 scopes, Gyrostats, etc., but the full study of the pheno- 
 mena requires higher mathematics than we can introduce 
 here. These are properly questions of Abstract Dynamics. 
 
INERTIA— CESTTEIFUGAL FOKCE. 99 
 
 120. Newton's Second Law of Motion is as 
 follows : — 
 
 Ghcmge of momentum is proportional to force, and iaJces 
 place in the direction in which the force acts. 
 
 When a body (however massive) is free, every force 
 (however small) produces its proper change of momentiun. 
 This used to be stated, under the name of Mohility, as a 
 characteristic property of matter. 
 
 As change of momentum alone is mentioned, it is clear 
 that Newton means that the effect of a force is independ- 
 ent of the previous motion of the body. Hence if a force 
 be uniform, as for instance is practically the case with a 
 falling body, the additional momentum produced by it 
 in one second will measure its amount. But if it be 
 variable, we must use (instead of the momentum produced 
 by it in one second) the rate at which momentum is pro- 
 duced by it. Thus the true measure of a force is the 
 rate of change of momentum ; or, to use the kinematical 
 term, the product of the mass of a body into the acceleror 
 tion of its velocity. 
 
 It is foimd that, in vacuo, all bodies acquire, per 
 second, an additional vertical velocity of about 32 -2 
 feet per second. This quantity (which varies with the 
 latitude, height above sea-level, etc.) is usually denoted 
 by the letter g. Hence, if M be the mass of a body, 
 its weight {i.e. the force which accelerates its fall) is 
 measured by the product M.g. 
 
 121. Now kinematics (as we saw in § 71) show us 
 that when a point moves uniformly in a circle, the 
 acceleration is directed inwards to the centre, and its 
 magnitude is the square of the speed multiplied into 
 the curvature of the path. Hence to keep a body, of 
 
100 PROPERTIES OF MATTER. 
 
 mass M, moving with uniform speed V in a circle of 
 radius E, a force whose magnitude is MV^/E, directed 
 towards the centre of the circle, must constantly act 
 upon it. 
 
 If a be the Angular Velocity in the circular path, i.e. 
 the angle described in unit of time by the radius drawn 
 to the moving body, we have obviously 
 
 V = Ra), 
 
 and the expression for the requisite force takes the form 
 MRw2. 
 
 122. Newton shows that, as an immediate consequence 
 of this law, we have the Law of OomposUion of Forces act- 
 ing at one point ; the so-called Parallelogram, or Triangle, 
 of Forces. This follows from the facts that the changes 
 of velocity produced in the same time, by different 
 forces acting on the same body, are (a) proportional to and 
 in the directions of the several forces, and (6) that the 
 effect of each force is independent of the simultaneous 
 action of others. Thus the problem is reduced to the 
 obvious kinematical composition of velocities (§ 69) . 
 
 123. We are now prepared to measure both Masses 
 and Forces. But, for this purpose, it is necessary to have 
 units in terms of which to measure. 
 
 The British unit of mass is the Standard Pownd, 
 whose amount was probably adopted in old times for 
 reasons of convenience, but is now fixed by law. 
 
 The French, or Metrical, unit of mass is the Kilo- 
 gramme, originally intended to be the mass of a cubic 
 decimfetre, or litre, of water at its maximum density 
 point, but, practically, defined by a platinum standard. 
 The Kilogramme is about 2 '20462 Pounds. 
 
INERTIA— CENTRIFUGAL FORCE. 101 
 
 124. The British Unit of Force is such that, when it 
 acts for one second on a mass of one pound, it produces 
 in it a speed of one foot per second. 
 
 The 0. G. S. Unit of Force (or Dyne), which is coming 
 into use for scientific measurements, is such that, when 
 it acts for one second on a mass of one gramme, it pro- 
 duces in it a speed of one centimetre per second. 
 
 The British Unit of Force is about 13,825 Dynes. 
 Since the speed acquired by a body faUing for one 
 second (im vacuo) is (§ 120) about 32 '2 feet per second, 
 the Weight of a Pound is about 445,165 Dynes. 
 
 125. Generally, if a definite force act upon any mass 
 for one second, it will generate in it a speed whose 
 magnitude is inversely as the mass. 
 
 Thus the comparison of masses, i.e. their measure- 
 ment in terms of some standard unit, becomes a perfectly 
 definite scientific process. 
 
 It may not be easy to carry out, and in fact it is not ; 
 at least by any very direct application of the principles 
 just explained. That, however, is another matter. No 
 one in his senses would object to the perfectness of 
 Euclid's process for dividing a straight line into a given 
 number of equal parts, on the ground that it is practi- 
 cally inapplicable when we try to carry it out. 
 
 What we have sought is an accurate, and an easily 
 intelligible, method of comparing masses. If it be not 
 easily workable in practice, we must find something more 
 workable ;— -just as we now use a screw dividing-engine 
 instead of Euclid's unrealisable straight lines, and still 
 more unrealisable parallel straight lines. 
 
 126. When we have measured the mass of a body, 
 and also its volume (§ 92), its a/eerage, or mean, density ■ 
 
102 PROPERTIES OF MATTER. 
 
 follows at once. If the body be homogeneous, this ia 
 its actual density throughout : — and whether it be homo- 
 geneous or not, its mean density is simply the average 
 amount of mass per unit of volume. 
 
 On account, chiefly, of the remarkable result of 
 Newton's (§ 34), we postpone all considerations regard- 
 ing the densities of various kinds of matter until we are 
 dealing with their specific gravities also. 
 
 127. The process of weighing is, as Newton showed 
 (§ 34), essentially a comparison of masses. So that our 
 measurement of mass is practically carried on by means 
 of the Bakmce, which is one of the most delicate and 
 accurate instruments of precision yet invented. 
 
 The processes for measuring force are not yet nearly 
 so accurate. Numerous instruments have been devised 
 for the measurement of special classes of forces, the great 
 majority depending upon elasticity of matter. Some of 
 the more important of these will be mentioned when we 
 require them ; but the reader must be reminded that the 
 true measwe of a force is the momentum it produces in 
 one second. 
 
 128. The first two laws of motion (applied with suffi- 
 cient mathematical resources) enable us to solve any 
 question whatever regarding the motion of a body, 
 treated as a mere particle, under the action of any given 
 forces. Conversely, they enable us, from the given 
 motion of a particle, to find the forces under which the 
 motion has taken place. But they do not suffice for 
 the calculation, of the motion of two or more particles 
 which mutually influence one another, whether by 
 gravitation, or cohesion, or by any physical mode of 
 attachment. Hence the necessity for the 
 
INERTIA— CENTRIFUGAL FORCE. 103 
 
 Third Law of Motion. 
 
 To every action there is always an equal and contrary 
 reaction ; or, the mutual actions of amy two bodies are always 
 egual and oppositely directed. 
 
 In modern speech Newton's jirst explanation of the 
 sense in which this statement is to be understood may 
 be simply expressed thus : — 
 
 Every action between two bodies is a Stress. 
 
 In this sense it is very closely connected with the 
 first law. For a system of two bodies, considered as one, 
 cannot set itself in motion. Even when the masses are 
 not connected in any way, the equality of action and 
 reaction involves transference of momentum between 
 them which leaves the motion of their Centre of Inertia 
 unaffected. It was by floating on water a magnet and a 
 piece of iron, both attached to the same light board, that 
 Newton proved the equality of action and reaction for 
 magnetic force. He also proved it by showing that, if 
 the gravitation of one part of the earth on the rest were 
 not exactly reciprocated, the earth (as a whole) would 
 alter its existing state of motion. 
 
 129. The term centre of inertia, employed above, re- 
 quires a few words of explanation. We assume the 
 following proposition, which can be established by very 
 elementary mathematics. 
 
 In every group of massive particles there is one 
 definite point, such that, whatever plane be drawn in 
 space, the distance of the point from that plane, multi- 
 plied by the sum of the masses, is equal to the sum of 
 the separate products formed by multiplying each mass 
 by its distance from the plane. It can, therefore, be 
 found by executing the requisite calculations for any 
 
104 PROPERTIES OE MATTER. 
 
 three planes whatever, provided no two of them are 
 parallel. This is the centre of inertia of the particles. 
 
 When the particles are severally acted on by forces, 
 it follows from Newton's Third Law that the centre of 
 inertia of the group moves as if the whole mass were 
 there concentrated, and acted on by all the forces simul- 
 taneously. This consideration greatly simplifies kinetical 
 problems connected with a rigid system or a group of 
 particles ; for it enables us to determine first, the niotion 
 of the centre of inertia as if it were a mere particle, and 
 then to study the motion relatively to that centre. 
 
 130. But Newton proceeds to point out that there is 
 a second sense in which the terms action and reaction in 
 the Third Law may be interpreted. In modem phrase 
 it may be expressed as 
 
 The activity of an agent {or the rate at which it does Work), 
 is equal to the counter-activity of the resistance. 
 
 Newton's statement of this second mode of interpret- 
 ing the Third Law has been shown to require compara- 
 tively little addition to make it a complete enunciation 
 of the Conservation of Energy (§ 7). 
 
 131. If two masses be connected by a spiral spring, 
 but be otherwise free, and if the spring remains stretched 
 to a constant amount, this can only be because the bodies 
 are revolving about one another. For, otherwise, the 
 stress in- the spring would have caused them to approach. 
 This is Newton's test of the existence of absolute rotation. 
 For, by the first law, the stress which certainly acts on 
 the masses must interfere with their states, and by the 
 third law it must do so in opposite directions. Each, 
 therefore, must be describing a curved path relatively to 
 the other, and this must of course be circular. 
 
INERTIA— CENTKIFUGAL FORCE. 106 
 
 Nothing is known, or even conceivable by the most 
 transcendental of metaphysicians, which could give us an 
 indication of absolute translation. 
 
 132. We conclude the chapter with a few additional 
 illustrations and explanations connected with inertia. 
 
 In § 119 we introduced, without explanation, the 
 iniportant term Moment of Inertia. This quantity is 
 defined, for any body, with reference to any assigned 
 axis. It is the sum of the products obtained by multi- 
 plying the mass of each small portion of the body by 
 the square of its distance from the axis. 
 
 Its use is twofold. If co be the angular velocity of 
 a rigid body about an axis, r the distance of the particle 
 whose mass is m from that axis, the speed of m is rca, 
 and the kinetic energy of rotation (half the product of 
 each part of the mass into the square of its speed) is 
 
 half the product of the moment of inertia into the square 
 of the angular velocity. 
 
 Again, the Moment of the momentum of a particle about 
 an axis is defined as the product of the momentum by 
 the shortest distance between the axis and the line of 
 motion of the particle. Hence the Moment of momentum 
 of m about the axis is mrw.r, and the whole Moment of 
 momentum of the body is 
 
 S(TOr^) . w = Iw, 
 
 the product of the moment of inertia into- the angular 
 velocity. 
 
 133. It is shown in treatises on Dynamics that the 
 effect of a pair of equal and opposite forces, whose lines 
 
106 PROPERTIES OP MATTER. 
 
 of action are different (called by Poinsot a Couple) is to 
 produce Moment of momentum in proportion to the time 
 it acts and to the moment of the couple. Hence, if Q 
 be the moment of the couple, at the angular velocity it 
 produces in time t, when its plane is perpendicular to 
 the axis above spoken of, 
 
 Iw = 2;(mr')ai = Qi, 
 whence 
 
 iI«2 = P(mriX = Q.-^=Qa, 
 
 where a is the angle through which the body has turned. 
 For ft) grows uniformly, and therefore its average value 
 during the time t was <»/2, so that the whole angle de- 
 scribed is <i)</2. 
 
 But if a force P act on a particle, of mass M, and pro- 
 duce in time t a speed v, we have 
 
 Ml) = VI. 
 The speed increases uniformly, so that its average value 
 is v/2, and therefore the space described is s = vi/2. 
 Henee, by multiplying both sides by v/2, we get 
 
 ii'M.v' = Ps. 
 
 It is obvious that in the former pair of equations the 
 quantities I and Q, co and a, play exactly the same parts 
 as do M and P, v and s, respectively, in the latter pair.. 
 
 This analogy shows, at least in part, the great con- 
 venience of the idea of the moment of inertia. 
 
 For special purposes we often write I in the form 
 MB, k being then the common distance from the axis at 
 which every one of the particles must be placed, so that 
 the whole may have the same moment of inertia as before. 
 It is called the Radms of Gyration. 
 
INERTIA— CENTKIFUGAL FORCE. 107 
 
 134. As an illustration of the application of the two 
 interpretations of the third law, suppose a fly-wheel to be 
 carefully mounted on friction rollers, and set in rotation 
 by the descent of a weight attached to a string wound 
 round its axle. 
 
 Let tt) be the angular velocity produced in the fly- 
 wheel when a length x of the cord has been unwound, a 
 the radius of the axle, M the mass of the appended 
 weight, I the moment of inertia of the wheel, and T the 
 stress in the cord. 
 
 Then the rate of increase of momentum of the mass 
 M is Mx (with Newton's notation, § 72). This must be 
 the measure of the force producing it, so that 
 
 M.x = Mg-T! . . . . (1.) 
 
 The rate of increase of moment of momentum of the 
 fly-wheel is la, which must measure the couple produc- 
 ing it. Hence 
 
 Iu = Ta (2.) 
 
 But aca is the amount of cord unwound per second, 
 i.e. the rate of descent of the weight. Thus 
 
 au = x (3.) 
 
 (1) and (2) are dynamical equations, (3) is kinemati- 
 cal. X, w, and T are to be found from the three. They 
 give 
 
 Mfflii + I ■ 
 
 If the wheel had no moment of inertia this would 
 become 
 
 X = g, 
 
 the ordinary equation of acceleration of a free falling body. 
 
108 PROPERTIES OF MATTER. 
 
 Hence, the only effect of the fly-wheel is to diminish 
 the effect of gravity on the weight in the proportion 
 Ma^ : (Ma^ + I). The measure of the stress on the 
 cord is ,,- 
 
 Ma^ + I 
 
 and it therefore remains the same throughout the motion. 
 
 135. But the solution of the same problem, by the 
 help of Newton's second interpretation of the third law, 
 is far more simple. 
 
 The rate at which the agent (the weight of the fall- 
 ing body) is doing work is, at any instant, 
 
 Wgis. 
 
 The rate at which energy is being gained by the fall- 
 ing body is Max. The rate at which energy is gained 
 by the fly-wheel is lead). 
 
 Hence 
 
 Mgx = Maa; + luw, 
 or by (3) 
 
 Mga? = Ma^x + Ix, . . . . (4.) 
 
 which is the same equation as before. 
 
 136. But, instead of reckoning rates of transference of 
 energy, we may still more simply proceed by the whole 
 amount of energy in the system (§ 7). 
 
 The falling body has hst Mgx, and has gained ^MxK 
 The fly-wheel has gained ^Ico^. Hence 
 
 Mgx = JMa2 + JI<^, 
 or by (3) 
 
 which is the fluent, or integral, of (4) 
 
INERTIA— CENTRIFUGAL FORCE. 109 
 
 137. If we consider these three solutions of the same 
 problem, we see that, while the stress between the 
 members of the system plays a prominent part in the 
 first, it is altogether unnoticed in the two latter. 
 
 This might, at first sight, tend to induce us to ignore 
 stress altogether ; and, undoubtedly, we can do so in all 
 cases, except when we study the condition of the intervening 
 mediwm, while energy is stored in any part of it ; or while 
 energy is being transferred through it from one part of the 
 system to another. The consideration of this part of the 
 subject is deferred to our chapters on Elasticity. 
 
CHAPTEE VII. 
 
 GRAVITATION. 
 
 138. Without preface we simply give a statement, 
 compounded from various parts of the Prindpia (espe- 
 cially the Third Book), which comprehends all the essen- 
 tials of NeT/vion's great generalisation. 
 
 Every particle of matter in the vmiverse attracts every other 
 particle with a force whose direction is that of the line joining 
 the two, and whose magnitvde is directly as the product of 
 their masses, and viwersely as the square of their dista/nce 
 from each other. 
 
 This statement divides itself, for proof, into a number 
 of separate heads ; as follows : — 
 
 (a) The Universality of Gravitation. 
 
 (J) The direction of the force between two particles. 
 
 (c) The proportionality of the force to the product of 
 the masses. 
 
 (d) The law of the inverse square of the distance. 
 Besides these more immediate questions the state- 
 ment also raises the questions — 
 
 (e) What do we mean by "attraction"? 
 (/) What is the cause of gravitation ? 
 
 And other matters of great importance naturally 
 
GRAVITATION. Ill 
 
 present themselves, such as, " What is the mass of the 
 Earth," etc. ? 
 
 These questions must be kept before us, so that we 
 may give to each of them (so far as our knowledge yet 
 extends, and so far as is consistent with the scope of 
 this work) a sufficient answer. (/) is still an open 
 question, for the attempts at answering it have not yet 
 been very successful, (a) of course can only be answered 
 either in an wpproxmate or in an indirect manner, because 
 we cannot (by our most delicate instruments) even prove 
 the existence of gravitation-attraction between two par- 
 tides of matter. Here, however, we tread (as will be 
 seen) on comparatively safe ground. 
 
 And the same may be said for (J), (c), and {d), because 
 the reasoning and experiment which sufficiently answer 
 (a) will be found here even more complete. («) will be 
 discussed along with (/). 
 
 139. (a) One strong argument for the universality of 
 gravitation is that the weight of a body is the sum of the 
 weights of its parts. This is, of course, a matter which 
 can be tested to a very great degree of accuracy by 
 means of the balance. Thus each particle of the body 
 contributes its share to the weight of the whole. 
 
 And the weight of a given quantity of matter does 
 not depend upon its form. A mass of gold retains 
 exactly the same weight when it is beaten out into the 
 finest leaf. Thus gravity acts as freely upon the particles, 
 when they are surrounded on all sides by the solid mass, 
 as when they are directly exposed by the beating. 
 
 In fact, it is quite easy to see that, were this not the 
 case, were it, in fact, possible to find a screen through 
 which gra/vity could not act, i.e. were it possible to interfere 
 
112 PROPERTIES OF MATTER. 
 
 with the universality of gravitation, we should also be 
 able to produce The Perpdual Motion : — an inexhaustible 
 source of new energy. This we know (§ 7) cannot be. 
 
 To show, however, that the above hypothesis would 
 lead to this result, we have only to think of a fly-wheel, 
 one part of which shall be screened from the earth's 
 attraction, the rest unscreened. Every part loses weight 
 as soon as it enters the shadow, as it were, of the screen, 
 and gains it again when it emerges. Thus the wheel 
 constantly gains energy from nothing. 
 
 The wheel would, in fact, become a treadmill : — work- 
 ing of itself, instead of by the hard labour of a gang of 
 prisoners climbing, without mounting, up one side. 
 
 140. (ft) continued. Newton attacked the question by 
 assuming the law of gravitation for the separate particles 
 of a body, and thence finding what should be the law 
 of attraction towards the body as a whole. He thus 
 arrived at two exceedingly beautiful results. 
 
 A spherical shell of uniform gravitating matter exerts no 
 attraction on a particle within it. 
 
 [For the proof of this, and of the succeeding proposi- 
 tion, we assume the following results of pure mathe- 
 matics : — 
 
 The area of a transverse section of a cone of small 
 angle is proportional to the square of its distance from 
 the vertex. 
 
 The measure of the spherical angle of such a cone is 
 the area it cuts off from the unit sphere whose centre is 
 its vertex; which is the same as the area of the transverse 
 section at unit distance from the vertex. 
 
 An oblique section has greater area than the trans- 
 verse section, at the same distance from the vertex, in 
 
GRAVITATION. 
 
 113 
 
 proportion to the secant of their incHnation to one 
 another.] 
 
 Take any point B, within the spherical shell, and 
 let it be the vertex of a double cone of exceedingly 
 small angle. This cuts out two minute areas on the 
 spherical surface, obviously at equal inclinations to the 
 axis of the cone. Hence their areas, and therefore their 
 masses, are as the squares of BP, BQ. But their attrac- 
 tions on B are inversely as the squares of BP, BQ. 
 Thus these attractions balance one 
 another. And the whole shell 
 may thus be divided into pairs 
 of parts, whose attractions exactly 
 balance one another on B. Hence 
 the proposition, which is obviously 
 true of any uniform shell, how- 
 ever thick, if only bounded by 
 concentric spheres. And it is 
 true, if the shell be made up of 
 concentric layers of different densities, provided the 
 density of each layer is uniform. 
 
 No other law than that of gravitation is capable of 
 giving this result. 
 
 141. ^ spherical shell of uniform gravitating matter 
 attracts an external particle as if its whole mass were con- 
 densed at its centre. 
 
 Let A be the external particle, C the centre of the 
 shell. Cut off OB, a third proportional to OA, CD ; and 
 divide the shell by small double cones whose vertices are 
 at B. , Let PBQ be such a cone. Then if m be its 
 spherical angle, the areas of the sections at P and Q are 
 
 BP^a sec CPB, BCi'a sec CPB, 
 I 
 
114 PROPERTIES OF MATTER. 
 
 and their attractions are 
 
 BP2. w sec CPB ^ BQ2. u sec CPB 
 
 IF ''- ^""^ W ''■ 
 
 ■where p is the surface-densiiy, i.e. the mass per unit area. 
 
 Flo. 12. 
 
 But the geometry of the figure shows us at once that 
 
 <CPB = <PAD = <QAD, and BP : AP : : CP : AC : : 
 
 BQ : AQ. Hence the elements at P and Q attract A 
 
 equally, and their resultant is therefore along AC. Its 
 
 value is „ „ 
 
 2CP^wp 
 
 AC2 • 
 
 in which the multiplier of a is constant ; i.e. each portion 
 of the shell produces a share, of the whole attraction 
 along AC, proportional to the angle it subtends at B. 
 
 The sum of all possible values of w is the area of the 
 surface of the unit hemisphere, i.e. 27r. Hence the whole 
 attraction is , ^pa 
 
 Now 47rCP^ is the area of the shell, so that this expres- 
 sion is merely Mass of shell 
 
 Square of distance from centre ' 
 and the proposition is proved. 
 
GRAVITATION. 115 
 
 It can at once be extended, as the former was, to 
 a mass made up of concentric shells of different densities, 
 provided each have the same density throughout. 
 
 No other law of force, except the law of the direct 
 distance, gives this result. 
 
 142. Hence a uniform spherical shell, or a mass made 
 up of uniform concentric shells, has a true Centre of 
 Gravity, so far as bodies external to it are concerned ; for 
 it attracts, and therefore is attracted by, all external 
 bodies, as if it were condensed in its centre. 
 
 It is only a limited class of bodies which have a true 
 centre of gravity in the sense just explained. When 
 such a point exists, it always coincides with the centre 
 of inertia, as we see at once by supposing the body 
 which attracts it to be so distant that its action on 
 different parts of the attracted body is in parallel lines, 
 and proportional simply to the relative masses : — and, 
 for many purposes, it is sufficiently accurate to assume 
 that the centre of inertia may be treated as a centre 
 of gravity. But we must beware of making too free a 
 use of this hypothesis. If, for instance, the earth had 
 a true centre of gravity, and were rotating about its 
 axis of greatest moment of inertia, there could be neither 
 Precession nor Nutation. 
 
 143. (a) mntirmed. Aimed with these results, Newton 
 was justified in dealing with spherical masses, such as 
 those of the sun and planets, as if each had been a mere 
 particle, condensed at its centre. And here he had the 
 benefit of the altogether extraordinary labours of Kepler ; 
 who, by sheer guessing, often of the wildest kind but 
 followed up by persevering calculation, had reduced to 
 a few simple statements the chief Idnematical results 
 
116 PROPERTIES OP MATTER. 
 
 deducible from the observations of Tycho Brahe. These 
 are given in Kepler's work, Be Motihus Stellce Martis, 
 Prague, 1609, and are now universally designated 
 
 Kepler's Laws. 
 
 I. Each planet describes an Ellipse (with comets this 
 may be any Conic Section) of which the Sun occupies 
 one focus. 
 
 II. The radius-vector of each planet describes equal 
 areas in equal times. 
 
 III. The square of the periodic time (in an elliptic 
 orbit) is proportional to the cube of the major axis. 
 
 144. (J) Newton showed that, as an immediate conse- 
 quence of II. above, the direction of the attraction of the 
 sun for a planet must be that of the line joining their 
 centres. 
 
 In fact, double the area described by the radius-vector 
 of a planet in one second is the moment of its velocity about 
 the sun's centre. But the moment of the resultant of 
 two velocities is the sum of their separate moments. 
 Hence, as the moment of the planet's velocity remains 
 the same, the moment of each successive increment which 
 it receives must be nil, i.e. these increments must be 
 directed towards the sun's centre. 
 
 We may prove this part also of the law of gravitation 
 by showing that, were it not true. The Perpetual Motion 
 would be attainable. But the reader may easily make 
 out this proof for himself. 
 
 145. (c) That the attraction varies directly as the pro- 
 duct of the masses will be proved at once if it be shown 
 to be proportional to one ^f the masses while the other 
 remains constant. For it must be remembered that, by 
 
6EAVITATI0IT. 117 
 
 the third law of motion (see § 128), gravitation-attraction 
 is mulml ; each of the two attracting bodies has as much 
 of a share in producing it as has the other. It is clear, 
 then, that the proof of this part of the law will be obtained 
 at once if we can show that the weights of bodies are, in 
 any and every one locality, proportional to their masses. 
 
 We have seen that the measure of a force is the 
 momentum it produces in one second. Submit a number 
 of bodies to the action of their own weights alone, each 
 will acquire in one second a momentum proportional to 
 its weight. But if the weight be proportional to the 
 mass, the momentum must also be proportional to the 
 mass, and thus the speed acquired must be the same 
 for all. That is, if they be under the action, each of its 
 own' weight alone, they will fall side by side through 
 any space whatever. Now this is known to be very 
 nearly the case when we let stones or bullets, or even 
 lumps of wood, fall ; while it is obviously not so vrith 
 feathers, paper, or gold leaf. But these exceptions show 
 at once why the trial is not a fair one. The falling 
 bodies are all resisted by the air, some only slightly, 
 others with forces not much less than their whole weights. 
 Hence, to make the experiment as nearly as possible free 
 from such interfering causes, Newton made the fall ex- 
 tremely slow, but in such a way that- it could be repeated 
 over and over again under precisely similar circumstances, 
 and therefore its period could be measured very exactly. 
 He used, as the bob of a simple pendulum, a light hollow 
 shell which could be filled successively with different 
 kinds of matter. 
 
 In Book II. sec. vi. prop. xxiv. of the Principia, he 
 proves that the mass of the bob of a simple pendulum 
 
118 PROPERTIES OF MATTER. 
 
 of given length is directly as its weight and as the square 
 of its time of oscillation in vacuo. And, in the 7th 
 Corollary to this proposition, we read : — 
 
 " Hence appears a method both of comparing bodies 
 one among another, as to the quantity of matter in each, 
 and of comparing the weights of the same body in differ- 
 ent places, to know the variation of its gravity. And, 
 by experiments made with the greatest accuracy, I have 
 always found the quantity of matter in bodies to be 
 proportional to their weight." 
 
 Thus gravity depends on the quantity, but in no way 
 on the qudity, of the matter in a body ; and it is in all 
 cases attractive. In these respects it stands in marked 
 contrast to magnetic forces. 
 
 146. id) An immediate deduction, from the first two of 
 Kepler's Laws, is that the Hodograph (§ 70) of a planet's 
 orbit is a circle. For the moment of the velocity, V, of P, 
 about S, is constant (§ 144). Let fall the perpendicular 
 SQ on the tangent at P, then Q hes on the circle whose 
 diameter is* the major axis of the orbit. V. SQ is con- 
 stant. But if QS cut the circle again in E, SE.SQ is 
 constant. Thus SR is proportional to V. Hence SR is 
 drawn from a fixed point S, in a direction perpendicular 
 to the motion of P, and its length is proportional to the 
 speed of P. The locus of E, the auxiliary circle, is 
 therefore the hodograph turned through a right angle. 
 The tangent at E, which is the direction of the accelerar 
 tion of SE, is therefore perpendicular to SP. [In fact 
 CE is parallel to SP, by a property of the ellipse.] The 
 magnitude of the acceleration of P is proportional to the 
 speed of E, i.e. proportional to the angular velocity of 
 the tangent at E, i.e. to the angular velocity of SP. But 
 
GRAVITATION. 
 
 119 
 
 the moment of P's velocity, about S, which is constant, 
 can also be expressed as the product of SP^ into the 
 angular velocity of SP. Hence the angular velocity of 
 
 SP, and therefore also the acceleration of'P, must be 
 inversely proportional to SP^. 
 
 147. The detailed investigation is easily given : 
 thus, if SP = r, <A'SP = 6, and if h be twice the area 
 described by SP in unit of time, 
 
 But SQ.T=A, 
 
 while SQ.SR=AS.SA' = BC2, 
 
 where BO is the semi-axis minor of the ellipse. 
 
 Thus SR = ?g'.V. 
 
 h 
 
120 PROPERTIES OF MATTER. 
 
 But, on the same scale, the acceleration of P is measured 
 by the velocity of E, which is CE. 6, or CA. 6. 
 Hence actual acceleration of P is 
 
 ±- CA fl = ^'-C-^' 1 
 BC2" ■ BC^.CA^ra" 
 
 Now twice the area of the ellipse is 27rBC.CA; and, 
 if T be the periodic time, it must also be A.T. Hence 
 
 Acceleration of P = — =5=— ^ -= . 
 
 Kepler's third law tells us that -=^ is the same for all 
 
 the planets. Hence we conclude that it is the sajm 
 gravitation, diminishing as the square of the distance 
 increases, which acts on each one of the planets. 
 
 148. The result of § 146 might at once have been ob- 
 tained from Kepler's third law. For if we suppose the 
 orbits of the planets to be circles (which they are ap- 
 proximately) that law gives 
 
 T2 oc R', 
 
 where T is the periodic time, E the radius of the circle. 
 But, if V be the planet's speed in its circular orbit, we 
 have the Unematical result 
 
 
 
 y2T2 
 
 OcR'i. 
 
 From the two 
 
 we 
 
 obtain 
 
 
 
 
 V2 
 
 R 
 
 -i 
 
 i.e. the accelerations are inversely as the squares of the 
 distances. But it is better to derive, as Newton did, 
 the law of inverse squartf*from the two first of Kepler's 
 laws ; and then the third gives us the further information 
 
GRAVITATION. 121 
 
 that every planet behaves exactly as any other would 
 do if substituted for it, ie. that the sun's gravity pays 
 no attention to the quality of matter. 
 
 149. Having found that, in these general matters at 
 least, the assumed law, of gravitation is in agreement 
 with the planetary motions, Newton turned to par- 
 ticulars, and the special one which he took as a test was 
 the moon's rotation about the earth. He says : — 
 
 " That the circumterrestrial force likewise decreases 
 in the duplicate proportion of the distances, I infer thus. 
 
 " Let us then assume the mean distance of the moon 
 60 semi-diameters of the earth, and its periodic time in 
 respect of the fixed stars 27^ 7^ 43" as astronomers 
 have determined it. And a body revolved in our air, 
 near the surface of the earth supposed at rest, by means 
 of a centripetal force which should be to the same force 
 at the distance of the moon in the reciprocal duplicate 
 proportion of the distances from the centre of the earth, 
 that is, as 3600 : 1, would (secluding the resistance of 
 the air) complete a revolution in l'' 24™ 27^ 
 
 "Suppose the circumference of the earth to be 
 123,249,600 Pwris feet, as has been determined by the 
 late mensuration of the French, then the same body, 
 deprived of its circular motion, and falling freely by 
 the impulse of the same centripetal force as before, 
 would, in one second of time, describe ISJ^- Paris feet. 
 
 "This agrees with what we observe in all bodies 
 about the earth. For by the experiments of pendulums, 
 and a computation raised thereon, Mr. Euygens has 
 demonstrated that bodies falling by all that centripetal 
 force with which (of whatever nature it is) they are 
 
122 PROPERTIES OP MATTER. 
 
 impelled near the surface of the earth, do, in one second 
 of time, describe 15^ Paris feet." 
 
 The comparatively accurate measurement, of the length 
 of a degree of latitude on the earth, by Picard was un- 
 doubtedly the cause which ultimately led to the publica- 
 tion of the Prmcipia, of which the fundamental proposi- 
 tions had been obtained nearly twenty years before. 
 For Newton, using the rough estimate of 60 miles to 
 a degree, had found that the moon's deflection by gravity, 
 in one second, from a rectilinear path, was not g g^^ ^ th 
 of the space through which a stone falls in one second 
 at the surface of the earth, and had in consequence put 
 his investigations aside, until he was led to resume them 
 by hearing the result of Picard's measures. 
 
 150. Having established the law of gravitation by 
 calculations founded mainly on Kepler's laws, Newton 
 proceeded to show that these laws could not themselves 
 be accurate. For a single spherical planet, revolving 
 about a spherical sun, the first two laws would still be 
 true, but a second planet would at once interfere with 
 this state of matters : — the orbits would no longer be 
 ellipses, and equal areas would no longer be described 
 in equal times. Again, the third law could never be 
 exactly true, even if the planets did not attract one 
 another, unless they contained each the same fraction of 
 the sun's mass. But the consideration of questions like 
 these belongs to Physical Astronomy, with which we 
 have nothing to do here. SuflS.ce it to say that Newton's 
 own magnificently -extended deductions', supplemented 
 as they have been by those of successive generations of 
 illustrious mathematicians, have verified already to a 
 very high degree of nicety the competence of the law of 
 
GRAVITATION. 123 
 
 gravitation to account for the excessively complex mo- 
 tions and perturbations observed in the solar system. 
 
 131. We have already (§118) adverted to the apparent 
 loss of weight by bodies at the equator. This loss, due 
 to the so-called Centrifugal Force, is, of course, directly 
 proportional to the mass of each body. But experiment 
 with the most delicate balances has shown that bodies 
 of any kind which equilibrate in one latitude equilibrate 
 in all. Hence their weights remain equal when, from 
 that of each, is subtracted an amount proportional to 
 the mass. This can only be if the weights are them- 
 selves proportional to the masses. Thus we have an 
 independent proof of the truth of clause (c) of Newton's 
 statement. 
 
 152. We can scarcely yet be said to have proof that 
 gravitation exists, as we know it, in stellar systems. For 
 the data, from which to calculate orbits of double stars, 
 have to be obtained under circumstances which do not 
 admit of more than rude attempts at approximation. We 
 know that there are hundreds of systems in which two or 
 more stars revolve about one another in away which leaves 
 no doubt that they axe physically connected. But the obser- 
 vations which have as yet been made have been applied, 
 not to prove that the relative orbits are consistent with 
 Kepler's laws but, to find the approximate forms of the 
 orbits, and thence the amounts of matter in the mutually 
 influencing bodies, on- the supposition that Kepler's laws 
 hold even in these remote systems. 
 
 Thus we cannot, at least for the present, look for 
 proof of the universality of gravitation in this direction. 
 But we have ample direct proofs that parts of the earth, 
 and not merely the earth, as a whole, exert gravitating 
 
124 PROPERTIES OF MATTER. 
 
 force. Some of these will be considered in the immedi- 
 ately succeeding sections. 
 
 153. The most direct of these (though not the earliest) 
 is what (though devised by Michell) goes by the name of 
 
 The Cavendish Experiment. 
 In this, by means of the elasticity of a wire, the attrac- 
 tion between two spheres of manageable size is not only 
 demonstrated, but measured. The following sketch 
 shows a horizontal section through the main parts of the 
 arrangement. 
 
 Two small balls, A and B, an inch or two in diameter, 
 are connected by a stiff, but very light, horizontal girder 
 or tube, which is suspended at its middle point (E) by a 
 long fine wire. The whole of this part of the apparatus 
 is enclosed in a case, carefully coated with tinfoil or gold- 
 leaf, to prevent (as far as possible) irregular heating and 
 consequent currents of air ; perhaps, also, slight electrifi- 
 cation. To the girder is attached a small mirror, whose 
 plane is vertical. A little glazed window in the case 
 allows any motion of the mirror to be measured by the 
 consequent deviations of a ray of light reflected by it 
 
 Outside the case are placed two equal, but much 
 more massive, spheres, usually balls of lead a foot or 
 more in diameter, so mounted that they can be made to 
 move (without jerk of any kind) from the positions Cj, 
 Dj to the positions Og, D2 and Og, D3, or back again, 
 at will. [In Comu's recently- constructed apparatus 
 there are four spherical iron vessels, of equal size, placed 
 once for all at Cj, C3, Dj, Dg, and so connected, two and 
 two, that Cj or Dg, and simultaneoiisly Dj or Cg, may be 
 filled with mercury, the other of each pair being left 
 empty. All four can be left empty when required.] 
 
GRAVITATION. 
 
 125 
 
 Cavendish, and all who have since made the experi- 
 ment, found that the apparatus was never at rest. In 
 
 -oB 
 
 D; 
 
 order to determine the equilibrium position it was 
 necessary, therefore, in all cases to measure the KmMs 
 of successive oscillations, and to compare the mean of 
 two successive deflections to one side, with the interven- 
 
126 PROPERTIES OF MATTER. 
 
 ing deflection to the other side. The time of each oscil- 
 lation was also carefully measured. 
 
 When the large masses were placed at Cj, Dg, in a 
 line perpendicular to the girder (i.e. each half-way 
 between its extreme positions), the oscillations were due 
 to torsion alone, and the couple required to twist the 
 suspending filament through a given angle coidd be 
 determined from the above observed data, taken along 
 with the length of the girder and the masses of the two 
 small balls. 
 
 When the masses were placed at Cj, Dj, within a 
 couple of inches of the small balls, the range of the 
 oscillation was completely altered. From the observa- 
 tions (made as before) the new position of equilibrium 
 could be calculated. A fresh set of observations was 
 then made with the balls at Cj, Dj, and then they were 
 shifted to C3, Dg. Thus is determined the deflection which, 
 would have been produced if the apparatus could have been 
 reduced to rest. 
 
 But from this deflection, and the ascertained coeffi- 
 cient of torsion of the wire, the force acting on each of 
 the small balls can be calculated. This is to be com- 
 pared with the weight of one of the small balls, and then 
 the question is, " What must be the mass of the earth 
 when it attracts a mass at its surface (i.e. 4000 miles 
 from its centre) with a force greater in a known ratio 
 than that with which the same mass is attracted by a 
 given spherical mass of lead, whose centre is placed at 
 a given distanced' The law of gravitation at once 
 enables us to write the requisite condition. The mass 
 of the earth, thus found, has only to be divided by its 
 volume (§126) to give the mean density. 
 
GRAVITATION. 127 
 
 The quantities compared in such a case, i.e. the 
 attractions, may be taken as approximately in propor- 
 tion to the radius and the mean density of the earth, 
 and of the leaden sphere, respectively. They are, 
 therefore (as the density of lead is 11 '3), in the ratio 
 4000 X 5280 X 5-5: 11-3; or 10'':! roughly. Hence, 
 to estimate correctly, to two significant figures only, the 
 earth's mean density, we require to measure a force of 
 the order of the hundred-millionth part of the weight of 
 the small ball. 
 
 154. The details of the necessary precautions, as well 
 as of the results of various repetitions of this experiment, 
 do not suit a work like this, and must be sought in the 
 original descriptions.^ 
 
 Cavendish's result for the mean density of the earth 
 was 5 '48 (the density of water being taken as unit) ; 
 Eeich obtained 5'49 ; Baily 5'67, since reduced (by the 
 recalculations of Oornu) to 5'55. Comu's own result 
 is 5-50. 
 
 It is very remarkable that Newton, in Book III. of 
 the Frincipia, prop, x, made the following guess : — 
 
 " Since the common matter of our earth, on the sur- 
 face thereof, is about twice as heavy as water, and a little 
 lower, in mines, is found about three, four, or even five 
 times more heavy, it is probable that the quantity of the 
 whole matter of the earth may be five or six times 
 greater than if it consisted all of water." 
 
 Every one of the experimental results, above given, 
 lies almost exactly half-way between the limits thus 
 
 1 Cavendish, Fhil. Trans., 1798. Baily, Mem. Ast. Soc, 1843. 
 Reich, Abhand. d. K. Sachs. Ges., 1852. Cornu, Gomptes Eendus, 
 1870-78. 
 
128 PROPERTIES OF MATTER. 
 
 assigned, and published, more than a century before even 
 the earliest attempt at direct determination was made. 
 
 155. Good results have been obtained by a modifica- 
 tion of this experiment, which enables the experimenter 
 to employ an ordinary balance ; an attracting sphere of 
 considerable mass being applied beneath a sphere attached 
 to one ana of the balance, and already counterpoised 
 (at a different level) by weights' in a scale-pan. Thus 
 the uncertainties of torsion are avoided. 
 
 156. Other methods, which have been employed for 
 the purpose, depend upon the comparison of the attrac- 
 tion exercised by a mountain, or by some other part of 
 the earth, with that of the whole earth, when both act 
 simultaneously, but in difierent directions, on the same 
 body. The first recorded trial of this method was made 
 by De la Condamine and others, among the Andes. It 
 was first carefully worked out by Maskelyne on a pro- 
 minent Perthshire mountain, and has consequently been 
 called 
 
 The Schehallien Experiment. 
 By geodetic measures, altogether uninfluenced by 
 gravitation, the actual distance between two stations, 
 one north the other south of the mountain, can be found, 
 and from it the difierence of their (geographical) latitudes. 
 But the true latitude of each station separately can be 
 determined by the usual astronomical methods. The 
 difference between the geographical and the true latitude 
 of each station depends upon the attraction of the moun- 
 tain for the plumb-line, or the trough of mercury, which 
 is used to determine the vertical. The station south of 
 the mountain (in the northern hemisphere) has its lati- 
 tude made less than the geographical, that to the north 
 
GKAVITATION. 129 
 
 made greater, by this action. Hence, if everything were 
 symmetrical on the two sides of the mountain, the 
 difference of the astronomically determined true latitudes 
 at the two stations would be greater than that of their 
 geographical latitudes by double the deviation produced 
 in the plumb-line by the mountain. 
 
 The mountain must next be contoured ; then studied 
 geologically, so as to determine the most probable dis- 
 tribution of matter in it; then the specific gravities of 
 samples of these kinds of matter must be determined. 
 Then a laborious calculation, of the species called quadror 
 tures, must be gone through to take account of the form 
 and density of the mass. Finally, the attraction of this 
 distribution of matter leads to an estimate of the amount 
 of matter in the earth, which produces an attraction 
 exceeding the former in a known ratio. 
 
 Maskelyne's ^ observations, developed successively by 
 Hutton^ and by Playfair,^ gave as result for the earth's 
 mean density 4'48 and 4'86. The great objection to this 
 method is the uncertainty under which we must remain 
 as to the internal structure, hot only of the mountain 
 itself but of the whole crust of the earth in its neigh- 
 bourhood. This cannot be got over completely. 
 
 157. The Harton Experiment was made by Airy in the 
 Harton pits. It consists in comparing the intensity of 
 gravity at the earth's surface with that at the bottom of 
 a mine : — the same pendulum being used successively at 
 the two stations.;, or, still better, two pendulums being 
 made to vibrate simultaneously, one at each station, but 
 now and again interchanged. This method, with the 
 help of modern electrical processes, is probably (so far 
 ' Phil Tram., 1775. ^ /j^.^ 1778. s jud., 1811. 
 
 K 
 
130 PEOPEETIES OF MATTER. 
 
 as exactness of measurement is contiemed) a really good 
 one. The intensity of gravity at the bottom of the 
 mine differs from that at the surface on two accounts. 
 Suppose a surface drawn inside the earth, but every- 
 ■ where at a depth equal to that of the mine. The earth 
 may be looked upon as made up of a core, and this skin, 
 as it were. Gravity at the top of the pit depends on 
 the combined attractions of both. At the bottom of the 
 pit the skin ceases to attract (by Newton's proposition, 
 § 140),' but we have come mwrer to the core. Hence the 
 observations enable us to compare the attraction of the 
 core with that of the skin. But we know the volume 
 of the skin, and it has to be assumed (and this is 
 the fatal defect of the method) that the skin is every- 
 wTwre of the mean density determined from examina- 
 tion of the various strata passed through in sinking . 
 the pit. 
 
 It is not, therefore, surprising that the result of this 
 experiment, 1 viz. 6-56, should differ very materially from 
 the consistent results of the various workers at the 
 Cavendish experiment. 
 
 158. It has been suggested that the alternate filling 
 and eniptying of an estuary or bay, at different- states of 
 the tide, might supply an excellent mode of measuring 
 the earth's mean density by means of observations of six- 
 hourly periodic changes of latitude. The contouring 
 required would be very easy ; the density of sea-water 
 is practically uniform, and there are places where the 
 whole rise of the tide sometimes amounts to 120 feet or 
 so. But this promising method seems not to have got 
 beyond the stage of suggestion. Yet it is the only one, 
 ' Phil. Trans., 1856. 
 
GRAVITATION. 131 
 
 besides the mere modifications of the Cavendish method, 
 which has not some inherent and fatal weakness. 
 
 159. (e) and (/) of § 138 above. That two pieces of 
 matter behave as if they attracted one another, accord- 
 ing to Newton's law, is certain. But it by no means 
 follows that they do so attract. All that we are entitled 
 to say, from the facts given above, is as follows : — • 
 
 The pmi of the energy of a system of two particles of 
 matter, of masses m and m', which depends upon their dis- 
 tance, r, from one another, is measwed ly 
 
 and this is not altered by the presence of other particles. 
 
 This, taken along with the conservation of energy 
 enables us fully to investigate the motions of any system 
 ' of gravitating masses. It represents, in fact, our whole 
 knowledge on the subject. And It is well to observe 
 that the statement is altogether free from even the 
 mention of the word force. 
 
 160. We may, however, briefly notice some hypotheses 
 which have been framed as to the mechanism on which 
 this fact depends. For Newton, in his celebrated Letters 
 to Bentley, expressly says : — 
 
 "You sometimes speak of gravity as essential and 
 inherent to matter. Pray do not ascribe that notion to 
 me ; for the cause of gravity is what I do not pretend 
 to know, and therefore would take more time to con- 
 sider of it." 
 
 "It is inconceivable that inanimate brute matter 
 should, without the mediation of something else which 
 is not, material, operate on and affect other matter with- 
 out mutual contact, as it must do if gravitation in the 
 
132 PROPERTIES OF MATTER. 
 
 sense of Epicurus be essential and inherent in it. . . 
 That gravity should be innate, inherent, and essential 
 to matter, so that one body may act upon another at a 
 distance through a vacuum, without the mediation of 
 anything else, by and through which their action and 
 force may be conveyed from one to another, is to me so 
 great an absurdity, that I believe no man who has in 
 philosophical matters a competent faculty of thinking, 
 can ever fall into it. Gravity must be caused by an 
 agent acting constantly according to certain laws ; but 
 whether this agent be material or immaterial, I have left 
 to the consideration of my readers.'' 
 
 161. When we come to deal with molecular forces we 
 shall find that small bodies, such as sticks, straws, air- 
 bubbles, etc., floating on water, are made to aggregate 
 themselves into groups by molecular tension in the water- 
 surface. Hence the idea that stress, in a medium filling 
 all space, might account for the apparent mutual attrac- 
 tion between bodies entirely surrounded by this medium. 
 Newton, in the Queries at the end of his Optics, speaks of 
 a possible explanation to be obtained by assuming that 
 dense bodies rarefy the ether surrounding them, to an 
 amount which is less as the distance is greater. 
 
 Clerk-Maxwell says on this point : — ^ 
 
 " To account for such a force by means of stress in an 
 intervening medium, on the plan adopted for electric and 
 magnetic forces, ... we must suppose that there is a 
 pressure in the direction of the lines of force, combined ' 
 with a tension in all directions at right angles to the 
 lines of force. Such a stress would, no doubt, account 
 for the observed effects of gravitation. We have not, 
 ^ Ency. Brit, ninth ed., Art. "Attraction." 
 
GKAVITATION. 133 
 
 however, been able hitherto to imagine any physical 
 cause for such a state of stress. It is easy to calculate 
 the amount of this stress which would be required to 
 account for the actual effects of gravity at the surface of 
 the earth. It would require a pressure of 37,000 tons' 
 weight on the square inch in a vertical direction, com- 
 bined with a tension of the same numerical value in all 
 horizontal directions. The state of stress, therefore, 
 which we must suppose to exist in the invisible medium 
 is 3000 times greater than that which the strongest steel 
 could support." 
 
 162. Other attempts have been made, with the view 
 of showing that waves, or pulsating motion, in a medium, 
 would have the effect of drawing immersed bodies 
 together. Again, Sir W. Thomson has shown that if 
 space be filled with an incompressible fluid, which comes 
 into existence in fresh quantities at the surface of every 
 particle of matter, at a rate proportional to its mass, 
 and is swallowed up at an infinite distance, or, if each 
 particle of matter constantly swallows up an amount pro- 
 portional to its mass, a constant supply being kept up 
 from an infinite distance, — in either case gravitation 
 would be accounted for. This is, however, virtually 
 a suggestion of a dynamical mode of producing the 
 diminution of pressure required in Newton's attempt at 
 explanation. 
 
 163. An attempt at explanation, from a totally differ- 
 ent point of view, was made by Le Sage in 1818. The 
 following account of it is taken from Clerk-Maxwell's 
 article, "Atom," already referred to : — 
 
 "The theory of Le Sage is that the gravitation of 
 bodies towards each other is caused by the impact of 
 
134 PEOPERTIES OF MATTER. 
 
 streams of atoms flying in all directions through space. 
 These atoms he calls ultramundane corpuscules, because 
 he conceives them to come in all directions from regions 
 far beyond that part of the system of the world which 
 is in any way known to us. He supposes each of them 
 to be so small that a collision with another ultramundane 
 corpuscule is an event of very rare occurrence. It is by 
 striking against the molecules of gross matter that they 
 d.ischarge their function of drawing bodies towards each 
 other. A body placed by itself in free space and exposed 
 to the impacts of these corpuscules would be bandied 
 about by them in all directions, but because, on the 
 whole, it receives as many blows on one side as on 
 another, it cannot thereby acquire any sensible velocity. 
 But if there are two bodies in space, each of them will 
 screen the other from a certain proportion of the corpus- 
 cular bombardment, so that a smaller number of corpus- 
 cules will strike either body on that side which is next 
 the other body, while the number of corpuscules which 
 strike it in other directions remains the same. 
 
 "Each body will therefore be urged towards the other 
 by the effect' of the excess of the impacts it receives on 
 the side farthest from the other. If we take account of 
 the impacts of those corpuscules only which come directly 
 from infinite space, and leave out of consideration those 
 which have already struck mundane bodies, it is easy to cal- 
 culate the result on the two bodies, supposing their dimen- 
 sions small compared with the distance between them. 
 
 " The force of attraction would vary directly as the 
 product of the areas of the sections of the bodies taken 
 normal to the distance and inversely as the square of the 
 distance between them. 
 
GRAVITATION. 135 
 
 " Now, the attraction of gravitation varies as the pro- 
 duct of the masses of the bodies between which it acts, 
 and inversely as the square of the distance between 
 them. If, then, we can imagine a constitution of bodies 
 such that the eifective areas of the bodies are propor- 
 tional to their masses, we shall make the 'two laws co- 
 incide. Here, then, seems to be a path leading towards 
 an explanation of the law of gravitation, which, if it can 
 be shown to be in other respects consistent with facts, 
 may turn out to be a royal road into the very arcana of 
 science. 
 
 " Le Sage himself shows that, in order to make the 
 effective area of a body, in virtue of which it acts as a 
 screen to the streams of ultramundane corpuscules, pro- 
 portional to the mass of the body, whether the body be 
 large or small, we must admit that the size of the solid 
 atoms of the body is exceedingly small compared with 
 the distances between them, so that a very small propor- 
 tion of the corpuscules are stopped even by the densest 
 and largest bodies. We may picture to ourselves the 
 streams of corpuscules coming in every direction, like 
 light from a uniformly illuminated sky. We may 
 imagine a material body to consist of a congeries of 
 atoms at considerable distances from each other, and we 
 may represent this by a swarm of insects flying in the 
 air. To an observer at a distance this swarm will be 
 visible as a slight darkening of the sky in a certain 
 quarter. This darkening will represent the action of 
 the material body in stopping the flight of the corpus- 
 cules. Now, if the proportion of light stopped by the 
 swarm is very small, two such swarms will stop nearly 
 the same amount of light, whether they are in a line 
 
136 PEOPERTIES OF MATTER. 
 
 with the eye or not, but if one of them stops an appreci- 
 able proportion of light, there will not be so much left 
 to be stopped by the other, and the effect of two swarms 
 in a line with the eye will be less than the sum of the 
 two effects separately. 
 
 " Now, we know that the effect of the attraction of 
 the sun and earth on the moon is not appreciably 
 different when the moon is eclipsed than on other occa- 
 sions when full moon occurs without an eclipse. This 
 shows that the number of the corpuscules which are 
 stopped by bodies of the size and mass of the earth, and 
 even the sun, is very small compared with the number 
 which pass straight through the earth or the sun without 
 striking a single molecule. To the streams of corpuscules 
 the earth and the sun are mere systems of atoms scattered 
 in space, which present far more openings than obstacles 
 to their rectilinear flight. 
 
 " Such is the ingenious doctrine of Le Sage, by which 
 he endeavours to explain imiversal gravitation. Let us 
 try to form some estimate of this continual bombardment 
 of ultramundane corpuscules which is being kept up on 
 all sides of us. 
 
 " We have seen that the sun stops but a very small 
 fraction of the corpuscules which enter it. The earth, 
 being a smaller body, stops a still smaller proportion of 
 them. The proportion of those which are stopped by a 
 small body, say a 1 lb. shot, must be smaller still in an 
 enormous degree, because its thickness is exceedingly 
 small compared with that of the earth. 
 
 " Now, the weight of the ball, or its tendency towards 
 the earth, is produced, according to this theory, by the 
 excess of the impacts of the corpuscules which come 
 
GEAVITATION. 137 
 
 from above over the impacts of those which come from 
 below, and have passed through the earth. Either of 
 these quantities is an exceedingly small fraction of the 
 momentum of the whole number of corpuscules which 
 pass through the ball in a second, and their difference 
 is a small fraction of either, and yet it is equivalent to 
 the weight of a pound. The velocity of the corpuscules 
 must be enormously greater than that of any of the 
 heavenly bodies, otherwise, as may easily be shown, they 
 would act as a resisting medium opposing the motion of 
 the planets. Now, the energy of a moving system is 
 half the product of its momentum into its velocity. 
 Hence the energy of the corpuscules, which by their 
 impacts on the ball during one second urge it towards 
 the earth, must be a number of foot-pounds equal to the 
 number of feet. over which a corpuscule travels in a 
 second, that is to say, not less than thousands of millions. 
 But this is only a small fraction of the energy of all the 
 impacts which the atoms of the ball receive from the 
 innumerable streams of corpuscules which- fall upon it 
 in all directions. 
 
 " Hence the rate at which the energy of the corpus- 
 cules is spent in order to maintain the gravitating pro- 
 perty of a single pound, is at least millions of millions 
 of foot-pounds per second." 
 
 164. One common defect of these attempts is, as Clerk- 
 Maxwell points out, that they all demand some prime- 
 mover, working beyond the limits of the visible universe 
 or inside each atom : creating or annihilating matter, 
 giving additional speed to spent corpuscles, or in some 
 other way supplying the exhaustion suffered in the pro- 
 duction of gravitation. Another defect is that they all 
 
138 PROPERTIES OF MATTER. 
 
 make gravitation a mere difference -effect, as it werej 
 thereby implying the presence of stores of energy abso- 
 lutely gigantic in comparison with anything hitherto 
 observed, or even suspected to exist, in the universe; 
 and therefore demanding the most delicate adjustments, 
 not merely to maintain the conservation of energy which 
 we observe, but to prevent the whole solar and stellar 
 systems from being instantaneously scattered in frag- 
 ments through space. 
 
 In fact, the cause of gravitation remains undiscovered. 
 
 165. The ordinary balance, as we have abeady seen, 
 merely tests equality of masses. To find the weight of a 
 body we must measure directly the earth's attraction for 
 it. This can be done, perfectly in principle but only 
 with a rude approximation to accuracy in practice, by 
 means of a Sp-ing- Balance, or by some other contriv- 
 ance which depends on the elastic resilience of a special 
 kind of matter. 
 
 By far the most accurate instrument for measuring 
 the intensity of gravity, from which, of course, the 
 weight of any body (whose mass is known) may be 
 immediately calculated, is the pendulum. 
 
 By means of a theorem of abstract dynamics we can 
 calculate the length of the simple pendulum, which will 
 vibrate in the same period as will a mass, of any form 
 and dimensions, freely supported in any assigned way 
 on a horizontal axis. This the reader must take for 
 granted.^ 
 
 The following expression, whose form is suggested by 
 the theory of the Figure of the Earth, and whose constants 
 have been determined and verified by pendulum observa- 
 
 ' Thomson and Tait's Elements of Nat. Phil., Appendix, § g. 
 
GRAVITATION. 
 
 139 
 
 tions made all over the world, gives approximately the 
 value of g (§ 120) at sea-level in any latitude X, 
 32-088 (1 + 0-00513 sin?X). 
 166. We conclude the chapter with a small table of 
 (approximate) Specific Gravities, or what is the same thing 
 (§ 36), Densities, and a few remarks suggested by it. 
 None of the numbers for solids can be given with any 
 great accuracy, for, even if the substance be pure, its 
 density may be altered to a considerable amount by the 
 processes through which it has passed in assuming the 
 state in which it is tested. Such a table as the present 
 must be looked on as affording materials for rough calcula- 
 tions only. When better results are required, specialdeterr 
 minations must be made for each substance dealt with. 
 
 Hydrogen . 
 Steam 
 
 0-00009 
 0-0008 
 
 Nitrogen . 
 
 0-00125 
 
 Air .... 
 
 0-00129 
 
 Oxygen 
 
 (These are at 1 atmosphe 
 Cork 
 
 0-00143 
 re.) 
 0-24 
 
 LitMum 
 
 0-59 
 
 Potassium . 
 
 0-86. 
 
 Gutta Percha 
 
 0-98 
 
 Water 
 
 1-00 
 
 Magnesium 
 
 Quartz 
 
 Aluminium 
 
 1-75 
 2-65 
 2-67 
 
 Granite 
 
 2-7 
 
 Marble 
 
 
 Slate 
 
 
 Glass 
 
 2-7 to 4-5 
 
 Basalt 
 
 2-9 
 
 Bromine . 
 
 3-0 
 
 Zinc 
 
 7-2 
 
 Tin ... . 
 
 7-3 
 
140 
 
 PROPERTIES OP MATTER. 
 
 Iron . 
 
 
 
 V-8 
 
 Nickel 
 
 
 
 8-7 
 
 Copper 
 Silver 
 
 
 
 8-9 
 10-6 
 
 Mercury- 
 Gold. 
 
 
 
 13-6 
 19-4 
 
 Platinum 
 
 
 
 21-5 
 
 Iridium 
 
 
 
 22-4 
 
 The chief additional remark suggested by the table is 
 that, not only are there bodies which, though liquid at 
 ordinary temperatures, are denser than the great majority 
 of solids, but that a comparatively moderate pressure, a 
 few hundred atmospheres, for instance, would (without 
 producing liquefaction) make the density of air or oxygen 
 greater than that of some solids ; so that, for instance, if 
 chemical action could be prevented, we might easily have 
 solid lithium floating upwards in compressed oxygen, as 
 a cork rises in water. 
 
 167. By considering the earth, for a moment, as a fluid 
 mass, it is easy (op hydrostatical principles) to calculate 
 the whole pressure across any plane section of it.^ This 
 is, of course, the resultant gravitation attraction between 
 the parts separated by the plane of section. Assuming 
 the result of § 154 for the mean density, we find that the 
 average attraction, per square foot, across a diametral 
 plane is about 18 X 10^ lbs. weight. The tenacity of 
 sandstone is about 72 X 10^ lbs. weight per square foot. 
 Thus gravitation is 25,000 times as effectual in keeping 
 the earth together, as would be its cohesion if it were 
 solid sandstone. Even if the earth were as tenacious as 
 steel, its cohesion across a diametral plane would be only 
 about 1 per cent of the attraction across it. 
 1 Tait, Proc. S.S.E., 1875. 
 
GRAVITATION. 141 
 
 As the cohesion between two halves of a globe is, 
 ceteris pdrihis, as the area of a diametral plane, i.e. as the 
 square of the radius, while the gravitation attraction is 
 as the sixth power of the radius directly, and as the 
 square of the radius inversely, a sphere of the earth's 
 mean density and of the tenacity of sandstone would 
 require to be of about 25 miles radius only, in order 
 that cohesion may be as effective as gravity in keeping 
 two hemispheres together. If the tenacity were that of 
 steel, the radius would be about 400 miles. 
 
 Hence the earth's strength depends almost wholly on 
 gravitation, while that of a stone, less than a mile or so 
 in diameter, depends almost wholly on cohesion, and the 
 more completely the smaller it is. 
 
CHAPTEE VIII. 
 
 PKELIMINAKY TO DEFORMABILITY AND ELASTICITY. 
 
 168. A SUBSTANCE is said to be elastic -wten, on being 
 left free, it recovers wholly or partially from a defor- 
 mation (§41). 
 
 This is sometimes given in another form : — a body is 
 said to be elastic when it requires the continued application 
 of stress to keep it deformed. But this is by no means 
 an equivalent of the former statement ; and, besides, it 
 usually introduces complications ; for in many substances 
 the force requisite to maintain a distortion becomes less 
 and less with the lapse of time; and the continued 
 application of a given distorting force often produces a 
 constantly increasing distortion. To this, and to another 
 curious property called the Fatigue of Elasticity, we wiU 
 recur, but we will for the present adhere to the first 
 definition. 
 
 Hence, as an introduction to this part of the subject, 
 we must inquire into the nature and mechanism of the 
 simpler kinds of deformation. 
 
 169. The term usually employed for deformation of 
 any kind is Strain. The treatment of strains is an entirely 
 geometrical, or (more properly) kinematical, question. 
 But when we inquire how a strain is produced in a 
 
DEFOKMABILITY AND ELASTICITY. 143 
 
 given piece of matter the question becomes a dynamical 
 one, and we are led to the notion of a system of equili- 
 brating forces, called a Stress. And we figure to our- 
 selves that every stress produces a corresponding strain, 
 which will be of greater or less amount as the specimen 
 of matter operated on is of more or less yielding quality. 
 
 It is sometimes convenient to speak of the property of 
 yielding to a particular stress, as when we speak of the 
 CompressiMlity of a substance ; sometimes it is more con- 
 venient to speak of the property of resistance to a stress, 
 as when we speak of a body's Rigidity. But the resist- 
 ance to a stress is measured by the reciprocal of the 
 amount of yielding (just as the electric resistance of a 
 wire is the reciprocal of its conducting power), so that 
 either of these numerical quantities is immediately 
 deducible from the other. 
 
 It will be seen shortly that if P be the measure of 
 
 any one kind of stress, and p that of the corresponding 
 
 strain (supposed small), experiment points to a general 
 
 relation of the form 
 
 P = Cp, 
 
 where C is a constant depending on the special substance, 
 and special form of stress. is obviously greater, the 
 smaller is the strain for a given stress ; and it therefore 
 measures the resistance of the substance to the particular 
 kind of stress denoted by P. 
 
 170. We shall not require for our elementary treat- 
 ment of the question more than the simplest portions of 
 the subject of strains, and shall therefore be concerned 
 with Homogeneous Strain only. 
 
 By this term it is implied that all originally similar, 
 equal, and similarly situated portions of a substance 
 
144 PEOPERTIES OF MATTER. 
 
 remain after the strain similar, equal, and similarly 
 situated, however their forms and dimensions may be 
 changed. Hence points originally in a straight Hne, or 
 in a plane, remain in a straight line, or in a plane. Also 
 equal parallel Unes remain equal parallel lines. There- 
 fore a parallelogram remains a parallelogram, an ellipse 
 remains an ellipse, a parallelepiped remains a parallel- 
 epiped, and an ellipsoid remains an ellipsoid. 
 
 171. Now suppose small, equal, and similarly situ- 
 ated cubes to be traced in the unstrained body. This 
 will be effected by three imagined series of equidistant 
 parallel planed, those of each series being perpendicular 
 to those of the other two. After the strain the cubes 
 become equal, similar, and similarly situated parallel- 
 epipeds. And it is clear that if one of the cubes, and 
 the corresponding parallelepiped, be given, everything 
 else can be determined. 
 
 But there is one special set, of three series of rect- 
 angular planes, with which it is best to commence. For 
 it is clear from what precedes that all originally spherical 
 portions of the body will become similar and similarly 
 situated ellipsoids. Also, if tangent planes be drawn to 
 a sphere, at the extremities of three diameters at right 
 angles to one another, i.e. so that any one of the tangent 
 planes is parallel to the plane containing the other two 
 diameters, this 'parallelism wiU be maintained after 
 deformation. Thus, every set of three mutually per^ 
 pendicular diameters of the sphere becomes a set of corir- 
 jugate diameters of the ellipsoid, and vice versd. Hence 
 the principal axes of the ellipsoid, which are conjugate 
 diameters perpendicular to one another, were also origin- 
 ally perpendicular to one another. 
 
DEFOEMABILITY AND ELASTICITY. 
 
 145 
 
 172. Thus every homogeneous strain may be looked 
 on as having been produced by uniform extensions, or 
 compressions, parallel to three mutually perpendicular 
 lines (the amounts parallel to each being generally 
 different), and a subsequent rotation of the whole as if 
 it were rigid. We shall not require to consider the 
 rotation, for we are concerned only with the deformation 
 which each small part suffers. 
 
 Thus we may simply inquire into the circumstances 
 under which an originally cubical portion of the sub- 
 stance becomes in general brick-shaped, without change 
 of the directions of its edges, The investigation presents 
 no grave difficulties when the strains are of finite magni- 
 tude, but we will, for simplicity, consider them as small. 
 
 173. There is one elementary form of strain which 
 we must specially consider, viz. that of the brick shape, 
 
 FlQ. 16. 
 
 formed from a cube by lengthening in a given ratio one 
 set of parallel edges, shortening a second set in the same 
 ratio, and leaving the third set unaltered. Here it is 
 obvious that the volume also remains unaltered. Let 
 the ratio of extension be 1 -]- Z : 1, that of contraction will 
 be 1 - ?; 1, on account of the smallness of the fraction 
 I in all the cases which we have to consider. Let the 
 figure represent one of those faces of the cube, of which 
 L 
 
146 PROPERTIES OF MATTER. 
 
 all the edges have been altered. The square inscribed 
 in that side is obviously distorted into a rhombus, of 
 which two of the angles are greater, and two less, than 
 right angles, by the same amount, 6 suppose. 
 Then 
 
 I ,/- \ ^"^"''2 
 
 1 + 
 
 i = tan J r 2 - SJ = 
 
 1 + tan 2 
 
 so that, as 6 is very small, 
 
 e = 2i. 
 
 174. Every equilibrating system of forces {i.e. every 
 stress) can be reduced to simple stresses, each consisting 
 of equal and opposite forces in the same line, that is, 
 thrusts, or tensions. Thus we have now to inquire what 
 thrusts or tensions will convert a cube of deformable 
 matter into an assigned brick shape. These must 
 evidently be spread uniformly over each of its surfaces, 
 for every one of any number of smaller equal cubes, into 
 which it may be supposed to be divided, suffers precisely 
 the same proportionate deformation. 
 
 And as we confine ourselves to very small deforma- 
 tions, any number of them may be superposed, without 
 interfering with one another — i.e. they may be succes- 
 sively inflicted in any order, with the same final result. 
 It is mainly for this reason that we restrict ourselves to 
 small strains. 
 
 175. The problem is too difficult for an elementary 
 work, unless the portion of matter dealt with be isotropic, 
 i.e. unless it possess exactly the same properties in all 
 directions, so that the effect of a given stress on a cube 
 of it is exactly the same however the cube be cut out of 
 the original material. 
 
DEFOEMABILITY AND ELASTICITY. 147 
 
 Hence we see that, far mles which hecome bride-shaped, 
 without change of direction of the edges, the thrusts or 
 tensions must each be perpendicular to the face on which it 
 acts. And we measure each by its amount per unit 
 area. 
 
 It is particularly to be remarked that, in all that 
 follows on this subject, it is understood that the body 
 operated on is kept at a definite temperatwe, alike through- 
 out its substance and throughout the whole period of the 
 operation. 
 
 The study of the heat developed by sudden applica- 
 tions of stress belongs entirely to Thermodynamics, which 
 we do not enter upon in this work. In fact -we confine 
 ourselves to Isothermals, and have nothing to do with 
 Adiabatics. 
 
 176. The simplest case of all, and that which alone 
 we require when we deal with fluids, is when the pres- 
 sure or tension is the same on each face of the cube. 
 Here the cube obviously remains a cube, but its edges 
 are diminished or increased in length. Let unit of edge 
 become 1 -/ (where / is very small) under pressure P 
 per square unit of each face ; what is called hydrostatic 
 pressure, pressui'e the same in all directions, and always 
 normal to the surface. Then the volume of unit cube 
 
 • becomes 1 - 3/. 
 
 The compressiUlity of an isotropic body is measured by 
 the ratio of the compression per unit volume to the hydrostatic 
 pressure applied. 
 
 Hence the compressibility is 3//P, and the Resistance 
 to compression (§ 169), usually called Jc, is P/3/, so that 
 /=P/3^. 
 
 177. When we deal with solids, in which the stress is 
 
148 PROPERTIES OF MATTER. 
 
 not necessarily of the nature of hydrostatic pressure, 
 some further considerations must be attended to. 
 
 We now assume, consistently with experiment (as 
 will afterwards be shown), that, if the strain produced 
 by any stress be small, the reversed stress will produce 
 exactly the reversed strain. 
 
 Suppose the pairs of opposite faces of a cube be called 
 A, B, and ; the edges joining the corners of each pair a, 
 b, c respectively. Then a tension P, per unit of area, on 
 the A faces will increase a in some definite ratio 1 + ^ : 1, 
 and diminish b and c in some common ratio l-q-A. Now 
 superpose ^pressure P, per unit area, on the B faces. This 
 will compress b in the ratio 1 - p:l, and extend a and 
 c in the ratio 1 +2:1- Hence the result of tension P 
 on the A faces and pressure P on the B faces is that a 
 is extended in the ratio I + p + q : I, b is compressed in 
 the ratio 1 -p - q:l, while the length of c is unaltered. 
 
 The efiect is, therefore, to change the form of the 
 section of the cube parallel to the C faces, but to leave 
 its area and the volume of the cube unaltered. This 
 strain is called a Simple Shear, and the corresponding 
 stress is called Shearing Stress. 
 
 178. It is usual, in defining Eigidity, to consider the 
 deformation produced in the unit cube by equal tangential 
 forces, applied to two pairs of its sides, in directions 
 parallel to the third pair of sides, as indicated in the 
 diagram below. These forces obviously constitute a 
 balancing system, or Stress. But it may be analysed 
 into a much simpler one. For, if we draw either diagonal 
 in the figure, the resultant of the forces applied to either 
 pair of faces on one side of it is easily seen to be P \/ 2, 
 in a direction perpendicular to the diagonal. But the 
 
DEFORMABILITY AND ELASTICITY. 
 
 149 
 
 length of the diagonal is >J% Hence the stress per- 
 pendicular to either diagonal plane is P per square unit. 
 And it is clearly a pressure perpen- 
 dicular to one diagonal plane, and a 
 tension perpendicular to the other. 
 It is therefore the system already 
 studied in § 177, and the effect on 
 the cube above is that studied in 
 § 1 7 3. We nowdefineas follows : — 
 
 The rigidity of an isotropic solid, 
 i.e. the resistance to change of 
 form under a stress such as that 
 in the above figure, is directly proportional to the tangential 
 force per unit a/rea, and inversely as the change of one of the 
 angles of thefigwre. 
 
 Hence, using the common designation, we- have 
 
 Rigidity = » = P/9, 
 or, by §§ 173, 177, p 
 
 P + ^ = ^ .... (1.) 
 
 179. But, by § 177, the effect of pressure P, applied 
 simultaneously to all the sides of the cube, would be to 
 reduce the lengths of the edges in the common ratio 
 
 or (approximately) 
 Hence (§ 176), 
 
 1:1 + p-2q. 
 
 180. From (1) and (2) we have at once 
 ^ \6re 9/fc/ 
 
 .IP 
 
 (2.) 
 
150 PROPERTIES OF MATTER. 
 
 These represent respectively the extension of one 
 set of edges of the unit cube, and the common contrac- 
 tion of the other two, when it is subjected to tension P 
 parallel to the former set. 
 
 These results might have been obtained, perhaps even 
 more simply, by assuming the existence of compressibility 
 with absolute rigidity, then assuming pliability with ab- 
 solute incompressibility, and superposing the effects. 
 But the logic of this process is more likely to puzzle the 
 beginner. 
 
 181. Hence the extension, per unit of length, of a 
 rod or bar, under tension P per square inch of its 
 cross-section, is 
 
 p 3k + n 
 9kn ■ 
 
 The corresponding diminution, per unit area, of cross- 
 section is 
 
 p Zk - 2n 
 9kn • 
 
 And thus the increase per unit volume is P/3^, a 
 result which we might have obtained directly in many 
 other ways. 
 
 Thus, in pulling out an india-rubber band with a 
 .given tension, we increase its volume by one-third of the 
 amount by which it would be diminished by hydrostatic 
 pressure of the same value. 
 
 From these formulae the result of the application of 
 any stress to an isotropic body can be calculated. 
 
 182. As an example, suppose we desire to find what 
 stress will produce extension of an isotropic bar or 
 cylinder unaccompanied by lateral change of any kind. 
 
 If we have tensions, P along, and P' in all directions 
 
DEFOKMABILITY AND ELASTICITY. 151 
 
 perpendicular to, the axis of the bar, we have for the 
 longitudinal extension (§ 177) 
 
 2P' 
 i^ - -p ?; 
 
 and for the extension in any radial direction 
 
 T'p P + F 
 P --^«- 
 
 The latter must vanish, by our assumed condition, so that 
 
 p,^ Fg _p 3fc-2m . 
 p - q Sic + in' 
 
 and the extension is 
 
 3k + in' 
 
 183. In the chapters which immediately follow, it 
 will be seen that to determine the compressibility of a 
 fluid we require (at least in all the ordinary modes of 
 experimenting) to know the distortion produced in the 
 vessel which contains it. 
 
 When the same hydrostatic pressure is applied sim- 
 ultaneously to the outside of the vessel and to its 
 
 contents, the correction for diminution of the interior 
 
 pv 
 is of course, ,§§ 176, 212, -^ : — where P is the pressure 
 
 per unit surface, V the interior volume, and k the re- 
 ciprocal of the compressibility of the material of the 
 vessel. This is to be added to the apparent compressi- 
 bility of the fluid. 
 
 But when the pressure on the vessel is mainly 
 internal (as in Andrews' experiments on carbonic acid), 
 or wholly external (as in glass manometers, § 233), the 
 correction is not so simple. It can, in every case, be 
 
152 PROPERTIES OF MATTER. 
 
 determined by means of the equations of § 180 ; but the 
 investigation even of symmetrical cases is beyond the 
 limits here imposed on us. We therefore merely state 
 the results for the forms of vessel most commonly used, 
 viz. tubes and bulbs. For simplicity we assume the 
 tubes to be cylindrical, and the bulbs to be spherical, 
 each being of uniform material and of equal thickness 
 throughout. The internal and external radii are, in 
 both cases, denoted by a^ and % respectively j and the 
 cylinders are supposed free to alter in length as well as 
 in cross-section. 
 
 Then the diminution per unit of content, by external 
 hydrostatic pressure P, is — 
 
 In cylinders ^^fir^,{^ + -^} 
 In spheres P ^rr;^ (^^+1^> 
 
 The increase per unit of content, by internal hydro- 
 static pressure P', is — 
 
 In cylinders P' ^^^, g +-J, ^). 
 
 In spheres P JT^^s (^^ +^3 te> 
 
 When there are simultaneous hydrostatic pressures out- 
 side and inside, the corresponding results, calculated 
 from these expressions, are to be simply superposed 
 (§ 174). 
 
 Thus, if P and P' be simultaneous and equal, we 
 have, alike in cylinders and spheres, for the diminution 
 of unit content, P/A as above. 
 
 When an exceedingly thick vessel is exposed to 
 
DEFORMABILITY AND ELASTICITY. 153 
 
 internal pressure only, the eifect on unit of its content 
 practically depends on its rigidity only, and is P'/w for 
 a cylinder, and 3P'/4w for a sphere. This is a very 
 striking result. 
 
 When such a vessel is exposed to external pressure 
 the result is — 
 
 For cylinders P (^ + ^)' 
 For spheres P (^ + J^) 
 
 This shows the fallacy of the too common notion that, 
 by- mating the bulb of a thermometer thick enough, we 
 enable it to "defy pressure"; as, for instance, when it is 
 to be employed to measure temperatures in a sounding 
 of 3000 or 4000 fathoms. 
 
 184. It is very interesting to study the cases of 
 heterogeneous strain presented by the walls of cylinders 
 and bulbs when the internal and external hydrostatic 
 pressures are different. The following data will show 
 the student the form of the strain -ellipsoid, i.e. the 
 ellipsoid into which a very small part of the wall, 
 originally spherical, is distorted. We give the formulae 
 for a cylinder under external pressure. Let the original 
 position of the centre of the little sphere be at a distance, 
 r (intermediate, of course, between (Sq and Sj), from the 
 axis. Then it is deformed into an ellipsoid, whose axes 
 are — (1) radial, (2) parallel to the axis of the cylinder, 
 (3) at right angles to these two. If we denote by 1 the 
 original radius of the little sphere, the semi-axes of the 
 ellipsoid are — 
 
 n) \_-p^l-(l-< I), 
 
154 PROPEETIES OF MATTEK. 
 
 ^^> ^ ^Oi^-VV^-fc r" in)' 
 (3) 1 -P "' „ ~. 
 
 These are, in order of increasing magnitude, (2), (3), (1). 
 The axes (2) and (3) are always reduced in length, but 
 the radial axis (1) -vnll be increased in length by the 
 strain provided r^ < ~a^. 
 
 In ordinary flint glass this condition becomes, ap- 
 proximately — 
 
 *-■ 32 • 
 
 So that the interior layers of a glass tube, exposed to 
 external pressure only, are always extended in the radial 
 direction. This extension is greatest at the interior 
 surface, and vanishes in the layer whose radius is about 
 \-%a^ If the external radius be greater than this, the 
 outer layers are radially compressed, and the more the 
 farther they lie beyond the limit of no extension. 
 
 185. The theory of the propagation of Waves, whether 
 of compression or of distortion, in an elastic body, is 
 beyond our limits; but we may make the statement 
 that, if we could set aside the effects of sudden stress 
 in producing changes of temperature, and thus altering 
 the coefficients of compressibility and rigidity (for this 
 question belongs properly to Thermodynamics), the rates 
 of propagation of waves of different kinds depend only 
 upon one or both of these coefficients {k and n), and 
 upon the density of the body. When the coefficients 
 are measured in terms of the weight of unit bulk of 
 the body, they are called Moduli. Hitherto we have 
 measured them in terms of pressure or tension, i.e. force 
 
DEFOKMABILITY AND ELASTICITY. 156 
 
 per unit area. But, if we measure the force by the 
 length of the column of the substance, of unit section, 
 whose weight it can just support, we obviously take 
 account of the weight of unit bulk. Now the theoretical 
 result (under the conditions above specified) is that the 
 speed of a wave is that which would be acquired by 
 a free body falling, under uniform gravity, through a 
 height equal to half the length of the modulus corre- 
 sponding to the particular kind of distortion which is 
 propagated. Thus the speed of sound in air or water 
 depends upon the value of h alone ; that of a shearing 
 wave, such as light and some forms of earthquake, on n. 
 alone. When a wave of extension is sent along a wire, 
 as (for instance) to set a distant railway signal, Young's 
 modulus (§ 224) comes in ; and, when we deal with 
 plane sound-waves in a solid, we must take the corre- 
 sponding modulus as given in § 182. 
 
CHAPTER IX. 
 
 COMPRESSIBILITY OF GASES AND VAPOUKS. 
 
 186. A VERY general proof of compressibility and of 
 elasticity of bulk is afforded at once by the fact that the 
 great majority of bodies are capable of transmitting 
 sound-waves. For the propagation of sound consists 
 essentially in the hcmding on by resilience, from layer to 
 layer of the medium, of a state of compression or dilata- 
 tion; the (small) disturbance of each particle taking 
 place to and fro in the direction in which the sound is 
 travelling. All ordinary sounds are propagated in air. 
 But the rate of passage of sound has been measured in 
 the water of the Lake of Geneva and elsewhere; and 
 miners are in the habit of signalling to one another by 
 the sounds (of taps with a pick) conveyed through solid 
 rock. 
 
 187. Compressibility, elasticity, and inertia of air 
 are all demonstrated by the action of an air-gun. Its 
 reservoir is charged, by means of a pump, with some 
 forty or sixty times the quantity of air which it would 
 contain at the normal pressure and temperature; the 
 moment the valve is thrust down, by the fall of the 
 hammer, a portion of the air is forced out by its elas- 
 ticity ; and this rapid stream, by its inertia, communi- 
 
COMPRESSIBILITY OF GASES AND VAPOURS. 157 
 
 cates motion to the bullet. The same thing is shown, 
 in a very beautiful form, by allowing the compressed air 
 to escape in a fine jet ; for a ball of cork can be suspended 
 in the jet, as a metal shell is suspended in a fountain-jet 
 of water, but .in this case without any visible support. 
 
 188. In 1662 Eobert Boyle published his Defence of 
 the Doctrine touching the Spring and Weight of the Air. 
 The following extract, especially, is still of great interest. 
 It occurs in Part II. chap. v. 
 
 "We took then a long Glass-Tube, which by a 
 dexterous hand and the help of Lamp was in such a 
 manner crooked at the bottom, that the part turned up 
 was almost parallel to the rest of the Tube, and the 
 Orifice of this shorter leg of the Siphon (if I may so call 
 the whole Instrument) being Hermetically seal'd, the 
 length of it was divided into Inches (each of which was 
 subdivided into eight parts) by a straight list of paper,_ 
 which containing those Divisions was carefully pasted all 
 along it : then putting in as much Quicksilver as served 
 to fill the Arch or bended part of the Siphon, that the 
 Merowry standing in a level might reach in the one leg 
 to the bottom of the divided paper, and just to the same 
 height or Horizontal line in the other ; we took care, by 
 frequently inclining the Tube, so that the Air might 
 freely pass from one leg into the other by the sides of 
 the Mercury, (we took (I say) care) that the Air at last 
 included in the shorter Cylinder should be of the same 
 laxity with the rest of the Air about it. This done, we 
 began to pour Quicksilver into the longer leg of the 
 Siphon, which by its weight pressing up that in the 
 shorter leg, did by degrees streighten the included Air : 
 and continuing this pouring in of Quicksilver till the Air 
 
168 PROPERTIES OF MATTER. 
 
 in the shorter leg was by condensation reduced to take 
 up but half the space it possess'd (I say, possess'd not 
 filVd) before ; we cast our eyes upon the longer leg of the 
 Glass, on which was likewise pasted a list of Paper care- 
 fully divided into Inches and parts, and we observed, not 
 without delight and satisfaction, that the Quicksilver in 
 that longer part of the Tube was 29. Inches higher than 
 the other. Now that this Observation does both very 
 well agree with and confirm our Hypothesis, vrill be easily 
 discerned by him that takes notice that we teach, and 
 Monsieur Paschall and our English friends Experiments 
 prove, that the greater the weight is that leans upon the 
 Air, the more forcible is its endeavour of Dilatation, and 
 consequently its power of resistance, (as other Springs 
 are stronger when bent by greater weights.) For this 
 being considered it wil appear to agree rarely-well with 
 the Hypothesis, that as according to it the Air in that 
 degree of density and correspondent measure of resistance 
 to 'which the weight of the incumbent Atmosphere had 
 brought it, was able to counterbalance and resist the 
 pressure of a Mercurial Cylinder of about 29. Inches, as 
 we are taught by the Torricellian Experiment ; so here 
 the same Air being brought to a degree of density about 
 twice as great as that it had before, obtains a Spring 
 twice as strong as formerly. As may appear by its being 
 able to sustain or resist a Cylinder of 29. Inches in the 
 longer Tube, together with the weight of the Atmos- 
 pherical Cylinder, that lean'd upon those 29. Inches of 
 Mercivry ; and, as we just now inferr'd from the Torrl- 
 cellian Experiment, was equivalent to them. 
 
 " We were hindered from prosecuting the tryal at that 
 time by the casual breaking of the Tube. But because 
 
COMPRESSIBILITY OF GASES AND VAPOURS. 159 
 
 an accurate Experiment of this nature would be of great 
 importance to the Doctrine of the Spring of the Air, 
 and has not yet been made (that I know) by 
 any man ; and because also it is more uneasie 
 to be made then one would think, in regard 
 of the difficulty as well of procuring crooked 
 Tubes fit for the purpose, as of making a just 
 estimate of the true place of the Protuberant 
 Mercury's surface ; I suppose it will not be un- 
 welcome to the Keader, to be informed that 
 after some other tryals, one of which we made 
 in a Tube whose longer leg was perpendicular, 
 and the other, that contained the Air, parallel 
 to the Horizon, we at last procured a Tube of 
 the Figure exprest in the Scheme; which 
 Tube, though of a pretty bigness, was so long, 
 that the Cylinder whereof the shorter leg of 
 it consisted admitted a list of Paper, which 
 had before been divided into 1 2. Inches and 
 their quarters, and the longer leg admitted 
 another list of Paper of divers foot in length, 
 and divided after the same manner: then 
 Quicksilver being poured in to fill up the 
 bended part of the Glass, that the surface of bIIHA 
 it in either leg might rest in the same Hori- 
 zontal line, as we lately taught, there was 
 more and more Quicksilver poured into the 
 longer Tube; and notice being watchfully 
 taken how far the Mercury was risen in , 
 that longer Tube, when it appeared to have ascended 
 to any of the divisions in the shorter Tube, the 
 several Observations that were thus successively made, 
 
160 
 
 PROPERTIES OF MATTER. 
 
 and as they were made set down, afforded us the 
 ensuing Table. 
 
 "A, Table of the Condensation of the Aik. 
 
 A. A. The number of 
 equal spaces in the 
 shorter leg, that con- 
 tained the same par- 
 cel of Air diversely 
 extended. 
 
 B. The height of the 
 Mercurial Cylinder 
 iu the longer leg, that 
 compress'd the Air 
 into those dimen- 
 sions. 
 
 C. The height of a Mer- 
 curial Cylinder that 
 counterbalanc'd the 
 pressure of the At- 
 mosphere. 
 
 D. The Aggregate of 
 the two last Columns, 
 B and C, exhibiting 
 the pressure sus- 
 tained by the in- 
 cluded Air. 
 
 E. What that pressure 
 should be according 
 to the Hypofhesis, 
 that supposes the 
 pressures and ex- 
 pansions to be in 
 reciprocal propor- 
 tion." 
 
 189. The form of apparatus employed by Boyle is 
 
 still recognised as by far the best for the purpose. - With 
 
 a few necessary modifications, to adapt it to difference 
 
 of circumstances, it was employed by Amagat^ in ,the 
 
 1 Annales de Chimie, 1880. 
 
 A. 
 
 A. 
 
 B. 
 
 0. 
 
 D. 
 
 E. 
 
 48 
 
 12 
 
 00 
 
 
 29]^ 
 
 29t\ 
 
 46 
 
 n^ 
 
 OItV 
 
 
 30A 
 
 30A 
 
 44 
 
 11 
 
 02i| 
 
 
 31^ 
 
 31il 
 
 42 
 
 101 
 
 04xV 
 
 
 33^^ 
 
 33|- 
 
 40 
 
 10 
 
 06A 
 
 
 35tV 
 
 35... 
 
 38 
 
 H 
 
 0714 
 
 
 37... 
 
 36if 
 
 36 
 
 9 
 
 lOA 
 
 
 39t\ 
 
 38| 
 
 34 
 
 8| 
 
 12^ 
 
 
 41^ 
 
 41^ 
 
 32 
 
 8 
 
 18t^ 
 
 3? 
 
 44t^ 
 
 43-1 
 
 30 
 
 n 
 
 I7if 
 
 -a 
 
 47tV 
 
 46- 
 
 28 
 
 7 
 
 21t\ 
 
 1 
 
 50t^ 
 
 50... 
 
 26 
 
 H 
 
 25A 
 
 ■S' 
 
 54t^ 
 
 53lt 
 
 24 
 
 6 
 
 291^ 
 
 oq 
 
 58i| 
 
 ^H. 
 
 23 
 
 5| 
 
 32A 
 
 .s 
 
 61A 
 
 60^ 
 
 22 
 
 5^ 
 
 34i# 
 
 'a 
 
 64,^ 
 
 63A 
 
 21 
 
 H 
 
 371.1 
 
 "^ 
 
 67tV 
 
 66|- 
 
 20 
 
 5 
 
 41 9 
 
 <J 
 
 70^1 
 
 70... 
 
 19 
 
 4f 
 
 45... 
 
 
 74A 
 
 73H 
 
 18 
 
 H 
 
 481f 
 
 
 77i| 
 
 77f 
 
 17 
 
 H 
 
 53il 
 
 
 82if 
 
 82^ 
 
 16 
 
 4 
 
 58A 
 
 
 87i# 
 
 87| 
 
 15 
 
 3| 
 
 63if 
 
 
 93A 
 
 93i 
 
 14 
 
 H 
 
 71 A 
 
 
 lOOxV 
 
 99^ 
 
 13 
 
 H 
 
 7811 
 
 
 107i| 
 
 107t^ 
 
 12 
 
 3 
 
 88,^ 
 
 
 117A 
 
 116| 
 
COMPRESSIBILITY OF GASES AND VAPOURS. 161 
 
 most' important recent experimental determinations of 
 the eflfects of great pressures on the volume of a gas. 
 
 Its action depends on the two hydrostatical principles 
 stated below, the truth of which we are here content to 
 assume.^ 
 
 In, a mass of fluid, at rest, the p-esswe (per square inch) 
 is the same at all points in any horizontal plane. 
 
 The change of pressure from one horizontal plame to another 
 is equal to the weight of a colwnn of the fluid, one square inch 
 in section, extending vertically between these planes. 
 
 From these it follows that the pressure of the gas 
 operated on, i.e. the pressure on the mercury surface at 
 A (Fig. 17) is the same as that at the same level, B, in 
 the other branch of the tube : — and this, again, exceeds 
 the pressure at C (the atmospheric pressure), by the 
 ■weight of a column of mercury of square inch section 
 and of height BO. 
 
 1 90. In his comments on this experiment Boyle says : — 
 
 "For the better understanding of this Experiment it 
 may not be amiss to take notice of the following particu- 
 lars : — 
 
 " 3. That we were two to make the observation to- 
 gether, the one to take notice at the bottom how the 
 Quicksilver rose in the shorter Cylinder, and the other 
 to pour it in at the top of the longer, it being very hard 
 and troublesome for one man alone to do both accurately. 
 
 " 6. That when the Air was so compressed, as to be 
 
 crouded into less than a quarter of the space it possess'd 
 
 before, we tryed whether the cold of a Linen Cloth dipp'd 
 
 ' Thomson and Tait, Elements of Natural Philosophy, §§ 692, 694. 
 
 M 
 
162 PROPERTIES OF MATTER. 
 
 in water would then condense it. And it sometimes 
 seemed a little to shrink, but not so manifestly as that 
 we dare build anything upon it. We then tryed likewise 
 whether heat would notwithstanding so forcible a corn- 
 pressure dilate it, and approaching the flame of a Candle 
 to that part where the Air was pent up, the heat had a 
 more sensible operation than the cold had before; so 
 that we scarce doubted but that the expansion of the Air 
 would notwithstanding the weight that opprest it have 
 been made conspicuous, if the fear of unseasonably break- 
 ing the Glass had not kept us from increasing the heat. 
 
 " And there is no cause to doubt, that if we had been 
 here fumishtwith a greater quantity of Quicksilver and a 
 very strong Tube, we might by a further compression of 
 the included Air have made it counterbalance the pres- 
 sure of a far taller and heavier Cylinder of Merowry. 
 For no man perhaps yet knows how near to an infinite 
 compression the Air may be capable of, if the compress- 
 ing force be competently increast. 
 
 "And to let you see that we did not (a little above) 
 inconsiderately mention the weight of the incumbent 
 Atmospherical Cylinder as a part of the weight resisted 
 by the imprisoned Air, we will here annex, that we took 
 care, when the Mercurial Cylinder in the longer leg of 
 the Pipe was about an hundred Inches high, to cause 
 one to suck at the open Orifice ; whereupon (as we ex- 
 pected) the Merawry in the Tube did notably ascend. . . . 
 And therefore we shall render this reason of it. That 
 the pressure of the incumbent Air being in part taken 
 off by its expanding it self into the Sucker's dilated chest ; 
 
COMPRESSIBILITY OF GASES AND TAPOUES. 163 
 
 the imprison'd Air was thereby enabled to dilate it self 
 manifestly, and repel the Mercwry that comprest it, till 
 there was an equality of force betwixt the strong Spring 
 of that comprest Air on the one part, and the tall Mer- 
 curial Cylinder, together with the contiguous dilated Air, 
 on the other part." 
 
 It is scarcely necessary to call attention to the truly 
 scientific caution with which Boyle thus gives his con- 
 clusions from this notable experiment. 
 
 191. Boyle's Law (as it is called in Britain) is now 
 stated in the extended form : — 
 
 The volume of a given mass of gas, Jcept at a given tempera- 
 twe, is inversely as the pressure} 
 In symbols this is merely 
 
 p)> = G (1.) 
 
 where C is a quantity depending upon the mass of gas, 
 and on its temperature. 
 
 From the definition of density as the quantity of 
 matter per unit of volume, we see at once that Boyle's 
 Law may be stated in the form — ■ 
 
 The density of a gas, at constant temperature, is propor- 
 tional to the pressure. 
 
 192. The compressibility follows at once. For a 
 small increase, tt, in the pressure, corresponds to a small 
 diminution, v, in the volume, .such that we still have 
 
 (p -I- 7r)(D - u) = C. 
 Neglecting the product of the two small quantities we 
 
 ^^^^ TV-pv=0. 
 
 ^ This Law usually goes by the name of Mariotte in foreign 
 books. See Aj^endix IV. 
 
164 PROPERTIES OF MATTER. 
 
 Here the change, per unit of volume, is v/v, so that 
 the compressibility (§ 88) is 
 
 T 1> p 
 
 The resistance to compression is therefore propor- 
 tional to the pressure. This result was obtained by a 
 graphic process in § 88. 
 
 193. So closely does air follow Boyle's Law through 
 all ordinary ranges of pressure, that it is constantly used 
 in Manometers for the direct measurement of pressure. 
 The manometer is, in its elements, merely a carefully 
 calibrated tube containing dry air, from whose volume 
 (when it is kept at constant temperature) the pressure 
 is at once calculated. 
 
 The chief defect of such manometers is that successive 
 equal increments of pressure produce gradually diminish- 
 ing eifects on the volume of the gas ; and thus the in- 
 evitable errors of observation become more serious, in 
 proportion to the quantity to be measured, as higher 
 pressures are attained. Various ingenious devices, such 
 as tubes of tapering bore, have been devised to remedy 
 this defect. In aU such modifications most careful 
 calibration is essential 
 
 194. All gases, at temperatures considerably above 
 what is called their critical point (§ 206), follow Boyle's 
 Law fairly through a somewhat extensive range of pres- 
 sures. But a gas, at a temperature under its critical 
 point, is really a vapour, and can be reduced (without 
 change of temperature) to the liquid state by the appli- 
 cation of sufficient pressure, if nuclei be present. The 
 compression of vapours will be treated farther on. 
 
 195.' So far, we have been dealing with the eflfects of 
 
COMPRESSIBILITY OF GASES AND VAPOUES. 165 
 
 pressure. But Boyle carried his inquiry into 
 the effects of diminution of pressure also. His apparatus 
 was of a very simple kind, though still useful, at least 
 for class illustration. The following extract, while highly 
 interesting, sufficiently describes his results and method : — 
 
 " A Table of the Rarefaction of the Air. 
 
 A. The number of equal spaces 
 at the top of the Tube, that 
 contained the same parcel of 
 Air. 
 
 B. The height of the Mercurial. 
 Cylinder, that together with 
 the Spring of the included 
 Air, counterbalanced the 
 pressure of the Atniosphere. 
 
 C. The pressure of the Atmos- 
 phere. 
 
 D. The Complement of B to C, 
 exhibiting the pressure sus- 
 tained by the included Air. 
 
 E. What that pressure should 
 be according to the Hypo- 
 
 A. 
 
 B. 
 
 C. 
 
 D. 
 
 E. 
 
 1 
 
 00^ 
 
 
 29f 
 
 29| 
 
 1* 
 
 lOf 
 
 
 19^ 
 
 19| 
 
 2 
 
 15| 
 
 
 14| 
 
 14| 
 
 3 
 
 20| 
 
 
 9| 
 
 9H 
 
 4 
 
 22| 
 
 
 n 
 
 
 5 
 
 24| 
 
 4) 
 
 5| 
 
 sp 
 
 6 
 
 24| 
 
 
 't 
 
 4ff 
 
 7 
 
 25| 
 
 1 
 
 4| 
 
 4i 
 
 8 
 
 26§ 
 
 3| 
 
 3|| 
 
 9 
 
 26| 
 
 H 
 
 3ii 
 
 10 
 
 26| 
 
 
 3§ 
 
 2fff 
 
 12 
 
 27i 
 
 1 
 
 2| 
 
 211 
 
 14 
 
 27| 
 
 § 
 
 H 
 
 2* 
 
 16 
 
 271 
 
 1 
 
 n 
 
 111 
 
 18 
 
 27| 
 
 za 
 
 n 
 
 1 47 
 I-T2 
 
 20 
 
 28-H 
 
 
 If 
 
 1 9 
 
 24 
 
 28f 
 
 
 i| 
 
 1 23 
 
 28 
 
 28| 
 
 
 If 
 
 1t^ 
 
 32 
 
 28| 
 
 
 If 
 
 OHI 
 
 " To make the Experiment of the debilitated force of 
 expanded Air the plainer, 'twill not be amiss to note some 
 particulars, especially touching the manner of making the 
 Tryal; which (for the reasons lately mention'd) we made 
 on a lightsome pair of stairs, and with a Box also hn'd with 
 Paper to receive the Mercwy that might be spilt. And 
 in regard it would require a vast and in few places pro- 
 
166 PEOPEEJIES OF MATTEE. . 
 
 curable quantity of Quicksilver, to employ Vessels of such 
 kind as are ordinary in the Torricellian Experiment, we 
 made use of a Glass-Tube of about six foot long, for that 
 being Hermetically seal'd at one end, serv'd our turn as 
 well as if we could have made the Experiment in a Tub 
 or Pond of seventy Inches deep. 
 
 "Secondly, We also provided a slender Glass-Pipe of 
 about the bigness of a Swan's Quill, and open at both 
 ends ; all along which was pasted a narrow list of Paper 
 divided into Inches and half quarters. 
 
 ■ • ■ I ■ 
 
 "Fourthly, There being, as near as we could guess, 
 little more than an Inch of the slender Pipe left above 
 the surface of the restagnant Mercury,, and consequently 
 unfiU'd therewith, the prominent orifice was carefiilly 
 clos'd with sealing Wax melted ; after which the Pipe 
 was let alone for a while, that the Air dilated a little 
 by the heat of the Wax, might upon refrigeration be 
 reduc'd to its wonted density. . . . 
 
 "Sixthly, The Observations being ended, we presently 
 made the Torricellian Experiment with the above men- 
 tion'd great Tube of six foot long, that we might know 
 the height of the Mercwrial Cylinder, for that particular 
 day and hour ; which height we found to be 29f Inches. 
 
 " Seventhly, Our Observations made after this manner 
 fumish'd us with the preceding Table, in which there 
 would not probably have been found the difference 
 here set down betwixt the force of the Air when ex- 
 panded to double its former dimensions, and what that 
 force should have been precisely according to the Theory, 
 but that the included Inch of Air receiv'd some little 
 
COMPRESSIBILITY OF GASES AND VAPOURS. 167 
 
 accession during the Tryal ; which this newly-mention'd 
 dilBference making us suspect, we found by replunging 
 the Pipe into the Quicksilver, that the included Air had 
 gain'd about half an eighth, which we guest to have 
 come from some little aerial bubbles in the Quicksilver, 
 contained in the Pipe (so easie is it in such nice Experi- 
 ments to miss of exactness)." 
 
 196. We must now state how far these results of 
 Boyle have been verified by modem experimenters, and in 
 what direction they are found to deviate from the truth. 
 But before we do so we must introduce a definition. 
 
 The unit usually adopted for the measurement of 
 pressure is called an Atmosphere, roughly 147 lbs. weight 
 per square inch. 
 
 Its definition is, in this country, the weight of a 
 column of mercury at 0° C, of a square inch in section, 
 and 29 '905 inches high ; the weighing to be reduced to the 
 value of gravity at the sea-level in the latitude of London. 
 
 The value of an atmosphere, in C.G-.S. units, is about 
 1,014,000 djmes per square centimetre. 
 
 197. It is to Eegnault that we owe the first really 
 adequate treatment of the subject, but the range of 
 pressures he employed was not very extensive. 
 
 Eegnault showed that air and nitrogen are, f or.'at least 
 the first twenty atmospheres, more compressed than if Boyle's 
 Law were true, but that hydrogen is less compressed. 
 
 Then batterer made an extensive series of experi- 
 ments at very high pressures (sometimes nearly 3000 
 atmospheres), whose result showed that air and nitrogen, 
 as well as hydrogen, are less compressible than Boyle's 
 Law requires, and deviate the more from it the higher 
 the pressure. 
 
168 PROPERTIES OP MATTER. 
 
 198. Andrews/ in his classical researches which 
 established the existence of the critical point, first gave 
 the means of explaining this very singular fact. We 
 will recur to it when we are dealing with vapours, but 
 we give a few of Andrews' data here. The way in 
 which the compressibility varies with pressure is obvious 
 from the curves in the diagram (§ 205), when interpreted 
 as in § 88. But from Andrews' tables of corresponding 
 volumes of air at 13°"1, and carbonic acid at 35°'5, 
 both subjected to the same pressure, we extract the 
 numbers in the two first columns : — 
 
 Carbonic Acid (Gas) at 35°'5 0. 
 
 pv for Carb. Acid. 
 
 356 
 246 
 239 
 239 
 242 
 250 
 
 Andrews points out that the deviation of air from 
 Boyle's Law is, even at the highest of these pressures, 
 inconsiderable. Taking the reciprocals of tKe volumes 
 of air, therefore, as measuring pressures with sufficient 
 accuracy, we form the third column of the table. This 
 shows that in carbonic acid, a few degrees above its 
 critical point, the deviation from Boyle's Law is like 
 that in air and nitrogen for the first 90 atmospheres, 
 and, after that, resembles that in hydrogen. Unfor- 
 tunately the bursting of the tubes prevented Andrews 
 from carrying the pressure beyond 108 atmospheres. 
 1 Phil. Trans., 1869. 
 
 cip. of Vol. 
 
 Reoip. of Vol. of 
 
 of Air. 
 
 Carbonic Acid. 
 
 81-28 
 
 228-0 
 
 86-60 
 
 351-9 
 
 89-52 
 
 373-7 
 
 92-64 
 
 387-9 
 
 99-57 
 
 411-0 
 
 107-6 
 
 430-2 
 
COMPKESSIBILITY OF GASES AND VAPOURS. 169 
 
 199; The remarkable researches of Amagat already 
 alluded to (§ 189) were carried out in a gallery of a deep 
 coal-pit, where the temperature remained steady for long 
 periods. The shorter branch of his apparatus terminated 
 in a very strong glass tube of small bore carefully cali- 
 brated. The longer branch was made of steel, and ex- 
 tended to a height of 330 mfetres (about 1000 feet) up 
 the shaft of the pit. A small but powerful pump was 
 employed to force mercury into the lower part of the 
 apparatus until it began to run out at one of a set of 
 stopcocks which were inserted at measured intervals 
 along the tall tube. Then a measurement of the volume 
 of the compressed gas was made, the stopcock closed, 
 and that next above it opened in turn for a measurement 
 at a higher pressure. 
 
 200. The following short table gives an idea of 
 Amagat's results ' for air at ordinary temperatures : — 
 
 Pressure in Atmospheres. 
 
 i)ii. 
 
 1-00 
 
 1-0000 
 
 si-ev 
 
 -9880 
 
 45-92 
 
 •9832 
 
 59-53 
 
 -9815 
 
 73-03 
 
 -9804 
 
 84-21 
 
 -9806 
 
 94-94 
 
 ■9814 
 
 110-82 
 
 -9830 
 
 133-51 
 
 -9905 
 
 176-lV 
 
 1-0113 
 
 233-68 
 
 1-0454 
 
 282-29 
 
 1-0837 
 
 329-18 
 
 1-1197 
 
 400-05 
 
 1-1897 
 
 ^ Ann. de CMmie, 1880 ; supplemented from Compies Bendus, 
 1884. 
 
iro PROPERTIES OF MATTER. 
 
 [As Amagat's pressure data were obtained direct from 
 a. column of mercury, they supply by far the most accu- 
 rate means of finding the unit for pressure gauges. Hence 
 it may be well to note that, at ordinary temperatures, 
 for a pressure of 152-3 atmospheres, or one ton-weight 
 per square inch, dry air almost exactly follows Boyle's 
 Law, i.e. it is reduced to 1/152'3 of its volume at one 
 atmosphere. Hence, practically, when dry air is com- 
 pressed to anything from 1/150 to 1/160 of its bulk 
 under .one atmosphere, Boyle's law may be used to 
 calculate the pressure.] 
 
 It is very difficult to assign with exactness the posi- 
 tion of the minimum value of pv, as inevitable errors of 
 observation rise to considerable importance when a 
 quantity varies very slowly ; but it may be put down 
 as corresponding to about 77 '6 atmospheres. 
 
 201. Amagat's direct measures with the mercury 
 column were made on the volume of nitrogen. But 
 when these were carefully made, once for all, the 
 nitrogen manometer was used in connection with a 
 similar instrument filled with some other gas. Thus the 
 relation of ^ to ^ was determined with accuracy for 
 hydrogen, oxygen, air (as above), carbonic oxide, car- 
 bonic acid, ethylene, etc. In a later paper ^ Amagat 
 has extended these results through a considerable range 
 of temperatures. For the numerical data we must refer 
 to the paper itself ; but we reproduce three of the most 
 important of his graphic representations of the results. 
 
 The diagram opposite consists of two parts. The 
 upper part shows the relation of pv to p, through a 
 range of about 80° C, for nitrogen, whose behaviour is 
 ' Annahs de Ohimie, xxii. 1881. 
 
Fio. 18. 
 
172 PROPERTIES OF MATTER. 
 
 typical of that of a large number of gases. The 
 mininmin value of pv is distinctly shown at every 
 temperature. The lower diagram exhibits the excep- 
 tional case of hydrogen, where all the curves are, 
 practically, straight lines. The pressure unit is a mfetre 
 of mercury, i.e. 100/76 atmospheres. 
 
 The diagram on the next page shows the correspond- 
 ing relations for carbonic acid, at temperatures above 
 its critical point ; as well as for liquid carbonic acid at 
 18° '2 0. In this last case the curve is given only for pres- 
 sures from 80 to 260 metres of mercury. This diagram 
 gives very valuable information. Especially it shows 
 the marked influence of change of temperature on the 
 pressure corresponding to the minimum value of pv. 
 Ethylene gives a diagram somewhat resembling this, but 
 the changes in the value of pv are so disproportionately 
 greater that its behaviour could not be satisfactorily 
 exhibited on a scale so restricted as a page of this book. 
 
 The reader should be reminded that, had the law of 
 Boyle been accurate, all of these curves would have been 
 simply horizontal straight lines. 
 
 202. There is, unfortunately, a considerable variety 
 of statement as to the relation between pressure and 
 volume in air and other gases, when they are consider- 
 ably rarefied. This is not to be wondered at, for the 
 experimental difficulties are extremely great. 
 
 The experiments of Mendeleeif gave a gradual descent 
 of value oipv, in air, from 
 
 1-0000 at 0-85 atm. 
 to 
 
 0"9655 at 0'019 atm. 
 
 These would tend to show that, at pressures lower than 
 
Fia. 19. 
 
174 PEOPERTIES OF MATTEE. 
 
 an atmosphere, air behaves as hydrogen does for pres- 
 sures above an atmosphere. 
 
 The experiments of Amagat do not show this result. 
 They rather seem to indicate th&tpv remains practically 
 constant for air, from one atmosphere down to at least 
 ■g-g-jfth of an atmosphere. 
 
 203. But the real difficulty in all such experiments 
 arises from the shortness of the colimin of mercury by 
 which the pressure must be measured. It is not easy 
 to see how this diflSculty can be obviated without intro- 
 ducing a chance of graver errors of another kind, due 
 for instance to vapour-pressure or to capillary forces. 
 
 We shall find, later, that a fair presumption from 
 Andrews' investigations would be that, in air and the 
 majority of gases, pv should increase (of course very 
 slightly) with diminution of pressure from one atmos- 
 phere downwards ; while (possibly) hydrogen may give 
 values of pv diminishing to a minimum, and then in- 
 creasing as the pressure is still farther reduced. 
 
 204. Passing next to the compressibility of vapours, 
 it would appear natural that we should specially consider 
 aqueous vapour, which is constantly present in the 
 atmosphere as superhmted, sometimes even as saimated, 
 steam. And we have for it the splendid collection of 
 experimental results obtained by Eegnault. But the 
 critical point of water vapour is considerably higher than 
 the range of temperature in Eegnault's work ; so that 
 we will deal chiefly with carbonic acid, for which we 
 have Andrews' data both above and below its critical 
 point, and which may be taken as affording a fair 
 example of the chief features of the subject. 
 
 205. Without further preface we give Andrews' dia- 
 
Pio. 20. 
 
176 
 
 PEOPERTIES OF MATTER. 
 
 gram, which will be easily intelligible after what haa 
 been said in § 88. It shows, in fact, how the figure in 
 that section, which is drawn from Boyle's Law, is 
 modified in the case of a true gas, and of a true vapour, 
 each within a few degrees of the critical temperature. 
 
 , [To save space, a portion of the lower part of the 
 diagram (containing the axis of -volumes) is cut away, 
 so that pressures, as shown, begin from about 47 atmos- 
 pheres. The dotted air-curves are rectangular hyper- 
 bolas, as in § 88, but the (unexhibited) axis of volumes is 
 their horizontal asymptote.] .■ ,-, 
 
 The critical temperature of carbonic acid is about 
 30°'9 C, so that the isothermals indicated by full lines 
 in the figure, and marked 13°'l and 21° '5 respectively, 
 belong to vapour or liquid, the others to gas. 
 
 Let us study, with Andrews' data, the values of the 
 product jj» for the isothermal of 13°'l 0. The following 
 table is formed precisely on the same principle as that of 
 § 198 for the isothermal of 35°-5 C. 
 
 Caebonio Acid (Vapour and Liquid) at 13°-1 C. 
 
 pv for Carb. Acid. 
 
 623 
 606 
 600 
 462 
 345 
 
 108 
 106 
 113 
 151 
 196 
 
 Eecip. of Vol. 
 
 Reoip. of Vol. of 
 
 of Air. 
 
 Carbonic Acid. 
 
 47-5 
 
 76-16 
 
 48-76 
 
 80-43 
 
 48-89 
 
 80-90 
 
 49-0 
 
 105-9 
 
 49-08 
 
 142-0 
 
 50-16 
 
 462-9 
 
 50-38 
 
 471-5 
 
 54-56 
 
 480-4 
 
 75-61 
 
 500-7 
 
 90-43 
 
 510-7 
 
COMPKESSIBILITY OF GASES AND VAPOURS. 177 
 
 206. Near to 49 atmospheres liquefaction commences, 
 the vapour being condensed to -rrst of its volunie at one 
 atmosphere, and we see that an exceedingly small increase 
 of pressure produces a marked change of volume. Had 
 it been possible to free the carbonic acid perfectly from 
 air, no additional pressure would have been required till 
 the whole was liquid, at about zhs^ of its original volume. 
 The numbers p) diminish, as in the case of air (but much 
 more rapidly), tiU the liquefaction begins : then they 
 ought to diminish exactly as the volume diminishes (the 
 pressure being constant) till complete liquefaction : after 
 which, of course, they begin to rise rapidly, as it is now a 
 liquid which is being compressed. 
 
 We need not give the experimental numbers for the 
 isothermal of 21° -5 0. ; but the cut shows that the stages 
 of the operation were much the same, only that the pres- 
 sure had to be raised over 60 atmospheres before liquefac- 
 tion began, and liquefaction was complete before the 
 volume had been reduced so far as at the lower temper- 
 ature. Thus the range of volume in which the tube 
 was visibly occupied partly by liquid, partly by saturated 
 vapour, and therefore (but for the trace of air) at con- 
 stant pressure, was shortened at each end. The dotted 
 line in the lower part of the figure, introduced by Clerk- 
 Maxwell, bounds the region in which we can have the 
 liquid in equilibrium with its vapour. This region 
 terminates at the critical isothermal, for above that 
 there can be neither vapour nor liquid. 
 
 But the properties of the gas, above the critical point, 
 maintain a certain analogy to those of the vapour and 
 liquid below it. For moderate pressures the gas has 
 properties analogous to the superheated vapour, ie. pv 
 
178 PROPERTIES OF MATTER. 
 
 diminishes with increase of pressure. For higher pres- 
 sures its properties are analogous rather to those of the 
 liquid, and pv increases with increase of pressure. Thus 
 there is in each isothermal of the gas a particular pres- 
 sure, for which pv is a minimum. This feature of the 
 isothermal becomes less marked as the temperature is 
 raised. [This, however, has been already exhibited 
 more fully on Amagat's diagram, p. 173.] We might 
 introduce a continuation, beyond the critical point, of the 
 left-hand portion of the dotted curve, which should pass 
 through the points on each isothermal at which pi) is a 
 minimum. This line would divide the wholly gaseous 
 region into two parts; that to its right, in which the 
 gas has properties somewhat resembling those of super- 
 heated vapour ; to the left, that in which its properties 
 resemble rather those of a liquid. 
 
 The apparently anomalous behaviour of hydrogen is 
 now to be explained by the fact that, at ordinary tem- 
 peratures and pressures, it is in that region of its gaseous 
 state which has more analogy with the liquid than with 
 the vaporous state. Thus it is probable that if hydrogen 
 be examined at sufficiently low pressure, and temperature 
 not far above its critical point, it also will show a mini- 
 mum value oipv. 
 
 207. The reduction of various gaseous bodies to the 
 liquid form was one of the earliest pieces of original 
 work done by Faraday. Some of them he liquefied by 
 cooling alone, many others by pressure alone; and he 
 pointed out that, in all probability, every gas could be 
 liquefied by the combined influences of cooling and pres- 
 sure, provided these could be carried far enough. 
 
 Thilorier prepared large quantities of liquid carbonic 
 
COMPRESSIBILITY OF GASES AND VAPOURS. 179 
 
 acid; and took advantage of the cooling produced by its 
 rapid evaporation, at ordinary pressures, to reduce it to 
 the solid state. 
 
 Cagniard de la Tour succeeded in completely evapor- 
 ating various liquids (including ether, and even water) 
 in closed tubes, which they half-filled whUe in the 
 liquid state. 
 
 It was Andrews' work, however, which first cleared 
 up the subject, and, as an early consequence of it, those 
 gases which had resisted all attempts to liquefy them 
 were, at the end of 1877, liquefied : — some, such as 
 hydrogen, solidified. These important results were 
 obtained by Pictet in Geneva ; and some . of them, 
 simultaneously and independently, by Cailletet in Paris. 
 
 Van der Waals, Clausius, and others, working from 
 thermodynamical principles and from the kinetic theory 
 of gases, have given formulse which accord somewhat 
 closely with the observed phenomena. But the treat- 
 ment of such matters is not for a work like this. Nor 
 have we anything here to do with the employment of 
 these liquefied gases for the production of exceedingly 
 low temperatures ; though, from the experimental point 
 of view, this application promises to be (for the present 
 at least) their most valuable property. 
 
CHAPTER X. 
 
 COMPRESSION OF LIQUIDS. 
 
 208. A GLIMPSE at the negative results of the early 
 attempts to compress water was given in § 98. The 
 problem is a difficult one, because (at least in the best 
 methods hitherto employed) the quantity really mea- 
 sured is the differeince of compressibility of the liquid 
 and the containing vessel. Hence it involves the com- 
 pressibility of solids also : — ^and this, as we shall find 
 (§ 231) is a very difficult problem indeed. The first to 
 succeed in proving the compressibility of water was 
 Canton,^ the value of whose work seems not to have 
 been fully appreciated. His second paper, in fact, has 
 dropped entirely out of notice. 
 
 Noting the height at which mercury stood in the 
 narrow tube of an apparatus like a large thermometer, 
 immersed in water at 50° F., the end of the tube being 
 drawn out to a fine point and open, he" heated the bulb 
 tiU the mercury fiUed the whole, and then, hermetically 
 sealed the tip of the tube. When the mercury was 
 cooled down to 50° it was found to have risen in the 
 capillary tube. This was due partly to expansion of 
 mercury, released from the pressure of the atmosphere, 
 1 Phil. Trams., 1762. 
 
COMPRESSION OF LIQUIDS. 181 
 
 partly to the compression of the bulb, due to one atmos- 
 phere of external pressure. Then he filled the same 
 apparatus with water, performed exactly the same 
 operations, and obtained a notably larger result. This, 
 of course, proves that water (if not also mercury) expands 
 when the pressure of the atmosphere is removed from it. 
 
 To get rid of the effect of unbalanced external pres- 
 sure, and thus (as he thought) to measure the full 
 amount of expansion, he placed his apparatus (with its 
 end open) in the receiver of an air-pump. He could 
 also place it in a glass vessel, in which the air was com- 
 pressed to two atmospheres. He observed that, on the 
 relief of pressure, the water rose in the stem, while on 
 increase of pressure it fell. He gives the proportional 
 change of volume per atmosphere, at 50° F. (10° C), as 
 1/21740 or 0-000046. He applied no correction for the 
 compressibility of glass, giving the completely fallacious 
 reason that he had obtained exactly the same results 
 from a thick bulb and from a thin one. [This, however, 
 proves the accuracy of his experiments.] His result, 
 considering its date, is wonderfully near the truth. 
 
 209. In a second paper, ^ published a couple of years 
 later, he made some specially notable additions to our 
 knowledge. For he says, referring to his first paper : 
 " By similar experiments made since, it appears that 
 water has the remarkable property of being more com- 
 pressible in. winter than in summer, which is contrary 
 to what I have observed both in spirits of wine and in 
 oil of olives ; these fluids are (as One would expect water 
 to be) more compressible when expanded by heat, and 
 less so when contracted by cold." 
 
 1 PMl. Tram., 1764, vol. liv. 261. 
 
182 PROPERTIES OF MATTER. 
 
 By repeated observations, at "opposite" seasons of 
 the year, he found that the effect of the " mean -weight 
 of the atmosphere" was, in millionths of the whole 
 volume — 
 
 At 34° F. At 64° F. 
 Water ... 49 44 
 
 Spirit of Wine . .60' 71 
 
 He also gives a table of compressibilities in miUionths 
 of the volume, per atmosphere of 2 9 "5 inches,' and of 
 specific gravities ; for difierent liquids, at 50° F. ; as 
 follows : — 
 
 
 Compressibility. 
 
 Spec. Gravity. 
 
 Spirit of Wine 
 
 . 66 
 
 846 
 
 on of Olives 
 
 48 
 
 918 
 
 Rain Water . 
 
 46 
 
 1000 
 
 Sea Water . 
 
 . 40 
 
 1028 
 
 Mercury ' 
 
 3 
 
 13595 
 
 and he observes that the compressions are not " in the 
 inverse ratio of the densities, as might be supposed." 
 
 He calculates from the result for sea water that two 
 miles of such water are reduced in depth by 69 feet 2 
 inches J the actual compression at that depth being 13 
 in 1000. This, of course, assumes that the compress- 
 ibility is the same at all pressures, which, as we shall 
 see immediately, is by no means the case. 
 
 210. Perkins, in 1820, made a set of experiments on 
 the apparent compressibility of water in glass, of a some- 
 what rude kind; but in 1826^ he gave some valuable 
 determinations, unfortunately defective because of the 
 inadequate measure of the pressure unit. Thus he did 
 not give accurate values of the compression, but he intro- 
 
 ^ " On the Progressive Compression of Water by high Degrees of 
 Force." — Phil. Trans. 
 
COMPRESSION OF LIQUIDS. 183 
 
 duced us to a higher problem : — how the compressibility 
 depends upon the amount of pressure. Perkins' results 
 are all for 50° F. (10° C), and are given in figures, as well 
 as in a carefully-executed diagram plotted by the graphic 
 method. His measurement of pressures depended upon 
 an accurate knowledge of the section of a plunger : — an 
 exceedingly precarious method : — and he estimated an 
 atmosphere at 14 lbs. weight only per square inch. It is 
 not easy to make out his real unit, especially as we know 
 nothing about the glass he used, but it seems to have 
 been about 1-5 times too great; i.e. when he speaks of 
 the effect of 1000 atmospheres he was probably applying 
 somewhere about 1500. Hence it is not easy to deduce 
 from his data anything of value as to the amownt of com- 
 pression. But the novel point, which he made out clearly, 
 is that (at 10° C.) the compressibility of water decreases, 
 quickly at first, afterwards more slowly, as the pressure 
 is raised. We obtain from Perkins' diagram the follow- 
 ing roughly approximate results, in which we have made 
 no attempt to rectify his pressure unit :-- 
 
 Pressure Compression of Water Average Com- True Com- 
 
 in in Millionths of pressibility per pressibility per 
 
 Atmospheres. Orig. Vol. Atmosphere. Atmosphere. 
 
 150 10,000 66 51 
 
 300 17,500 58 48 
 
 900 43,400 48 39 
 
 and from a further isolated statement we obtain 
 2,000 83,300 42 
 
 211. Orsted's improvement in the experimental 
 method (1822) consisted chiefly in applying pressure, as 
 in Canton's process, in such a way that the effects of 
 pressures up to 40 or 50 atmospheres can be read off at 
 every stage of the pressure. 
 
ISi 
 
 PROPERTIES OF MATTER. 
 
 The liquid operated on fills the bulb and the greateir 
 part of the stem of the apparatus (called a Piezometer) ; 
 it is bounded by a short column of mercury which 
 separates it from the water-contents of a strong glass 
 cylinder, in which the pressure is 
 produced by forcibly screwing in a 
 piston or plug. The little column 
 of mercury often moves in the water, 
 so that this arrangement is defective. 
 [In the improved form of the instru- 
 ment, the piezometer itself is filled 
 with water and inverted, so that its 
 open end, which is rwt widened (as 
 in the cut), dips into a small capsule 
 containing mercury.] As in Canton's 
 apparatus, the stem of the piezo- 
 meter is carefully calibrated and 
 divided into parts corresponding to 
 equal volumes, and the cubic content 
 of the bulb is determined. Hence 
 the ratio of the content of one 
 division of the tube to the whole 
 content of bulb and stem is found. 
 
 When pressure is applied, the 
 mercury index is seen to descend; 
 and, as the pressure is increased, the index descends 
 nearly in proportion to the pressure. The pressure 
 is easily measured with sufficient accuracy (from 
 Boyle's law) by the observed change of volume of air 
 contained in a very uniform tube, closed at the top, and 
 immersed in the water of the compression vessel. 
 
 Orsted verified Canton's result that "the compress- 
 
 Fia. 21. 
 
COMPRESSION OF LIQUIDS. 185 
 
 ibility of water diminishes with rise of temperature, 
 and suspected that the rate of diminution becomes less 
 as the temperature is raised ; but he did not obtain 
 Perkins' result. In fact he states that at any one 
 temperature the compression is the same, per atmos- 
 -phere, up to 70 atmospheres. 
 
 212. Orstedj.and too many who have followed him, 
 held the opinion that, if the bulb of the piezometer 
 were^ry thin, it would suffer no perceptible change of 
 internal volume by equal interior and exterior pressures. 
 That this (like the somewhat similar notion of Canton) 
 is a fallacy, we see at once from the consideration of the 
 effect of hydrostatic pressure on a solid (§ 176). If we 
 suppose the solid to.be divided into an infinite number 
 of equal cubes, these would be changed into equal but 
 smaller cubes, in consequence of compression. The 
 strained and the unstrained vessel may therefore be com- 
 pared to two vaults of brickwork, similar in every respect 
 as to number and position of bricks, but such that the 
 bricks in the one are all larger in the same ratio than 
 those in the other. From this point of view it is clear 
 that the interior content of the bulb is diminished just 
 as if it had, itself, been a solid sphere of glass. 
 
 Thus the numbers obtained from the piezometer must 
 all be corrected by adding the percentage compression of 
 glass under the same pressure. 
 
 213. Another fallacy much akin to this, and which is 
 still to be found in many books, is that by filling the 
 bulb of the piezometer partly with glass, partly with 
 water, and making a second set of experiments, we shall 
 be able to obtain a second relation between the compress- 
 ibihties of glass and of water ; and that, therefore, we 
 
186 PROPERTIES OF MATTER. 
 
 shall be able to calculate the value of each by piezometer 
 experiments alone. What we have said above shows 
 that this process comes merely to using a piezometer 
 with a smaller internal capacity ; and therefore gives no 
 new information. 
 
 If we had a substance which we knew to be incompress- 
 ible, and were partly to fill the cavity of the piezometer 
 with this, we should be able to get the second relation 
 above spoken of. 
 
 In fact the piezometer gives differences of compress- 
 ibility only ; so that, for absolute determinations with 
 it, we must have one substance whose compressibility is 
 known by some other method. 
 
 214. Eegnault's^ apparatus, though managed by a 
 master-hand, was by no means faultless in principle. 
 For pressure was applied alternately to the outside and 
 to the inside of his piezometer, and then simultaneously 
 to both. There are great objections to the employment 
 of external or internal pressure alone, at least in such 
 delicate inquiries as these. For, unless a number of 
 almost unrealisable conditions are satisfied by the appa- 
 • ratus, the theoretical methods (which must be employed 
 in deducing the results) are not strictly applicable. They 
 are all necessarily founded on some such suppositions as 
 that the bulbs are perfectly cylindrical, or spherical, and 
 that the thickness of the walls and the elastic coeflBcients 
 of the material are exactly the same throughout. These 
 requirements can, at best, be only approximately fulfilled ; 
 and their non-fulfilment may (in consequence of the 
 argeness of the effects on the apparatus, compared with 
 that on its contents) entail errors of the same order as 
 ^ M4m. de TAcad. des Sciences, 1847. 
 
COMPKESSION OF LIQUIDS. 187 
 
 the, whole compression to be measured. Jamin has tried 
 to avoid this difficulty by measuring directly the increase 
 of (external) volume, when a bulb is subjected to internal 
 pressure ; but, even with this addition to the apparatus, 
 we have still to trust too much to the accuracy of the 
 assumptions on which the theoretical calculations are 
 based. 
 
 Finding that he could not obtain good results with 
 glass vessels, Eegnault used spherical bulbs of brass and 
 of copper. With these he obtained, for the compress- 
 ibility of water, the value 
 
 '000048, per atmosphere 
 
 for pressures from one to ten atmospheres. The temper- 
 ature is, unfortunately, not specially stated. 
 
 215. Grassi,^ working with Kegnault's apparatus, 
 made a number of determinations of compressibihty of 
 different liquids, all for small ranges of pressure. 
 
 He verified Canton's specially interesting result, viz. 
 that water, instead of being (like the other substances, 
 ether, alcohol, chloroform, etc., on which he experimented) 
 more compressible at higher temperatures, becomes less 
 compressible. Here are a few of his numbers. 
 
 Temperature C. 
 
 Compressibility 
 per Atmosphere. 
 
 0°0 
 
 0-0000502 
 
 1°5 
 
 515 
 
 4°-0 
 
 499 
 
 10°-8 
 
 480 
 
 18°-0 
 
 462 
 
 26°-0 
 
 455 
 
 34°-5 
 
 453 
 
 53°-0 
 
 441 
 
 ^ Ann. de Chimie, xxxi., 1851. 
 
188 • PEOPERTIES OF MATTER. 
 
 These numbers, when exhibited graphically, show 
 , irregularities too great to be represented by any simple 
 formula. 
 
 Grassi assigns, for searwater at 17°"5 C, 0'94 of the 
 compressibility of pure water, and gives 0'00000295 per 
 atmosphere as the compressibility of mercury. But he 
 asserts that alcohol, chloroform, and ether have their 
 average compressibility, from one to eight or nine atmos- 
 pheres, at ordinary temperatures, considerably greater 
 than the compressibility for one atmosphere. As this 
 result was shown by Amagat to be erroneous, little con- 
 fidence can be placed in any of Grassi's determinations. 
 
 Amagat ^ gave, among others, the following numbers 
 
 for ether: — 
 
 Temperature C. Pressure in Average Compression 
 
 '^ Atmospheres. per Atmosphere. 
 
 13°-7 11 0-000168 
 
 13°-7 33 0-000152 
 
 100° 11 0-000560 
 
 100° 33 0-000474 
 
 Thus the diminution of compressibility with increase of 
 pressure is much more marked at the higher temperature. 
 216. A very complete series of determinations of the 
 compressibility of water (for a few atmospheres of pressure 
 only), through the whole range of temperature from 
 0° C. to 100° C, has recently been made by Pagliani and 
 Vincentini.^ Unfortunately, in their experiments pres- 
 sure was applied to the inside only of the piezometer, so 
 that their indicated results have to be diminished by from 
 40 to 50 per cent. The effects of heat on the elasticity 
 of glass are, however, carefully determined, a matter of 
 
 1 Ann. de Chimie, 1877. 
 2 Sulla Compressibilitd, dei Liquidi, Torino, 1884. 
 
COMPRESSION OF LIQUIDS. 189 
 
 absolute necessity when so large a range of temperature 
 is involved. But in these experiments one datum (the 
 compressibility of water at 0° 0.) has been assumed from 
 Grassi. The results show that the maximum of com- 
 pressibility, indicated by Grassi as lying between 0° C. 
 and 4° C, does not exist. The following are a few 
 of the numbers, which show a temperature effect much 
 larger than "that obtained by Grassi : — 
 
 CompressibUity 
 of Water. 
 
 0-0000503 
 496 
 450 
 403 
 389 
 389 
 398 
 409 
 
 Thus, about 63° 0. water appears to have its minimum 
 compressibility. The existence of a minimum does seem 
 to be proved, but the remarks above show that its 
 position on the temperature scale is somewhat uncertain. 
 
 217. Cailletet^ worked at much higher pressures, 
 and gave the following values (in which we assume, 
 from Eegnault, 0-00000184 for the compression of glass 
 per atmosphere ; Cailletet making no correction, as he 
 says there exist no data for the purpose) : — 
 
 AtmosDheres ■*-™'^»ge Compresaion 
 Atmospneres. ^^^ Atmosphere. 
 
 Water at 8° C. . . . 705 0-0000469 
 
 Sulphuric etter at 10° 0. . 630 0-0001458 
 
 Bisulphide of carton at 8° 0. 607 0-0000998 
 
 Sulphurous acid at 14° C. . 606 0-0003032 
 
 ' Comptes Bendus, 1872. 
 
 Temperature C 
 
 0° 
 
 -0 
 
 2* 
 
 -4 
 
 15° 
 
 -9 
 
 49° 
 
 -3 
 
 61° 
 
 •0 
 
 66° 
 
 •2 
 
 77° 
 
 -4 
 
 99° 
 
 -2 
 
190 PROPERTIES OF MATTER. 
 
 Thus, according to Cailletet, Orsted was right as to 
 the non-diminution of compressibility of water at higher 
 pressures ; and Perkins' results are fallacious. 
 
 Tait^ (assuming the compressibility of glass as 
 0-0000025 per atmosphere) gives, for temperatures from 
 6° C. to 15° C, and pressures from 150 to 500 atmospheres, 
 the following empirical expression for the average com- 
 pressibility of water — ' 
 
 '"■^^^ = 0-0000489 - 0-00000025* - 0-0000000067^, 
 
 where v is the volume at t° C. and p atmospheres, v^ at 
 f C. and one atmosphere. This formula agrees through- 
 out very closely with experiment, but the pressure unit 
 is somewhat uncertain. 
 
 He has also given 0-925 as the relative compressibility 
 of sea-water to pure water within these ranges of tem- 
 perature and pressure. This is, of course, not affected 
 by uncertainty of the pressure unit. 
 
 218. From the results of Andrews already given 
 (§ 205) we find the following roughly approximate 
 values of the 
 
 COMPEESSIBILTTT OF LIQUID CARBONIC AciD AT 13°-1 0. 
 
 Pressure in 
 
 True 
 
 Compressibility 
 
 Atmospheres. 
 
 per Atmosphere. 
 
 50 
 
 
 0-0059 
 
 60 
 
 
 0-00174 
 
 10 
 
 
 0-00096 
 
 80 
 
 
 0-00066 
 
 90 
 
 
 0-00044 
 
 ' Proc. R.S.E., 1883-84. 
 
COMPKESSION OF LIQUIDS. 191 
 
 showing very great, but very rapidly decreasing, com- 
 pressibility. As already explained, Andrews has pointed 
 out that part of this, especially for the lower pressures 
 in the table, is due to the trace of air which, in spite 
 of every precaution, was associated with the carbonic 
 acid. 
 
CHAPTER XI. 
 
 COMPRESSIBILITY AND RIGIDITY OF SOLIDS. 
 
 219. In the two preceding chapters we had to deal with 
 bodies practically homogeneous (except in the case of 
 vapour in presence of liquid) and perfectly isotropic ; 
 and, besides, devoid of elasticity of form, while possess- 
 ing perfect elasticity of volume. Hence the determinar 
 tion of (apparent) compressibility for any definite sub- 
 stance of these kinds depended for accuracy only on 
 the care 'and skill of the experimenter, and on the 
 adequacy of the apparatus employed. 
 
 When we deal with solids the circumstances are very 
 different. It is rarely the case that we meet with a 
 solid which is more than Wj^oximately homogeneous. 
 Some natural crystals, such as fluor spar, Iceland spar, 
 etc., are probably very nearly homogeneous; so are 
 metals such as gold, silver, lead, etc., when melted and 
 allowed to cool very slowly. To produce homogeneous 
 glass (especially in large discs, for the object-glasses of 
 achromatic telescopes) is one of the most difficult of 
 practical problems. On the other hand, crystalline bodies 
 are essentially non-isotropio ;- so is every substance, 
 crystalline or not, which shows "cleavage." 
 
 And further, very small traces of admixture or im- 
 
COMPKESSIBILITY AND RIGIDITY OF SOLIDS. 193 
 
 purity often produce large effects on the elastic, as well 
 as on the thermal and electric, qualities of a solid. body. 
 Think, for instance, of the differences between various 
 kinds of iron and steel, or of the pwrposely added impuri- 
 ties in the gold and sUver used for coinage. Very slight 
 changes in the manipulation, by which wires or rods are 
 drawn from the same material, may make large differ- 
 ences in their final state — differences by no means entirely 
 to be got rid of by heating and annealing, etc' The 
 whole question of " temper " is still in a purely empirical 
 state. Besides, we must remember that every solid has 
 its limits of elasticity, to which attention must be care- 
 fully paid. Thus we can give only general or average 
 statements as to the amount of compressibility or rigidity 
 of any solid, in spite of the labour which Wertheim and 
 many others have bestowed on the subject. 
 
 220. In an elementary work we 'cannot deal, even 
 partially, with the properties of non -isotropic bodies. 
 The necessary mathematical basis of the investigation, 
 though it has been marvellously simplified, is quite 
 beyond any but advanced students. And. the experi- 
 mental study of the problem has been carried out for 
 isolated cases only. Hence we limit ourselves, except in 
 a" few special instances, to homogeneous, isotropic, 
 solids. 
 
 221. On the other hand, the compression or distor- 
 tion produced in a solid by any ordinary stress is usually 
 very small. This consideration tends to simplify our 
 work ; for, as a rule, small distortions may be regarded 
 as strictly superposable. Thus we may calculate, inde- 
 pendently, the effects of each of the simple stresses to 
 which a solid is subjected. 
 
 o 
 
194 PROPERTIES OF MATTER. 
 
 Our warrant for this must of course be obtained 
 experimentally. It was first given by Hooke. 
 
 In 1676^ he published the following as one of "a 
 dedmate of the centesme of the Inventions, etc." — 
 
 " 3. The true Theory of Elasticity or Springiness, a/nd a 
 particular Explication thereof in several Subjects in which it 
 is to be found: And the way of computing the velocity of 
 Bodies moved by them. ceiiinosssttUU." 
 
 The key to this anagram was given in 1678^^ in the 
 words : — 
 
 " About two years since I printed this Theory in an 
 Anagram at the end of my Book of the Descriptions of 
 Helioscopes, viz. cdivnosssttim, id est, Ut tensio sic vis ; 
 That is. The Power of any Spring is in the same propor- 
 tion with the tension thereof : That is, if one power 
 stretch or bend it one space, two will bend it two, and 
 three wiU bend it three, and so forward. Now as the 
 Theory is very short, so the way of trying it is very 
 easie." 
 
 He then shows how to prove the law in various ways : 
 — ^with a spiral spring drawn out ; a watch spring' made 
 to coil or uncoil ; a long wire suspended vertically and 
 stretched ; and a wooden beam fixed (at one end) in a 
 horizontal position, and loaded. 
 
 The above extracts sufficiently show in what sense 
 Hooke intended the words Temm and Vi& to be under- 
 
 1 A Description of Helioscopes, <Ssc., made hy Hobert Hooke, 
 Postscript, p. 31. 
 
 ^ Lectnires de Potentia Sestitutiva, or of Spring, p. [1], This is- 
 a very curious pampMet, containing some remarkably close antici- 
 pations of modem theories, especially Synchronism and its results, 
 and the Kinetic Theory of Oases. The first is foreign to our present 
 subject, the second will be considered later (§ 322). 
 
COMPKESSIBILITY AND RIGIDITY OF SOLIDS. 195 
 
 stood : — and his law is now usually stated in the (some- 
 what amplified) form, 
 
 Distortion is proportional to the distorting Force, 
 or, still more definitely, 
 
 Strain is proportional to Stress. 
 In the latter form we have made anticipatory use of it 
 in Chap. VIII. and elsewhere. 
 
 222. A very general proof of the accuracy of this law 
 is easily to be obtained in the case of bodies which can 
 be made to produce a musical sound — a tuning-fork, for 
 instance. For, if the pitch of the note (i.e. the number 
 of vibrations per second) do not alter as the sound grows 
 fainter, the vibrations must be isochronous, and the 
 elastic resilience therefore proportional to the distortion. 
 (See § 72.) 
 
 223. The ordinary experimental illustrations of 
 Hooke's Law are given, very much as he originally 
 gave them, by ; — 
 
 1. A rod or wire, fixed vertically and stretched by 
 appended weights ; or a rod or column compressed by 
 weights laid on its upper end. 
 
 2. A wire stretched horizontally and extended by 
 weights suspended at its middle point. 
 
 3. A bar or plank fixed horizontally at one end and 
 loaded with weights at the other. 
 
 4. A plank with its ends resting on trestles and loaded 
 at the middle. 
 
 5. A spiral spring, forming a helix of smail step, 
 compressed or extended by weights. 
 
 6. A wire or rod, fixed at one end and twisted at the 
 other. 
 
 The mere mention of these methods is sufficient, 
 
198 PROPERTIES OP MATTER. 
 
 ■without further illustration, to suggest the means by 
 which the requisite measurements can be carried out. 
 They will be considered in detail, but not in the above 
 order. 
 
 In all these cases experiment shows that (within 
 certain limits, which will be afterwards discussed) the 
 distortion is proportional to the distorting force. 
 
 1 and 2 are mere varieties of one experiment. 
 The same may be said of 3 and i, which are examples 
 of a somewhat more complex form. And 5 and 6, 
 though at first sight very unlike, are practically one 
 problem. Besides, they are of a simpler character than 
 either of the other pairs, for they involve the coeflBcient 
 of rigidity alone; the others involve both coeflScients. 
 But 1 and 2, on the other hand, are simpler than the 
 rest, on a difi'erent account, viz. that they involve homor 
 geneous strain. 
 
 224. Ymmg's Modulus, as it is called, is determined from 
 the stretching of a rod or wire by appended weights. 
 its defined by Young, its measure is the ratio, of the 
 simple stress required to produce a small shortening or 
 elongation of a rod of unit section, to the proportionate 
 change of length produced. Its value is, as we see by 
 § 181, in terms of the rigidity and the resistance to. 
 compression, g^j,^ 
 
 3k + n 
 
 For bodies like india-rubber, in which k is large in 
 comparison with n, its value is nearly 3w. Hence the 
 pulling out of an india-rubber band is almost entirely 
 due to change of form, and therefore the area of a cross 
 section is diminished in nearly the same proportion asr 
 that in which the band is lengthened. 
 
COMPRESSIBILITY AND RIGIDITY OF SOLIDS. 197' 
 
 A piece of good cork suggests, though it does not 
 realise, the conception of a solid in which n shall be very 
 large in comparison with k ; and for such a body Young's 
 modulus would be nearly 9^. Traction or pressure, in 
 any direction, would expand or contract a body of this 
 kind nearly equally in all directions. In cork the effect 
 is confined mainly to the dimension operated on. 
 
 From such considerations we see that Young's modulus, 
 though comparatively easy of measurement, is not the 
 simple quantity which it at first appears to be; and that, 
 in fact, it may have the same numerical value in each of 
 two bodies which differ widely from one another, alike 
 in rigidity and in compressibility. 
 
 225. The following table gives approximate values 
 (§ 166) of Young's modulus for some common materials ; 
 the unit being 10'' grammes' weight per square centi- 
 mfetre : — 
 
 Young'3 I 
 
 modulus, „, , . . Tenaci 
 
 Gold 
 
 86 
 
 0-27 
 
 Silver 
 
 76 
 
 0-3 
 
 Copper (hard) 125 
 
 0-4 
 
 Copper (annealed) 110 
 
 0-3 
 
 Iron . 
 
 180 
 
 0-6 
 
 Steel . 
 
 240 
 
 0-8 
 
 Oak . 
 
 10 
 
 0-1 
 
 Teak . 
 
 17 
 
 0-1 
 
 Fir . 
 
 12 
 
 0.07 
 
 Glass . 
 
 40 to 60 
 
 0.06 
 
 To convert these numbers (as they stand in the table) 
 into the common reckoning of pounds' weight per square 
 inch, it suffices to multiply them by about 142,000 instead 
 of by 10^. To convert to 0. G. S. units, i.e. dynes per 
 square centimetre, multiply by 9 'SI X 10^. 
 
198 PROPERTIES OF MATTER. 
 
 226. A second column (to the same units) has been 
 added to the above table, to give an indication of the 
 Tenacity of each of the materials specified. This means 
 the utmost longitudinal stress -which (when cautiously 
 applied) a rod or wire can endure without rupture. It 
 has no direct connection with Young's modulus, nor with 
 either of the coefficients of elasticity, for a substance has 
 usually to be straiued far beyond its limits of elasticity 
 before rupture takes place, and the dimensions of the 
 cross section are also much reduced. The uncertainty 
 of the amount of this quantity, even in different speci- 
 mens taken from the same piece of matter, leads to our 
 giving it usually to one significant figure only. 
 
 227. Young's treatment of the subject of elasticity is 
 one of the few really imperfect portions of his great 
 work.i He gives the value of his modulus for water, 
 mercury, air, etc. ! It is not easy to understand what 
 he really meant by calling the quantity above described 
 " the " modulus of elasticity. It is, no doubt, a quantity 
 of great value in practical engineering : — in many cases 
 the only elastic datum required. Yet Young speaks of 
 rigidity, etc., in a way which is scarcely Compatible with 
 the idea of one modulus only. But the subject was in a 
 state of great confusion tUl long after his time, mainly 
 in consequence of an unwarranted conclusion (due to 
 Navier and Poisson) that there is a necessary numerical 
 ratio between rigidity and resistance to compression. 
 In fact, what was called Poisson's ratio, that of the lateral 
 shrinking, to the longitudinal extension, of a bar or rod 
 under tension, was supposed to be necessarily equal to 
 1/4. This gives (§ 180) p = iq, or 3k = 5n. 
 
 ^ Lecbwres on Natural Philosophy, 1807. 
 
COMPRESSIBILITY AND RIGIDITY OF SOLIDS. 199 
 
 The erroneousness of this conclusion was first pointed 
 out by Stokes,! and his paper, followed by the investiga- 
 tions of de St. Venant' and Sir W. Thomson,* has put the 
 whole subject in a new and clear light. The former has 
 given complete solutions of a number of interesting cases, 
 many of which are very valuable in practical applications. 
 Thomson, besides giving the theory with extreme gener- 
 ality, has also specially developed the application of 
 Thermodynamics * to the subject. 
 
 228. We will now consider the pure Torsion of a 
 cylindrical rod or wire, as eniployed, for instance, in 
 the Cavendish experiment (§ 153). 
 
 This is a very simple problem if the cylinder be truly 
 circular, and of perfectly homogeneous isotropic material 
 For it is clear that equal and opposite couples, applied 
 perpendicularly to its ends, will simply make successive 
 transverse slices, of equal thickness, rotate about the 
 axis each by the same amount less than the one before it. 
 
 The length of the cylinder cannot increase under 
 torsion, for a reversal of the couples (which is practically 
 the same arrangement) would shorten it, and nee versd. 
 Neither can its radius increase, for exactly the same 
 reason. Nor can a transverse section become curved 
 at any part. Thus the volume remains unchanged, and 
 the coefficient of rigidity alone is involved. 
 
 Consider a thin annular portion of the solid bounded 
 
 by transverse sections at a very small distance, t, from 
 
 one another, and by concentric cylinders of radii r, and 
 
 r + t. We may subdivide this into cubes, of side t, 
 
 ^ Owmb. Phil. Tram., 1845. 
 
 2 MSm. des Savans Mrwngers, 1855. See also Thomson and 
 Talt's Nai. Phil., §§ 699, etc. ' Phil. Trans., 1854. 
 
 * Quarterly Math. Journal,[1855. 
 
200 PROPERTIES OF MATTER, 
 
 by planes through the axis, making angles tjr with one 
 another. 
 
 Let <^ be the twist per unit length of the cylinder, 
 t^ is the angle by which one of 
 our parallel sections has rotated 
 relatively to the other, and r. t(j)/t, 
 or r<j>, is the change of angle in 
 each of the little cubes. Hence, 
 if P be the tangential force per 
 unit of area (§ 178) 
 P = nr<j>. 
 The moment, about the axis, of the tangential force 
 on the cube is therefore 
 
 Ft", r = mrtV- 
 [Note here, for an ulterior purpose, that r^f^ is the 
 moment of inertia (§ 132) of the area of the face of the 
 cube about the axis.] 
 
 But the number of cubes is lirrjl, so that the whole 
 moment is 'i'nivfit^. 
 
 This is the couple required to twist a- circular cyhnder 
 of radius r, and very small thickness t, through the 
 angle ^ per unit of length. 
 
 To find the result for a solid cylinder of radius E, we 
 must put dr for t, and integrate. The result is 
 
 |»R^0. 
 
 Hence the twist produced, per unit of length in a 
 cylinder, is directly as the twisting couple; inversely 
 as the rigidity and as the fourth power of the radius. 
 
 229. This suggests an obvious and direct experi- 
 mental process for determining the rigidity of homo- 
 
COMPRESSIBILITY AND RIGIDITY OF SOLIDS. 201 
 
 geneous isotropic substances. There are two diflBculties, 
 of a formidable character, in the way of its application : 
 first, the obtaining a homogeneous isotropic material, and 
 secondly, the making it into a circular. cylinder. It is 
 . clear that very small irregularities of form, or errors in 
 the estimate of the radius, may give rise to large errors 
 in the calculation of the rigidity, since the fowrth power 
 of the radius is directly involved in the calculations. 
 And it is probable that the mode of manufacture of the 
 cylinder (especially if it be drawn) may render its other- 
 wise isotropic material markedly non- isotropic. Hence 
 the following numbers are given as mere approximations. 
 The unit is, again, 10^ grammes' weight per square 
 centimetre (§ 225). 
 
 Approximate Eigidity 
 
 (n). 
 
 Glass 
 
 15 to 25 
 
 Biass .... 
 
 35 
 
 Iron (wrought) . 
 
 79 
 
 Iron (cast) 
 
 55 
 
 Steel .... 
 
 85 
 
 Copper .... 
 
 45 to 50 
 
 These values are for ordinary temperatures. As 
 the temperature is raised, the rigidity is found steadily 
 to diminish. 
 
 230. When a spiral spring is drawn out, it is pretty 
 clear to every one that there is unbending, for the curv- 
 ature becomes less as the helix is lengthened. And 
 the following simple experiment shows that this flexure 
 is accompanied by torsion. Coil up a strip of sheet 
 india-rubber, as in the cut, and pull out the inner end. 
 It assumes the form sketched. The portion puUed out 
 straight is twisted merely; the coiled part is merely 
 
202 
 
 PKOPERTIES OF MATTER. 
 
 bent; the intermediate portion is partly bent and partly 
 twisted. Every coil puUed out gives one complete turn 
 of twist. If we make lemks on the strip, as below, 
 then, on pulling out the first, we find two complete 
 turns of twist, but on pulling out the second there is no 
 
 LFIO. 24. 
 
 twist, one of the kinks giving a right-handed, the other 
 a left-handed, complete tiim of twist. 
 
 When the spring is very flat, i.e. has a very small 
 step, the principal effect of a moderate extension is 
 mere torsion; and the investigation is of a character 
 precisely the same as that in the preceding section. The 
 somewhat more complex combination, of torsion and 
 flexure simultaneously, will be adverted to later. 
 
 231. Theoretically speaking, we can of course deduce 
 the resistance to compression from the (known) values of 
 the rigidity and of Young's modulus ; and it is in this 
 way that most data on the subject have been obtained. 
 But especially in cases where Young's modulus is not 
 very far from threefold the rigidity (as, for instance, in 
 
COMPEESSIBILITY AND KIGIDITY OF SOLIDS. 203 
 
 india-rubber), the inevitable errors in the determination 
 of these might lead to enormously greater errors in the 
 calculated value of h 
 
 The method which was incidentally employed by 
 Eegnault in his measurements of the compressibility 
 of liquids, consisted in applying pressure externally, 
 internally, and externally and internally, to a species 
 of piezometer containing water. The results of § 183 
 show that (supposing it cylindrical, and unit pressure 
 applied) its internal volume must have been altered, in 
 these three cases respectively, by the following fractions 
 of its whole amount : — 
 
 
 1 
 k 
 
 The algebraic sum of the two first is of course equal 
 to the third. But the quantities measured in the two 
 latter cases were both less than those stated above by the 
 fractional change of volume of the water. The relation, 
 therefore, stiU holds, and furnishes a test of the accuracy 
 of the experiments. But it reduces the number of 
 independent equations to two, from which there are 
 three coefficients of elasticity to be determined. Hence 
 Eegnault also had to fall back on the employment of 
 Young's modulus. 
 
 232. Probably the best, certainly the most direct, 
 method is that adopted by Buchanan,^ in which the 
 length of a rod is very carefully measured while it is 
 1 Trans. B.S.E., 1880. 
 
204 PROPERTIES OF MATTER. 
 
 under hydrostatic pressure, and also while free. The 
 linear contraction so determined is (if the, material be 
 homogeneous and isotropic) one-third of the compression 
 (§ 176). Unfortunately his published measures are con- 
 fined to one particular kind of glass. The special 
 merit of this method is that, provided the rod be of 
 isotropic material, the regularity of its cross section is 
 of no consequence. 
 
 Thus we can give only a few roughly approximate 
 numbers for this property also. They are given in the 
 same units as the preceding. 
 
 Approximate Eesistance to Compression (k). 
 
 Glass 20 to 40 
 
 Copper .... 160 
 
 Iron (wrought) . . 150 
 
 Steel 185 to 200 
 
 It is greatly to be desired that more, and more accurate, 
 data should be obtained in this matter. 
 
 233. Though, as we have seen, we can give only 
 general and somewhat vague numerical data, there is 
 practical unanimity on the part of experimenters that, 
 witMn the limits of elasticity, Hooke's law is very closely 
 followed. Hence, although it is necessary to measure 
 the elastic coeflScients for each specimen of each sub- 
 stance we employ, once that measurement is effected 
 we can trust to it as giving the special qualities of the 
 material through a range which, in glass, steel, etc., is 
 often very wide. One excellent example is to be found 
 in the substitution of glass or steel for air or nitrogen 
 in the construction of instruments for measuring hydro- 
 static pressure. 
 
COMPRESSIBILITY AND RIGIDITY OF SOLIDS. 205 
 
 The first to introduce this principle seems to have 
 been Parrot,^ whose MaUromkire was merely an ordinary 
 thermometer, with a bulb thick enough to stand great 
 pressure. Keeping it immersed in water at a constant 
 temperature, and applying great pressures, he found 
 that the diminution of capacity of the bulb was almost 
 exactly proportional to the pressure. 
 
 Instruments working on the same principle have 
 since been introduced, in ignorance of Parrot's work, by 
 many investigators. Bourdon gauges, aneroid barometers, 
 etc., are merely special instances. 
 
 Sudden application of pressure produces temperature- 
 changes, but these instruments (in Parrot's form at least) 
 may be made practically insensible to such changes by 
 the simple expedient of nearly filling the bulb (which, 
 for this purpose, should be cylindrical) with a piece of 
 glass tube closed at each end.^ The mercury in the 
 bulb is thus greatly reduced in quantity, and therefore 
 the temperature efiects in the stem are very small, while 
 the instrument is still as ready as ever to indicate 
 changes of volume. 
 
 The dimensions and thickness of such an instrument, 
 for any special purpose, can be easily calculated from the 
 formulae of § 183 ; and the unit of pressure can be deter- 
 mined for it, by a single comparative experiment, with 
 the aid of Amagat's table of compression of air (§ 200). 
 
 There is great advantage in using simultaneously two 
 instruments of this kind, in one of which the thickness is 
 
 ^ "Experiences de forte Compression sur Divers Corps," Mim. 
 de l'Acad6mie Impiriale des Sciences de St. Petersbowrg, 1833. 
 
 ^ Tait, Rqiort on, (he Pressure Errors of the OJmllenger Ther- 
 mometers, 1881. 
 
206 PROPERTIES OF MATTER. 
 
 considerably greater (in comparison with the diameter) 
 than in the other. For, so long as their indications 
 agree, loth may be trusted as following Hooke's law 
 very accurately; 
 
 234 The limit of pressure measurable by means of 
 these instruments depends upon the resistance of a glass 
 or steel tube to crushing by external pressure. From a 
 series of experiments, made for the purpose, Tait ^ has 
 calculated that ordinary lead glass (in the form of a tube 
 closed at each end) gives way when the distortion of 
 the interior layer amounts to a shear of about 1 i ^\(y , 
 coupled with a compression of about -g^- Hence 
 even a very thick tube of such glass cannot resist 
 more than .about 14 tons' weight per •square inch (2130 
 atmospheres) of external pressure. No corresponding 
 experiments seem yet to have been made for steel. 
 
 235. We now come to the case of bending of a rod or 
 bar. Here we have no such simple problem as in the 
 case of the torsion of a cylinder, and must consequently 
 assume the solution as given by mathematical investiga- 
 tion. This shows us that, so long as the radius of 
 curvature is large in comparison with the thickness of 
 the bar in the plane of bending, the line passing through 
 the centre of inertia of each transverse section, the 
 elastic central line as it is called, is bent merely, and not 
 extended nor shortened. 
 
 The flexural rigidity of the bar, in any plane through 
 the central line, is directly as the couple in that plane, 
 which is required to produce a given amount of curvature 
 in the central Hne. Its amount may easily be calculated 
 by means of the following considerations. Let the 
 ' Proc. 11.S.K, April 18, 1881. 
 
COMPRESSIBILITY AND RIGIDITY OF SOLIDS. 207 
 
 figure represent a section of the cylinder, C its centre 
 of inertia, CD a line in it perpendicular to the plane of 
 bending, and let the centre of curvature of the bending 
 lie towards E. Then obviously all lines parallel to the 
 axis of the bar on the E-ward side of CD are compressed, 
 all towards the other side extended ; each in proportion 
 to its distance from CD and 
 to the curvature. If we con- 
 template a transverse slice, of 
 small thickness t, we see that 
 its thickness remains un- 
 changed along CD, is dimin- 
 ished on the E-ward side of 
 that line, and increased on 
 the ■ other. The thickness 
 at the small area A becomes ^"'- ^^■ 
 
 t (l + — \ where r is the radius of bending. This 
 
 requires a tension A —m, where m is Young's modulus. 
 The moment of this about CD is — 
 
 A.AB^™. 
 r 
 
 Hence the sum of all such, i.e. the moment of the 
 bending couple, is - multiplied by the moment of inertia 
 
 of the. area of the section about CD. Now through C, 
 in the plane of the section, there are two principal axes 
 of inertia, in directions at right angles to one another. 
 Hence, except in the cases of " Kinetic Symmetry " of 
 the section (as when it is circular, square, equilateral- 
 triangular, etc.), there are two principal flexural rigidities, 
 a maximum and a minimum, in planes (through the 
 
208 PROPERTIES OF MATTER. 
 
 axis) perpendicular to one another. If the rigidities in 
 these planes be called Rj and Eg, the flexural rigidity in 
 a plane inclined at an angle to that of Ej is — 
 
 Rjcos^.e + R2sin2.e. 
 
 [Compare § 228, in which the corresponding case of 
 torsion-rigidity was shown to depend upon the moment of 
 inertia of the area of the section about the elastic central 
 line. This is the third principal axis of the area at its 
 centre of inertia.] 
 
 236. It appears from last section that flexure (within 
 moderate limits) is, practically, as regards any very small ■ 
 portion of the substance, the same thing as longitudinal 
 extension or compression, and therefore gives us no 
 simple information as to the elastic coefficients of the 
 substance. But it has very important practical appli- 
 cations, and therefore we devote some sections to the 
 more common cases. 
 
 The principal moments of inertia of the area of a 
 rectangle, sides 2a and 26, about axes through its centre 
 and in its plane, are da'S/S and iab^/S. Multiplied by 
 m,. they represent the flexural rigidities of a plank in ■ 
 planes parallel to its breadth, and to its thickness re- 
 spectively. These, multipUed by the. bending curvature, 
 give the couple required to produce and to maintain it. 
 
 237. The Elastic Cwrve of James Bernoulli, celebrated 
 in the early days of the diflerential calculus, is a particular 
 case of the bending of a wire or plank, in which the 
 flexural rigidity in the plane of bending is the same 
 throughout, and a simple stress (§ 128) alone is applied. 
 
 The obvious condition is that the curvature at each 
 point is directly proportional to the distance from the 
 
 i 
 
COMPRESSIBILITY AND RIGIDITY OF SOLIDS. 209 
 
 line in which the stress acts. For the investigation of 
 'the equation of the curve from this condition, and for 
 drawings of its various forms, the reader must be referred 
 to works on Ahskact Dynamics /■* but we figure here the 
 special case which corresponds to a stretched uniform 
 
 wire, of infinite length, with a single kink upon it. 
 This wiU be referred to in Chap. XII. below. 
 
 The investigation of the bending of planks, variously- 
 supported, and under various loads, is a somewhat 
 generalised form of the question of the elastic curve. 
 The principles involved in its solution are simple, and 
 almost obvious; but the mathematical treatment of it 
 would lead us too much out of our course. So would 
 that of the problem of the eflfect of a couple applied 
 anyhow to one end of a cylindrical or prismatic wire, 
 of any form of section, the other end being fixed. The 
 wire, in such a case, takes generally the form of a 
 circular helix. The extreme particular cases are — (1) 
 when the wire is in the plane of the couple, and there 
 is bending only ; (2) when the wire is perpendicular to 
 the plane of the couple, and there is twist only. 
 
 238. The results hitherto given are all approximate 
 
 ^ See, for instance, Thomson and Tail's Nat. Phil., toL i. 
 part ii. p. 148. 
 
 P 
 
210 PROPERTIES OF MATTER. 
 
 only, and depend upon the radius of bending being large 
 compared with the thickness of the wire or bar in the 
 plane of flexure. Those given in § 2 28, for torsion, may be 
 applied, under a similar restriction, to cases in which the 
 section of the wire or bar is not circular. The mathe- 
 matical treatment of the exad solution of such problems 
 is of too high an order of difficulty for the present work ; 
 but some of its results, alike interesting and important, 
 may be easily understood. A few of them wiU now be 
 given, but the reader must be referred to the works 
 already cited (§ 227) for a more complete account. 
 
 239. Thus, in the flexure of a uniform bar into a 
 circular arc, we saw (§ 235) that each fibre is extended 
 or compressed to an amount depending on its distance 
 from the plane passing through the centres of inertia of 
 its transverse sections (wMle it is straight), and per- 
 pendicular to the plane of bending. But this involves 
 (§ 177) uniform compression or extension of the trans- 
 verse section of the fibre to an amount proportional to 
 the extension or shortening of its length. Hence, if the 
 section of the unbent bar be divided into equal indefinitely 
 small squares, each of these wiU remain a square after 
 'bending. From this we can obtain an approximate idea 
 of the change of shape of the transverse section. 
 
 Consider the annexed figure, which represents a series 
 of concentric circles, whose radii increase in a slow 
 geometrical ratio, intersected by radii making with one 
 another equal angles such that the arcs into which any 
 one circle is divided are equal to the difi^erence between 
 its radius and that of the succeeding circle. When the 
 circles and radii are infinitely numerous, all the little 
 intercepted areas are squares. The sides of the squares 
 
COMPRESSIBILITY AND RIGIDITY OF SOLIDS. 211 
 
 along CD are obviously greater than those of the squares 
 along AB by quantities proportional to AO. Those of 
 the squares along EF are less than those of the squares 
 along AB by quantities proportional to AE. The figure 
 CDFE must therefore represent the 
 distorted form of the cross section 
 of a beam, originally rectangular, 
 and bent in a plane through OGr 
 (and perpendicular to the plane of 
 the figure). The side of the beam 
 which is concave in the plane of 
 flexure is convex in a direction per- 
 pendicular to the plane of flexure ; 
 that which is convex in the former 
 plane is concave in the latter. The 
 cause is, of course, the 'transverse 
 swelling of the fibres on the side to- 
 wards G, the centre of bending, and 
 the diminution of section of those 
 on the other side of the bar. It is 
 sufficiently accurate to assume that 
 AB, which is unchanged in length, was originally mid- 
 way between the faces of the bar. 
 
 If OG- be the radius of flexure, the ratio of the 
 extension of one of the fibres which pass through a point 
 of EF to its original length is AE/OG. Its lateral con- 
 traction in all directions must therefore be (§ 180) 
 
 3A-2m 
 
 AE/OG. 
 
 2 (3fc+») 
 
 But it is also obviously AE/OH. Hence 
 2(3ft + n)OG = (Sk - 2m)0H. 
 
212 
 
 PROPERTIES OF MATTER. 
 
 Thus the point H is determined, and the approximate 
 solution is complete. A, square bar of vulcanised indiar 
 rubber shows these results very clearly. 
 
 240. In the case of torsion of a cylinder whose 
 section is not circular, plane transverse sections do not 
 remain plane. The following figure gives de St. 
 Venant's result for an elliptic cylinder. It represents the 
 
 FlQ. 28. 
 
 contour lines of the distorted section made by planes 
 perpendicular to the axis. They are equilateral hyper- 
 bolas (as in § 88), the common asymptotes being the axes 
 of the section. The torsion is applied in the positive 
 direction to the end of the cylinder above the paper; 
 and the full lines represent distortion upwards; the 
 dotted,- downwards. 
 
 241. Coulomb, who first attacked the torsion problem, 
 was led (by an indirect and unsatisfactory process) to 
 the result above, viz. that the torsional rigidity is pro- 
 portional to the moment of inertia of the area of the 
 transverse section about the elastic central line. This is 
 true only in circular cylinders or wires. It gives too 
 large a value for all other forms of section. From de 
 St. Venant's paper we extract the following data. The 
 first numbers express the ratio of the true torsional 
 
COMPEESSIBILITY AND EIGIDITY OF SOLIDS. 213 
 
 rigidity to the estimate by Coulomb's rule. The second 
 numbers show the ratio of the torsional rigidity to that 
 of a cylinder, of the same sectional area, but circular. 
 
 Equilateral Triangle. 
 
 Square. 
 
 0-600 
 
 0-843 
 
 0-725 
 
 0-883 
 
 The torsional rigidity of an elliptic cylinder, a and h 
 being the semi-axes of the transverse section, is 
 
 ""^T6^- 
 
 When J = fl! we have, !of course (as in § 228), 
 
 2 
 
 242. From these and like results we are led to 
 see that projecting flanges, which add greatly to the 
 flexural rigidity of a rail or girder, are practically of 
 no use as regards resistance to torsion. 
 
 Another of de St. Venant's important results is that 
 the places of greatest distortion in twisted prisms are 
 the parts of the boundary nearest to the axis. 
 
 Near a re-entrant angle in the boundary of the 
 section there are usually infinite stress and infinite 
 strain, whether the stress .be such as to produce torsion 
 or bending. Hence the reason for the practical rule of 
 always rounding oif such angles, when they cannot be 
 entirely dispensed with. 
 
 243. Still keeping to statical experiments, we have 
 to consider briefly the limits of elasticity. 
 
 When a solid is strained beyond a certain amount, 
 which depends not merely on its material but upon its 
 state and the mode of its preparation, one of two things 
 
214 PROPERTIES OF MATTER. 
 
 occurs. Either it breaks, and is said to be brittle, or it 
 becomes permanently distorted, and is said to he plastic. 
 Dififerent kinds of steel, or the same steel differently 
 tempered, give excellent instances. Some have qualities 
 superior to those of the best iron, others are more brittle 
 than glass. 
 
 244. When a body has been permanently distorted, 
 as, for instance, a copper wire which has received a few 
 hundred twists per foot, it has new limits of elasticity 
 (within which Hooke's law again holds, though with 
 altered coefficients) j but the elasticity, at all events for 
 distortions of the same kind, is usually of a very curious 
 character, inasmuch as the strain produced by a stress 
 will, in general, no longer be exactly reversed by reversal 
 of the stress. In fact the body has been rendered non- 
 isotropic; and, so far as this problem has yet been 
 treated (though that does not amount to much), it is of 
 the order of questions which we cannot enter on in this 
 volume. 
 
 The limits of elasticity vary so much, even in different 
 specimens of the same material, that no numbers need 
 here be given. Every one who has occasion to take 
 account of these limits must determine them for himself 
 on the materials he is about to employ. 
 
 245. A curious fact, showing that elasticity may remain 
 dorrmmf, as it were, is exhibited by sheet indiarrubber. 
 When it has been wound in strips, under great tension, 
 on a stout copper wire, and has been left in that condi- 
 tion for years, it appears to harden in its state of strain, 
 and can be peeled off like a piece of unstretched gutta- 
 percha. But, if it be placed in hot water, it almost 
 instantly springs back to its original dimensions. The 
 
COMPRESSIBILITY AND RIGIDITY OF SOLIDS. 215 
 
 experiment may be made, but with less perfect results, 
 in a few minutes, by merely putting the strained india- 
 rubber into a mixture of snow and salt. 
 
 246. Excellent instances, illustrative of the possibility 
 of arrangements giving peculiar kinds of non-isotropy, 
 are furnished by many manufactured articles, such as 
 woollen or linen cloth, wire-gauze, etc., in which Young's 
 modulus is large for strips cut parallel to the warp or 
 woof, but small for strips cut diagonally. Still more 
 curious is wire-gauze in which the meshes are rhombic. 
 Another suggestive instance is a strip formed of wire 
 
 Via. 30. 
 
 knotted as in Fig. 29, in which the flexure and torsion 
 rigidities for any bending or twist, and its reverse, are 
 markedly different. Similarly a coat -of -mail made of 
 rings, each three joined as in the first figure (30 above), 
 is perfectly flexible ; as in the second figure, nearly rigid. 
 247. Kinetic processes for determining coeflScients of 
 elasticity are often based upon the pitch of the note 
 given out by a vibrating body. We do not give any of 
 these, as they belong properly to the subject of Sound. 
 
216 PROPERTIES OF MATTER. 
 
 All require an exact determination of pitch, and (except 
 in the very simplest case, that, of stretched wires, as those 
 of a piano) require, for their comparison with the other 
 experimental data, higher mathematics than we can 
 introduce here. 
 
 248. There is, however, one kinetic process of a very 
 simple character (we have already adverted to it while 
 describing the Cavendish experiment, § 153) by which 
 the rigidity of a body is determined from torsional 
 vibrations. 
 
 The wire to be experimented on is firmly fixed at its 
 upper end, and supports a mass whose weight is suffi- 
 cient to render it straight, but not so great as to produce 
 any sensible efiect on its rigidity. The moment of 
 inertia of this mass may be caused to have any desired 
 value by making the whole into a transverse slice of a 
 hollow circular cylinder of sufficient radius, which can 
 be very accurately turned and centred on a lathe. The 
 wire must be attached to a light cross bar, so as to lie 
 in the axis of the cylindrical vibrator. 
 
 If N represent the torsional rigidity of the wire, I its 
 length, and (j> the angle through which the vibrator has 
 been turned, the elastic couple is 
 
 The rate at which work is done against the elastic 
 forces is , 
 
 But this must be equal to the rate at which the appended 
 mass loses energy of rotation, i.e. (§ 135) 
 
 -10^ 
 
COMPRESSIBILITY AND RIGIDITY OF SOLIDS. 217 
 if I be its moment of inertia. Hence 
 
 This shows that the oscillations are of the simple 
 harmonic character, and that the period is 
 
 or, if the wire be of circular section (§ 228), 
 
 / SttIZ 
 
 V »R*" 
 
 In this expression all the factors are known, with the 
 exception of n, which can therefore be determined. 
 
 The chief difficulties in the application of this process 
 are the iinding exactly the radius of the wire, and the 
 ensuring that its substance is really isotropic. 
 
 249. The solution just given is accurate only if all the 
 circumstances have been taken into account. But a very 
 few' trials, with wires of different metals, show that the 
 range of vibration diminishes at every oscillation, and 
 with some metals much more rapidly than with others. 
 This cannot, therefore, be wholly due to the resistance 
 of the air. Part of it, at least, is undoubtedly due to 
 the dissipation of energy, by thermal effects of change 
 of form, which occur even when the elasticity is perfect. 
 This, however, is beyond our province. But a large part, 
 with metals like zinc much the greater part, is due to 
 internal viscosity. 
 
 250. So long as we deal with steel, iron, silver, -etc., 
 and keep to torsions weU within the limits of elasticity, 
 the arc of oscillation is found to diminish in simple 
 
218 PROPERTIES OF MATTER. 
 
 geometrical progression. This points to a resistance to 
 the motion, partly due to air acting on the suspended 
 mass, partly to thermal effects and to viscosity in the 
 wire itself, but, on the whole, proportional to the rate of 
 motion, i.e. the rate of distortion. 
 
 Thus the equation of § 248 takes the form 
 
 .^ + A0 + ji ^ = 0. 
 
 The solution of the problem in this case is, therefore, 
 of the nature of that 'given in § 74 above ; and we see 
 that, if the diminution of the arc of oscillation (per 
 vibration) is large, the periodic time will be perceptibly 
 increased. Thus the direct determination of n, by the 
 mode of calculation given in § 248, would necessarily 
 lead to underestimation of its value. 
 
 The logarithmic decrement of the arc of vibration gives 
 us K, the time of vibration gives us m, and then we have 
 
 whence N, and therefrom n, can be found. 
 
 251. All this part of our subject is still very imperfectly 
 worked . out We have already seen (§ 50) that even 
 brittle bodies may be completely changed in form by 
 small but persistent forces. And there is no doubt that 
 all elastic recovery in solids is. gradual, so that, for 
 instance, in the torsion vibrations which we have just 
 considered, even when there is no sensible viscous 
 resistance, the middle point of the range does not coin- 
 cide with the original untwisted position of the wire. 
 It is always shifted towards the side to which torsion 
 was applied, and to a greater extent the longer the wire 
 
COMPRESSIBILITY AND RIGIDITY OF SOLIDS. 219 
 
 has been kept twisted before being allowed to vibrate. 
 With every vibration, however, it creeps slowly back 
 towards the original undisturbed position, but usually 
 comes to rest before reaching it. But, even after the 
 oscillations have ceased, the wire still continues to 
 untwist, more and more slowly, sometimes not even 
 approximately reaching its' undisturbed position till hours 
 or even days have passed. 
 
 When viscous resistance is considerable these results 
 are usually still more marked; and Sir W. Thomson^ 
 has discovered the very curious additional fact that this 
 molecular friction becomes greatly increased by keeping 
 the wire oscillating for days together. He has pushed 
 this process so far with one of two similar wires that, 
 whereas, in that which had been made to vibrate only a 
 few times, the arc of oscillation became reduced to half 
 in 100 vibrations, the (equal) arc of that whose elasticity 
 had been " fatigued " fell to half after 44 or 45 vibrations 
 only. 
 
 252. These phenomena are seen in a more striking 
 form when we dispense with oscillation. Thus, for 
 example, suppose the wire to be kept twisted through 
 90° to the right for six hours, then for half an hour 
 90° to the left, and be then so gradually let go that there 
 is no oscillation. When it is left to itself it turns slowly 
 towards the right, gradually undoing part of the effect 
 of the more recent twist, then stops, and twists still 
 more slowly to the left, thus undoing part of the quasi- 
 pernianent effect of the earliertwist. Thus the behaviour 
 of such a wire, strictly speaking, is an excessively com- 
 plex one, depending, as it were, upon its whole previous 
 ^ Proc. B.S., 1865. 
 
220 PROPERTIES OF MATTER 
 
 history ; though, of course, the trace left by each stage 
 of its treatment is less marked as the date of that stage 
 is more remote. This subject has of late attracted great 
 attention in Germany, and, under the name Elastische 
 Nachmrhimg, has been the object of numerous researches 
 by Wiedemann, Kohlrausch, Boltzmann, etc. 
 
 253. Clerk-Maxwell ^ has given a sketch of a theory 
 of this peculiar action, from which we quote the 
 following : — 
 
 "We know that the molecules of all bodies are in 
 motion. In gases and liquids the motion is such that 
 there is nothing to prevent any molecule from passing 
 from any part of the mass to any other part; but in 
 solids we must suppose that some, at least, of the mole- 
 cules merely oscillate about a certain mean position, so 
 that, if we consider a certain group of molecules, its 
 configuration is never very different from a certain stable 
 configuration, about which it oscillates. 
 
 " This will be the case even when the solid is in a 
 state of strain, provided the amplitude of the oscillations 
 does not exceed a certain limit, but if it exceeds" this 
 limit the group does not tend to return to its former 
 configuration, but begins to oscillate about a new con- 
 figuration of stability, the strain in which is either zero, 
 or at least less than in the original configuration. 
 
 " The condition of this breaking up of a configuration 
 must depend partly on the amplitude of the oscillations, 
 and partly on the amount of strain in the original 
 configuration ; and we may suppose that different groups 
 of molecules, even in a homogeneous solid, are not in 
 similar circumstances in this respect. 
 
 1 "Constitution of Bodiea," ^Ency. Brit., ninth edition. 
 
COMPRESSIBILITY AND RIGIDITY OF SOLIDS. 221 
 
 " Thus we may suppose that in a certain number of 
 groups the ordinary agitation of the molecules is liable 
 to accumulate so much that every now and then the 
 configuration of one of the groups breaks up, and this 
 whether it is in a state of strain or not. We may in 
 this case assume that in every second a certain proportion 
 of these groups break up, and assume configurations 
 corresponding to a strain uniform in all directions. 
 
 "If all the groups were of this kind, the medium 
 would be a viscous fluid. 
 
 " But we may suppose that there are other groups, 
 the configuration of which is so stable that they will not 
 break up under the ordinary agitation of the molecules 
 unless the average strain exceeds a certain limit, and 
 this limit may be different for different systems of these 
 groups. 
 
 "Now if such groups of greater stability are dissemi- 
 nated through the substance in such abundance as to 
 build up a solid framework, the substance will be a solid, 
 which will not be permanently deformed except by a 
 stress greater than a certain given stress. 
 
 "But if the solid also contains groups of smaller 
 stability and also groups of the first kind which break 
 up of themselves, then when a strain is applied the 
 resistance to it will gradually diminish as the groups 
 of the first kind break up, and this will go on till the 
 stress is reduced to that due to the more permanent 
 groups. If the body is now left to itself, it will not at 
 once return to its original form, but will only do so 
 when the groups of the first kind have broken up so 
 often as to get back to their original state of strain. 
 
 " This view of the constitution of a sohd, as consisting 
 
222 PROPERTIES OF MATTER. 
 
 of groups of molecules some of which are in different 
 circumstances from others, also helps to explain the state 
 of the soHd after a permanent deformation has been 
 given to it. In this case some of the less stable groups 
 have broken up and assumed new configurations, but it 
 is quite possible that others, more stable, may still 
 retain their original configurations, so that the form of 
 the body is determined by the equilibrium' between 
 these two sets of groups ; but if, on account of rise of 
 temperature, increase of moisture, violent vibration, or 
 any other cause, the breaking up of the less stable 
 groups is facilitated, the more stable groups may again 
 assert their sway, and tend to restore the body to the 
 shape it had before its deformation." 
 
 254. There remains one specially complex kinetical 
 case of elastic reaction, i.e. the eflfects of Collision. 
 According to Newton, the "rules of the congress and 
 reflection of hard bodies" were discovered about the 
 same time by Wren, Wallis, and Huygens. Wallis had 
 the priority, then followed Wren. But Wren " confirmed 
 the truth of the thing " by pendulum experiments (see 
 Appendix IV). By "hard bodies" are meant such as 
 rebound from one another with the same relative velocity 
 as they had before collision. Newton goes onto describe 
 his own mode of experimenting on the subject, how he 
 allowed for the resistance of the air, etc., and proceeds 
 as follows : — 
 
 " By the theory of Wren and . Huygens, bodies abso- 
 lutely hard return one from another with the same 
 velocity with which they meet. But this may be 
 aflSrmed with more certainty of bodies perfectly elastic. 
 In bodies imperfectly elastic the velocity of the return 
 
COMPRESSIBILITY AND RIGIDITY OF SOLIDS. 223 
 
 is to be diminished together with the elastic force; 
 because that force (except when the parts of bodies are 
 bruised by their congress, or suflfer some such extension 
 as happens under the strokes of a hammer) is (as far as 
 I can perceive) certain and determined, and makes the 
 bodies to return one from the other with a relative 
 velocity, which is in a given ratio to that relative 
 velocity with which they met. This I tried in balls of 
 wool, made up tightly and strongly compressed . . . the 
 balls, always receding one from the other with a relative 
 velocity, which was to the relative velocity with which 
 they met as about 5 to 9. Balls of steel returned with 
 almost the same velocity : those of cork with a velocity 
 something less ; but in balls of glass the proportion was 
 about 15 to 16." 
 
 255. So far as spherical bodies are concerned, there 
 is yet but little to add to Newton's results, for the 
 problem of the deformation and elastic rebound of two 
 impinging spheres has not yet been worked out. These 
 results, however, were fuUy confirmed by the careful 
 and instructive experiments of Hodgkinson.^ 
 
 But recent inquiries have shown that Newton's use 
 of the term "perfectly elastic" is not correct, for two 
 bodies may be perfectly elastic, and yet not rebound 
 from one another with the relative velocity of their 
 approach. This happens, in an easily intelligible manner, 
 when a bell or other body capable of vibrations is struck 
 by a hammer. 
 
 In modem phraseology the ratio of the" i'elative 
 velocity of recoil to that of collision is called the Coeffi- 
 cient of Bestitution. It is not directly a coefficient of. 
 ' Brit. Ass. Report, 1834. 
 
224 PROPERTIES OF MATTER. 
 
 elasticity, for it depends to some extent upon the sizes 
 and shapes of the impinging bodies, as well as upon 
 the materials of which they consist. 
 
 256. It is clear that, so far as direct impact of spheres is 
 concerned (where the whole motion of each of the masses 
 is in one common line), the third law of motion, along 
 with the value of the coefficient of restitution, suffices 
 for the calculation of the entire circumstances of the 
 motion after impact. 
 
 For, if M, M', be the masses of the spheres, v, v' and 
 v^, v'^ their velocities before and after impact, and e the 
 coefficient of restitution, the third law gives 
 
 M»i + M'«i' = M» + MV, 
 
 while the elastic property gives 
 
 Vi - Vi = e{v-i/) ; 
 
 so that V and v' are fully determined. 
 
 The subject will be found fuHy discussed in most 
 treatises on Dynamics. For illustrations of cases in 
 which, even with perfect elasticity, the coefficient of 
 restitution is necessarily less than unity, the reader may 
 consult Thomson and Tait's Natural Philosophy, § 304. 
 
CHAPTEE XII. 
 
 COHESION AND CAPILLARITY. 
 
 257. A SOMEWHAT pedantic nomenclature has intro- 
 duced the terms Cohesion and Adhesion in senses dis- 
 tinct from one another. Thus contiguous parts of a 
 piece of glass, or of a drop of water hanging from it, are 
 said to cohere, while the water is said to adhere to the 
 glass. Such pedantry usually tends to produce confu- 
 sion, as will be seen at once if we try to state in its 
 language how the parts of a lump of granite, or of a drop 
 of mixed alcohol and water, are kept together. We wiU 
 therefore use, indiscriminately, whichever of the words 
 happens to present itself when we require one of them. 
 
 258. We have already referred to the molecular forces 
 which are practically alone efficient in keeping together 
 the particles of a solid of moderate dimensions (§ 167), 
 and to the resolidification (by pressure) of powdered 
 .graphite (§ 53). We have studied the elasticity of fluids 
 and of solids, and have also made some remarks on 
 the tenacity of solids (§ 226). A few other instances of 
 cohesion between the particles of solids may now be 
 noticed, but the subject is one on which no exact informa- 
 tion can be expected in the present state of science. 
 The one characteristic of these forces, and that which 
 Q 
 
226 PROPERTIES OF MATTER. 
 
 specially contrasts them with gravitation, is that they 
 are insensible at sensible distances. 
 
 259. Two masses of marble, on each of which a true 
 plane surface has been worked, will, when these surfaces 
 are brought firmly together, even in vacuo, adhere so as to 
 overcome the weight of either (unless it be great in com- 
 parison with the area of contact), so that one remains 
 suspended from the other. Barton, early in the century, 
 made a set of cubes of copper whose sides were so very 
 true that when a dozen of theni were piled on one 
 another the whole series adhered together when the 
 upper one was lifted. If a small plane surface be scraped 
 bright on each of two pieces of lead, and these be pressed 
 together (with a slight screwing motion) they adhere 
 almo'st as if they formed one mass. The processes of 
 gilding, silvering, nickelising, etc., and their results, are 
 known to all. So are the properties of lime, glue, and 
 other cements, all depending on the molecular forces in 
 solids. 
 
 260. Nor are we in a much better position when we 
 seek the force of adhesion between a solid and a liquid. 
 For, in the great majority of cases, the liquid wets the 
 soM. Thus, when we suspend a plate of the solid 
 horizontally from one scale -pan of a balance, and try 
 what amoimt of weights we must put into the opposite 
 pan so as just to detach the plate from a liquid surface, 
 in the majority of cases it is the liquid which is torn 
 asunder. Unsatisfactory as they are from this point 
 of view, such experiments are also very tedious and 
 difficult. 
 
 It is, however, worth while to mention that a force 
 of about 60 grains' weight is required to draw a square 
 
COHESION AND CAPILLARITY. 227 
 
 inch of glass (wetted) from the surface of water ; while, 
 if the plate be carefully cleaned and dried, nearly three 
 times as great a force is required to separate it from 
 clean mercury. When a square inch of amalgamated zinc 
 is used it requires more than 500 grains' weight to 
 remove it from mercury. Here, however, as in the case 
 of glass and water, it is the liquid which is torn asunder. 
 These numbers depend to a great extent upon the 
 care with which the experiment is conducted, and must 
 be looked on as the merest approximations to the values 
 of a quantity whose precise nature we cannot define. 
 
 261. The case is quite different with the phenomena 
 which we have next to consider. With them accurate 
 measurements, of a perfectly deiinable quantity, are in 
 general possible. These are the phenomena due to the 
 Surfaee-Tension of liquids. We owe the idea to Segner 
 (1751), but its development and application are due 
 mainly to Young. The more recent theoretical advances 
 in the subject were made chiefly by Laplace and Gauss. 
 [For a sketch of the history of this subject the reader is 
 referred to Clerk-Maxwell's article, " Capillary Action," in 
 the ninth edition of the Ency. Brit.'] 
 
 262. As soon as we recognise, as a fact, the extremely 
 short distance at which these powerful molecular forces 
 are sensible, we see that there must be an essential 
 difference in state between parts of a liquid close to the 
 surface and others in the interior of the mass. For if we 
 describe, round any particle of the liquid as centre, a 
 sphere whose radius is the utmost range at which the 
 molecular forces are sensible, the only parts of the 
 liquid which act directly on that particle are those 
 contained within the sphere. So long as the sphere 
 
228 PROPERTIES OP MATTER. 
 
 lies wholly within the liquid the forces on the particle 
 must obviously balance one another. [At least it 
 must be approximately so, unless the distance from 
 particle to particle is comparable with the radius of 
 the sphere. We know of no liquid for which this is the 
 case.] But when part of the sphere lies outside the 
 liquid surface, i.e. when the distance of the particle from 
 the surface is less than the range of the molecular f orcesj 
 we can no longer make this assertion. 
 
 Hence we should expect to find peculiarities in the 
 surface-iilm whose thickness is approximately equal to 
 this molecular range. [If liquids were not, happily, but 
 slightly compressible, the reasoning above would lead to 
 the result that the " peculiarities " should extend to a 
 distance from the surface somewhat greater than the 
 radius of the sphere of action of the molecular forces.] 
 We must appeal to experiment or observation to find 
 their nature in each particular case. And here, as we 
 shall soon see, a multitude of well-known facts comes at 
 once to our assistance. But we must first examine, after 
 Gauss, the theoretical conditions a little more closely. 
 
 263. An important theorem of Dynamics is that, for 
 stable equilibrium of a system, the potential energy of 
 the whole must be a minimum. It is easy to see that, 
 so long as we consider molemla/r forces alone, the amount of 
 energy of the liquid mass can vary only with the extent 
 of the surface, but we may formally prove it as below. 
 
 Let the energy be e^ per unit mass of the interior 
 liquid, and e per unit mass for a layer of the skin, of 
 surface S, thickness t, and density p. Then, if M be the 
 whole mass of the liquid, and E its whole potential 
 energy, we have, by summing the energy of the interior 
 
COHESION AND CAPILLARITY. 229 
 
 mass and of the successive layers of skin (which have 
 all the same superficies so long as their curvature is 
 finite, in consequence of the shortness of range of the 
 molecular forces) — 
 
 E = (M - S. 2</))e„ + S.'Stpe 
 = Me„ + S.2*/)(e - e„). 
 
 Thus, in consequence of this property of the skin, the 
 whole energy is increased or diminished by a quantity 
 which is directly proportional to S. The multiplier of 
 S depends upon the nature and the temperature of the 
 liquid, and on the nature of the substance which is in 
 contact with its free surface, only. 
 
 Hence, when e is greater than «q everywhere through- 
 out the film, as we find happens when a water surface 
 is exposed to air, S tends to the smallest value compatible 
 with the conditions. But when e is less than «q, as when 
 water is in contact with glass, S tends to take as large a 
 value as possible. If part be exposed to air, and part be 
 in contact with glass or other substance, the final result 
 is more complex, but involves the same principles. 
 
 264. Still keeping to the theoretical view of the sub- 
 ject, let us consider what is implied in the tendency of 
 the liquid surface to become as small as possible. It 
 must behave, only in an incomparably more' perfect 
 manner, like an elastic membrane (such as a sheet of ' 
 india-rubber) which has been stretched by equal tensions 
 in all directions. But while the tension of such a mem- 
 brane becomes less as it is permitted to shrink more and 
 more, the liquid film has still as great a tendency to 
 shrink, however small its surface may have become. 
 Thus it must be under a definite Surface-Tension. 
 
 If T be this tension across a line of unit-length on 
 
230 PROPERTIES OF MATTER. 
 
 such a liquid surface, it is easy to see that the -work 
 required to stretch a rectangle, whose sides are a and h, 
 into another whose sides are a and h + l, is 
 
 Ta. Z = T.aZ, 
 
 i.e. T multiplied by the increase of area. This quantity 
 T must, therefore, be the multiplier of S in the expres- 
 sion (of last section) for the whole energy of a mass of 
 liquid. 
 
 265. When the liquid is drawn out into a film, as in 
 blowing a soap-bubble, the tension of this film is 
 practically 2T ; so long, at least, as the whole thickness 
 of the film is greater than twice the molecular range. 
 Thus it may be regarded as consisting of a layer of 
 interior water, with two surface-skins. 
 
 2()6. We now come to the observed facts which are to 
 be compared with these indications of theory. But, first, 
 we assume the mathematical theorem that the sphere is, 
 of all surfaces, that which, with a given content, has the 
 smallest superficies. 
 
 Whenever a drop of liquid is left free from all but 
 its own molecular forces, we find it assumes a perfectly 
 spherical form. By far the most rigorous proof of this 
 is afforded by the rainbow. Exceedingly slight devia- 
 tions from perfect sphericity of the falling drops would 
 suffice entirely to alter the character of this phenomenon. 
 Greater deviations would altogether prevent its occur- 
 rence. 
 
 The manufacture of small-shot, where a true spherical 
 form is important, is another good example. A shot- 
 tower, as it is called, is merely a gigantic shower-bath, 
 in which the liquid employed is melted lead, which is 
 
COHESION AND CAPILLARITY. 231 
 
 slightly alloyed, mainly for the purpose of making it 
 less viscous while liquid. The majority of the falling 
 drops solidify in forms very nearly spherical before they 
 reach the water-bath which is employed to break their 
 fall. 
 
 The rounding off of the sharp edges of a broken piece 
 of sealing-wax, as soon as it melts in a flame, is 
 another example ; and it was in consequence of the 
 almost perfect sphericity of the little bead, formed on- 
 the end of a glass fibre which is held in a flame for a 
 short time, that such beads were used by the early 
 microscopists as singk magnifiers. By the use of these, 
 Leuwenhoeck and others anticipated many of the results 
 which are now shown by means of the splendid modem 
 achroniatic object-glasses of compound microscopes. 
 
 267. An ingenious method of guarding a liquid from 
 the action of any but molecular forces was devised by 
 Plateau. He simply placed a mass of oil in a mixture 
 of alcohol and water of the same density as the oil. 
 Here the mass assumes a perfectly spherical form. 
 When this arrangement is altered by evaporation, the 
 mixture becomes denser from the top downwards. The 
 globe of oil becomes flattened, because its lower parts tend 
 to rise and its upper portions to sink, being immersed 
 respectively in parts of the liquid denser and less dense 
 than the oil. This mass of oil can be made to fulfil 
 definite boundary conditions, by bringing into contact 
 with it various frames of wire, &c., all thoroughly oiled 
 beforehand. 
 
 268. If a large drop of water be laid on a clean glass 
 plate, it spreads itself over a considerable surface. 
 [Theoretically it should wet the whole surface.] Now 
 
232 PROPERTIES OF MATTER. 
 
 let ether vapour, which is heavier than air, be poured 
 upon the middle of the water surface; or let it be 
 touched by a glass rod moistened with alcohol ; or even 
 hold the point of a red-hot poker close to it. In all 
 these cases the effect is to reduce, locally, the surface- 
 tension, so that, as it were, a weak part of the surface 
 film is produced, and this is pulled out over the surface 
 by the greater contractile tension of the unaffected parts 
 of the skin. The water, in fact, often retreats on all 
 sides from the affected part, leaving the central portion 
 of the glass uncovered. The effect is precisely similar 
 to that which is produced in a stretched sheet of india- 
 rubber where one part is either thinner than the rest, 
 or has been slightly heated by a flame. 
 
 The surface-tension of a drop of mercury is greatly 
 altered when it is (electrically) " polarised." Eemark- 
 able phenomena of this kind were described by StrethUl 
 Wright in 1860.-^ Lippmann has employed this surface 
 effect in the construction of a very sensitive electrometer. 
 269. The phenomenon called the "tears of strong 
 wine," first explained by J. Thomson, is anothei example. 
 When the sides of a drinking glass have been moistened 
 with strong wine, we observe that the liquid film soon be- 
 comes corrugated. The ridges are formed of the portions 
 from which most alcohol has evaporated, and which, there- 
 fore, have the greatest surface-tension. As these slide, 
 by gravity, down the sides, we see them now and then, 
 stop, and even retract, when they come to a part where 
 there is more alcohol, and therefore less surface-tension. 
 By far the best example, however, is furnished by 
 some of the less viscid oils. A few drops, let fall on the 
 ' Phil. Mag., xix. p. 129. 
 
COHESION AND CAPILLAKITY. 233 
 
 surface of a quiet pool, seem almost to flash out over tlie 
 surface, showing in the most brilliant manner the inter- 
 ference colours of thin plates. 
 
 Another striking instance is furnished by- a piece of 
 camphor when placed on water. It usually dissolves 
 more rapidly at some one place than at others, thus 
 relatively weakening the surface-tension of the water in 
 that place, and is consequently dragged about with con- 
 siderable rapidity, and apparently in the most capricious 
 manner. Similar results, but of a more complex and 
 often much more violent nature, are obtained with a 
 pellet of potassium or of sodium. 
 
 270. It is the same when we deal with a double skin, 
 as in a soap-bubble. When a soap-film is lifted by the 
 mouth of a glass funnel previously wetted with the 
 solution, it rapidly runs up the cone, and fixes itself 
 finally at the place where the area of the tross section is 
 least. 
 
 Van der Mensbrugghe has devised a very beautiful 
 and instructive experiment of this kind, viz. placing on 
 a soap-film, lifted by a large wire ring, a short endless 
 silk thread, thoroughly wetted with the soap solution. ■ 
 As soon as the film is broken inside the coil of the 
 thread, the thread is stretched out into an exact circle 
 bounding the hole in the film. 
 
 When a soap-bubble is blown at the mouth of a 
 funnel, and the neck is left open, we see it shrink faster 
 and faster, expelling the contained air, which is thus 
 proved to be at greater than atmospheric pressure. 
 Faraday succeeded in blowing out a candle by the air 
 thus expelled. 
 
 The soap glycerine solution invented by Plateau, by 
 
234 
 
 PROPERTIES OF MATTER. 
 
 rendering soap-films permanent for hours together, enables 
 us to study the phenomena of surface-tension even more 
 simply and accurately than can be done by the help 
 of his earlier method (§ 267). His great work^ on these 
 subjects forms a storehouse of interesting and important 
 experiments, all the. more remarkable that they were 
 devised by one who had the misfortune of being per- 
 manently disabled from seeing them. The reader must 
 be referred to these volumes for other examples than the 
 few for which we can find room. 
 
 271. It is very instructive to observe the mode in 
 which many problems, some of extreme mathematical 
 difficulty, are at once solved practically by experiments 
 with these soap-films. 
 
 The whole physical part of the phenomena depends 
 upon the fact that the' film takes the fmm of smallest swper- 
 fides consistent with the conditions. 
 
 272. Here it becomes neces- 
 sary to consider the curvature of 
 the film, and the way in which 
 it depends upon the pressures 
 to which the sides are exposed. 
 A very simple investigation gives 
 us all we require in this matter. 
 Suppose a tape or band, of unit 
 breadth, while under tension T, 
 be wrapped transversely round 
 a cylinder of radius E, and 
 let p be the pressure which 
 it produces on unit surface of the cylinder. Consider n 
 
 1 Statigue expirwrwrdale et tMorique des Liquides sowmis mux 
 Smiles Forces MoUeulaires. Paris, 1873. 
 
COHESION AND CAPILLARITY. 235 
 
 very small portion of its length, AB, subtending an 
 angle 9 at the centre, 0, of the cylinder. This portion 
 of the band is kept in equilibrium by the tensions, T, at 
 its ends, and the reaction^. AB of the cylindrical surface. 
 Eesolving these forces in the direction 00, bisecting 
 the angle AOB, we have 
 
 2Tsin| =:p.AB=p.K8; 
 
 which, when 6 is very small, becomes 
 
 T 
 
 Thus the band requires, for its support in the cylindrical 
 form, an excess of pressure on the concave, over that on 
 the convex, side, of = per unit of surface. 
 
 273. In the case of a soap-film, or of the surface-film of 
 a liquid generally, there may be simultaneous curvatures 
 in two planes at right angles to one another. The 
 effects of these are to be simply superposed, as they are 
 independent. Let Ej be the radius of the second cur- 
 vature, then, as the film exerts equal tension in all 
 directions, the diflference of pressures on its sides is, per 
 square unit, ,-, ■, . 
 
 ^E + Si)- 
 
 This expression must be doubled in the case of a soap- 
 bubble, for (§ 265) it has too surface-films. 
 
 By a well-known geometrical theorem, due to Euler, 
 the quantity in brackets, i.e. the sum of the curvatures 
 in two planes at right angles to each other, and both . 
 passing through the normal to a surface at a particular 
 point, is independent of the aspects of these planes. 
 
236 PROPERTIES OF MATTER. 
 
 Hence it is convenient to choose R and Ej as the 
 prmcipd radii of curvature of the film. 
 
 AVhen, as a purely mathematical problem, we seek 
 the characteristic of the surfaces of least area which 
 satisfy given boundary conditions, we are led to the 
 condition that the sum of the cwrvatwres at any point is 
 constant. This agrees with the physical result. 
 
 2.74. Thus, when a soap-film is exposed to equal 
 pressures on its two sides, it must satisfy the given 
 boundary conditions, and possess the further property 
 that, at every point of its surface, 
 
 s+i = °: 
 
 i.e. whatever be its curvature in any normal section,* it 
 must have an equal and opposite curvature in the 
 normal section perpendicular to the first. 
 
 Such must, therefore, if we neglect the (very slight) 
 disturbing effects of gravity, be the form of a soap-film 
 exposed on both sides to the air. Thus if we lift such 
 a film on a flat loop of wire it assumes a plane surface ; 
 but, by bending the boundary, we can make it assume 
 forms of marked curvature. In all its forms, however, 
 the sum of the curvatures at each point is nil. And 
 the same is the case, however ramified, linked, or 
 knotted the wire frame may be, provided only that 
 there is no air imprisoned at any place. 
 
 275. If we imprison a quantity of air by the film, as, 
 for instance, by forming it between the rims of two 
 equal funnels, and closing the neck of each with a finger, 
 we have in general difierent pressures outside and inside ; 
 and then we have « i i 
 2T~R ■*■ El 
 
COHESION AND CAPILLARITY. 
 
 237 
 
 where p is the constant difference of pressures. By 
 altering the relative position of the funnels, as by shift- 
 ing one sidewise out of the line of symmetry, or by 
 making it rotate (otherwise than 
 about its axis of symmetry), we 
 can throw the film into extra- 
 ordinary shapes ; all of them, 
 however, possessing the fundar 
 mental property of constant sum 
 of the curvatures at each point, 
 selves with a brief notice of the results of gradually 
 withdrawing the funnels from one another, while keeping 
 their axes of sjmimetry in one line. 
 
 Thus we may begin with the film as a quasi-spherical. 
 
 But we content our- 
 
 <[Z>= 
 
 Fig. 33. 
 
 or even spherical surface, having both its curvatures 
 moderate (Fig. 1). As we withdraw the funnels from 
 one another the longitudinal curvature diminishes, and 
 the transverse increases to the same amount, till at last 
 the longitudinal curvature vanishes altogether, and the 
 film becomes cylindrical (Fig. 2). StiU further separat- 
 
238 PEOPEETIES OF MATTER. 
 
 ing them, the film takes an hour-glass form as in Fig. 3, 
 where the increasing curvature of the transverse section 
 is now balanced by a gradually increasing negative curv- 
 ature in the longitudinal section. At a certain limit 
 this state of the film becomes unstable, and the positive 
 and negative curvatures near the middle both rapidly 
 increase, till the walls at that part collapse into a mere 
 neck of water, which is ruptured, and leaves a pro- 
 tuberant film on each of the funnels. By a little 
 dexterous manipulation these may easily be made to 
 reunite into the original form. 
 
 276. The facts we have just described show us the 
 nature of the process by which a complete soap-bubble 
 is detached from a funnel, always leaving a film on the 
 funnel ready to produce a second. This process can 
 easily be studied by completing the blowing of the bubble 
 with coal-gas, after it has been commenced with air, 
 and watching it detach itself in virtue of the hghtness 
 of its contents. 
 
 277. When two complete soap-bubbles are made to 
 unite, the tendency of the liquid film is to contract, 
 that of the (compressed) air inside is to expand. It 
 becomes a curious question to find which of these actu- 
 ally occurs. 
 
 Let their radii, when separate, be E and En and let 
 them form, when united, a bubble of radius r. Then, if 
 n be the atmospheric pressure, the original pressures in 
 the bubbles were ,m ,m 
 
 n + ^andn + ^; 
 
 while that in the joint bubble is 
 
 r 
 
COHESION AND CAPILLAKITY. 239 
 
 By Boyle's Law the densities are as the pressures. 
 Hence, expressing that no air is lost, we have 
 
 °^ n(R3 + E^s _ ^) + 4T(E'i + Ri^ -r^) = 0. 
 
 If V be the diminution of the whole volume occupied by 
 the air, S that of the whole surface of the liquid film, 
 this condition gives at once 
 
 3nv + 4TS = 0. 
 
 As n and T are both positive, V and S must have 
 opposite signs. Hence the surface, as a whole, shrinks, 
 and the contained air, as a whole, increases in volume, 
 simultaneously. But the work done by the expanding 
 gas is only about two-thirds of that done by the contract- 
 ing film. 
 
 It is worthy of notice that, as is easily proved, the air 
 .in a soap-bubble of any finite radius would, at atmos- 
 pheric pressure, fill a sphere of radius greater than 
 before by the constant quantity 4T/3n. 
 
 278. As a practical illustration of the use of these 
 formulae, let us apply them to a stationary steam-boiler 
 of the usual cylindrical form, with the ends portions of 
 spheres. If E be the radius of the cylinder, Rj that 
 of each end, and P the excess of internal over external 
 pressure, the tension is 
 
 Across a generating line, RP, 
 
 ■p2p 
 
 Parallel to a generating line, = JEP, 
 
 Across any line on the end, iRjP. 
 
 Thus, if the boiler-plate be equally tenacious in all 
 directions, there is no danger of the ends being blown 
 
240 
 
 PROPEETIES OF MATTER. 
 
 off, for the boiler will rather tear along a generating line. 
 And, to make the- ends as strong as the sides, they 
 require only half the curvature. 
 
 Thus, also, we see why stout glass tubes, if of small 
 enough bore, are capable of resisting very great internal 
 pressure, when, as in Andrews' experiments on carbonic 
 acid, they are exposed only to atmospheric pressure 
 outside. 
 
 279. We are now prepared to consider the phenomena 
 properly called Capillary, as having been detected in 
 tubes of very fine bore. 
 
 When a number of clean glass tubes, each open at 
 both ends, are partially immersed in a large dish of 
 water, we observe that (in apparent deviation from the 
 hydrostatic laws, § 189) the water rises in each to a 
 
 <-> 
 
 FlQ. 34. 
 
 higher level than that at which it stands outside. Also 
 we notice that this rise is greater the finer the bore of 
 the tube. The cut shows the phenomenon in section. 
 
 Perform the same experiments with mercury instead 
 of water, and we find that the liquid stands at lower 
 levels inside than outside each tube, and that this 
 depression is greater the finer the bore of the tube. 
 
COHESION AND CAPILLARITY. 241 
 
 Turn the above cut upside down, and it will correspond 
 to this effect. 
 
 280. But a closer inspection at once shows the 
 immediate cause of the phenomena. The water surface 
 inside each tube is always concave outwards, that of the 
 mercury convex ; and the curvature of either is greater 
 the finer is the bore of the tube. 
 
 Eemember the surface-tension of the liquid, and the 
 consequent excess of pressure on the concave side, over 
 that on the convex side, which is necessary (§ 272) 
 for its equilibrium, and we see at once that the water 
 immediately under the surface-film must have a less pres- 
 sure than that of the atmosphere to which its concave side 
 is exposed. Thus, hydrostatically (§ 189), it belongs to a 
 higher level than the undisturbed water, whose surface 
 is plane, and the pressiire in which (immediately under 
 the surface) is equal to the atmospheric pressure.^ 
 
 As the surface curvature is greater in the finer tubes, 
 so the higher rise of water in these is a direct hydrostatic 
 consequence. 
 
 The convexity of the mercury surface, on the other 
 hand, requires immediately under the film a pressure 
 exceeding that of the atmosphere by an amount propor- 
 tional to the sum of its curvatures. Thus we see why 
 the mercury stands at a lower level in the tube than 
 outside it. 
 
 281. It only remains that we should account for the 
 
 ' In some theories of capillary action, especially that of Poisson, 
 it is supposed that the interior of a mass of liquid, even when free 
 from atmospheric pressure or gravitation action, is necessarily at a 
 very high pressure in consequence of molecular action. But were 
 it so, it would not alter the conclusion above, as this part of the 
 pressure does not depend on the form of the liquid surface. 
 E 
 
N 
 
 A 
 
 B 
 
 C 
 
 Fia. 35. 
 
 242 PROPERTIES OF MATTER. 
 
 concavity of the water surface, and the convexity of 
 that of the mercury. 
 
 In the annexed sections of a concave and of a convex 
 surface, in which a tangent, AB, is drawn to the liquid 
 
 film, where it meets the side of 
 D the tuhe at B, the angle ABC of 
 
 the wedge of liquid is obviously 
 
 less than a right angle for the 
 p concave surface, and greater than 
 
 a right angle for the convex. 
 
 Hence the problem is reduced 
 ^ to the determination of this 
 
 angle, called the Angle of Contact. 
 That this angle must have a definite value for each 
 liquid, in contact with each particular solid, appears at 
 once from the consideration that, in the immediate neigh- 
 bourhood of B, the gravitational or other external forces, 
 acting on a very small portion of the liquid, are incom- 
 parably less intense than the molecular tensions. Hence 
 the equilibrium of that portion (tangentially to the solid) 
 will depend upon the surface-tensions along BA, BO, BD 
 alone. The directions of two of these, and the magnitudes 
 of all three, are determinate, whatever two fluids (even 
 when one is gaseous) are in contact with each other and 
 with the solid (§ 263). BA, therefore, will ultimately 
 assume such a direction that the surface-tension along it 
 will, when resolved in CD, just balance the difierence 
 between the tensions in BD and BC. Hence, if that 
 in BD is the greater, the angle of contingence will be 
 acute ; if that in BC be the greater, it wUl be obtuse. 
 
 282. In the case of mercury and clean glass, exposed 
 to air, the angle of contact is 
 
COHESION AND CAPILLARITY. 243 
 
 140° (Young), 135° (Gay-Lussao), 128° 52' (Quincke), 132° 2' 
 (Bashforth). 
 
 With water and clean glass in air the angle vanishes 
 entirely ; but when the glass is not clean it may reach 
 (and even surpass) 90°. When it is exactly 90° there is 
 no curvature of the water surface inside the capillary 
 tube, and it therefore stands at the level of the undis- 
 turbed water outside.^ 
 
 283. We may now complete the explanation of the 
 behaviour of a liquid in a capillary tube as follows : — 
 When the rise (or depression) exceeds several diameters 
 of the tube, the curvature is practically the same over 
 the whole free surface, which is therefore approximately 
 spherical. In mercury, because of the finite angle of 
 contact, it forms only a segment less than a hemisphere ; 
 in water it is a complete hemisphere. 
 
 In the former case the radius is directly proportional 
 to that of the tube, in the latter it is equal to it. In 
 both cases, therefore, the relief of pressure, and conse- 
 quently the rise or depression of the liquid, is inversely 
 as the radius of the tube. This agrees with the (long- 
 known) results of experiment. 
 
 284. We may make, in a very simple manner, due to 
 Dr. Jurin, a calculation of the capillary elevation, which 
 is applicable to wider tubes than those spoken of in last 
 section. Suppose the radius of the tube to be r, p the 
 density of the liquid, a its angle of contact, T the tension 
 of the surface-film, and h the mean height to which it is 
 
 ' One of Gay-Lussac's methods for determining this angle when 
 it is finite must he at least indicated here. If the liquid he intro- 
 duced gradually into a small glass sphere (from below) there will 
 be one position in which its surface is plane. By measuring this 
 position the angle can be at once calculated. 
 
244 PROPERTIES OF MATTER. 
 
 elevated. [This mean height is taken such that the 
 volume of the liquid actually raised would, if the surface 
 ■were not curved, fill the length h of the tube.] Then 
 the vertical component of the whole tension round the 
 edge of the film is obviously 
 
 27rrT cos a. 
 But this supports the weight 
 
 of liquid, (virtually) filling a length h of the tube. Equat- 
 ing these quantities we obtain, after reduction, 
 
 2Tcosa 
 
 h = * 
 
 rgp 
 
 When a> ^, A is negative, and the liquid is depressed. 
 
 All the quantities here are easy to measure except T 
 
 and a. Hence, if a can be found by a separate process, 
 
 T is at once determined. In the case of water in glass 
 
 we have cos a = \, so that the above relation gives T 
 
 - directly. 
 
 285. The following values of T are given by Quincke. 
 
 Ea,ch datum in the table belongs to the film at the 
 
 common surface of the substances whose names are in 
 
 the same line and column with it. 
 
 
 Air. 
 
 "Water. 
 
 Mercury. 
 
 Water 
 
 81 
 
 — 
 
 418 
 
 Mercuiy 
 
 540 
 
 . 418 
 
 — 
 
 Alcohol 
 
 25-5 
 
 — 
 
 . 399 
 
 The unit here is one dyne per centimetre. To reduce 
 •to grains' weight per inch divide by 25. Thus we may 
 easily calculate, from the formula of last section, that 
 water rises a little more than half an inch in a glass tube 
 whose bore is i^rth inch in diameter. 
 
COHESION AND CAPILLAKITY. 245 
 
 286. In the Atmometer, which is merely a ball of un- 
 glazed clay with a. glass stem, the whole filled with 
 water and inverted in a vessel of mercury, not only is 
 the reduction of pressure by the fine concave surfaces 
 of water in the pores sufficient to keep a column of 
 3 or 4 feet of water supported, but, as evaporation 
 proceeds, mercury rises to take the place of the water, 
 sometimes to 23 inches or more. Thus these pores can 
 sustain (virtually) a column of 26 feet of water; and 
 the consequent great concavity of the surface of the 
 water renders it eminently fit (§ 291) as a nucleus for 
 the deposition of vapour.^ The process has not, so far 
 as we know, been pushed to its limit. 
 
 287. Such data enable us easily to calculate the force 
 with which a boy's " sucker " is pressed against a stone. 
 Suppose we have two plates of glass, 6 inches square, 
 with a film of water between them whose thickness is 
 ■g-girth of an inch. The force required to pull one per- 
 pendicularly from the other, in which case the free water 
 surface round the edges will take a (cylindrical) curvature 
 of radius Toirth of an inch, would be the weight of a 
 six-inch prism of water about 5 inches high, i.e. between 
 6 and 7 pounds' weight. If the film were of half that 
 thickness (at the edges) the force required would be 
 double. Thus, as J. Thomson has pointed out, two flat 
 slabs of ice, resting side by side with a film of water 
 between them, are pressed together with a force which 
 may much exceed the weight of either. When a mere 
 drop of water is placed between two very true glass 
 planes the pressure produced forces them closer together, 
 thus increasing itself, not only by the enlargement of the 
 
 1 Proc. M.S.E., Febraary 16, 1885. 
 
246 PROPERTIES OF MATTER. 
 
 wetted surfaces, but by increasing the curvature round 
 the edges. The pressure producible in this way is very 
 great. On the other hand, a few small drops of mercury, 
 interposed between the plates, form an exceedingly 
 perfect elastic cushion. 
 
 288. There are many common phenomena whose 
 explanation is easily traced to the action of capillary 
 forces. Thus air-bubbles, sticks, and straws floating on 
 still water, appear to attract one another; and gather 
 into groups, or run .to the edge of the containing vessel. 
 This is always the case with any two bodies, each of 
 which is wetted by the water, and it is also true when 
 neither is wetted. But when one of the bodies is wetted, 
 and the other is not, they behave as if they repelled one 
 another. The explanation is easily given : — either from 
 the point of view of the various forces called. into play 
 by the displacement of the water, or (more simply) by 
 the consideration of the whole energy of the liquid as 
 depending on the relative position of the floating bodies 
 (§ 263). 
 
 A needle, or even a (very small) pellet of mercury, 
 may easily be made to float on water. The hydrostatic 
 condition requires merely a depression of the surface, 
 so that the water displaced may be equal in weight to 
 the floating body ; but, that this displacement may take 
 place, the angle of contact must be made greater than 
 90°, which is at once ensured if the needle be very 
 slightly greased. Thus we explain how water-flies run 
 on the surface of a pool. 
 
 In the same way we can explain why a piece of wood 
 is not wetted when it is dipped into water whose surface 
 is covered with lycopodium seed ; and why mercury can 
 
COHESION AND CAPILLARITY. 247 
 
 be poured in considerable quantity into a bag of gauze 
 or cambric without escaping through the meshes. 
 
 An air-bubble in water assumes a spherical form, even 
 when it is in contact with the side of a glass vessel, and 
 a very small globule of mercury laid on glass becomes 
 almost spherical. But an air-bubble on the side of a glass 
 vessel containing mercury is flattened out, while a drop of 
 water on clean glass spreads itself out indefinitely. In all 
 these cases the angle of contact at once explains the result. 
 ■ The difiiculty of obtaining a clean surface of water or 
 mercury depends upon the great surface-tension of these 
 liquids relatively to that of the majority of other 
 substances. 
 
 289. The form of section of a (cylindrical) liquid sur- 
 face, in contact with a plane solid surface, is easily 
 deduced from the hydrostatic principle that the elevation 
 (or depression) at any point is proportional to the relief 
 (or increase) of pressure, i.e. to the (one) curvature. 
 Hence it must be the curve of flexure (§ 237) of a very 
 long elastic wire, with a kink in it, under the action of 
 tensions at its ends ; for at every point of that curve the 
 curvature is proportional to the distance from the line in 
 which the stress acts. Hence we can at once find the 
 form in which the hquid surface meets a plane solid face, 
 whether it be vertical or not, by drawing the correspond- 
 ing elastic curve and taking account of the inclination of 
 tjie' plate and of the angle of contact. 
 
 290. The surface-tension of liquids diminishes with 
 rise of temperature. And Andrews showed that, as liquid 
 carbonic acid is gradually raised to its critical tempera- 
 ture, the curvature of its surface in a capillary tube 
 gradually diminishes. 
 
248 PROPERTIES OF MATTER. 
 
 291. W. Thomson 1 showed that there is a definite 
 vapour-pressure, for each amount of curvature of a liquid 
 surface, necessary to equilibrium. It is less as the sur- 
 face is more concave, greater as it is less concave or more 
 convex. Hence precipitation of water -vapour wUl, 
 ceteris paribus, take place more rapidly the more concave 
 or the less convex is the surface of that already de- 
 posited. Thus, as Clerk-Maxwell pointed out, the larger 
 drops in a cloud must grow at the expense of the smaller 
 ones. The explanation of these curious facts is given 
 by the kinetic theory much in the same way as is 
 that of the effect of the curvature of the discs of a 
 
 So great a pressure of vapour would be necessary 
 for the existence of very small globtdes of water (in the 
 nascent state of cloud, as it were), that, as Aitken has 
 shown, condensation cannot commence in free air without 
 the presence of dust-nuclei. The more numerous these 
 are, the smaller is the share of each, and thus we have 
 various kinds of fog, mist, and cloud. 
 
 292. Many extremely curious phenomena, due to 
 surface-tension, have been investigated by various ex- 
 perimenters, especially Tomlinson. . Thus different kinds 
 of oils can be distinguished from one another, or the 
 purity of a specimen of a particular oil may be ascer- 
 tained, by the form which a drop takes when let fall on 
 a large, clean water surface. In some cases a drop of 
 oil does not spread entirely over a liquid surface, but 
 forms a sort of lens. The angles at which its faces meet 
 one another, and the surface of the liquid, are then to 
 be determined by the triangle of forces. 
 1 Froc. R.S.E., 1870. 
 
COHESION AND CAPILLAEITY. 249 
 
 Again, when a drop of an aqueous solution of a salt, 
 say permanganate of potash or some other highly- 
 coloured substance, is allowed slowly to descend in 
 water, it at first takes the form of a vortex-ring, bounded, 
 of course, by a film of definite surface-tension. But, as 
 diffusion proceeds, this film becomes weaker at certain 
 places (just as in the case of wine, § 269) and the ring 
 breaks into segments, each of which is (as it were) a new 
 drop, which behaves as the original drop did, though 
 somewhat less vigorously. Thus we have a very curious 
 appearance, almost resembling the development of a 
 polyp ; the number of distinct individuals being mark- 
 edly greater in each successive generation. With a drop 
 of ink these developments take place so fast that the 
 eye can scarcely follow them. 
 
 293. We may now say a word or two as to the ex- 
 treme limits at which the molecular forces are sensible. 
 It is not at aU remarkable that the various estimates 
 differ widely from one another, for they are all obtained 
 by processes more or less indirect. They all agree, how- 
 ever, in giving small values. Experiments of Plateau 
 on soap-films, and of Quincke on the behaviour of water 
 and thinly-silvered glass, give only about 1/500,000 of 
 an inch. It is probable that the limits vary somewhat 
 with the nature of the substance experimented on ; and 
 the question is certainly connected, iu no remote manner, 
 with the differences in the c/ritical temperatures (§ 194) of 
 various substances. 
 
 294. The separation into drops, of a liquid column 
 slowly escaping from a small hole in the bottom of a vessel, 
 can be studied by examining it by the light of electric 
 sparks rapidly succeeding one another. It is a phenomenon 
 
250 PROPERTIES OP MATTER. 
 
 similar to that which we have described in § 275, when 
 a cylindrical film is drawn out between two funnels. 
 When the liquid is a very viscous one, as treacle or 
 Canada -balsam, the viscosity interferes with this effect 
 of instability ; and such liquids can, like melted glass, 
 be drawn into fine continuous threads. This property 
 sometimes gets the special name of Viscidity. 
 
 295. The propagation of ripples, as Sir W. Thomson 
 , showed,^ is also due mainly to surface-tension. The 
 proof is given by the fact that the shorter the ripples the 
 faster they run, whilfi ordinary oscillatory waves in deep 
 water, propagated by gravitation, run faster the longer 
 they are. In two similar ripples, of different wave- 
 lengths, the forces are independent of the lengths, the 
 ranges are directly as the lengths, and the masses of 
 water are as the squares of the lengths, of the ripples. 
 H^nce the rates of propagation are inversely as the square 
 roots of the lengths. In similar oscillatory gravitation 
 waves, on the contrary, the forces are as the squares of 
 the lengths, the ranges as the lengths, and the masses as 
 the squares of the lengths, and the rate is directly pro- 
 portional to the square root of the wave-length. Thus 
 very short ripples run almost entirely by surface-tension, 
 while long ripples and short waves run partly by gravity 
 partly by surface-tension. Thomson has shown that the 
 limit between waves and ripples in water, which is the 
 slowest-moving surface disturbance, has about 2/3 inch 
 as its wave-length, and runs at a speed of 9 inches per 
 second. Every shorter disturbance runs mainly by sur- 
 face-tension, and may be called a ripple; every longer 
 one runs mainly by gravitation, and may be called a wave. 
 1 Phil. Mag., 1871, u. p. 375. 
 
COHESION AND CAPILLARITY. 251 
 
 It is probable that very accurate determinations of surface- 
 tension may be obtained by measurement of the lengths 
 of ripples produced when the stem of a vibrating tuning- 
 fork (of known pitch) is immersed in a liquid. -"^ 
 
 296. When a solid is exposed to a gas or vapour, a 
 film is deposited on its surface which, in many cases, 
 introduces confusion in weighings, etc. Thus, if a dry 
 platinum capsule be carefully weighed, then heated to 
 redness and weighed again immediately after it has cooled, 
 it is found to be lighter. If left exposed to the air it 
 gradually recovers its former weight. In so far as this 
 effect is a purely surface one, it is increased in propor- 
 tion as the surface of a given mass of the solid is in- 
 creased. Thus "spongy" platinum, -as it is caUed — i.e. 
 platinum in a state of very minute division (obtained by 
 reducing it by heat from some of its salts) — exhibits the 
 phenomenon to a notable extent. Dobereiner showed 
 that a jet of hydrogen can be set on fire, by the heat 
 developed when it is blown against spongy platinum 
 which has been exposed to the air. The platinum is 
 heated to redness by the combination of the oxygen 
 film, already condensed on its surface, with the hydro- 
 gen which suffers condensation in its turn. 
 
 Another remarkable form of experiment, analogous to 
 this, consists in heating a platinum wire to incandescence 
 in the flame of a Bunsen lamp, turning off and then 
 immediately turning on again the supply of gas ; for the 
 wire remains permanently red-hot in the explosive mix- 
 ture of air and coal-gas ; without, however, reaching a' 
 high enough temperature to inflame it again. 
 
 The amount of siirface really exposed by finely porous 
 1 Proc. B.S.E., 1875, p. 485. 
 
252 PROPERTIES OF MATTER. 
 
 bodies, especially (as Hunter^ showed) cocoa-nut charcoal, 
 is enormous in comparison with their apparent surface ; 
 and in consequence they are able to absorb (as it is called) 
 quantities of gas altogether disproportionate to their 
 volume. Even ordinary charcoal, when heated red-hot 
 (to drive out the gases already condensed in its pores) 
 and allowed to cool in an atmosphere of carbonic acid 
 gas, absorbs from sixty to eighty times its volume of the 
 gas. If it be then introduced into a tube full of mercury 
 it can be made, by heating, to disgorge this gas, which 
 it reabsorbs as it cools. 
 
 The student may easily imderstand the immense 
 addition to the surface of a body, which is caused either 
 by pores or by fine division, if he reflect that a cube, 
 when sliced once parallel to each of its pairs of faces, 
 obviously has its whole surface doubled. 
 
 297. There is another form of action, analogous^ to 
 this, produced by certain finely divided substances, such 
 as Peroxide of Manganese. When a stream of oxygen, 
 containing ozone, is passed through the powder it emerges 
 as oxygen alone. The ozone has been reduced to the 
 form of oxygen by what is called Catalysis; the oxide of 
 manganese is practically unaltered. 
 
 298. What is called the solution of a gas in a liquid 
 is, in many respects, analogous to the condensation on 
 the surface (or in the pores) of a solid. 
 
 The empirical laws of this subject, originally given 
 by Henry and by Dalton, have been verified for moderate 
 ranges of pressure by Bunsen. 
 
 According to Henry, when a solution of a gas is in 
 equilibrium with the gas itself, the amount dissolved in 
 1 Chem. Soc, Journal, 1865-72. 
 
COHESION AND CAPILLARITY. 253 
 
 unit volume of the liquid is proportional to the pressure 
 of the gas. The coefficient of proportionality diminishes 
 rapidly with rise of temperature. 
 
 To this Dalton added that each constituent of a 
 gaseous mixture is dissolved exactly as if the others had 
 not been present. 
 
 It appears that the heat disengaged in solution is 
 always greater than that due to the mere liquefaction of 
 the gas. Hence the phenomenon is, to a considerable 
 extent, of a chemical character ; and thus we are prepared 
 to find great differences in the absorption of the same 
 gas by dififerent liquids. Thus carbonic acid is 2 '5 times 
 more soluble in alcohol than in water; and 1"8 times 
 more soluble in water at 0° 0. than in water at 15° C. 
 
CHAPTER XIII. 
 
 DIFFUSION, OSMOSE, TRANSPIRATION, VISCOSITY, ETC. 
 
 299. Though we cannot mark a special group of the 
 particles of any one liquid or gas, so as to enable us to 
 see how they gradually mix themselves with the others, 
 we have almost perfect assurance that they do so. This 
 assurance is based partly upon the relative behaviour of 
 two miscible liquids, or two gases, put in contact with 
 one another; partly upon the results of the kinetic 
 theory, which have been found fully to explain at least 
 the greater number of the phenomena ordinarily exhibited 
 by gases. Thus, altogether independent of the convec- 
 tion currents due to differences of temperature, there 
 goes on, in every homogeneous liquid or gas, a constant 
 transference of each individual particle from place to 
 place throughout the mass. In homogeneous solids, at 
 least, it seems probable that there is no such transference, 
 but that each particle has a mean, or average, position 
 relatively to its immediate neighbours, from which it 
 suffers only exceedingly small displacements. 
 
 300. True diffusion, which is much more rapid in 
 gases than in Liquids, is essentially a very slow process 
 compared with those convection processes which are 
 
DIFFFSION, OSMOSE, TKANSPIKATION, ETC. 255 
 
 mainly instrumental in securing the thorough inter- 
 mixture of the various constituents of the air or of 
 dissolved salts with the ocean water. For its careful 
 study, therefore, great precautions are required, with a 
 view to the preservation of uniformity of temperature, 
 as the only mode of preventing convection currents. 
 We will suppose that these precautions have been 
 taken. 
 
 If, by means of a tube (fitted with a stop-cock) which 
 is adjusted at the bottom of a tall glass cylinder nearly 
 full of water, we cautiously introduce by gravity a 
 strong solution of some highly-coloured salt (such as 
 bichromate of potash), the solution, being denser than 
 the water, forms a layer at the bottom of the vessel. If 
 we watch it from day to day we find that, in spite of 
 gravity, the salt gradually rises into the water column, 
 which now shows an apparently perfectly continuous 
 gradation of tint from the still undiluted part of the 
 solution up to the as yet uncontaminated water above. 
 The result irresistibly suggests an analogy with the state 
 of temperature of a bar of metal which is exposed to a 
 source of heat at one end. The analogy would be almost 
 complete if we could prevent loss of heat by the sides 
 of the bar ; for experiment has shown that, just as the 
 flux of heat is from warmer to colder parts, and (ceteris 
 pwribus) proportional to the gradient of temperature, 
 so the diffusion of the salt takes place from more to less 
 concentrated solution, and at a rate at least approximately 
 proportional to the gradient of concentration. This is, 
 possibly, not quite the case at first, when there are 
 exceedingly steep gradients of concentration, for then 
 (see § 292) there is something akin to a surface-film 
 
256 PEOPEETIES OF MATTER. 
 
 which for a time behaves somewhat like that between 
 two liquids which do not mix. This is beautifully 
 shown by rough stirring of the contents of a vessel with 
 parallel glass sides, in which there is a layer of strong 
 brine with clear water above it ; especially if a horizontal 
 beam of sunlight, from a distant aperture in the shutter 
 of a dark room, be made to pass through the vessel, 
 and be received on a sheet of paper placed a few 
 inches behind. However rough the stirring, if it be 
 not too long continued, the mixture is soon seen to 
 settle into layers of different densities; and time is 
 required before diffusion does away with the steep 
 gradients of concentration which have been produced 
 between the layers. These effects can be produced 
 again and again in the same mixture, and show how 
 very much more rapid is the mixing when aided by 
 rough mechanical processes than when left entirely to 
 the slow but sure effects of diffusion. The effect of the 
 stirring is to produce immensely extended surfaces of 
 steep gradient of concentration all through the mixture, 
 and thus greatly to accelerate the natural action of diffu- 
 sion, to which the final result of uniform concentration 
 is really due. 
 
 301. The first accurate experiments on this subject 
 are due to Graham,^ who employed various very simple 
 but effective processes. He showed that while the rate 
 of diffusion varied considerably with the substances 
 employed, these could be ranged in two classes, Colloids 
 and Crystalloids, the members of the first class having 
 very small diffusivity compared with those of the 
 second. Thus he found that the times employed for 
 ^ Chemical <md Physical Researches, collected and reprinted, 1876. 
 
DIFFUSION, OSMOSE, TRANSPIEATION, ETC. 257 
 
 equal amounts of diffusion in "water were relatively as 
 follows : — 
 
 Hydrochloric Acid 
 
 . 1 
 
 Common Salt 
 
 . 2-33 
 
 Sugar 
 
 . 7 
 
 Albumen . 
 
 . 49 
 
 Caramel . 
 
 . 98 
 
 He also verified that the rate of diffusion of any one sub- 
 stance is proportional to the gradient of concentration, 
 and added the important fact that rise of temperature 
 has a marked effect in accelerating the process. 
 
 302. The subject has since been elaborately investi- 
 gated by various experimenters, and absolute values of 
 diffusivity have been calculated from their experiments 
 as well as from those of Graham. 
 
 Following the analogy with heat-conduction, we may 
 define, after Fourier's method, as follows : — 
 
 The diffusivity of one substance in another is the 
 number of units of the substance which pass in unit of 
 time through unit of surface, across which the gradient 
 of concentration is unit of substance per unit of volume 
 per unit of length. 
 
 If we use the C. G. S. system, in which unit of length 
 is a centimetre, unit of mass a gramme, and unit of time 
 a second, the numbers obtained would be exceedingly 
 small, so that the C. Gr. S. system is departed from in 
 practice to the extent of making a day the unit of time. 
 With this we have, according to Stefan's calculations 
 from Graham's results : — 
 
 Temperature 0. 
 
 5° 1-74 
 
 5° 0-76 
 
 10° 0'91 
 
 Hydrocliloric Acid 
 Common Salt 
 
258 PROPERTIES OF MATTER. 
 
 
 Temperature 0. 
 
 Sugar 
 
 9° 0-31 
 
 Albumen . 
 
 13° 0-06 
 
 Caramel 
 
 10° 0-05 
 
 The meaning'of this is that, for instance at 10° C, in 
 water which so holds common salt in solution that there 
 is one gi-amme per cubic centimfetre more in each hori- 
 zontal stratum, than in the stratum one centimetre 
 above it, the upward progress of the salt is at the rate 
 of 0-91 gramme through each square centimetre per day. 
 [Solutions of common salt differing by whole grammes 
 of salt per cubic centimfetre are, of course, only a pleasant 
 fiction of the C. G. S. system.] 
 
 303. Tick, Voigt, Hoppe-Seyler, H. Weber, and many 
 others, have greatly extended Graham's work; some 
 using his process (with slight variations), others employ- 
 ing processes depending upon special physical results 
 (such as rotatory polarisation or electromotive force) due 
 to the salt which is diffusing. It is probable that very 
 good measures may be obtained, though the method 
 would be laborious, by using a narrow tank with parallel 
 glass sides (as in § 300), and observing, from time to 
 time, the greatest refraction suffered by any part of a 
 horizontal beam of sunlight transmitted through the 
 heterogeneous liquid, the tank having been originally 
 half filled (as in § 300) with a strong solution of a salt, 
 under pure water. ^ Sir W. Thomson introduced a 
 rough-and-ready method by letting down into the 
 diffusion column a number of glass beads, containing 
 more or less of air, and therefore having, each as a whole, 
 different mean densities, and observing from day to day 
 1 Tait, "On Mirage," Trans. B.S.K, 1881. 
 
DIFFUSION, OSMOSE, TKANSPIKATIOW, ETC. Z59 
 
 the position of the stratum in which each floated in 
 equilibrium. This method would probably be the best 
 of all, could we only make the beads small enough, so 
 as not to trench upon too many strata at once, and 
 <!Ould we also make certain that neither air-bubbles nor 
 crystals should develop upon them. The latter condi- 
 tion, however, is practically unattainable. 
 
 304. It seems that the idea of comparing diffusion 
 with heat -conduction was originally propounded by 
 BerthoUet before Fourier published his great investiga- 
 tions on the latter subject ; but Fick was the first to 
 revive and develop it in more recent times. 
 
 The physical explanation of the cause of diffusion of 
 liquids in one another, or of solids in a liquid, is vastly 
 more complex and difficult than that of the diffusion of 
 gases, though, in some of their coarser features, the 
 first two of these are closely analogous to the last. In 
 the words of Clerk-Maxwell: "It is easy to see that if 
 there is any irregular displacement among the molecules 
 of a mixed liquid, it must, on the whole, tend to cause 
 each component to pass from places where it forms a 
 large proportion of the mixture to placeswhere.it is 
 less abundant. It is also manifest that any relative 
 motion, of two constituents of the mixture will be 
 opposed by a resistance arising from . the encounters 
 between the molecules of these components. The value 
 of this resistance, however, depends, in liquids, on more 
 complicated conditions than in gases, and for the present 
 we must regard it as a function of all the properties of 
 the mixture at the given place — that is to say, its tem- 
 perature and pressure, and the proportions of the differ- 
 ent components of the mixture." 
 
260 PROPERTIES OF MATTER. 
 
 305. The interdififusion of gases is thus, theoretically, 
 a simpler question than that of liquids ; and has heen 
 developed, from the basis of the kinetic theory of gases, 
 into an almost complete explanation of the observed 
 phenomena. We cannot here introduce the mathe- 
 matical part of the investigation, as it involves analysis 
 of a kind foreign to the range of an elementary book ; 
 but we simply state that the equations ultimately arrived 
 at are of precisely the same form as those obtained by 
 Fourier for heat-conduction in a homogeneous isotropic 
 solid. This part of the theory we owe mainly to Clerk- 
 Maxwell. The experimental part has been well supplied 
 by Loschmidt.^ The following numbers give an idea of 
 his values of interdiflfusivity of pairs of gases, in a mix- 
 ture at a pressure of one atmosphere. We have pre- 
 served only two significant figures, though the measures 
 (in 0. G. S. units) were given to four. 
 
 Carbonic Acid and Air . . . . 0"14 
 
 Oxygen and Hydrogen . . . .0-72 
 
 Carbonic Acid and Hydrogen . . . 0*55 
 
 Carbonic Acid and Carbonic Oxide . . fl-14 
 
 Carbonic Oxide and Hydrogen . . . 0'64 
 
 According to the theory, as given by Maxwell, these 
 quantities should be nearly in inverse proportion to the 
 geometrical mean of the densities (at one atmosphere) 
 of the two gases. 
 
 306. Keturning to the consideration of liquids, we 
 meet with certain very curious phenomena when two 
 miscible liquids are separated by a membrane such as a 
 diaphragm or septum of bladder, or of parchment paper. 
 These are usually arranged under the general title of 
 ^ SUzimgs-Berichte d. Kais, Ac. eu Wicn, 1870. 
 
DIFFUSION, OSMOSE, TKANSPIEATION, ETC. 261 
 
 Osmose, sometimes pedantically divided into endosmose 
 and exosmose. The first careful examination of them 
 we owe to Dutrochet, but the earliest observation 
 recorded is due to NoUet in the first half of last century. 
 
 The main phenomenon, of which all the others are 
 more or less complicated varieties, is simply that difier- 
 ent liquids pass at different rates through a porous mem- 
 brane. Thus NoUet immersed in water a vessel full of 
 alcohol, tightly closed by a piece of bladder, and was 
 surprised to find that the contents soon increased to such 
 an extent as almost to burst the bladder. He then 
 filled, the vessel with water, tied on the bladder, and 
 immersed the whole in alcohol, when the reverse effect 
 was obtained; the contents of the vessel diminished 
 and the bladder was forced inwards. Strange to say, 
 after so good a commencement, he contented himself 
 with recording the two observations. 
 
 307. The phenomenon is so obviously connected with 
 many processes which go on in living bodies, whether 
 vegetable or animal, that it has attracted the attention 
 of physiologists as well as of physicists, and an immense 
 mass of observations on various forms of it has already 
 been accumulated. 
 
 Its theoretical explanation is much more complex 
 than that of ordinary liquid diffusion, because it is found 
 that the material of the septum plays an important, 
 sometimes a paramount, part in determining the rate, 
 and even the direction, in which the osmose with two 
 given liquids shall proceed. 
 
 It is undoubtedly a case of ordinary diffusion, com- 
 plicated by the molecular actions between the material 
 of the septum and the various liquids employed. Thus 
 
262 PROPERTIES OF MATTER. 
 
 there need be no more reason for surprise that a liquid, 
 such as the sap in plants, should be osmotically raised 
 to great heights against gravity, than that water should 
 rise in a capillary tube, or that bichromate of potash 
 should (§ 300) diffuse upwards in a column of water. 
 
 308. Something very similar to osmose can be obtained 
 by ordinary diffusion, when horizontal strata of two liquids 
 are separated by a stratum of a third liquid of intermediate 
 density. Sometimes one or other of the extremes alone 
 passes through the intermediate layer, sometimes both 
 diffuse into it. A beautiful method of gradually develop- 
 ing chemical actions which, on the large scale, would 
 produce dangerous explosions, is thus suggested. When 
 nitric acid, water, and alcohol are the three liquids, the 
 chemical action takes place slowly where the two ex- 
 treme liquids meet, as they diffuse towards one another 
 through the water-septum. 
 
 309. Though the theory is but imperfectly understood , 
 the practical applications of osmose have been developed 
 to an important extent. Of these we need here mention 
 only the process of Dialysis, due to Grraham. The dis- 
 tinction between Colloids and Crystalloids, in their 
 behaviour as regards a porous -septum, is even more 
 marked than in direct liquid diffusion. Hence, when a 
 mixture of colloids and crystalloids, in solution, is placed 
 on one side of a bladder or a piece of parchment paper, 
 and pure water on the other side, it is practically the 
 crystalloids alone which pass through the septum into 
 the water. If the" colloids be originally in enormous 
 excess, one repetition of the process on- the mixture 
 which has passed through the septum is sufficient to 
 separate the crystalloids almost entirely from the colloids. 
 
DIFFUSION, OMOSE, TRANSPIEATION, ETC. 263 
 
 This process is of very great importance as an 
 auxiliary to chemical analysis in medico-legal questions ; — 
 for the more common of the violent poisons are with few 
 exceptions crystalloids, and can be easily and almost com- 
 pletely separated, by dialysis, from the large admixture 
 of colloids in which they are usually found in the viscera; 
 
 310. Graham, in bis extensive series of experiments 
 on the passage of gases through various solids with holes 
 or pores, recognised several quite distinct processes, each 
 with its own laws. 
 
 When a gas is -maintained at constant pressure on one 
 side of a very thin non-porous plate, which has a small hole 
 in it, there being vacuum at the other side, the process of 
 passage is called Effusion. This may be looked on as at 
 least roughly analogous to the passage of a liquid through 
 the orifice. The closer consideration of it belongs to 
 Thermodynamics, The work done on unit volume as it 
 passes out is directly as the pressure, the kinetic, energy 
 acquired is measured by the density and the square of 
 the speed of effusion conjointly. Hence, under the same 
 pressure, the speed of effusion is inversely as the square 
 root of the density. This result was very nearly realised 
 in Graham's experiments ; witness the following : — 
 
 Time of Effusion. 
 
 Theoretical Time 
 
 Air . 
 
 1-0 
 
 1-0 
 
 Nitrogen . 
 
 0-984 
 
 0-986 
 
 Oxygen . 
 
 1-050 
 
 1-051 
 
 Hydrogen 
 
 0-276 
 
 0-263 
 
 Carbonic Acid . 
 
 1-197 
 
 1-236 
 
 The only discrepancies which call for notice are with 
 hydrogen and carbonic acid. But Graham was able to 
 show from the results of another of his series of experi- 
 
264 PROPERTIES OF MATTER. 
 
 ments, that these discrepancies are due to the fact that 
 the perforated plate was not infinitely thin, and that the 
 aperture therefore behaved like a very short capillary 
 tube. This explanation is fully borne out by the fact 
 that the discrepancies are in opposite directions for these 
 two gases, and that this characteristic difference is required 
 by the mode of explanation. 
 
 From these experiments Graham concluded that the . 
 law of this process is analogous to that of diffusion with- 
 out a septum. Bunsen has applied the result to the 
 construction of a very excellent instrument for measuring 
 the density of a gas. 
 
 311. Transpiration is the name given by Graham to the 
 passage of a gas under pressure through a capillary tube. 
 The results obtained were of a much more complex 
 character than in the case of effusion ; and the law of the 
 process, so far as it could be ascertained by experiment 
 alone, was of a different form. Capillary tubes, varying 
 in length from 20 feet down to a few inches, were em- 
 ployed. It was found that the material of the tube had 
 no influence ; hence it has been suggested that the tube 
 becomes lined with a film of the gas, and that the key to 
 the difficulties of the problem is mainly connected with 
 viscosity. The rate of transpiration of hydrogen is only 
 double of that of nitrogen, while that of carbonic acid 
 is much greater than that of oxygen : — 
 
 Limiting Transpiration Times 
 in very fine Capillaries. 
 
 Oxygen .... 1-000 
 
 Air 0-901 
 
 Nitrogen and Carbonic Oxide . 0-875 
 
 Hydrogen .... 0-437 
 
 Carbonic Acid . . . 0-727 
 
DIFFUSION, OMOSE, TEANSPIEATION, ETC. 265 
 
 The two last results show the foundation of the explana- 
 tory remark towards the end of last section. 
 
 The following are some of Graham's comments on 
 this very curious subject : — 
 
 " The times of oxygen, nitrogen, carbonic oxide, and 
 air, are directly as their densities, or equal weights of 
 these gases pass in equal times. Hydrogen passes in 
 half the time of nitrogen, or twice as rapidly for equal 
 volumes. The result for carbonic acid appears at first 
 anomalous. It is that the transpiration time of this gas 
 is inversely proportional to its density, when compared 
 with oxygen. It is to be remembered, however, that 
 carbonic acid is a compound gas, containing an equal 
 volume of oxygen. The second constituent carbon 
 which increases the weight of the gas, appears to give 
 additional velocity to the oxygen in the same manner 
 and to the same extent as increased density from pres- 
 sure, or from cold (as I believe I shall be able to show), 
 increases the transpiration velocity of pure oxygen itself. 
 A result of this kind shows at once the important bear- 
 ing of gaseous transpirability, and that it emulates a 
 place in science with the doctrines of gaseous densities 
 and combining volumes. 
 
 "The circumstance that the transpiration time of 
 hydrogen is one-half of that of nitrogen, indicates that the 
 relations of transpirability are even more simple in their 
 expression than the relations of density among gases. 
 In support of the same assertion may be adduced the 
 additional fact; that binoxide of nitrogen, although 
 differing in density, appears to have the same transpir- 
 ation time as nitrogen. Protoxide of nitrogen and 
 carbonic acid have one transpiration time, so have 
 
266 PROPERTIES OF MATTER. 
 
 nitrogen and carbonic oxide, as each pair has a common 
 density." 
 
 312. When one gas is separated from another, or from 
 a vacuum, by a septum of compressed graphite (§ 53), the 
 law, and even the rate, of passage come to be very nearly 
 the same as those of ordinary gaseous diifusion. Thus 
 gases pass through such a septum at rates inversely as 
 the square roots of their densities, as in efiusion. If the 
 septum is made of plaster of Paris, the results become 
 partially complicated by transpiration. This source of 
 confusion is practically non-existent when the septum is 
 made of " biscuit- ware," as it is technically called ; and 
 the same may be said of all the finer kinds of uhglazed 
 earthenware. Here the pores are so fine that, as Graham 
 says, the action ceases to be molar and becomes molecular. 
 Each particle acts, as it were, on its own account. Hence, 
 when a mixture of two gases of difierent densities is 
 placed on one side of such a septum, the less dense gas 
 passes in greater percentage than the denser, and we 
 have Atmolysis : — a mode of separating difierent gases 
 somewhat akin to dialysis (§ 309). There are few physical 
 experiments more striking and suggestive than the 
 simple one of surrounding, with an atmosphere of coal 
 gas, the bulb (made of unglazed clay) of an arrangement 
 like a large ordinary air-thermometer. The rapidity with 
 which the gases pass through the bulb is extraordinary. 
 
 313. But when the septum is made of caoutchouc the 
 process of penetration is quite different. The septum 
 now acts as a colloidal body, ' not as a porous one ; 
 and the gas combines in an imperfect chemical manner 
 with the matter of the septum, in which it diffuses 
 (in the ordinary sense of the term), until it reaches 
 
DIFjrUSION, OMOSE, TEANSPIRATION, ETC. 267 
 
 the other side and is set free. Thus the small toy- 
 balloons of thin india-rubber, when originally filled 
 with hydrogen, soon collapse. On the other hand, 
 when they are blown with air and then immersed in an 
 atmosphere of hydrogen, they rapidly swell and burst. 
 
 The same phenomenon is beautifully shown by blowing 
 a soap-bubble with carbonic acid gas. For the gas dis- 
 solves in the liquid film, diffuses through it, and escapes 
 into the air, so that the bubble soon collapses. Similarly, 
 an ordinary soap-bubble made to float on carbonic acid 
 gas expands, gradually sinks, and finally bursts. 
 
 A good instance of gaseous diffusion is afforded by 
 evaporation of water, or other liquid, at temperatures 
 below the boiling point, when air is present. For the pro- 
 cess goes on until the vapour in contact with the liquid 
 has a pressure determined solely by the temperature, and 
 by the curvature of the liquid surface. When a layer of 
 vapour of the proper pressure has once formed at the 
 surface, the resistance to its diffusion is so considerable 
 that, unless there be wind or convection-currents, the rate 
 of evaporation is reduced to that of diffusion ; and vapour 
 is formed (at the liquid surface) only as fast as that which 
 is already formed is able to get away. By weighing, 
 from time to time, a-test-tube of known length, which has 
 a layer of liquid at the bottom and is open at the top, 
 fair measures of the rate of diffusion in air, of vapours 
 heavier than air, can be obtained. 
 
 314. Very curious results have been obtained by 
 Deville and Troost with reference to the rapid passage 
 of various gases through heated cast-iron. Carbonic 
 oxide is one of these, and as this is a highly poisonous 
 gas, the matter is one of great importance in relation to 
 
268 PROPERTIES OF MATTER. 
 
 the heating of rooms by stoves. They also showed that 
 highly heated platinum is freely pervious to hydrogen. 
 Graham's researches on the behaviour of palladium with 
 respect to hydrogen have afforded the means of obtaining 
 similar effects even at temperatures far below red-heat ; 
 and, quite recently, v. Helmholtz and Eoot have proved 
 that platinum is pervious to hydrogen even at ordinary 
 temperatures. Thus the question is one of importance, 
 not alone from the sanitary point of view, nor from the 
 point of view of its purely scientific explanation, but 
 also from the very important point of view of the con- 
 struction of gas-thermometers for the measurement of 
 high temperatures, in which the recipient must necessarily 
 be made of some practically infusible metal. The whole 
 of this part of the subject, however, has a specially 
 chemical interest, so that we are not called on to discuss 
 it further. 
 
 315. We have already employed the word Viscosity in 
 two somewhat different applications. In our general 
 discussion of common terms (§ 37) we spoke of it as 
 applied to liquids, and also, by parity of results, to gases. 
 But in § 249 we used it as denoting a property possessed 
 even by the most elastic of solids. 
 
 We must now consider, more carefully, its application 
 to fluid motion. 
 
 And, first, as regards liquids. Questions such as 
 were briefly touched upon in § 37 belong, in their full 
 development, where eddies present almost insuperable 
 difficulties, to Hydrokinetics, and are therefore not to be 
 treated farther in this work. But the passage of a liquid, 
 under pressure, through a capillary tube, is (so far as 
 it is amenable to elementary mathematical treatment) 
 
DIFFUSION, OMOSE, TEANSPIEATION, ETC. 269 
 
 part of our subject. So is the torsional vibration of a 
 disc, in its own plane, when it is immersed in a fluid, 
 especially when, as in Clerk -Maxwell's experiments on 
 gases, other two discs are fixed near and parallel to it, 
 on opposite sides. So far as liquids are concerned, 
 these forms of experiment were carefully worked out 
 by Poiseuille and by Coulomb respectively, and have 
 since been extended, with various modifications, 
 by V. Helmholtz, 0. E. Meyer, etc. Later, Graham 
 (as we have just seen), Clerk-Maxwell, and many others, 
 have applied one or other of these forms of experiment, 
 more or' less modified, to the determination of the 
 viscosity of gaseous bodies. 
 
 316. Before going farther, we must define precisely 
 what we mean l)y viscosity ; and the definition wiU, of 
 course, show how it is to be measured. 
 
 Suppose a layer of fltdd, of unit thickness, to fill the 
 interval between two plane surfaces of indefinite extent 
 to which the fluid adheres. When there is rectilinear 
 relative motion of these surfaces, with unit speed, the 
 tangential force on either per unit of surface is the 
 measure of the viscosity. 
 
 Hence, if, in Fig. 1, § 38, v be the speed at depth y, 
 the tangential stress per unit surface of the layer at that 
 
 depth is — 
 
 dv 
 
 ' dy 
 
 where k is the viscosity. The dimensions of Viscosity, 
 therefore, difier from those of Eigidity (§ 178) simply 
 by the time unit; ie. as the dimensions of velocity 
 difler from those of acceleration. This may be seen 
 at a glance from the equation of § 250. 
 
270 PROPERTIES OF MATTER. 
 
 The establishment of a simple working definition, 
 such as that above, leads at once to the formation of the 
 proper equations of motion in all problems of this kind. 
 The process is precisely the same as that adopted by 
 Fourier in his definition of Heat Conductivity ; and it is 
 curious to see how all who have, ia modern times, 
 treated viscosity without using Fourier's method, have 
 fallen into the vague and misleading methods of Fourier's 
 predecessors. 
 
 317. If, with this definition, we investigate the 
 motion of a liquid in a capillary tube, when it has be- 
 come steady, we are led to the result (fuUy borne out 
 by the experiments of PoiseuHle^) that, ceteris pa/ribus, 
 the discharge in a given time is proportional to the 
 fourth power of the radius of the bore. 
 
 For the solution of the problem we assume that the 
 motion is everywhere parallel to the axis of the tube, 
 and with speed dependent only on the distance from it. 
 Let K be the viscosity, and consider the tangential stress 
 (per unit length of the tube) on the surface of a cylin- 
 drical layer of liquid of radius r, concentric with the 
 tube. If V be the speed of that layer, the amount of 
 the stress is (by the definition above) 
 
 271'rK-r 
 dr- 
 
 Hence the difierence of the tangential forces on the 
 surfaces of a cylinder of liquid of thickness Br is 
 
 2«#, 
 dr 
 
 {^th 
 
 But, as the motion is not accelerated, this must be equal 
 1 M^. des Savans j^cmgers, IX., 1846. 
 
DIFFUSION, OHOSE, TKANSPIRATION, ETC. 271 
 
 and opposite to the difference of the pressures on the 
 ends of the cylinder, which is 
 
 where x is measured parallel to the axis. This must 
 obviously be independent of x, and, as the motion is 
 always very slow under the conditions, p is approxi- 
 mately independent of r. Hence 
 
 ^= -a, 
 
 dx 
 
 a constant, whose obvious value is the difference of 
 pressures at the ends, divided by the length, of the tube. 
 Thus we have the very simple equation 
 
 d/ dv\ 
 This gives 
 
 dr 2/c ' 
 
 the integration constant being zero, because otherwise 
 we should have finite tangential stress on an infinitely 
 small filament along the axis. 
 
 Thus, if r, be the radius of the bore of the tube, and 
 if we assume that v is nil when r = ?•„, 
 
 The whole volume of liquid which passes in unit of 
 time through each cross section is 
 
 
 'i.-trf\'odr = ^j[\i? - r'ydr = ' 
 318. This expression enables us at once to calculate the 
 
272 PEOPEETIES OF MATTER. 
 
 values of k from experiments such as those of Poiseuille. 
 Their agreement with the formula ahove is very close 
 throughout, the bores, lengths, and pressures being 
 varied within wide limits. The most remarkable addi- 
 tional feature which PoiseuiUe recognised is the rapid 
 diminution of viscosity of water with rise of temperature. 
 In C. Gr. S. units his experiments give, approximately, 
 
 Temperature C, 
 
 Coefift. of Viscosity. 
 
 0° 
 
 0-018 
 
 lo- 
 
 0-013 
 
 go" 
 
 0-010 
 
 This means that there is a tangential stress of 0-018 
 dynes per square centimetre on each of two parallel 
 planes, one centimetre apart, when one is moving relatively 
 to the other at the rate of one centimetre per second, 
 and when the interspace is filled with water at 0° C. 
 
 It is well to note that from 0° to 10° 0. the viscosity 
 of water falls ofi" at the rate of 2-8 per cent per degree. 
 Compare this with the corresponding increase of the 
 rate of diffusion of common salt in water (§ 302), which, 
 by Graham's results, is about 4 per cent per degree of 
 temperature. 
 
 319. The oscillation method of Coulomb is trouble- 
 some in the complex mathematical details to which it 
 leads, and even more so in the experimental precautions 
 which it requires. It has been carefully worked out, 
 from both points of view, by many different physicists, 
 including especially 0. E. Meyer. Modifications of it 
 have been employed by v. Helmholtz and others. The 
 theoretical part of the investigation, which is very com- 
 plex, was first developed by Stokes, who applied it to 
 the reduction of pendulum experiments. 
 
DIFFUSION, OMOSE, TEANSPIEATION,' ETC. 273 
 
 The results of Meyer are about a sixth greater than 
 those of Poiseuille. Those of v. Helmholtz and v. 
 Pietrowski,^ in which liquids were contained in an 
 oscillating sphere, were complicated by finite slipping, 
 which led to a new problem. Their result for water is 
 about one-fourth greater than that of Poiseuille. 
 
 According to Schottner,^ the coefficient of viscosity of 
 glycerine (in C. G. S. units) sinks, from about 42 at 3° 0., 
 to little more than 8 at 20° C. 
 
 320., The Viscosity of a gas can be calculated, at least 
 approximately, from the Kinetic theory, for it is easy 
 to see that it must depend upon the transference of 
 momentum by the interchange of particles between two 
 contiguous layers of the gas which have relative velocity. 
 
 Clerk -Maxwell, who first gave the theory of this 
 subject, found that the viscosity is independent of the 
 density in each particular gas, and increases with rise of 
 temperature, being directly proportional to the absolute 
 temperature. His experiments on air gave (in 0. G. S. 
 units) the value 
 
 0-000,000,685(274 + t) 
 
 where t is the temperature centigrade. He also showed 
 that in oxygen the viscosity is greater than in air, and 
 in carbonic acid less. In hydrogen it is about half as 
 great as in air. [Compare with Graham's results above, 
 §§ 310, 311.] This subject has since been elaborately 
 investigated by Meyer, Kundt and Warburg, and many 
 others, but the exact law of the temperature-variation 
 is still uncertain. 
 
 vr. der K. Ac. in Wioii, 1860. ^ Ibid, 1878, p. 686. 
 
CHAPTEE XIV. 
 
 AGGREGATION OF PARTICLES. 
 
 321. This chapter must be a very short one, because, 
 though experimental facts are to be had in profusion, the 
 subject, as a whole, has not yet been raised to the higher 
 level of Science from that of the mere preliminary "beetle- 
 hunting or crab-catching stage." Parts of it are already 
 much farther advanced. The geometry of crystalline 
 forms has been very fully developed and systematised. 
 The physical properties of the aggregate have been 
 scientifically developed, as we have seen, mainly from 
 the basis of Hooke's Law for solids and liquids, and of 
 Boyle's Law for gases; and the formulation of these 
 laws has enabled us to discuss, with something like a 
 secure foothold, the deviations from them. 
 
 But the mode of formation of the aggregate of particles, 
 at all events in solids and liquids, is a question of much 
 greater difficulty. We still require, in fact, a Kepler to 
 co-ordinate the facts, before there can be a chance for a 
 Newton to group them under some simple but all-embrac- 
 ing statement. 
 
 All that our plan permits us to do is to point out 
 briefly how far the Ptolemy and the Copernicus, as well 
 as the Tycho Brahe, of this subject have marshalled the 
 
AGGREGATION OF PARTICLES. 275 
 
 materials for the coming Kepler. The Newton will be 
 later in appearing. 
 
 322. In the case of gases a real step to explanation 
 has been taken, but a great part of the very elements of 
 the Kinetic Theory is still obscure and difficult. The 
 earliest suggestion of it is commonly attributed to D. 
 Bernoulli (1738), but the following passage from Hooke's 
 Pamphlet De Potentid Bestitutivd (which was alluded to 
 in the footnote to § 221 above) shows that essentially the 
 same ideas had been published long before. 
 
 " In the next place for fluid bodies, amongst which 
 the greatest instance we have is air, though the same be 
 in some proportion in all other fluid bodies. 
 
 " The air then is a body consisting of particles so small 
 as to be almost equal to the particles of the Hetero- 
 geneous fluid medium incompassing the earth. It is 
 bounded but on one side, namely, towards the earth, 
 and is indefinitely extended upward being only hindered 
 from flying away that way by its own gravity (the cause 
 of which I shall some other time explain.) It consists 
 of the same particles single and separated, of which 
 water and other fluids do, conjoyned and compounded, 
 and being made of particles exceeding small, its motion 
 (to make its baUance with the rest of the earthy bodies) 
 is exceeding swift, and its Vibrative Spaces exceeding 
 large, comparative to the Vibrative Spaces of other ter- 
 restrial bodies. I suppose that of the Air next the 
 Earth in its natural state may be 8000 times greater 
 than that of Steel, and above a thousand times greater 
 than that of common water, and proportionably I suppose 
 that its motion must be eight thousand times swifter than 
 the former, and above a thousand times swifter than the 
 
276 PROPERTIES OF MATTER. 
 
 latter. If therefore a quantity of this body be inclosed 
 by a solid body, and that be so contrived as to conipress 
 it into less room, the motion thereof (supposing the heat 
 the same) will continue the same, and consequently the 
 Vibrations and Occursions will be increased in reciprocal 
 proportion, that is, if it be Condensed into half the space 
 the Vibrations and Occursions will be double in number : 
 If into a quarter the Vibrations and Occursions will be 
 quadruple, etc. 
 
 " Again, if the conteining Vessel be so contrived as 
 to leave it more space, the length of the Vibrations will 
 be proportionably inlarged, and the number of Vibrations 
 and Occursions will be reciprocally diminished, that is, if 
 it be suffered to extend to twice its former dimensions, 
 its Vibrations will be twice as long, and the number of its 
 Vibrations and Occursions will be fewer by half, and conse- 
 quently its endeavours outward will be also weaker by half. 
 
 " These Explanations wiU serve mutatis mutandis for 
 explaining the Spring of any other Body whatsoever." 
 
 The modern revival of the theory is due to Herapath ; 
 and, a little' later, to Joule, who was the first to make 
 definite calculations as to the speed of the particles of a 
 gas necessary for the production of the observed pressure 
 at different temperatures. Kronig is constantly cited, 
 especially in German works, as having advanced the 
 theory ; but the only novelty which his paper ^ seems to 
 contain is the somewhat startling, and certainly as yet 
 imverified, result that the weight of a gas when in motion 
 is only half what it is when the gas is at rest. [He for- 
 gets to take account of the additional impulse due to 
 restitution, § 255.] 
 
 ^ Pogg. Ann., 1856, xcix. p. 319. 
 
AGGEEGATION OF PARTICLES. 277 
 
 The first approach to a thorough treatment of the 
 theory was made by Clausius, who took account of the 
 mutual impacts of the particles, and who introduced the 
 statistical method of treatment. He was followed by 
 Clerk-Maxwell, and by Boltzmann, and among the three 
 the theory was rapidly developed. 
 
 It is already competent, as is shown in works on Heai 
 (to which the theory now properly belongs) to explain 
 fully many of the properties of gases ; but it still labours 
 under an unsurmounted difficulty, viz. the explanation 
 of the diversity of values of the ratio of the two specific 
 heats (at constant pressure and at constant volume) in 
 various groups of gases. The difficulty is probably due 
 to our ignorance of the interior mechanism of the 
 particles of gases; and it has been greatly enhanced 
 by an unwarranted application of the Theory of Prob- 
 abilities, on which the statistical method is based. But 
 the examination of such questions is foreign to our 
 work. 
 
 323. The key to the explanation of the liquid state 
 is undoubtedly to be sought in connection with Andrews' 
 grand discovery of the Critical Temperature (§ 194). 
 ■Whether something akin to this does, or does not, hold 
 with reference to the relation between the solid and liquid 
 states, is a problem which does not appear to have been 
 attacked. It presents, undoubtedly, most formidable 
 experimental difficulties, which are heightened by the 
 well-known fact that there are solids whose melting point 
 is lowered by pressure. But here, again, we trench on 
 the domain of Heat. 
 
 324. The essential difiierence between the solid and 
 the liquid states of any kind of matter lies in the fact 
 
278 PEOPEETIES OF MATTER. 
 
 that any distorting stress, however small, if only persist- 
 ently applied, produces iinite change of arrangement 
 among the particles of a liquid, whereas it can in general 
 but inflnitesimally alter the relative position of contiguous 
 particles of a solid (unless, of course, it be sufficient to 
 produce rupture of the mass). It is barely conceivable 
 that there can be in any case an abrupt transition from 
 one of these states to the other ; and, in fact, in the great 
 majority of cases at least, there is known to be a gradual 
 transition from the solid to the liquid states, as the 
 temperature is raised in the vicinity of the melting point, 
 so that there is continuous passage from the state of very 
 plastic solid to that of very viscous liquid. The same 
 thing is observed in various colloidal bodies such as 
 isinglass and other jellies, when made up with different 
 amounts of water. In this connection the student 
 should again read the remarkable statement of Clerk- 
 Maxwell in § 253 above. 
 
 Especially on the subject of crystallisation do we 
 appear to get some light from such a view of inter- 
 molecular action. For it would seem that an essential 
 requisite to the formation of a homogeneous crystal must 
 be the comparative freedom of each particle from the in- 
 fluence, direct or not, of all besides those in its immediate 
 vicinity. That there is, even in the most homogeneous 
 crystals, still at least a trace of what Maxwell calls 
 the more stable groups, is probably indicated by the 
 existence of Cleavage Planes, which are not in general 
 parallel to the more prominent faces of the crystal. In 
 fact, as Sorby has shown, cleavage planes analogous to 
 those of slate-rocks can be developed in the majority of 
 plastic solids by the application of pressure-stress in a 
 
AGGBEGATION OF PARTICLES. 279 
 
 direction perpendicular to them. It is thus that slates 
 are formed from a deposit of mud. The proof is obvious 
 from the fact that the planes of cleavage are often in- 
 clined at large angles to those .of stratification. But we 
 offer these remarks merely as a suggestion, which our 
 limits prevent us from developing. 
 
 In this connection it may be well to mention an acute 
 remark of Le Roux^ to the effect that annealing appears 
 to preserve the state of isotropy characteristic of fusion. 
 He was led to this by remarking that a glass of borate 
 of magnesia, which was ordinarily transparent, became 
 like porcelain or, rather, hke white marble when very 
 quickly cooled. 
 
 325. Whether all soUds tend to acquire ultimately a 
 crystalline structure or not, we' can at least seek to find 
 the forms which they are likely to assume in cases (such 
 as that of slow deposition from a state of solution) where 
 each particle is free to choose its position of least poten- 
 tial energy relative to those already deposited. This 
 was long ago very weU worked out by Hauy, though (of 
 course) without any reference to potential energy. And 
 the agreement, of the results of such hypothetical calcu- 
 lations, with the observed forms of natural and artificial 
 crystals, shows that we have really made a step in the 
 right direction (though a ve/ry short one) to the explana- 
 tion of these singular and beautiful results of the action 
 of molecular forces. 
 
 326. The simplest case is that in which the separate 
 particles behave as if they were spherical, i.e. as if they 
 exerted equal resultants of molecular and thermal action 
 (at the same distance) in all directions. Here we can 
 
 1 Gomptes Rendus, 1867, p. 126. 
 
280 PEOPEETIES OF MATTER. 
 
 call to our assistance the analogy of well-known results 
 as to the piling of shot, etc. 
 
 There are two ways in which we may suppose the 
 first plane layer to be laid downj i.e. in square order, 
 or in equilateral triangular order. Again there are two 
 ways in which the next layer- may be (symmetrically) 
 deposited, i.e. particle above particle, or particle above 
 the middle point of each square, or of each equilateral 
 triangle. The two latter cases, however, are really the 
 same arrangement, so far as the relative positions of 
 contiguous particles are concerned; as we see at once by 
 looking at one of the faces of a four -sided pyramid 
 built up on a square base. For in the face the order is 
 triangular. Or, if we remove one edge from the three- 
 sided pyramid built on an equilateral triangular base, 
 we find the particles in the plane of replacement to be 
 in square order. 
 
 We find these two forms, apparently so different, in 
 many species both of natural and of artificial crystals. 
 Thus the regular oktahedron, which is merely two four- 
 sided pyramids built on opposite sides of the same square 
 base, and its hemihedral form, the regular tetrahedron, 
 which is the three-sided pyramid built on an equilateral 
 triangular base, are both met with in .various substances 
 belonging to the Cubic System. The measured angles 
 between the several pairs of faces are found to agree 
 exactly with those of the geometrical solids. 
 
 327. An imperfectly developed oktahedron, when the 
 imperfection is symmetrical, may assume various forms, 
 all of which are met with- in actual crystals. Thus, 
 suppose we remove, layer after layer, symmetrically, 
 from the summits of the oktahedron. The new faces 
 
AGGEE6ATI0N OF PAETICLES. 281 
 
 thus produced form parts of the faces of a cube ; and 
 we may obtain the cube complete by continuing the 
 process till the last trace of the oktahedral faces has 
 been removed. 
 
 Replace, symmetrically, the edges either of the oktahe- 
 dron or of the cube, and w6 produce a set of parts of the 
 faces of what is called a Ehombic Dodekahedron (Fig. 40). 
 Many natural crystals assume symmetrical forms con- 
 taining faces belonging respectively to the oktahedron, 
 the cube truncating its summits, and the rhombic dode- 
 kahedron replacing its edges. 
 
 Instead of replacing each edge by a single plane 
 equally inclined to the faces which meet in that edge, 
 suppose we level it symmetrically by two planes. We 
 shall now no longer have a form with fixed angles {i.e. 
 invariable shape) as in the cases previously mentioned, 
 but one which (though its general character is deter- 
 mined) will have its form dependent on the inclination 
 of the bevelling planes to the faces which meet in the 
 edge bevelled. Here, again, the geometrical results are 
 found to fit the forms actually presented in nature. 
 
 It is an interesting and instructive work for the 
 student to verify these statements by operating on a 
 lump of soft chalk, or (still better) modeller's clay, by 
 means of a broad, but thin, bladed knife. 
 
 328. This is not the place to enter into crystallo- 
 graphic details, but we may introduce the following 
 statement, which includes all the above results, and 
 which is found, at all events, to accord fully with the 
 general appearance of any of the very numerous crystals 
 belonging to what is called the cubic system. 
 
 Any three axes which meet, but which do not lie in 
 
282 PROPERTIES OF MATTER. 
 
 one plane, being chosen, the equation of any plane 
 whatever, which does not pass through the origin, can be 
 written in the form 
 
 hx + kg + Iz = 1. 
 
 Here h, h, and I are finite quantities, the reciprocals of 
 the distances from the origin at which the plane meets 
 the axes respectively. 
 
 To obtain all the plane faces of any one simple form 
 of crystal, all we have to do is to give to h, h, I in the 
 above expression all admissible values in succession. 
 
 329. When we take the cubic system, of which alone 
 we have yet spoken, symmetry shows that if the three 
 axes be taken as the Hnes joining pairs of opposite 
 summits of the oktahedron, obtained as in § 326 above, 
 these will possess properties absolutely alike. Thus 
 sjnnmetry shows that the numbers h, h, I may be arrcmged 
 in amy order, for what is true of any one of the axes is 
 true of each of the others. Similar considerations show 
 that each of h, k, I, independent of the others, Trmy he 
 either posUive or negative. These quantities are always 
 found, in actual crystals, to have their ratios rigorously 
 expressible in small Whole Numbers. 
 
 They can be arranged in six diflferent orders ; and, as 
 any one of these orders has eight possible arrangements 
 of signs, there are forty -eight symmetrically arranged 
 plane triangular faces on the most general simple form 
 of this system. This form is found in many natural 
 crystals, and is called a hexakis-oktahedron (Fig. 36). 
 
 When two oih,Jc,l are equal, it is usual to divide the 
 forms into two groups according as the third is less or 
 greater than the others. Both classes have' twenty -four 
 
AGGREGATION OF PARTICLES. 
 
 283 
 
 faces only, but in the first they are triangular, in the 
 second quadrilateral. These are, respectively, the triakis- 
 oktahedron (Fig. 37) and the eikositetrahedron (Pig. 38). 
 Intermediate to these there is the case when all three 
 are equal, and we have the oktaiedron itself. 
 
 There is also a limiting case when the third vanishes. 
 This is the rhombic dodekahedron (Fig. 40); and of 
 course it also forms the limiting case of the series when 
 one vanishes and the others are (generally) unequal, 
 called the tetrakis-hexahedron (Fig. 39). 
 
 When two vanish we have the cube. 
 
 The following figure shows the Hexakis-Oktahedron, 
 an octant (lying among the positive axes of x, y, z), being 
 specially lettered for reference, and we can easily see 
 how it degrades into the other forms. 
 
 Thus if zGy, and therefore ykz and zBx, become 
 straight lines {i.e. if the planes marked a and 6, c and d, 
 e and/, coincide in pairs), the figure becomes the triakis- 
 oktahedron (Fig. 37). This degrades into the oktahedron 
 if D be in the plane xyz {i.e. if a, b, c, d, e, f, are all one 
 plane) : and into the rhombic dodekahedron if DC be 
 perpendicular to the plane of xy {i.e. if a, b, a', .b', are all 
 one plane). 
 
284 
 
 PROPERTIES OF MATTER. 
 
 Again, if the planes / and a, b and c, d and e, coincide 
 in pairs, we have the eikositetrahedron (Fig. 38). 
 
 If ADB be parallel to the plane of xy, etc., the angle 
 ADB is a right angle, the planes / and/', a and a', h and 
 b', etc., coincide in pairs, and the figure is the tetrakis- 
 hexahedron (Fig. 39), which becomes the cube when/', /, 
 
 Pig. 38. 
 
 Fig. 39. 
 
 Fig. 40. 
 
 a, a', etc., are all in one plane; and the rhombic dodeka- 
 hedron when a, a', b and b' are all in one plane. 
 
 330. Six faces of the rhombic dodekahedron are 
 parallel to the line joining D with the origin, and are 
 situated symmetrically round it. If, then, these faces 
 be extended in their own planes without any alteration 
 of the two groups of three forming the other faces, the 
 whole will become a regular hexagonal prism with ends 
 consisting each of three equal rhombic faces with their 
 greater angles in contact. This is found, by measurement, 
 
AGGREGATION OF PARTIOLES. 285 
 
 to be the form of a bee's cell. And a series of tHese 
 dodekahedra (all equally and similarly distorted from 
 equal rhombic dodekahedra) can (like them) be so packed 
 together as to fill space without leaving interstices. 
 
 331. If, instead of building up a mass of equal 
 spheres, we use similar, equal, and similarly situated ellip- 
 soids of revolution, we must make corresponding altera- 
 tions in our rules for h, h, I above. If the chief axes of 
 the ellipsoids be perpendicular to the layers of particles, 
 the X, y, « axes are still a rectangular system, but h (say) is 
 no longer interchangeable with k or with I. For h is 
 now a small integral multiple of a parameter which 
 depends on the chief axis of the ellipsoid, while k and I 
 are similar multiples of the (equal) parameters of the 
 other two axes. This greatly reduces the number of 
 possible faces in the simple forms. 
 
 If the particles are similar, equal, and similarly 
 situated ellipsoids, touching one another in a layer by 
 the ends of two of their axes, the x, y, z system is still 
 rectangular, but no two of h, h, I are interchangeable. 
 
 In any other arrangement, which is capable of giving 
 a homogeneous whole, the simplest x, y, z system is no 
 longer rectangular; and the- ellipsoids, though still 
 similar, equal, and similarly situated, may come in con- 
 tact (in threes) in an infinity of different ways. 
 
 There is no known form of crystallised matter whose 
 separate faces cannot be exactly accounted for by results 
 based on these premises, though there are many cases of 
 hemihedry, etc., in which the faces geometrically deter- 
 mined for a simple form present themselves only in 
 selected groups. 
 
 332. In what we have said above, the only assumptions 
 
286 PEOPERTIES OF MATTEJt. 
 
 made (for the purpose of explaining homogeneity) were 
 that the particles grouped were themselves iequal, similar, 
 and similarly situated ; and that the arrangement of its 
 neighbours round it was exactly the same for each 
 particle. 
 
 But it is easy to conceive that very different results 
 may be obtained, even with identical materials, according 
 to circumstances. Every one who has seen water which 
 is prevented only by currents from becofiiing solid ice, 
 and which is full of excessively small ice crystals, may 
 easily imagine a state of things in which the particles 
 (which, otherwise, would have been deposited one by 
 one to form a crystal) may arrange themselves in very 
 small, but similar and equal, groups before they are 
 deposited. Thus the aggregates, above contemplated, 
 may be built up, not directly of individual particles, but 
 of other less complex though (among themselves) similar 
 and equal sets of aggregated particles. Here again we 
 come back to the same idea as that in the quotation from 
 Clerk-Maxwell (§ 253), and may employ it for the pur- 
 pose of explaining the existence of cleavage planes, etc. 
 
 333. The aggregations we have considered have been 
 such as take place freely ; but if we consider what is 
 likely to happen under circumstances of temporary or 
 permanent stress, as, for instance, in a Rupert's drop, 
 or any other melted mass of which a portion is cooled 
 and solidified more suddenly than the rest, we see that 
 we cannot expect a result in which the potential energy 
 of the whole shall be as small as possible; and are, 
 therefore, prepared to find that such bodies, unless care- 
 fully annealed, are essentially in unstable equilibrium. 
 
APPENDIX I. (§ 18). 
 
 Hypotheses as to the Constitution of Mattee. 
 By Professor Flint, D.D, 
 
 1. All material substances are infinitely divisible into 
 parts of the same nature as themselves and as complex, 
 even qualitatively, as themselves. 
 
 2. All material substances are divisible into ultimate 
 indivisible homogeneous parts as complex as the wholes. 
 
 One or other of these two hypotheses (it is, perhaps, 
 impossible to determine which) is attributed by Lucretius 
 to Anaxagoras, whose real opinion, however, was prob- 
 ably the one which foUows. 
 
 3. All material substances are formed from a primi- 
 tive matter, " in which all things were together, infinitely 
 numerous, infinitely little," and of which each infinitely 
 little part was infinitely complex. 
 
 4 All material substances result from the combination 
 of a few kinds of material elements, each of which is 
 composed of particles like to itself, e.g., earth of earthy 
 particles, water of aqueous particles, air of aerial particles. 
 — This was thehypothesis of the Hindu Kanada, the Greek 
 Empedocles, and a host of medieval physicists. 
 
 5. All material substances are states or stages of one 
 
288 PROPERTIES OF MATTER. 
 
 primitive matter or element, e.g. of water or air. — The 
 hypothesis of Thales, Anaximenes, etc. 
 
 6. All material substances are divisible into ultimate 
 indivisible parts, " strong in solid singleness," which 
 have no qualitative but only quantitative differences, and 
 which variously aggregate through motion in a void. 
 
 This is the atomic hypothesis as taught by Democritus, 
 Epicurus, etc. 
 
 7. All material substances are divisible into element- 
 ary substances which are subdivisible into molecules, and, 
 ultimately, into atoms possessed of distinctive qualitative 
 as well as quantitative diiferences. — ^Eecently, and prob- 
 ably still, the ordinary chemical hypothesis. 
 
 8. All material substances are divisible into so-called 
 elementary substances composed of molecular particles 
 of the same nature as themselves, but these molecular 
 particles are complicated structures consisting of congre- 
 gations of truly elementary atoms, identical in nature 
 and differing only in position, arrangement, motion, etc., 
 and the molecules or chemical atoms are produced from 
 the true or physical atoms by processes of evolution 
 under conditions which Chemistry has not yet been alble 
 to reproduce.— Hypothesis of H. Spencer, etc. 
 
 9. All material substances are composed of atoms, 
 not hard and solid and on that account indivisible, but 
 the rotatory rings or infinitesimal whirls of an incom 
 pressible frictionless fluid, supposed to be homogeneous 
 and perfect, but the nature of which is not otherwise 
 described ; and all the difiierences of material substances 
 a,re due to the characters and behaviow of their component 
 rings or whirls. — The hypothesis of Sir William Thomson. 
 
 10. The matter which is the object of the senses is 
 
APPENDIX. 289 
 
 the product of a world -building power moulding in 
 accordance with eternal ideas an uncreated substratum, 
 the "receptacle" and "nurse" of "forms," but itself 
 devoid of form and definite attributes. — Plato's hypo- 
 thesis. 
 
 11. The matter which is an object of sense is a 
 synthesis of form with a primary matter which is merely 
 capacity and passivity — a synthesis produced by a for- 
 mative cause, which. must be both an efficient and final 
 cause. — Aristotle's hypothesis. 
 
 12. Impenetrability is the essence of matter. — Hypo- 
 thesis of various physicists. 
 
 13. Extension, not impenetrability, is the essence of 
 matter. "Give me extension and motion and I will 
 construct the world." — Descartes. 
 
 14. Material things are " modes" of extension, which 
 is one of the only two discoverable " attributes '' of the 
 one " Substance." — Spinoza. 
 
 15. Matter in its ultimate constitution consists of 
 metaphysical poinds which give rise to sensible matter by 
 states of efibrt (conatus) transitional from rest to motion. 
 — The hypothesis of Vico. See my " Vico." 
 
 16. The ultimate elements of matter are indivisible 
 points without extension, but surrounded by spheres of 
 attractive and repulsive force which alternate according 
 to the distance of these points up to a certain degree of 
 remoteness. — Hypothesis of Boscovich. 
 
 17. The physical universe is constituted by the un- 
 conscious perceptions of a vast collection of unextended 
 spiritual forces or monads, endowed with a power of 
 spontaneous development and with something of the 
 nature of desire and sentiment : and the properties which 
 
 V 
 
290 PEOPERTIES OF MATTER. 
 
 physical science ascribes to the ultimate elements of 
 matter are the modes under which the reciprocal actions 
 of the monads appear to sense. — The hypothesis of 
 Leibnitz. 
 
 18. Matter is a mental picture in which "mind-stuff" 
 is the thing represented, and mind-stuff is constituted by 
 feelings which can exist by themselves, without forming 
 parts of a consciousness, but which are also woven into 
 the complex form of human minds. — The hypothesis of 
 Clifford. 
 
 19. Matter apart from perception has no existence; 
 physical phenomena are essentially sensations or ideas; 
 " bodies " are groups or clusters of actual or expected 
 sensations arranged according to so-called laws of nature 
 in which is manifest the working of the divine mind. — 
 Berkeley's hypothesis. 
 
 20. Matter is simply an appearance to sense, without 
 anything real in it. — The Hindu hypothesis of Maya, the 
 Eleatic hypothesis of non-being, etc. 
 
 21. Matter is "the permanent possibility of sensations." 
 —J. a MiU. 
 
 22. " Die Materie ist Dasjenige, wodurch der Wille, 
 der das innere Wesen der Dinge ausmacht, in die Wahr- 
 nehmbarkeit tritt, anschaulich, Sidttbar wird. In diesem 
 Sinne. ist also die Materie die blosse Sichtbarkeif des 
 Willens, oder das Band der Welt als Wille mit der Welt 
 als Vorstellung. Die Materie ist durch und durch 
 Causalitat." — Schopenhauer. 
 
 23. Matter is constituted by forces which are out- 
 goings or manifestations of the Divine Will. 
 
 24. Matter is not objectified Will but objectified 
 thought. 
 
APPENDIX. 291 
 
 25. Matter is Nature's self-externality in its most 
 universal form with a tendency to self-internality or 
 individuation shown in the nis'us of gravitation, and 
 nature is the Idea in the form of otherness, or self- 
 alienation. — ^Hegel. 
 
 APPENDIX II. (i 29, 101). 
 
 From the article " Atom," by Clerk-Maxwell, 
 Ency. Brit, 9th ed. 
 
 Atom (aro/to?) is a body which cannot be cut in two. 
 The atomic theory is a theory of the constitution of 
 bodies, which asserts that they are made up of atoms. 
 The opposite theory is that of the homogeneity and 
 continuity of bodies, .and asserts, at least in the case 
 of bodies having no apparent orgaliisation, such, for 
 instance, as water, that as we can divide a drop of 
 water into two parts which are each of them drops of 
 water, so we have reason to believe that, these smaller 
 drops can be divided again, and the theory goes on to 
 assert that there is nothing in the nature of things to 
 hinder this process of division fsom being repeated over 
 and over again, times without end. This is the doctrine 
 of the infinite divisibility of bodies, and it is in direct 
 contradiction with the theory of atoms. 
 
 The atomists assert that after a certain number of 
 such divisions the parts would be no longer divisible, 
 because each of them would be an atom. The advocates 
 of the continuity of matter assert that the smallest con- 
 ceivable body has parts, and that whatever has parts 
 may be divided. 
 
292 PROPERTIES OP MATTER. 
 
 There are thus two modes of thinking about the 
 constitution of bodies, which have had their adherents 
 both in ancient and in modern times. They correspond 
 to the two methods of regarding quantity — the arith- 
 metical and the geometrical. To the atomist the true 
 method of estimating the quantity of matter in a body 
 is to count the atoms in it. The void spaces between 
 the atoms count for nothing. To those who identify 
 matter with extension, the volume of space occupied by a 
 body is the only measure of the quantity of matter in it. 
 
 Of the different forms of the atomic theory, that of 
 Boscovich may be taken as an example of the purest 
 monadism. According to Boscovich matter is made up 
 of atoms. Each atom is an indivisible point, having 
 position in space, capable of motion in a continuous path, 
 and possessing a certain mass, whereby a certain amount 
 of force is required to produce a given change of motion. 
 Besides this the atom is endowed with potential force, 
 that is to say, that any two atoms attract or repel each 
 other with a force depending on their distance apart. 
 The law of this force, for all distances greater than say 
 the thousandth of an inch, is an attraction varying as the 
 inverse square of the distance. For smaller distances 
 the force is an attraction for one distance and a repulsion 
 for another, according to some law not yet discovered. 
 Boscovich himself, in order to obviate the possibility of 
 two atoms ever being in the same place, asserts that the 
 ultimate force is a repulsion which increases without 
 limit as the distance diminishes without limit, so that 
 two atoms can never coincide. But this seems an 
 unwarrantable concession to the vulgar opinion that 
 two bodies cannot co- exist in the same place. This 
 
APPENDIX. 293 
 
 opinion is deduced from our experience of the behaviour 
 of bodies of sensible size, but we have no experimental 
 evidence that two atoms may not sometimes coincide. 
 For instance, if oxygen and hydrogen combine to form 
 water, we have no experimental evidence that the mole- 
 cule of oxygen is not in the very same place with the 
 two molecules of hydrogen. Many persons cannot get 
 rid of the opinion that all matter is extended in length, 
 breadth, and depth. This is a prejudice of the same 
 kind with the last, arising from our experience of bodies 
 consisting of immense multitudes of atoms. The system 
 of atoms, according to Boscovich, occupies a certain 
 region of space in virtue of the forces acting between 
 the component atoms of the system and any other 
 atoms when brought near them. No other system of 
 atoms cam occupy the same region of. space at the same 
 time, because, before it could do so, the mutual action 
 of the atoms would have caused a repulsion between the 
 two systems insuperable by any force which we can 
 command. Thus, a number of soldiers with firearms 
 may occupy an extensive region to the exclusion of the 
 enemy's armies, though the space filled by their bodies 
 is but small. In this way Boscovich explained the 
 apparent extension of bodies consisting of atoms, each of 
 which is devoid of extension. According to Boscovich's 
 theory, all action between bodies is action at a distance. 
 There is no such thing in nature as actual contact 
 between two . bodies. When two bodies are said in 
 ordinary language to be in contact, all that is meant 
 is that they are so near together that the repulsion 
 between the nearest pairs of atoms belonging to the two 
 bodies is very great. 
 
294 PROPERTIES OF MATTER. 
 
 Thus, in Boscovich's theory, the atom has continuity 
 of existence in time and space. At any instant of time 
 it is at some point of space, and it is never in more than 
 one place at a time. It passes from one place to another 
 along a continuous path. It has a definite mass which 
 cannot be increased or diminished. Atoms are endowed 
 with the power of acting on one another by attraction 
 or repulsion, the amount of the force depending on the 
 distance between them. On the other hand, the atom 
 itself has no parts or dimensions. In its geometrical 
 aspect it is a mere geometrical point. It has no exten- 
 sion in space. It has not the so-called property of 
 Impenetrability, for two atoms may exist in the same 
 place. This we may regard as one extreme of the 
 various opinions about the constitution of bodies. 
 
 The opposite extreme, that of Anaxagoras — the 
 theory that bodies apparently homogeneous and con- 
 tinuous are so in reality — is, in its extreme form, a 
 theory incapable of development. To explain the 
 properties of any substance by this theory is impossible. 
 We can only admit the observed properties of such 
 substance as ultimate facts. There is a certain stage, 
 however, of scientific progress in which a method corre- 
 sponding to this theory is of service. In hydrostatics, 
 for instance, we define a fluid by means of one of its 
 known properties, and from this definition we make the 
 system of deductions which constitutes the science of 
 hydrostatics. In this way the science of hydrostatics 
 may be built upon an experimental basis, without any 
 consideration of the constitution of a fluid as to whether 
 it is molecular or continuous. In like manner, after the 
 French mathematicians had attempted, with more or less 
 
APPENDIX. 295 
 
 ingenuity, to construct a theory of elastic solids from the 
 hypothesis that they consist of atoms in equilibrium 
 under the action of their mutual forces, Stokes and 
 others showed that all the results of this hypothesis, so 
 far at least as they agreed with facts, might be deduced 
 from the postulate that elastic bodies exist, and from the 
 hypothesis that the smallest portions into which we can 
 divide them are sensibly homogeneous. In this way the 
 principle of continuity, which is the basis of the method 
 of Fluxions and the whole of modern mathematics, may 
 be applied to the analysis of problems connected with 
 material bodies by assuming them, for the purpose of 
 this analysis, to be homogeneous. All that is required 
 to make the results applicable to the real case is that the 
 smallest portions of the substance of which we take any 
 notice shall be sensibly of the same kind. Thus, if a 
 railway contractor has to make a tunnel through a hill 
 of gravel, and if one cubic yard of the gravel is so like 
 another cubic yard, that for the purposes of the contract 
 they may be taken as equivalent, then, in estimating the 
 work required to remove the gravel from the tunnel, he 
 may, without fear of error, make his calculations as if 
 the gravel were a continuous substance. But if a worm 
 has to make his way through the gravel, it makes the 
 greatest possible difference to him whether he tries to 
 push right against a piece of gravel, or directs his course 
 through one of the intervals between the pieces ; to him, 
 therefore, the gravel is by no means a homogeneous and 
 continuous substance. 
 
 In the same way, a theory that some particular 
 substance, say water, is homogeneous and continuous 
 may be a good working theory up to a certain point. 
 
296 PROPERTIES OF MATTER. 
 
 but may fail 'when we come to deal with quantities so 
 minute or so attenuated that their heterogeneity of 
 structure comes into prominence. Whether this hetero- 
 geneity of structure is or is not consistent with homo-, 
 geneity and continuity of substance is another question. 
 The extreme form of the doctrine of continuity is 
 that stated by Descartes, who maintains that the whole 
 universe is equally full of matter, and that this matter 
 is all of one kind, having no essential property besides 
 that of extension. All the properties which we perceive 
 in matter he reduces to its parts being movable among 
 one another, and so capable of all the varieties which we 
 can perceive to follow from the motion of its parts 
 (Princvpia, ii. 23). Descartes' own attempts to deduce 
 the different qualities and actions of bodies in this way 
 are not of much value. More than a century was 
 required to invent methods of investigating the con- 
 ditions of the motion of systems of bodies such as 
 Descartes imagined. 
 
 A cube, whose side is the 4000th of a millimetre, 
 may be taken as the minimum visibile for observers of 
 the present day. Such a cube would contain from 60 
 to 100 million molecules of oxygen or of nitrogen; but 
 since the molecules of organised substances contain on an 
 average about 50 of the more elementary atoms, we may 
 assume that the smallest organised particle visible under 
 the microscope contains about two million molecules of 
 organic matter. At least half of every living organism 
 consists of water, so that the smallest living being visible 
 under the microscope does not contain more than about 
 a million organic molecules. Some exceedingly simple 
 
APPENDIX. 297 
 
 organism may be supposed built up of not more than a 
 million similar molecules. It is impossible, however, to 
 conceive so small a number sufficient to form a being 
 furnished with a whole system of specialised organs. 
 
 Thus molecular science sets us face to face with 
 ■physiological theories. It forbids the physiologist 
 from imagining that structural details of infinitely small 
 dimensions can furnish an explanation of the infinite 
 variety which exists in the properties and functions of 
 the most minute organisms. 
 
 A miscroscopic germ is, we know, capable of develop- 
 ment into a highly organised animal. Another germ, 
 equally microscopic, becomes, when developed, an animal 
 of a totally diiferent kind. Do all the differences, infinite 
 in number, which distinguish the one animal from the 
 other, arise each from some difference in the structure of 
 the respective germs ? Even if we admit this as possible, 
 we shall be called upon by the advocates of Pangenesis 
 to admit still greater marvels. For the microscopic 
 germ, according to this theory, is no mere individual, 
 but a representative body, containing members collected 
 from every rank of the long-drawn ramification of the 
 ancestral tree, the number of these Aembers being amply 
 sufficient not only to furnish the hereditary characteristics 
 of every organ of the body and every habit of the animal 
 from birth to death, but also to afford a stock of latent 
 gemmules to be passed on in an inactive state from germ 
 to germ, till at last the ancestral peculiarity which it 
 represents is revived in some remote descendant. 
 
 Some of the exponents of this theory of heredity have 
 attempted to elude the difficulty of placing a whole 
 world of wonders within a body so small and so devoid 
 
298 PROPERTIES OF MATTER. 
 
 of visible structure as a germ, by using the phrase 
 structureless germs. ^ Now, one material system can 
 differ from another only in the configuration and motion 
 which it has at a given instant. To explain differences 
 of function and development of a germ without assuming 
 diflferences of structure is, therefore, to admit that the 
 properties of a germ are not those of a purely material 
 system. 
 
 Coincidences observed, in the case of several terrestrial 
 substances, with several systems of Hnes in the spectra of 
 the heavenly bodies, tend to increase- the evidence for 
 the doctrine that terrestrial substances exist in the 
 heavenly bodies, while the discovery of particular lines 
 in a celestial spectrum which do not coincide with any 
 line in a terrestrial spectrum does not much weaken the 
 general argument, but rather indicates either that a sub- 
 stance exists in the heavenly body not yet detected by 
 chemists on earth, or that the temperature of the heavenly 
 body is such that some substance, undecomposable by 
 our methods, is there spHt up into components unknown 
 to us in their separate state. 
 
 We are thus led =to believe that in widely-separated 
 parts of the visible universe molecules exist of various 
 kinds, the molecules of each kind having their various 
 periods of vibration either identical, or so nearly identical 
 that our spectroscopes cannot distinguish them. We 
 might argue from this that these molecules are alike in 
 all other respects, as, for instance, in mass. But it is 
 sufficient for our present purpose to observe that the 
 
 ' SeeF. Galton, "On Blood Relationship," Proc. Boy. Soc, June 
 13, 1872. : 
 
APPENDIX. 299 
 
 same kind of molecule, say that of hydrogen, has the 
 same set of periods of vibration, whether we procure the 
 hydrogen from water, from coal, or from meteoric iron, 
 and that light, having the same set of periods of vibra- 
 tion, comes to us from the sun, from Sirius, and from 
 Arcturus. 
 
 The same kind of reasoning which led us to believe 
 that hydrogen exists in the sun and stars, also leads us 
 to believe that the molecules of hydrogen in all these 
 bodies had a common origin. For a material system 
 capable of vibration may have for its periods of vibration 
 any set of values whatever. The probability, therefore, 
 that two material systems, quite independent of each 
 other, shall have, to the degree of accuracy of modern 
 spectroscopic measurements, the same set of periods of 
 vibration, is so very small that we are forced to believe 
 that the two systems are not independent of each other. 
 When, instead of two such systems, we have innumer- 
 able multitudes all having the same set of periods, the 
 argument is immensely strengthened. 
 
 Admitting, then, that there is a real relation between 
 any two molecules of hydrogen, let us consider what this 
 relation may be. 
 
 We may conceive of a mutual action between one 
 body and another tending to assimilate them. Two 
 clocks, for instance, will keep time with each other if 
 connected by a wooden rod, though they have different 
 rates if they were disconnected. But even if the pro- 
 perties of a molecule were as capable of modification as 
 those of a clock, there is no physical connection of a 
 sufficient kind between Sirius and Arcturus. 
 
 There are also methods by which a large number of 
 
300 PROPERTIES OF MATTER. 
 
 bodies differing from each other may be sorted into sets, 
 so that those in each set more or less resemble each 
 other. In the manufacture of small shot this is done 
 by making the shot roll down an inclined plane. The 
 largest specimens acquire the greatest velocities, and are 
 projected farther than the smaller ones. In this way the 
 various pellets, which differ both in size and in round- 
 ness, are sorted into different kinds, those belonging to 
 each kind being nearly of the same size, and those which 
 are not tolerably spherical being rejected altogether. 
 
 If the molecules were originally as various as these 
 leaden pellets, and were afterwards sorted into kinds, 
 we should have to account for the disappearance of all 
 the molecules which did not fall under one of the very 
 limited number of kinds known to us ; and to get rid of 
 a number of indestructible bodies, exceeding by far the 
 number of the molecules of all the recognised kinds, 
 would be one of the severest labours ever proposed to a 
 cosmogonist. 
 
 It is well known that living beings may be grouped 
 into a certain number of species, defined with more or 
 less precision, and that it is diflBcult or impossible to find 
 a series of individuals forming the links of a continuous 
 chain between one species and another. In the case of 
 living beings, however, the generation of individuals is 
 always going on, each individual differing more or less 
 from its parents. Each individual during its whole life 
 is undergoing modification, and it either survives and 
 propagates its species, or dies early, accordingly as it is 
 more or less adapted to the circumstances of its environ- 
 ment. Hence, it has been found possible to frame a 
 theory of the distribution of organisms into species by 
 
APPENDIX. 301 
 
 means of generation, variation, and discriminative destruc- 
 tion. But a theory of evolution of this kind cannot be 
 applied to the case of molecules, for the individual mole- 
 cules neither are bom nor die, they have neither parents 
 nor offspring, and so far from being modified by their 
 environment, we find that two molecules of the same 
 kind, say of hydrogen, have the same properties, though 
 one has been compounded with carbon and buried in the 
 earth as coal for untold ages, while the other has been 
 "occluded" in the iron of a meteorite, and after un- 
 known wanderings in the heavens has at last fallen into 
 the hands of some terrestrial chemist.' 
 
 The process by which the molecules become distributed 
 into distinct species is not one of which we know any 
 instances going on at present, or of which we have as 
 yet been able to form any mental representation. If 
 we suppose that the molecules known to us are built up 
 each of some moderate number of atoms, these atoms 
 being all of them exactly alike, then we may attribute 
 the limited number of molecular species . to the limited 
 number of ways in which the primitive atoms may be 
 combined so as to. form a permanent system. 
 
 But though this hypothesis gets rid of the difficulty 
 of accounting for the independent origin of different 
 species of molecules, it merely transfers the difficulty 
 from the known molecules to the primitive atoms. How 
 did the atoms come to be all aUke in those properties 
 which are in themselves capable of assuming any value ? 
 
 If we adopt the theory of Boscovich, and assert that 
 the primitive atom is a mere centre of force, having a 
 certain definite mass, we may get over the difficulty about 
 the equality of the mass of all atoms by laying it down 
 
302 PROPERTIES OF MATTER. 
 
 as a doctrine which cannot be disproved by experiment, 
 that mass is not a quantity capable of continuous increase 
 or diminution, but that it is in its own nature discon- 
 tinuous, like number, the atom being the unit, and 
 all masses being multiples of that unit. We have no 
 evidence that it is possible for the ratio of two masses 
 to be an incommensurable quantity, for the incommen- 
 surable quantities in geometry are supposed to be traced 
 out in a continuous medium. If matter is atomic, and 
 therefore discontinuous, it is unfitted for the construc- 
 tion of perfect geometrical models, but in other respects 
 it may fulfil its functions. 
 
 But even if we adopt a theory which makes the 
 equality of the mass of different atoms a result depend- 
 ing on the nature of mass rather than on any quantita- 
 tive adjustment, the correspondence of- the periods of 
 vibration of actual molecules is a fact of a different order. 
 
 We know that radiations exist having periods of vibra- 
 tion of every value between those corresponding to the 
 limits of the visible spectrum, and probably far beyond 
 these limits on both sides. The most powerful spectro- 
 scope can detect no gap or discontinuity in the spectrum 
 of the light emitted by incandescent lime. 
 
 The period of vibration of a luminous particle is there- 
 fore a quantity which in itself is capable of assuming any 
 one of a series of values, which, if not mathematically 
 continuous, is such that consecutive observed values differ 
 from each other by less than the ten-thousandth part of 
 either. There is, therefore, nothing in the nature of time 
 itself to prevent the period of vibration of a molecule from 
 assuming any one of many thousand different observable 
 values. That which determines the period of any par- 
 
APPENDIX. 303 
 
 ticular kind of vibration is the relation which subsists 
 between the corresponding type of displacement and the 
 force of restitution thereby called into play, a relation 
 involving constants of space and time as well as of mass. 
 
 It is the equality of these space- and time-constants 
 for all molecules of the same kind which we have next 
 to consider. We have seen that the very different cir- 
 cumstances in which different molecules of the same 
 kind have been placed have not, even in the course of 
 many ages, produced any appreciable difference in the 
 values of these constants. If, then, the various processes 
 of nature to which these molecules have been subjected 
 since the world began have not been able in all that 
 time to produce any appreciable difference between the 
 constants of one molecule and those of another, we are 
 forced to conclude that it is not to the operation of any 
 of these processes that the uniformity of the constants 
 is due. 
 
 The formation of the molecule is therefore an event 
 not belonging to that order of nature under which we 
 live. It is an operation of a kind which- is not, so far 
 as we are aware, going on on earth or in the sun or the 
 stars, either now or since these bodies began to be 
 formed. It must be referred to the epoch, not of the 
 formation of the earth or of the solar system, but of the 
 establishment of the existing order of nature, and tUl 
 not only these worlds and systems, but the very order 
 of nature itself is dissolved, we have no reason to expect 
 the occurrence of any operation of a similar kind. 
 
 In the present state of science, therefore, we have 
 strong reasons for believing that in a molecule, or if not 
 in a molecule, in one of its component atoms, we have 
 
304 PROPERTIES OF MATTER.' 
 
 something which has existed either from eternity or at 
 least from times anterior to the existing order of nature. 
 But besides this atom, there are immense numbers of 
 other atoms of the same kind, and the constants of each 
 of these atoms are incapable of adjustment by any pro- 
 cess now in action. Each is physically independent of 
 all the others. 
 
 Whether or not the conception of a multitude of 
 beings existing from all eternity is in itself self-contra- 
 dictory, the conception becomes palpably absurd when, 
 we attribute a relation of quantitative equahty to all 
 these beings. We are then forced to look beyond them 
 to some common cause or common origin to explain 
 why this singular relation of equality exists, rather than 
 any one of the infinite number of possible relations of 
 inequality. 
 
 Science is incompetent to reason upon the creation of 
 matter itself out of nothing. We have reached the 
 utmost limit of our thinking faculties when we have 
 admitted that, because matter cannot be eternal and 
 self-existent, it must have been created. It is only when 
 we contemplate not matter in itself, but the form in 
 which it actually exists, that our mind finds something 
 on which it can lay hold. 
 
 That matter, as such, should have certain fundamental 
 properties, that it should have a continuous existence in 
 space and time, that all action should be between two 
 portions of matter, and so on, are truths which may, for 
 aught we know, be of the kind which metaphysicians 
 call necessary. We may use our knowledge of such 
 truths for purposes of deduction, but we have no data 
 for speculating on their origin. 
 
APPENDIX. 305 
 
 But the equality of the constants of the molecules is 
 a fact of a very diflferent order. It arises from a parti- 
 cular distribution of matter, a collocation, to use the ex- 
 pression of Dr. Chalmers, of things which we have no 
 difficulty in imagining to have been arranged otherwise. 
 But many of the ordinary instances of collocation are 
 adjustments of constants, which are not only arbitrary 
 in their own nature, but in which variations actually 
 occur ; and when it is pointed out that these adjustments 
 are beneficial to living beings, and are therefore instances 
 of benevolent design, it is rephed that those variations 
 which are not conducive to the growth and multiplica- 
 tion of living beings tend to their destruction, and to 
 the removal thereby of the evidence of any adjustment 
 not beneficial. 
 
 The constitution of an atom, however, is such as to 
 render it, so far as we can judge, independent of all the 
 dangers arising from the struggle for existence. Plaus- 
 ible reasons may, no doubt, be assigned for believing 
 that if the constants had varied from atom to atom 
 through any sensible range, the bodies formed by aggre- 
 gates of such atoms would not have been so weU fitted 
 for the construction of the world as the bodies which 
 actually exist. But as we have no experience of bodies 
 formed of such variable atoms this must remain a bare 
 conjecture. 
 
 Atoms have been compared by Sir J. Herschel to 
 manufactured articles, on account of their imiformity. 
 The uniformity of manufactured articles may be traced 
 to very different motives on the part of the manufacturer. 
 In certain cases it is foimd to be less expensive as regards 
 trouble, as well as cost, to make a great many objects 
 
 X 
 
306 . PEOPERTIES OF MATTER. 
 
 exactly alike than to adapt each to its special require- 
 ments. Thus, shoes for soldiers are made in large 
 numbers without any designed adaptation to the feet of 
 particular men. In another class of cases the uniformity 
 is intentional, and is designed to make the manufactured 
 article more valuable. Thus, Whitworth's bolts are 
 made in a certain number of sizes, so that if one bolt is 
 lost, another may be got at once, and accurately fitted 
 to its place. The identity of the arrangement of the 
 words in the different copies of a document or book is a 
 matter of great practical importance, and it is more 
 perfectly secured by the process of printing than by that 
 of manuscript copying. 
 
 In a third class not a part only but the whole of the 
 value of the object arises from its exact conformity to a 
 given standard. Weights and measures belong to this 
 class, and the existence of many well-adjusted material 
 standards of weight and measure in any country furnishes 
 evidence of the existence of a system of law regulating 
 the transactions of the inhabitants, and enjoining in aU 
 professed measures a conformity to the national standard. 
 
 There are thus three kinds of usefulness in manufac- 
 tured articles — cheapness, serviceableness, and quantita- 
 tive accuracy. Which of these was present to the mind 
 of Sir J. Herschel we cannot now positively affirm, but 
 it was at least as likely to have been the last as the first, 
 though it seems more probable that he meant to assert 
 that a number of exactly similar things cannot be each 
 of them eternal and self -existent, and must therefore 
 have been made, and that he used the phrase "manu- 
 factured article " to suggest the idea of their being made 
 in great numbers. 
 
•APPENDIX. 307 
 
 APPENDIX III. (§ 97). 
 
 TuM vero ex eo inventionis ingressu duas dicitur 
 fecisse massas aequo pondere, quo etiam fuerat corona, 
 unam ex auro alteram ex argento. Cum ita fecisset, vas 
 amplum ad summa labra implevit aqua ; in quo demisit 
 argenteam massam : cuius quanta magnitudo in vase 
 depressa est, tantum aquse effluxit. Ita exempta massa, 
 quanto minus factum fuerat, refudit sextario mensus, ut 
 eodem modo, quo prius fuerat, ad labra eequaretur. Ita ex 
 eo invenit, quantum [ad certwm\ pondus argenti ad certam 
 aquse mensuram responderet. Cum id expertus esset, 
 turn auream massam similiter pleno vase demisit, et ea 
 exempta, eadem ratione mensura addita invenit ex aqua 
 non tantum defluxisse sed \tantwrn\ minus, quanto minus 
 magno corpore eodem pondere auri massa esset quam 
 argenti. Postea vero repleto vase in eadem aqua ipsa 
 corona demissa, invenit plus aquae defluxisse in coronam, 
 quam in auream eodem pondere massam : et ita ex eo, 
 quod plus defluxerat aquse in corona quam in massa, 
 ratiocinatus deprehendit argenti in auro mixtionem et 
 manifestum furtum redemptoris. — ^Vitruvius, De Archir 
 tedurd, Lib. IX., Prcefatio. 
 
 APPENDIX IV. (§ 191). 
 
 Note on a Singular Passage in the "Pkincipia." 
 By Professor Tait.^ 
 
 In the remarkable Scholium, appended to his chapter 
 
 on the Laws of Motion, where Newton is showing what 
 
 1 Froc. E.S.I!., January 19, 1885. 
 
308 PROPERTIES OF MATTER. 
 
 Wren, Wallis, and Huygens had done in connection with 
 the impact of bodies, he uses the following very peculiar 
 language : — 
 
 "Sed et Veritas comprobata est a D. Wrermo coram 
 Begid Societate per experimentum Pendulorum, quod etiam 
 Clarissimus Ma/riottus Libre integro exponere mox dig- 
 natus est.'' 
 
 The last clause of this sentence, which I had occasion 
 to consult a few days ago, appeared to me to be so sar- 
 castic, and so unlike in tone to all the context, that I 
 was anxious to discover its full intention. 
 
 Not one of the Commentators, to whose works I had 
 access, makes any remark- on the passage. The Transla- 
 tors differ widely. 
 
 Thus Motte softens the clause down into the trivial 
 remark "which Mr. Mariotte soon after thought fit to 
 explain in a treatise entirely on that subject." 
 
 The Marquise du Chastellet (1756) renders it thus : — 
 ". . . . mais ce fut Wrerm, qui les confirma par des' 
 Experiences faites avec des Pendules devant la Soci6t6 
 Eoyale : lesquelles le c61febre Mariotte a rapport^es depuis 
 dans un Trait6 qu'U a compos6 expr^s sur cette matifere." 
 
 Thorp's translation (1777) runs : — 
 " which the very eminent Mr. Mariotte soon after thought 
 fit to explain in a treatise entirely upon that subject." 
 
 Finally, Wolfers (1872) renders it thus :— 
 "der zweite zeigte der Societat die Eichtigkeit seiner 
 Erfindung an einem Pendelversuche, den der beriihmte 
 Mariotte in seinem eigenen Werke aus einander zu setzen, 
 fur wiirdig eraohtete." 
 
 Not one of these seems to have remarked anything 
 singular in the language employed. But when we con- 
 
APPENDIX. 809 
 
 suit the "entire book" in which Mariotte is said by 
 Newton to have " expounded " the result of Wallis, and 
 which is entitled Trait4 de la Percussion m Choc des Corps, 
 we find that the name of Wallis is not once mentioned 
 in its pages ! From the* beginning to the end there is 
 nothing calculated even to hint to the reader that the 
 ' treatise is not wholly original. 
 
 This gives a clue to the reason for Newton's sarcastic 
 language ; whose intensity is heightened .by the contrast 
 between the Clarissimus which is carefully prefixed to the 
 name of Mariotte, and the simple B. prefixed, not only 
 to the names of Enghshmen like Wren and Wallis, but 
 even to that of a specially distinguished foreigner like 
 Huygens. 
 
 Newton must, of course, like all the scientific men of 
 the time (Mariotte included), have been fully cognisant 
 of Boyle's celebrated controversy with Linus, which led 
 to the publication, in 1662, of the Defence of the Doctrine 
 touching the Spring and Weight of the Air. In that tract, 
 Part II. Chap, v., the result called in Britain Boyle's Law 
 is established (by a very remarkable series of experi- 
 ments) for pressures less than, as well as for pressures 
 greater than, an atinosphere ; and it is established by 
 means of the very form of apparatus stiU employed for 
 the purpose in lecture demonstrations. Boyle, at least, 
 claimed originality, for he says in connection with the 
 difficulties met with in the breaking of his glass tube, — 
 
 " .... an accurate Experiment of this nature would 
 be of great importance to the Doctrine of the Spring of 
 the Air, and has not been made (that I know) by any 
 man. . . ." 
 
 In Mariotte's Discows de la Natwe de I' Air, published 
 
310 PROPERTIES OF MATTER. 
 
 FOURTEEN years later than this work of Boyle, we find 
 no mention whatever of Boyle, though the identical form 
 of apparatus used by Boyle is described. The whole 
 work proceeds, as does that on Percussion, with a calm 
 ignoration of the labours of the majority of contemporary 
 philosophers. 
 
 This also must, of course, have been perfectly well 
 known to Newton : — and we can now see full reason for 
 the markedly peculiar language which he permits himself 
 to employ with reference to Mariotte. 
 
 What was thought of this matter by a very distin- 
 guished fereign contemporaiy, appears from the treatise 
 of James Bernoulli, Be Gravitate ^theris, Amsterdam, 
 1683, p. 92. 
 
 " Veritas utriusque hujus regulse manifesta fit duobus 
 curiosis experimentis, ab lUustr. Dn Boylio hanc in rem 
 factis, quae videsis in Tradatu ejus contrh Linum, Gap. V., 
 cui duas Auctor subjunxit Tabulas pro diversis Conden- 
 sationis et Earefactionis gradibus." 
 
 In order to satisfy myself that Newton's language, 
 taken in its obvious meaning, really has the intention 
 which I could not avoid attaching to it, I requested my 
 colleague Professor Butcher to state the impression which 
 it produced on him. I copied for him the passage above 
 quoted, putting A. for the word Wrenno, and B. for 
 Mariottus ; and I expressly avoided stating who was the 
 writer. Here is his reply : — 
 
 " I imagine the point of the passage to be something 
 of this kind (spealdng without farther context or acquaint- 
 ance with the Latinity of the learned author) : — 
 
 " A established the truth by means of a (simple) ex- 
 periment, before the Eoyal Society ; later, B thought it 
 
APPENDIX. 311 
 
 worth his while to write a whole book to prove the same 
 point. 
 
 " I should take the tone to be highly sarcastic at B's 
 expense. It seems to suggest that B was not only clumsy, 
 but dishonest. The latter inference is not certain, but 
 at any rate we have a hint that B took no notice of A's 
 discovery, and spent a deal of useless labour." 
 
 This conclusion, it will be seen, agrees exactly with 
 the complete ignoration of Wren by Mariotte. 
 
 When I afterwards referred Professor Butcher to the 
 whole context, in my copy of the first edition of the 
 Principia, and asked him whether the use of Glwrissimus 
 was sarcastic or not, he wrote — 
 
 " I certainly think so. Indeed, even apart from the 
 context, I thought the Olari-ssimm was ironical, but there 
 can be no doubt of it when it corresponds to D. Wren.'' 
 
 In explanation of this I must mention that, when I 
 first sent the passage to Professor Butcher, I had copied it 
 from Horsley's sumptuous edition ; in which the Ds are 
 omitted, while the Clarissimus is retained. 
 
 Ahke in France and in Germany, to this day, the Law 
 in question goes by the name of Mariotte. The following 
 extracts, from two of the most recent high-class text- 
 books, have now a peculiar interest. I have put a word 
 or two of each in italics. These should be compared 
 with the dates given. 
 
 "Diese Frage ist schon friihzeitig untersucht und 
 zwar fast gleicheeitig von dem franzosischen Physiker 
 Mariotte (1679) und den englisohen Physiker Boyle 
 (1662)." Wiillner, Lehrbuch der Experimentdphysik, 
 1882, § 98. 
 
 "La loi qui r6git la compressibility des gaz k tern- 
 
312 PEOPERTIES OF MATTER. 
 
 p6rature constante a 6t6 trouv6e presque smudtanMient par 
 Boyle (1662) en Angleterre et par Mariotte (1676) en 
 France ; toutefois, si Boyle a public le premier ses ex- 
 periences, il ne sut pas en tirer r6nonc6 clair que donna 
 le physician frangais. C'est done avec quelque raison 
 que le nom de loi de Mariotte a pass6 dans I'usage." 
 VioUe, Cows de Physique, 1884, § 283. 
 
 On this I need make no remark further than quoting 
 one sentence from Boyle, where he compares the actual 
 pressure, employed in producing a certain compression 
 in air, with "what the pressure should be according to 
 the Hypothesis, that supposes the pressures and expansions 
 to be in reciprocal proportion.'' M. VioUe has probably 
 been misled by the archaic use of "expansion" for volume. 
 
 It must be said, in justice to Mariotte, that he does 
 not appear to have claimed the discovery of any new 
 facts in connection either with collision or with the 
 effect of pressure on air. He rather appears to write 
 with the conscious infallibility of a man for whom nature 
 has no secrets. And he transcribes, or adapts, into his 
 writings (without any attempt at acknowledgment) what- 
 ever suits him in those of other people. He seems to 
 have been a splendidly successful and very early example 
 of the highest class of what we now call the Paper- 
 Scientists. Witness the following extracts from Boyle, 
 with a parallel citation from Mariotte of fourteen years 
 later date at least. The comparison of the sponges had 
 struck me so much, in Mariotte's work, that I was 
 induced to search for it in Boyle, where I felt convinced 
 that I should find it. 
 
 " This Notion may perhaps be somewhat further ex- 
 plain'd, by conceiveing the Air near the Earth to be such 
 
APPENDIX. 313 
 
 a heap of little Bodies, lying one upon another, as may- 
 be resembled to a Fleece of Wooli. For this (to omit 
 other likenesses betwixt them) consists of many slender 
 and flexible Hairs; each of 'which, may indeed, like a 
 little Spring, be easily bent or rouled up ; but wUl also, 
 like a Spring, be still endeavouring to stretch itself out 
 again. For though both these Haires, and the -illreal 
 Corpuscles to which we liken them, do easily yield to 
 externall pressures ; yet each of them (by virtue of its 
 structure) is endoVd with a Power or Principle of Selfe- 
 Dilatation ; by virtue whereof, though the hairs may by 
 a Mans hand be bent and crouded closer together, and 
 into a narrower room then suits best with the Nature of 
 the Body, yet, whils't the compression lasts, there is 
 in the fleece they composeth an endeavour outwards, 
 whereby it continually thrusts against the hand that 
 opposeth its Expansion. And upon the removal! of the 
 external pressure, by opening the hand more or less, the 
 compressed Wooll doth, as it were, spontaneously expand 
 or display it self towards the recovery of its former more 
 loose and free condition till the Fleece hath either regain'd 
 its former Dimensions, or at least, approached them as 
 neare as the compressing hand, (perchance not quite 
 open'd) will permit. The power of Selfe-Dilatation is 
 somewhat more conspicuous in a dry Spunge compress'd, 
 then in a Fleece of Wooll. But yet we rather chose to 
 employ the latter, on this occasion, because it is not like 
 a Spunge, an intire Body ; but a number of slender and 
 flexible Bodies, loosely complicated, as the Air itself 
 seems to be.'' 
 
 And, a few pages later, he adds : — 
 
 " .... a Column of Air, of many miles in height. 
 
314 PEOPEETIES OF MATTER. 
 
 leaning upon some springy Corpuscles of Air here below, 
 may have weight enough to bend their little springs, and 
 keep them bent : As, (to resume our former comparison,) 
 if there were fleeces of WooU pil'd up to a mountainous 
 height, upon one another, the hairs that compose the 
 lowermost Locks which support the rest, would, by the 
 weight of all the "Wool above them, be as well strongly 
 compress'd as if a Man should squeeze them together in 
 his hands, or imploy any such other moderate force to 
 compress them. So that we need not wonder, that upon 
 the taking off the incumbent Air from any parcel of the 
 Atmosphere here below, the Corpuscles, whereof that 
 undermost Air consists, should display themselves, and 
 take up more room than before." 
 
 Mariotte (p. 151). "On pent comprendre k pen prfes 
 cette difference de condensation de I'Air, par I'exemple 
 de plusieurs 6ponges qu'on auroit entass6es les unes sur 
 les autres. Car il est Evident, que celles qui seroient 
 tout au haut, auroient leur entendue naturelle : que 
 celles qui seroient immddiatement au dessous, seroient 
 un pen moins dilatdes; et que celles qui seroient au 
 dessous de toutes les autres, seroient tr6s-serr6es et con- 
 dens6es. II est encore manifeste, que si on 6toit toutes 
 celles du dessus, celles du dessous reprendroient leur 
 6tendue naturelle par la vertu de ressort qu'elles ont, et 
 que si on en 6toit seulement une partie, elles ne repren- 
 droient qu'une partie de leur dilatation." 
 
 Those curious in such antiquarian details wiU probably 
 find a rich reward by making a careful comparison of 
 these two works ; and in tracing the connection between 
 the Liber integer, and its fons et origo, the paper of Sir 
 Christopher Wren. 
 
APPENDIX. 315 
 
 Condorcet, in his Eloge de Mwriotte, says : — " Les lois 
 du choc des corps avaient 6t6 trouv6es par une ni6ta- 
 physique et par une application d'analyse, nouvelles 
 I'une et I'autre, et si subtiles, que les demonstrations de 
 ces lois ne pouvaient satisfaire que les grands math6- 
 maticiens. Mariotte chercha k les rendre, pour ainsi 
 dire, populaires, en les appuyant sur des experiences, 
 etc." i.e. precisely what Wren had thoroughly done before 
 him. 
 
 " Le discours de Mariotte sur la nature de I'air ren- 
 ferme encore une suite d'experiences int6ressantes, et 
 qui etaient absolument neuves." This, as we have seen, 
 is entirely incorrect. 
 
 But Gondorcet shows an easy way out of all questions 
 of this kind, however delicate, in the words : — " On ne 
 doit aux morts que ce qui peut gtre utile aux vivants, la 
 y6rit6 et la justice. Cependant, lorsqu'il reste encore 
 des amis et des enfants que la v6rit6 peut affliger, les 
 6gards deviennent un devoir ; mais au bout d'un sifecle, 
 la vanit6 peut seule Itre bless6e de la justice rendue aux 
 morts." 
 
 Thus it is seen that even the turn of one of Newton's 
 phrases serves, when rightly viewed, to dissipate a wide- 
 spread delusion ; — and that while Boyle, though perhaps 
 he can scarcely be said to have been " born great," cer- 
 tainly "achieved greatness"; the assumed parent of La 
 loi de Mariotte (otherwise Mariottesches Gesetz) has as 
 certainly had "greatness thrust upon" him. 
 
INDEX. 
 
 Note. — Thefigwres lelow refer to sections, not to pages. 
 
 AOOELEEATION, 70, 120. 
 
 Adiabatics, 175. 
 
 Altitude, 76. 
 
 Angle of Contact, 281. 
 
 Annealing, 324, 333. 
 
 Atmolysis, 312. 
 
 Atmometer, 286. 
 
 Atmosphere, 196. 
 
 Atom, 23 ; also Appendix II. 
 
 vortex, 18, 27. 
 Axioms, 1.' 
 Azimntli, 76. 
 
 Balance, 35, 127. 
 
 spring, 118, 165. 
 Billion, 104. 
 Boiler, strength of, 278. 
 Bookworm, 89. 
 
 Capillaeitt, 279-295. 
 
 Casting, 92. 
 
 Catalysis, 297. 
 
 Cavendish experiment, 153-155. 
 
 Centre of inertia, 129. 
 
 of gravity, 142. 
 Centrifugal Force, 117, 151. 
 Cleavage, 219, 324, 332. 
 Clouds, suspension of, 40. 
 Cohesion, 63, 257-297'. 
 Col, 84. 
 
 Collision, 254-256. 
 Colloids, 100, 301, 324. 
 Colour, 42. 
 
 Compressibility, 88, 169. 
 
 measure of, 176. 
 
 of gases and vapours, 186- 
 207. 
 
 of liquids, 208-218. 
 
 of solids, 219, etc. 
 Condensation, surface, 296. 
 Conservation of matter, 5. 
 
 of energy, 7. 
 Constraint, 77. 
 Contact, angle of, 281. 
 Contour lines, 80-88. 
 Coronie, 103. 
 
 Corpuscules, ultra-mundane, 163. 
 Couple, 133. 
 Critical temperature, 194, 198, 
 
 206, 323. 
 Crystallisation, 324. 
 Crystalloids, 301. 
 Crystals of cubic system, 326. 
 Curvature, 89. 
 Curve, 68. 
 
 equation of, 77. 
 
 elastic, 237. 
 
 Deusitt, 36. 
 
 mean, 126. 
 
 mean, of earth, 153-158. 
 
 surface-, 141. 
 Diagram of Energy, 88. 
 Dialysis, 309. 
 Diftasion, 299-309, 313. 
 Dilution, 106. 
 
818 
 
 PROPERTIES OF MATTER. 
 
 Distance-Action, 10, 
 Diving-bell, 94. 
 DiTisibility, 101-107. 
 Ductility, 35. 
 Dyne, 124. 
 
 Eaeth, rotation of, 112. 
 
 mean density, 163-158. 
 
 figure of, 165. 
 
 how kept together, 167. 
 Effusion, 310. 
 Elastic curve, 237. 
 
 recovery, 252-253. 
 Elasticity, 41, 168, 221. 
 
 dormant, 245. 
 
 fatigue of, 168, 251. 
 
 limits of, 219, 233, 243. 
 Elatdromfetre, 233. 
 Energy, 4, 6. 
 
 conservation of, 7, 130. 
 
 diagram of, 88. 
 
 kinetic, 14, 132. 
 
 potential, 13, 14, 263. 
 
 transformation of, 9. 
 Equilibrium, stable, 263. 
 Equipotential, 87. 
 Expansibility, 88. 
 Extension, 21, 58. 
 
 Fatigue of Elasticity, 158, 251. 
 Fluxions, 72. 
 Force, 11. 
 
 centrifugal, 117. 
 
 parallelogram of, 122. 
 
 measurement of, 123. 
 
 unit of, 124. 
 
 activity of, 130. 
 
 molecular, 26, 258. 
 
 resolution of, 76. 
 Freedom, 77. 
 Friction, 38. 
 
 Gases, compressibility of, 186. 
 
 liquefaction of, 207. 
 
 kinetic theory of, 55, 207. 
 Glen Eoy, Parallel Roads of, 80. 
 Gold-beating, 47. 
 
 • -leaf, thickness of, 102. 
 Gradient, 82. 
 
 Grained structure, 26, 107. 
 Gravitation, 138-164. 
 
 universality of, 139-143, 152. 
 
 energy of two masses, 159. 
 
 cause of, 160-164. 
 Gravity, 138. 
 
 centre of, 142. 
 
 measure of, in different lati- 
 tudes, 165. 
 
 specific, 36, 166. 
 
 Hardness, Scale of, 56. 
 Harmonic Motion, 71. 
 Harton experiment, 157. 
 Head, 6, 14. 
 Heat, 62. 
 
 Hemihedry, 326, 331. 
 Hodograph, 70. 
 
 of planet's orbit, 146. 
 Homogeneous, 219. 
 Hydrophane, 99. 
 Hydrostatic Laws, 189. 
 
 pressure, 176. 
 
 IMIT, 85. 
 Impact, 254-6. 
 Impenetrability, 21,191-97. 
 Impurities, purposely.added, 219. 
 Inertia, 9, 108. 
 
 moment of, 119, 132. 
 
 centre of, 128, 129. 
 Isobars, 86. 
 Isothermals, 86, 175. 
 Isotropic, 175, 219. 
 
 Kink, 230, 237. 
 
 Kinetic theory, 33, 55, 107, 322. 
 
 Latitode, 67, 156. 
 Law, Boyle's, 191. 
 
 Hooke's, 221. 
 Laws, Kepler's, 144. 
 
 Newton's, 108, 120 128, 
 Lemniscate, 87. 
 Levelling, 80. 
 Levigation, 101. 
 Lightning, 33. 
 
 Limits of elasticity, 233, 243. 
 Liquefaction of gases, 207. 
 
INDEX. 
 
 319 
 
 Liquids, compressibility of, 208- 
 
 218. 
 Longitude, 67. 
 
 Machine, 48. 
 Malleability, 45. 
 Manometer, 193, 233. 
 Mass, 34. 
 Mass, measurement of, 123. 
 
 unit of, 123. 
 Matter, 2, 4. 
 
 conservation of, 5. 
 
 definitions of, 18. Also 
 Appendix I. 
 Million, 104. 
 Mobility, 120. 
 Modulus, 185. 
 
 Young's, 224. 
 Molecular forces, 26, 258. 
 
 limit of, 293. 
 Moment of Inertia, 119, 132. 
 
 of momentum, 132. 
 
 of couple, 133. 
 Moon, 33, 35, 110, 149. 
 ■Motion, First Law of, 108-119. 
 
 Second Law of, 120-127. 
 
 Third Law of, 128-137. 
 
 proof of Laws of, 110. 
 
 the Perpetual, 139. 
 Musk, 104. 
 
 NoN-iaoTBOPlc, 220. 
 Nutation, 119, 142. 
 
 Objeotivitt, 2, 5, 6, 7, 15. 
 Opacity, 51. 
 Origin, 62. 
 Osmose, 306-308. 
 
 Paktiole, of matter, 29. 
 of gas, 33, 55, 107. 
 Pendulum, 115. 
 Perpetual Motion, the, 139. 
 Piezometer, 211. 
 Plasticity, 35, 50. 
 Porosity, 98-100. 
 Potential, 87. 
 Pound, 123. 
 
 Precession, 119, 142. 
 Pressure, 65, 88, 189. 
 Properties, specific, 36. 
 
 Quadrillion, 104. 
 
 Eadius-Veotok, 65, 69. 
 Radius of gyration, 133. 
 Eegistering instruments, 68. 
 Restitution, coeificient of, 255. 
 Rigidity, 169, 178, 229. 
 
 flexural, 235. 
 Ripple, 33, 295. 
 Rotation, 116, 119. 
 
 absolute, 131. 
 
 SoHEHALLiEN, experiment, 156. 
 Senses, 1. 
 
 muscular, 12. 
 Shear, 37. 
 
 Shell, attraction of spherical 140- 
 141. 
 
 Soap-bubble, 270-277. 
 Sodium vapour, 105. 
 Solution of gas, 298. 
 Sound, 186. 
 Soundings, 79. 
 Space, 57. 
 
 tridimensional character of, 
 58. 
 
 of no dimensions, 60. 
 
 of one dimension, 61-65. 
 
 of two dimensions, 67-74. 
 
 of three dimensions, 75-88 
 " of four dimensions, 89. 
 Specific Properties, 36 
 Speed, 64. 
 
 Spiral, logarithmic, 73. 
 Stars, 33. 
 Strain, 169. 
 
 homogeneous, 170-175. 
 
 of spheres and cylinders by 
 pressure, 183-184. 
 Stream-line, 83. 
 Stress, 128, 137, 169, 177. 
 Sucker, 287. 
 Sumomit, 86. 
 
320 
 
 PROPERTIES OF MATTER. 
 
 Surface, equation o^ 78. 
 -condensation, 296. 
 -density, 141. 
 -tension, 261-278, 290. 
 
 Tabasheeb, 99. 
 Tears of strong wine, 269. 
 Temper, 219. 
 Temperature, 52. 
 
 absolute, 55. 
 
 critical, 194, 198, 206, 323. 
 Tenacity, 226. 
 Tension, surface-, 261-278. 
 Time, 57, 66. 
 Torsion, 228, 240. 
 Transformation of energy, 9. 
 Translation, absolute, 131. 
 Translucency, 51. 
 Transparency, 51. 
 Transpiration, 311, 314. 
 Tridimensional character of space, 
 58. 
 
 Vapoub-Peesstire on curved sur- 
 face, 291. 
 Vector, Radius, 65, 69. 
 
 Angle, 65, 69. 
 Velocity, 69. 
 
 angular, 65, 72. 
 
 components of, 69, 76. 
 Viscidity, 294. 
 Viscosity of liquids, 37, 315. 
 
 of solids, 249. 
 
 oi gases, 320. 
 Volume, 58. 
 
 measure of, 93. 
 Vortex atom, 18, 27. 
 
 Watercourse, 85. 
 Watershed, 85. 
 Waves, 185, 295. 
 Weight, 14, 34. 
 Wire-drawing, 49. 
 Work, 13, 88. 
 
 rate of doing, 130. 
 
 THE END. 
 
 Printed 3)/ R. & R. Clark, Edinhirgk. 
 
By THE Same. 
 In crown 8vo, cloth, price 6s. 
 
 LIGHT 
 
 CONTENTS.— Introductory and Historical — Preliminary 
 Statements — Sources of Ligtt — Theories of Propagation of 
 LigM — Eectilinear Propagation of Light — Speed of Light — 
 Behaviour of Light at the Common Surface of two Homo- 
 geneous Media — ^Reflection of Light — Refraction of Light — 
 Caustics — Refraction in a Non- Homogeneous Medium — 
 Absorption and Fluorescence — ^Preliminary Remarks on the 
 Undulatory Theory — Interference — Double Refraction and 
 Polarisation — Radiation and Spectrum Analysis.: — Appendix 
 — Index. 
 
 " For masterly clearness and cautious aoouraoy of exposition, Professor 
 Tait is not surpassed by any scientific writer of our time. ... It is,, 
 indeed, a complete exposition of the subject in its varied aspects, and 
 ought to satisfy all but the most advanced students." — Saturday Review. 
 
 "Professor Tail's new book on 'Light' is one of the most suggestive 
 handbooks that the student of physics could possess. It is designed for 
 private reading, because the curriculum at Edinburgh does not admit of 
 an adequate exposition of every department of physios in eUoh session. 
 The sixteen chapters are further divided into 328 paragraphs, so that the 
 subject is treated with more exactness than is usual in a non-mathematical 
 work suitable for general reading. The term light is used in an objective 
 sense, so as to exclude the subjective field of physiological optics ; and, for 
 obvious reasons, the construction of telescopes, microscopes, and spectro- 
 scopes is omitted. The author begins by describing vision in its charac- 
 teristics of definition, stereoscopic effect, colour, and the duration of 
 sensation of light, the sources of light ; and then succeed certain of 
 Newton's queries extracted from the end of his ' Optics, ' to give an idea 
 of some ways in whicb modern discoveries were anticipated. The theories 
 of propagation of light known as the corpuscular theory and the undula- 
 tory theory are explained, with a consideration of rectilinear propagation 
 of light and its speed. Then succeed discussions of reflection and refrac- 
 tion, absorption and fluorescence, the undulatory theory, interference, 
 double refraction, polarisation, radiation and spectrum analysis. The 
 hook is beautifully printed ; it contains a few diagrammatic illustrations." 
 — Westminster Seview.