Electromagnetic Theory, Vol. 2 (1899) — part 2 of 31

169 The Rotational Analogy, with Electric Energy Kinetic, ex-

tended to a Conducting Dielectric by means of Translational Friction ... •.. ••> ... ••• ... ... 253

160 [Sept. 2, 1892.] Symmetrical Linear Operators, direct and in-

verse, referred to the Principal Axes ... ... ... ... 256

161 Geometrical Illustrations. The Sphere and Ellipsoid. Inverse

Perpendiculars and Maccullagh's Theorem .„ ... ... 269

162 Internal Structure of Linear Operators. Manipulation of

several when Principal Axes are Parallel 262

OONTBNTS.

ti£CTION. I'AGK.

163 iSept, 16, 189£.] Th«ory of DisplMement in an Eolotropio

Dieleofcrio. The Solution fnr ft Point-Source... 264

164 Theory of the BeUtive Motion of Eleotrificfttion and the

Medium. The Solution for a Point-Source in steedy Recti*

linear Motion. The Equilibrium Surfaces in general ... 269 166 [^"ycpt. 30, 189 J.] Theory of the Kelative Motion of Mag-

netification and the Medium 274

166 Theory of the Relative Motion of Magnetisation and the

Medium. Inereaaed Induction as well as Eolotroinc Die- ^vuImuico ••• ••• 277

167 [Oct. n, 189J.] Theory of the RektiTe Motion of Electric

Currents and the Medium 281

168 The General Linear Operator 285

169 The Dyadical Structure of Linear Operatorn ... ... ... 285

170 Hamilton's Theorem 287

171 [Nov. 18, 189£.] Hamilton's Cubic and the Invariants con-

ottTQodl ••• ••■

172 The Inversion of Linear Operators 293

173 Vector Product of a Vector and a Dyadic. The Differentiation

of Linear Operators ... ... ... ... ... ... 295

174 [Dec. 9, 189 J.] Summary of Method of Vector Analysi.s ... 297

175 Unsuitabiiity of Quaternions for Physical Needs. Axiom : —

Onoe ft Vector, always a Vector ... 301

CHAPTER IV.

THEORY OF PLANE ELECTROMAGNETIC WAVES.

(Pages 306 to 466.) sionov. PAos.

176 [Dee, SO, ISSISJ] Action at a Distance vertut Intermediate

Agency. Contrast of New with Old Views about Electricity 306

177 General Notions about Electromagnetic Waves. Qeneration of

Spherical Waves and Steady States ... ... ... ... 310

178 [Jan. 6, 1893.] Intermittent Source producing Steady States

and Electromagnetic Sheets. A Train of S.H. Waves ... 514

179 Self-contained Forced Electromagnetic Vilmtions. Contrast

with Static Problem 316

180 Relations between E and H in a Pure Wave. Effect of Self-

induction. Fatuity of Mr. Preece's " KR law " 320

181 [Jan. ii7y 1893.] Wave-Fronta ; their Initiation and Progress 321

182 Effect of a Non-Conducting Obstacle on Waves. Also of a

Heterogeneous Medium 323

183 Efitetof Eolotropy. Optical Wave-Surfiees. Electromagnetic

versus Elastic Solid Theorise 326

184 A Perfsct Conductor is a Perfect Obstructor, but does not

absorb the Energy of Electromagnetic Wsves 328

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eacnoN. PAcn.

185 [Feb. 3J^ 1893.] Conducton at Low Temperatarw 330

186 E<}uilibrinm of Radiation. The MeMi Flux of Energy ... 331

187 The Mean PreMure of Radiation ... ... ... 334

188 Emisaivity and Temperature 335

189 [Marek 10, 189S,} Intemtl Obttroctum and Suporfidal Coa-

doctioo ... ... ... ... ... ... 337

190 The Effect of a Perfect Conductor on External Dittarbanoes.

Reflection and Conduction of Waves ... ... 340

191 [March 24, 1893.'] Tlie Effect of Conducting Matter in

Diverting External Induction ... ... ... ... 344

192 Parenthetical liemarks on Inductiou, Magnetisation, Indue-

tavity and fikiaoeptibflitgr — 349

193 {AprH 7, ISBS,"] Eilbet of a Thin Plana Conducting Sheet on

a Wave. Persistence of Induction and Loss of Displaoement 355

194 The Persistence of Induction in Plane Strata, and in generaL

Also in Cores and in Linear Circuits .. ... ... ... 357

195 {April 21, 1S9.1] The Laws of Attenuation of Total Displace-

ment and Total Induction by Electric and Magnetic Conduc- tfluftott ••• ••• ••• ••• ••• 360

196 Th» Laws of Attenuation at the Front of a Wvn, due to

Kloctric and Magnetic Conductance ... ... ... ... 364

197 The Simple Propagation of Waves in a Distortionless Con-

ducting Medium ... ... ... ... ... ... 366

198 [May J, ISili.] The Transformation by Conductance of an

Elastic Wave to a Wave of Diffusion. Oeneration of Tails. Distinct EflRMits of Eleetric and Magnetic Conductance ... 369

199 Application to Waves along Straight Wires 374

200 [May 2(1, 1893.] Transformation of VariaUea from Electric

and Magnetic Force to Voltage and Gaussage 378

201 Transformation of the Circuital Equations to the Forms in-

volviuK Voltage and Gaussage ... ... ... ... 381

202 [June 0, 1893.] The Second Circuital Equation for Wires in

Terms of V and C when Penetration is Instantaneous ... 386

203 The Second Circuital Equatiim whm Penetration is Kot In-

sUntaneotts. Resistance Operators, and tii^ Definite Meaning ... ... ... ... ... ... ... ... 390

204 Simjily Periodic Waves Easily Treated in Case of Imperfect

Penetration 393

205 [July 7, 1893.] Long Waves and Short Waves. Identity of

Speed of and Guided Waves 396

206 The Guidance of Waves. Usually Two Guides. One suflScient,

though with Leas. Possibility of Guidance within a Single Tulie 399

207 Interpretation of Intermediate or Terminal Conditions in the

Exact Theory ... ... .. ... ... ... ... 401

208 [Aug. 2o, 1893.] The Spreading of Charige and Current in a

long Circoit, and their Attenuation 403

OONTSNIB.

xxi.

SECTION. PAOB.

209 The DiatortionlesH Circuit. No limiting Distance set by it

whm the AMmmlOaa fa Ignored 409 •

210 [SepL 16, 189S.} The two Eztreme Kinds of Diflboon in one

'l^ll COk'^ •■• ••• 4H

211 The Effect of varying the Fonr Line-ConrtentB m regudt Du-

tortion and Attenuation ... ... 413

212 The Beneficial Effect of Leakage in Submarine Cables... ... 417

213 [Oct. 6, 1893.] Short History of Leakage Effects on a Cable

C^iFcui^ ••• ••• •■• «•• ••■ 420

214 Kiplanetion of Anomeloiu Effacte. Artificiel Leeks 424

216 {OeL 20, 1893.] Self-induction imparts Momentum to Waves,

and that carries them on. Analogy with a Flexible Cord ... 429

216 Self-induction combined with Leaks. The Bridge System of

Mr. A. W. Heaviside, and suggested Di.stortionless Circuit . . . 433

217 [yov. 3, 1S93.] Evidence in favour of Self-induction. Con-

dition of Flnt-ClesB Telqihony. Importanoe of the Magnetic JEt^MM^Cftoott • • • *•• ••• vat 437

218 Various ways, good end bed, of inoneeing the Inductence of

Circuits 441

219 [Xov. 17, lS9o.] Effective Resi-stance and Inductance of a

Combination when regarded as a Coil, and Effective Con- ductance and Permittance when regarded as a Condenser . . . 446

220 Inductive Leeks epplied to Sttbmerine CeUee 447

221 Qenerel Theory of Trensmission of Weves along » CSrcuit with

or without Auxiliary Devices ... 449

222 Applicatimi of above Theoiy to Inductive Leekanoe 453

APPENDIX B. A Obattfational and Elbotromaoitrio Ahaloqt.

Part I. [Jidf 14, 1893,] 456

I^urt IL [Avg, 4t 1893,} 463

CORUF.CTIOSS

... xxii.

CORRECTIONS.

Page 82, equation (26). In case of eolotropy, add to the right member the term 8a, the scalar product of the torque S, per unit volume, and the spin a.

Page 186, § 125. I regret to have misrepresented Dr. C. V. Burton's notion of a moving strain-figure. He does it entirely by conservative elastic forces, and my objection does not apply.

Page 188, equation (148). On the right side, for D read D, three times.

Page 248. The sectional number is 158a.

CHAPTER 1.

INTRODUCTION.

§ 1. Preliminary Bemarh, — The main object of the series of artioles of which this is the first, is to continue the work en- titled *' Electromagnetic Induction and its Propagation," com- menced in The JSUctrieian on January 3, 1885, and continued to the 46th Section in September, 1887, when the great pressure on space and the want of readers appeared to necessitate its abrupt discontinuance. (A straggler, the 47th Section, appeared December 31, 1887.) Perhaps there were other reasons than those mentioned for the discontinuance. We do not dwell in the Palace of Truth. But, as was mentioned to me not long since, " There is a time coming when all things shall be found out." I am not so sanguine myself, believing that the well in which Truth is said to reside is really a bottomless pit.

The particular branch of the subject which I was publishing in the summer of 1887 was the propagation of electromagnetic waves along wires through the dielectric surrounding them. This is itself a large and many-sided subject. Besides a general treatment, its many-sidedness demands that special cases of interest should receive separate full development. In general, the mathematics required is more or less of the charac- ter sometimes termed transcendental. This is a grandiloquent word, suggestive of something beyond human capacity to find out ; a word to frighten timid people into believing that it is all speculation, and therefore unsoimd. I do not know where transcendentality begins. You can find it in arithmetic. But never mind the word. What is of more importance is the fact that Uie interpretation of transcendental formuleo is sometimes

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Tery laborious. Now the real object of true naturalists, in Sir W. Thomson's meaning of the word, when they employ mathematics to assist them, is not to make mathematical exer- cises (though that may be necessary), but to find out the con- nections of known phenomena, and by deductive reasoning, to obtain a knowledge of hitherto unknown phenomena. Any- thing, therefore, that aids this, possesses a value of its own wholly apart from immediate or, indeed, any application of the kind commonly termed practical. There is, however, practicality in theory as well as in practice. The very useful word " practi- cian " has lately come into use. It supplies a want, for it is evident the moment it is mentioned that a practician need not be a practical man ; and that, on the other hand, it may happen occasionally that a man who is not a practician may still be quite practicaL

§ 2. Now, I was so fortunate as to discover, during the examination of a practical telephonic problem, that in a certain case of propagation along a conducting circuit through a con- ducting dielectric, the transcendentality of the mathematics automatically vanished, by the distorting effects on an electro- magnetic wave, of the resistance of the conductor, and of the eonduotance of the dieleotric, being of opposite natures, so that they neutralised one another, and rendered the circuit non- distortional or distortionless. The mathematics was reduced, in the main, to simple algebra, and the manner of transmission of disturbances could be examined in complete detail in an elementary manner. Nor was this all. The distortionless circuit could be itself employed to enable us to understand the inner meaning of the transcendental cases of propagation, when the distortion caused by the resistance of the circuit makes the. mathematios more difficalt of interpretation. For instance, by a study of the distortionless circuit we are enabled to see not only that, but also why, self-induction is of such great import- ance in the transmission of rapidly-varying disturbances in preserving their individuality and preventing them from being attenuated to nearly nothing before getting from one end of a long circuit to the other ; and why copper wires are so success- ful in, and iron wires so prejudicial to effective, long-distance telephony. These matters were considered in Sections 40 to

INTRODUCTION. 3

»

45 (June, July, August, 1887) of the work I have referred to, and Sections 46, 47 contain further developments.

§ 3. But that this matter of the distortionless circuit has, -directly, important practical applications, is, from the purely scientific point of view, a mere accidental circumstance. Per- haps a more valuable property of the distortionless circuit is, that it is the Royal Road to electromagnetic waves in general, ■especially when the transmitting medium is a conductor as well as a dielectric. T have somewhat developed this matter in the PhU, Mag,, 1888-9. Fault has been found with these articles -that they are hard to read. They were harder, perhaps, to write. The necessity of condensation in a journal where space di so limited and so valuable, dealing with all branches of physi- cal science, is imperative. What is an investigator to do, when h» can neither find acceptance of matter in a comparatively •elementary form by journals of a partly scientific, partly tech> nical type, with many readers, nor, in a more learned form, by a f>urely scientific journal with comparatively few readers, and little space to spare? To get published at all, he must con- <ien8e greatly, and leave out all explanatory matter that he possibly can. Otherwise, he may be told his papers are more fit for publication in book form, and are therefore declined.

There is a third course, of course, viz., to keep his investiga- tions to himself. But that does not answer, in a general way, though it may do so sometimes. It is like putting away seed in a mummy case, instead of planting it, and letting it take its •chance of growing to a useful plant. There is nothing like publication and free criticism for utility. I can see only one good excuse for abstaining from publication when no obstacle presents itself. You may grow your plant yourself, nurse it -carefully in a hot-house, and send it into the world full-grown. But it cannot often occur that it is worth the trouble taken. As for the secretiveness of a Cavendish, that is utterly inex- •cusable ; it is a sin. It is possible to imagine the case of a man being silent, either from a want of confidence in himself, or from disappointment at the reception given to, and want of appreciation of, the work he gives to the world ; few men have an unbounded j)ower of persistence; but to make valuable dis-

<K)verie8, and to hoard them up as Cavendish did, without any

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ELECTROMAQNETIC THEORY.

cn. I.

Talid reason, seems one of the most oriminal acts such a man could be guilty of. This seems strong language, but as Prof. Tait tells U8 that it is almost oriminal not to know several foreign languages, which is a veiy venial offenoe in the opinion of others, it seems necessary to employ strong language when the crimi- nality is more evident. (See^ on this point, the article in 27u Electrician, November 14, 1890. It is both severe and logical.)

g 4. I had occasion, just lately, to use the word " naturalist.'' The matter involved here is worthy of parenthetical considez» tion. Sir W. Thomson does not like " physicist," nor, T think,

"scieutist" either. It must, however, be noted that the naturalist, as at present generally understood, is a student o. living nature only. He has certainly no exclusive right to 8( excellent a name. On the other hand, the physicist is i student of inanimate nature, in the main, so that he has m exclusive right to the name, either. Both arc naturalists. Bu^ their work is so different, and their type of mind also s< different, that it seoras very desirable that their names shouU be differentiated, and that " naturalist," comprehending both should be subdivided. Could not one set of men be induced to caL themselves organists? We have organic chemistry, and organisms md organic science ; then why not organists 1 Perhaps, how ever, organists might not care to be temporarily confounde( with those members of society who earn their living by setting cylinder in rotatory motion. If so, there is another good name, viz., vitalist, for the organist, which would not have any ludic rous association. Then about the other set of men. Arc thev not essentially students of the properties of matter, and there fore materialists? That ''materialist" is the right name it obvious at a glance. Here, however, a certain suppositioui* evil association of the word might militate against its adoption But this would be, I think, an unsound objection, for I do no think there is, or ever was, such a thing as a materialist, m the supposed evil sense. Let that notion go, and the valuabh word " materialist " be put to its proper use, and be dignifiec by association with an honourable body of men.

Buffon, Cuvier, Darwin, were typical vitalises. Newton, Faraday, Maxwell, were typical materialists-

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5

All were naturalists. For my part I always admired the old- fashioned term "natural philosopher." It was so dignified, and raised up visions of the portraits of Count Rumford, Young, Herschel, Sir H. Davy, &C., usually highly respectable-looking •elderly gentlemen, with very large bald heads, and much "Wrapped up about the throats, sitting in their studies ponder- ing calmly over the secrets of nature revealed to them by their •experiment. There are no natural philosophers now adays. How" is it possible to be a natural philosopher when a Salvation Army baud is performing outside ; joyously, it may be, but not most melodiously ? But I would not disparage their work ; it may be far more important than bis.

§ 5. Returning to electromagnetic waves. Maxwell's in- imitable theory of dielectric displacement was for long gene- rally regarded as a speculation. There was, for many years, an Almost complete dearth of interest in the unverified parts of Maxwell's theory. Prof. Fitzgerald, of Dublin, was the most prominent of the very few materialists (if I may use the word) ■who appeared to have a solid faith in the electromagnetic theoiy of the ether ; thinking about it and endeavouring to arrive at an idea of the nature of diverging electromagnetic waves, and how to {nroduce them, and to calculate the loss of energy by radiation. An important step was then made by Poynting, establishing the formula for the flow of energy. Still, however, the theory wanted experimental proof. Three years ago electromagnetic waves were nowhere. Shortly after, they were everywhere. This was due to a very remarkable and unexpected event, no lees than the experimental disoovexy by Hertz, of Karlsruhe {now of Bonn), of the veritable actuality of electromagnetic waves in the ether. And it never rains but it pours ; for whilst Hertz with his resonating circuit was working in Germany {where one would least expect such a discovery to be made, if one judged only by the old German electro-dynamic theories), Lodge was doing in some respects similar work in England, in connec- lion with the theory of lightning conductors. These researches, followed by the numerous others of Fitzgerald and Trouton, J. J. Thomson, <fec., have dealt a death-blow to the electro- dynamic speculations of the Weber-Clausius type (to mention only the first and one of the last), and have given to Maxwell's

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ELECTROMAGNETIC THEORY.

CU. I,

tlieory just what was wanted in its higher parts, more experi- mental basis. The interest excited has been immense, and the theorist can now write about electromagnetic waves witliout in- curring the reproach that he is working out a mere paper theory. The speedy recognition of Dr. Hertz by the Royal Society is a very nnusual testimony to the value of his researches.

At the same time I may remark that to one who had care- fully examined the nature of Maxwell's theorj-, and looked into- its consequences, and seen how ratiomdly most of the }>heuo- mena of electromagnetism were explained by it, and liow it furnished the only approximately satisfactory (paper) theory of light known ; to such a one Hertz's demonstration came as a matter of course — only it came rather unexpectedly.

^ 6. It is not by any means to be concluded that Maxwell sjiolls Finality. There is no finality. It cannot even be accu- rately s;iid that the Hertzian waves prove Maxwell's dielectric theory completely. The observations were very rough indeed^ wheti compared with the refined tests in other parts of electrical science. The important thing proved is that electroni.iuuctic waves in the ether at least approximately in accordance with Maxwell's theory are a reality, and that the Faraday-Max well ian method is the correct one. The other kind of electroilynamic speculation is played out completely. There will be j'lonty of room for more theoretical speculation, but it must now be of the Maxwelliaa type, to be really useful.

§ 7. In what is to follow, the consideration of electromagnetic waves will (perhaps) occupy a consideraljle space. How much de}»cnds entirely upon the reception given to the articles. Mathematics is at a discount, it seems. JN'evertheless, as the subject is intrinsically a mathematical one, I shall not scruj)le to employ the appropriate methods when required. The reader whose scientific hori/on is bounded entirely by commercial con- siderations may as well avoid these articles. Speaking without prejudice, matter more to his taste may perhaps be found under the heading Trade Notices,* Sunt qiios curricido.

I shall, however, endeavour to avoid investigations of a com- plex character j also, when the methods au<i terms used are not

• Referring to The Electrician, in which this work first appeared.

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INTRODUCTION.

7

genenolly known I shall explain them. Considering the lapse of time since the discontinuance of E.M.T. and its P. it would be absurd to jump into the middle of the subject all at once. It must, therefore, be gradually led up to. I shall, therefore, in the next place make a few remarks upon mathematical investi- gations in general, a subject upon which there are many popular delusions current, even amongst people who, one would thinks should know better.

§ 8. There are men of a certain type of mmd who are nevtt' wearied with gibing at mathematics, at mathematioians, and at mathematical methods of inquiry. It goes almost without say> ing that these men haye themselves little mathematical bent. I believe this to be a general fact ; but, as a fact, it does not ezplaud very well their attitude towards mathematicians. The reason seems to lie deeper. How does it come about, for in- stance, that whilst they are themselves so transparently ignorant of the real nature, meaning, and effects of mathemati^ investi- gation, they yet lay down the law in the most confident and self-satisfied manner, telling the mathematician what the nature of his work is (or rather is not), and of its erroneousness and inutility, and so forth 1 It is quite as if they knew all about it.

It reminds one of the professional paradoxers, the men who want to make you believe that the ratio of the cuoumference to the diameter of a circle is 3, or 3*125, or some other nice easy number (any but the right one) ; or that the earth is flat, or that the sun is a lump of ice ; or that the distance of the moon is exactly 6 miles 500 yards, or that the speed of the current varies as the square of the length of the line. They, too, write as if they knew all about it ! Plainly, then, the anti-mathematician must belong to the same class as the paradoxcr, whose characteristic is to be wise in his ignorance, whereas the really wise man is ignorant in his wisdom. But this matter may be left for students of mind to settle. What is of greater importance is that the anti-mathematicians some- times do a deal of mischief. For tlicrc are many of a neutral frame of mind, little acquainted themselves with mathematical methods, who are sutiicicntly impressible to be easily taken in by the gibers and to be prejudiced thereby ; and, should they possess some mathematical bent, they may be hindered

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KLBCTBOMAGNSnO THBORT.

cn. I

by their prejudice from giving it fair development. We cannot all be Newtons or Laplaces, but that there is an immense amount of moderate niiathematical talent lying latent in the average man I regard as a fact ; and even the moderate development implied in a working knowledge of simple alge- braical equations can, with common-sense to assist, be not only the means of valuable mental discipline, but even be of commercial importance (which goes a long way with some people), should one's occupation be a branch of engineering for example.

§ 9. " Mathematics is gibberish." Little need be said aboat this statement. It is only worthy of the utterly illiterate.

What is the use of it ? It is all waste of time. Better be doing something useful. Why, you might be inventing a new dynamo in the time you waste over all that stuff." Now, similar remarks to these I have often heard from fairly intelli- gent and educated people. They don't see the use of it, that is plain. That is nothing ; what is to the point is that they oon- olude that it is of no use. For it may be easily obsenred that the parrot-cry " What's the use of it V* does not emanate in a hmnble spirit of inquiry, but on the oontraiy, quite the reverse. Tou can see the nose turn up.

But what is the use of it| then t Well, it is quite certain that if a person has no mathematical talent whatever he had really better be doing something useful," that is to say, something else than mathematics, (inventing a dynamo^ for instance,) and not be wasting his Ume in (so to speak) trying to force a crop of wheat on the sands of the sea-shore. This is quite a personal question. Every mind should receive fair development (in good directions) for what it is capable of doing fturly well. People who do not cultivate their minds have no conception of what they lose. They become mere eating and drinking and money-grabbing machines. And yet they seem happy ! There is some merciful dispensation at work, no doubt

*< Mathematics is a mere machine. You can't get anything out of it that you don't put in first. You put it in, and then just grind it out again. You can't discover anything by mathematics, or invent anything. You can't get more than a pint out of a pint pot." And so forth«

INTRODUCTION.

9

It is scarcely credible to the initiated that such statements could be made by any person who could be said to have an in- tellect. But I have heard similar remarks from really talented men, who might have fair mathematical aptitude themselves, though quite undeveloped. The fact is, the statements coutain at once a profound truth, and a mischievous fallac}'. That the fallacy is not self-evident affords an excuse for its not being perceived even by those who may (perhaps imperfectly) recog- nise the element of truth. But as regards the truth men- tioned, I doubt whether the caviller has generally any distinct idea of it either, or he would not express it so contemptuously along with the fallacy.

§ 10. By any process of reasoning whatever (not fancy) you cannot get any results that are not implicitly contained in the material with which you work, the fundamental data and their connections, which form the hasis of your inquiry. Tou may make mistakes, and so arrive at eironeous results from -the most correct data. Or the data may be faulty, and lead to eironeous conclusions by the most correct reasoning. And in general, if the data be imperfect, or be only true within cer- tain limits hardly definable, the results can have but a limited application. Now all this obtains exactly in mathematical reasoning. It is in no way exempt from the perils of reasoning in general. But why the mathematical reasoning should bj singled out for condemnation as mere machine work, dependent upon what the machine is made to do, with a given supply of material, is not very evident. The cause lies deep in the nature of the anti-mathematician ; he has not recognised that all reasoning must be, in a sense, mechanical, else it is not sound reiisoning at all, but vitiated by fancy.

Mathematical reasoning is, fundamentally, not different from reasoning in general. And as by the exercise of the reason discoveries can be made, why not by mutliematical reasoning? Whatever were Newton and the long array of mathematical materialists who followed him doing all the time? Making discoveries, of course, largely assisted by their mathe- matics. I say nothing of the pure mathematicians. Their discoveries are extensions of the field of mathematics itself — a perfectly limitless field. I refer only to students of Nature on

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ELECTBOMAGNBnO ISBORT.

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its material side, who haye employed mathematics expressly for the purpose of making disooveriea. Some of the unmathe- matical believe that the mathematician is merely engaged in oountiDg or in doing long sums; this probably arises from reminiscences of their schooldays, when they were flogged over fractions. Now this is only a part of his work, a some- times necessary and very disagreeable part, which he would willingly hand over to a j)roperly trained computator. This part of the work only concerns the size of the effects, but it is the effects themselves to which I refer when I speak of discoveries.

§11. Mathematics is reasoning about quantities. Even if qualities are in question, it is their quantities that are subjected to the mathematics. If there be something which cannot be reduced to a (|uantity, or more generally to a definite function, no matter how complex and involved, of any number of other quantities which can be measured (either actually, or in imagination), then that something cannot be accurately reasoned about, because it is in part unknown. Not unknown in the sense in which a quantity is said to be unknown in algebra, when it is virtually known because virtually expressible in terms of known quantities, but literally unknown by the absence of sufficient quantitative connection with the known. Thus only the known can be accurately reasoned about. But this in- cludes, it will be observed, everything that can be deduced horn the known, without appeal to the unknown. The un- known is not necessarily unknowable ; fresh knowns may make the former unknowns become also known. The distinction is a very important one. The limits of human knowledge are ever shifting. But there must be an ultimate limits because we are a part of Nature, and cannot go beyond it. Beyond this limit, the Unknown becomes the Unknowable, which it is of little service to discuss, though it will always be a favourite subject of speculation. But whatever is in this Universe can be (or might be) found out, and therefore does not belong to the unknowable. Thus the constitution of the middle of the sun, or of the ether, or the ultimate nature of magnetisation, or of universal gravitation, or of life, are not unknowable ; and this statement is true, even though they should never be dis-

INTRODUCTION.

11

covered. There are no insorutables in Nature. By Faith only can we go beyond — as far and where we please.

Human nature, or say a man, is a highly complex quantity. We are compelled to take him in parts, and consider this or that quality, and imagine it measured and brought into proper connection with other qualities and external influences. Yet a man, if we only knew him intimately enough, could be , formularised, and have his whole life-history developed. Even the universe itself, if every law in action were thoroughly known, could have its history, past) present^ and to come, formularised down to the minutest particulars, provided no discontinuity or special act of creation occur. But even the special act of dreation could be formularised, and its effects deduced, if we knew in what it consisted. And special acts of creation might be going on continuously, involving continuous changes in the laws of nature, and could be formularised, if the acts of creation were known, or the — so to speak — ^law of the discontinuities. The case is somewhat analogous to that of impressed forces acting upon a dynamical system. The behaviour of the system is perfectly definite and formularisable so long as no impressed forces act, and ceases to be definite if unknown impressed forces act. But if the forces be also known, tiicii the course of events is afrain definitely formularis- able. The assumption of a special act of creation, either now or at any time, is merely a confession of ignorance. Wc have no evidence of any sucli discontinuities. We cannot prove that there have never been any; nor can we prove that the sun will not rise to-morrow, or that the clock will not wind itself up again when the weight has run down.

12. Nearly all the millions, or rather billions, of human beings who have i)eopled this earth have been content to go tiirou;::h life taking things as they found them, and without any desire to understand what is <i:oiuir on around them. It is exceedingly remarkable that the scientific spirit (asking liow it is done), which is so active and widespread at the present day, should be of such recent origin. With a few exceptions, it hardly existed amongst the Ancients (who would be more appro- priately termed the Youngsters). It is a very encouraging fact for evolutionists, leading them to beheve that the evolution of

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ELECTROMAUNtXIC TtiEORY.

CH. I.

tnan is not played out ; but that man is capable, intellectually, ■of great development, and that the general standard will bo far higher in the future than at present.