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volume. The subject is brought into touch with other determination of physical constants such as are playbranches of the science. Thus, under hydrocarbons, ing an increasingly important role in the study of we read a little about thermochemistry; under alde- structure. The descriptions are clear and full, and hvdes there is a reference to autoxidation ; under acids the photographic illustrations are masterpieces of their there are a few words about steric hindrance; under kind. Altogether the book is probably the most comethereal salts (a rather antiquated term) a short plete among those of home manufacture on the account is given of mass action, and so forth. In subject of practical organic chemistry that has yet addition to this there are separate chapters on appeared. laboratory methods, stereochemistry, the sugars, (3) This modest little volume, which is one of the dynamic isomerism, heterocyclic compounds, and the University Tutorial Series, should form an excellent physiological properties of organic compounds. introduction to the study of organic chemistry, and

That the subjects are treated rather broadly than if the process of practical instruction can be carried deeply seems no serious defect. They are sufficient on concurrently with theoretical teaching, as the for the general reader, who is provided with elaborate author does with his own class, nothing better could references if he desires to extend his knowledge. In be desired. He takes a few of the commonest organic conclusion, we would direct the author's attention to substances and uses them, as they can easily be used, a few inaccuracies which have been noticed, and which to illustrate quite a large variety of chemical operamight be modified or corrected in a future reprint. tions and products. If the substance of the book can The two isomeric dimethylethylenes, which are stated be assimilated in the course of four school terms, as to be known in only one form, have been prepared by the author states, we think that both teacher and J. Wislicenus (p. 310); the molecular weight of student should be satisfied with the result. May we triphenylmethyl has been determined, and corresponds suggest that the name of Wurtz should be spelt to the double formula (p. 423); Fischer and Slimmer without the diæresis and Senderens without an a? were unsuccessful in effecting an asymmetric synthesis (4) One turns from the intricacies of a modern (p. 301); it is incorrect to state that propylene and treatise on organic chemistry to Wurtz's classical hydrobromic acid give exclusively isopropyl bromide memoir on the glycols with the same sense of relief (p. 45).

that one listens to the simple melody of an early We would also suggest the following :-Thiele's composer after the confused sounds of a modern hypothesis requires amplification to be understood orchestral symphony. Short and simple though it is, (p. 46); it is very questionable if the explosiveness of it is difficult to overrate the far-reaching results of a compound depends upon its breaking up into stable Wurtz's research. It was not merely the discovery molecules, for many silver salts share with silver of a new class of alcohols and organic oxides, or an oxalate this property, whereas a substance like platinic extension of Williamson's water type. It afforded for chloride does not explode; the statement that ethyl | the first time clear experimental evidence of the existand methyl cannot exist in the free state because ence of what were then termed “polyatomic they contain one of the carbon affinities unsaturated ” | radicals. To quote Wurtz's own words :(p. 27) is inconclusive, especially as triphenylmethyl " The main result, which, in my opinion, is derived is referred to later as possibly existing (pp. 36, 423); from these synthetic experiments, is not the discovery without some qualification it is misleading to

of the new compound, glycol-there are enough new

say that Dumas's theory of types “was especially de- compounds in organic chemistry--- it is not even the veloped by Gerhardt,” and “received the support of with its preparation which have been successfully

synthesis of glycerine nor the difficulties connected Williamson and Wurtz” (p. 17). In the first place, overcome; but it is the manner of the formation of Williamson originated the idea of Gerhardt's types, glycol and the antecedent reactions which made it which were simple inorganic compounds in which possible; it is the conversion of the allyl compound hydrogen could be replaced by radicals. They were

by which the iodide passed into glycerine. All these intended to denote chemical behaviour and not relas experiments, which were directed to the same end, tionships. Ether had no generic relationship to acetic of chlorine or bromine can replace two atoms of silver,

have shown that an organic group united to 2 atoms anhydride, though they belonged to the same type. and are therefore equivalent to two atoms of hydrogen, Dumas's types, on the other hand, were organic sub- and that an organic group united to three atoms of stances which were intended to show relationships chlorine or bromine can replace three atoms of silver produced by substitution rather than chemical and is equivalent to three atoms of hydrogen." behaviour.

The theory of polyatomic radicals, like ethylene and (2) The “ Practical Organic Chemistry" of Sud- glyceryl, soon developed into that of the polyatomic borough and James is rather a laboratory handbook elements or the theory of valency, upon which the or book of reference than a course of practical instruc- whole fabric of modern organic chemistry rests. tion. As stated in the preface, examples are given of Wurtz himself held perfectly clear views on the difdifferent types of operations. These types are grouped ferent valency of the elements. In his address to the together. Thus, there is a chapter on the preparation Chemical Society in London in 1862 on ethylene oxide, of hydrocarbons, one on alcohols, another on halogen he points out that as ethylene oxide represents a compounds, acids, esters, nitro-compounds, sulphonic diatomic radical united to oxygen, so many of the acids, and so forth. In addition, there is a pre- metals may be regarded as diatomic elements. The liminary chapter on organic analysis and molecular- paper is well worth re-reading, and is not by any weight determinations, and, at the end of the volume, means the least interesting addition to the Klassiker. a number of useful examples of analyses and the

J. B. C.

66

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Pp.
Co.;

viii + 351.

(1) THE

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PHYSICS FOR THE LECTURE ROOM AND tric waves, and to the appendix on ship's magnetism. LABORATORY.

Both these portions are very well done, though

would have thought them fairly strong meat for those (1) The Elements of Electricity and Magnetism. A who are making a “ first systematic study of the Text-book for Colleges and Technical Schools. By subject.” However, whether a student takes them Prof. W. S. Franklin and Barry Macnutt.

in completely in his first study or not, he will be very (New York : The Macmillan

glad to find them here ready to hand when required. London: Macmillan and Co., Ltd., 1908.) Price (2) The second of the volumes under review “ repre75. net.

sents primarily an attempt to secure a satisfactory (2) A Short University Course in Electricity, Sound articulation of the laboratory and class-room phases and Light.

By Dr. Robert A. Millikan and J. of instruction in physics.” Expressed otherwise, it Mills. Pp. V+389. (Boston and London : Ginn

consists of a description of laboratory work, each and Co., n.d.) Price 8s. 60.

experiment being preceded by as much theory as is (3) Naturlehre für höhere Lehranstalten auf Schuler- necessary to make a complete logical exposition of the

übungen gegrundet. By Dr. Friedrich Danneman. subject under study. We think that this plan is an Teil ii. Physik. Pp. vii + 204. (Hanover and

excellent one; and it has been very satisfactorily Leipzig : Hahnsche Buchhandlung, 1908.) Price carried out. Of course, it will be understood that the 3.50 marks.

theoretical part is not sufficient to replace a text-book (4) The Elementary Theory of Direct Current Dynamo dealing specially with the theory.

Electric Machinery. By C. E. Ashford and E. W. E. Although electrostatics is introduced in the first Kempson. Pp. vii + 120. (Cambridge : University chapter, electric capacity is not defined until later, Press, 1908.) Price 3s. net.

when it can be measured by means of a ballistic gal. (5) Electrical Laboratory Course for Junior Students. vanometer. There may be something to be said for By R. D. Archibald and R. Rankin. Pp. vi +95. this, but we think that the course of experiments (London : Blackie and Son, Ltd., 1908.) Price would be considerably improved and the student would IS. 6d. net.

get a more vivid idea of what capacity is if experi1)

THE order in which the elements of electricity and ments were added on parallel plate condensers used

magnetism are presented in the first volume along with a gold-leaf electroscope or an electrostatic under notice is :-(a) Electric current; (b) magnetism;

voltmeter. (c) electrostatics; (d) electric waves. This order is one

In sound, a series of experiments on diffraction is which does not make the exposition perfectly happy diffraction, a fact which prepares us for the exclusive

. included. The experiments on light begin with Thus it does not seem natural when it is found neces. sary to refer provisionally to the measurement of cur

use of the wave-method in proving the general

The final rents by their magnetic effect (p. 7) prior to any state- phenomena of reflection and refraction. ment as how magnetic effects themselves

chapter is on radio-activity, and contains some simple

The book measured. Surely the natural order is to take mag- experiments on uranium and thorium salts. netism before considering the electric current, even

is altogether a most excellent manual. though it may be preferred to deal with both before (3) We find in our third volume a well-selected series considering the phenomena of electrostatics. The of very elementary experiments in the whole round author passes naturally and easily to the exposition of physics suitable particularly for school use. Though of the last-named phenomena, and as many prefer the subject is dealt with satisfactorily as a rule, it is this order this portion may certainly be commended to

not beyond criticism. The diagram of the paths of them.

rays in a microscope would be improved if the rays The author is convinced that “elementary science represented as passing through the eyepiece were the instruction must be made to touch upon the things

same as those transmitted through the objective. The

" would be

velocity of electricity of everyday life if it is to be effective." This sentence experiment on the " may be taken as the keynote to the entire book. Thus best left out of such an elementary book; the stateelectric resistances are usually represented as electric

ment that electricity travels with the velocity of light lamps. Those who are accustomed to abstract think is, of course, absurd. ing may smile at these concrete representations; but

(4) We are in entire agreement with the authors of it must be remembered that this is only an elementary this book, that in the training of an electrical engineer book, and it must be admitted that much of the diffi

there should be included a knowledge of the theory culty which many junior students feel is connected of the subject built up logically from first principles, with the unreality of the subject as it appears to them. each step being illustrated with the help of some piece We commend the book for endeavouring, in this and of machinery or practical appliance of a general and other ways, to make the subject more real than it simple rather than an elaborate or necessarily up-tousually is.

date type. The present volume is intended to be used More attention is given than is customary in an only as a note-book accompanying a course of experielementary course to phenomena connected with recent mental lectures. The authors are to be congratulated discoveries, such as kathode rays, radio-activity, electric on the excellence of their little manual. The diagrams waves, &c. We conclude that in America a junior in particular are very carefully designed. course is in some respects more advanced than with (5) The last of this group of text-books covers an

This remark applies most to the chapter on elec- elementary first year's evening course and part of a

are

to

us.

was

our own.

second year's course. The first method of proving the has been instrumental in rendering familiar to the inverse square law for magnetic poles will not con

students of this branch of science will be found among

them. vince. However, putting aside an occasional criticism of this kind, we think that the book will well serve

One small slip we notice in connection with the

matter of protein nomenclature. The initiation of the its purpose of replacing manuscript instruction sheets

new system of terminology which is now being in a junior laboratory.

adopted for the albuminous substances is wrongly attributed to the British Medical Association. It was

really a committee of the Physiological and Chemical OUR BOOK SHELF.

Societies of this country which set the ball rolling. Théorie des Corps déformables. By E. et F. Cosserat.

The mistake is, however, a pardonable one, seeing Pp: vi+226. "(Paris : A. Hermann et Fils, 1909.) that it was at the meeting of the British Medical AssoPrice 6 francs.

ciation held at Toronto in 1906 that the opportunity The authors, who are well known by their writings on

os presenting the subject to our American colleagues general elastic theory, here reprint in separate form an

taken advantage of. The success that has appendix contributed by them to M. Chwolson's attended this effort to secure uniformity of nomen" Traité de Physique." The kinematical and

clature amongst English-speaking people has been dynamical theories of the flexible line, the flexible very gratifying; the American system, adopted under surface, and the deformable three-dimensional medium

the auspices of the American Physiological Society are discussed in turn in great detail. The dynamical and the American Society of Biological Chemists, standpoint adopted is that of the principle of action, differs in only small and unimportant details from

W. D. H. which forms, in the authors' opinion, the only satisfactory basis for the “ deductive " exposition of the Behind the Veil in Bird-land. Some Nature Secrets subject. In each case the most general form of the revealed by Pen and Camera. By Oliver G. Pike, function representing the “ action” is sought which with a number of pen sketches by E. R. Paton. is consistent with the necessary invariantive relations. Pp. 106. (London : The Religious Tract Society, This procedure is, of course, not altogether new, and 1908.) Price ios. 6d. net. an expert, turning over the pages, will recognise much Since the Keartons, some years ago, showed what that in one form or another is familiar to him. The splendid results could be achieved by an intelligent use treatment is necessarily somewhat abstract, and is of the camera as an aid to the study of natural history, mathematically very elaborate, Cartesian methods a host of nature-photographers has arisen, but only a being followed throughout. To many readers the very few have attained the high standard of merit set long train of general investigations, unrelieved by a by the founders of this branch of photography. Mr. single application, may prove deterrent; but the R. B. Lodge and Miss E. L. Turner in this country, authors at all events claim that their procedure has Schillings in Germany, and H. K. Job in America have never before been carried out so resolutely and com- in some respects even surpassed the Keartons; while pletely, and may justly pride themselves on the mathe- in this display of resource and dogged persistence matical elegance of their work. Apart from its other in the most trying circumstances they stand unqualities, the treatise has a distinct value as a book of rivalled. reference, and furnishes a whole arsenal of formulæ Mr. Pike in this rather pretentious volume has which may save trouble to future writers.

given some very excellent photographs, but the The book begins with a kind of philosophical intro- “Nature Secrets revealed by Pen and Camera which duction to which the authors attach great importance. he promises in his title-page are conspicuous by their This requires to be read in conjunction with a previous absence. His pages contain hardly one single new treatise, which has also appeared in the French fact, but a great deal that is banal. He solemnly edition of M. Chwolson's work. Those who adopt in

in writing of the kestrel, that “ The its fullest extent the empirical view of mechanics will first summer rose, a delicate pink amidst the surperhaps consider that too much weight is attached to rounding green, is a greater picture of spring than discussions of this kind. The historical references ever the sunlit could be "-which statement are, however, interesting, and fairly complete. The contains a great deal of truth !—“ and," he conauthors are indeed exceptionally well read in the tinues, “a kestrel hovering over a meadow, yellow history of their subject, and admirably conscientious with summer's flowers, tells us a deeper story than in their citation of authorities. In their preface they the eagle soaring over a wind-swept moor.” We fail promise a subsequent treatment of the theories of heat to grasp why this should be so. and electricity from a similar standpoint.

“ Bird-land's veil” is constantly being “ lifted up"

for him, like the drop-scene at the theatre, and on the Practical Physiological Chemistry. A Book designed stage appear blackbirds, which tell him the story for Use in

Courses in Practical Physiological of the leaves and flowers,” and wrens, which reveal Chemistry in Schools of Vedicine and of Science.

“ the secrets of the hedgerows," while skylarks, to By Prof. Philip B. Hawk. Second edition, revised complete the illusion, like the celebrated Grigolati and enlarged. Pp. xvi + 447. (London J. and A.

troupe in the pantomime, fly to and fro across the Churchill, 1909.) Price 16s. net.

stage, and sing “happy songs "'! Perfectly charmPROF. Hawk's text-book falls into the front ranking!

W. P. P. with the numerous additions and improvements which An Account of the Deep-sea Asteroidea collected by have been introduced into the new edition. It is not the R.I.M.S.S. Investigator.By Prof. René only a practical guide, and, as such, should be found Koehler. Pp. 143; 13 plates. (Calcutta : Indian in all physiological laboratories, but forms a very Museum, 1909.) Price 12 rupees. complete, readable, and up-to-date account

of our

This substantial contribution to the material of the present knowledge of the chemical side of physiology. echinoderm “system " consists of 126 pages of

A special feature has been made of the illustrations, minute description, and nine pages of general remarks. which are beautifully executed, and most of which | It is a continuation of certain reports of a preliminary will be new to workers in physiological chemistry. and incentive character published many years ago by The crystalline forms of the many protein derivatives the naturalists and pioneers of the Indian Marine which the work of Emil Fischer and his colleagues Survey, but, except that some doubtful identifications

assures

us,

sea

the rays.

are disposed of and some errors criticised, it does not tensile, shock, bending, hardness, and torsion tests. incorporate that earlier work.

Chapter ii. shows the influence of annealing and of In the descriptive part of the memoir thirty-nine cold work. Chapter iii. is devoted to " étirage, species are enumerated, of which thirty are regarded drawing, defined as “an operation which has for its as new, and are exhaustively described. The general object the completing of work done by rolling and remarks refer to eighty-eight species--the thirty-nine giving to the metal a cross-section which cannot be species treated by the author, and forty-nine species obtained by rolling," after the manner of wire-drawing dealt with in the earlier reports-and furnish the evi- | (**tréfilage," chapter iv.), which is a special case cí dence of the author's main conclusions. These con drawing where the cross-section is circular. Chapter v. clusions are that the deep-sea starfish of the Bay of gives a short account of methods of straightening Bengal and Arabian Sea are much more Phanerozonia • dressage").

A. McW. than Cryptozonia, and that their geographical affinities, so far as they can be discerned at all, are ex

Nutrition and Evolution. By Hermann Reinheimer. clusively Indo-Pacific, with a slight Hawaiian touch. Pp. xii +. 284. (London: John M. Watkins, 1909.)

Of the new species described by Prof. Koehler, five Price 6s. net. are separated as types of new genera. These are This is an essay on the importance of nutrition as a Johannaster, which is placed with very justifiable hesi- factor in evolution, and the author is in good comtation among the Plutonasteridæ, for some of its char- pany. For was it not Claude Bernard who said, acters suggest a pentagonasterid connection; Phidi- * l'évolution, c'est l'ensemble constant de ces alternaaster, which seems scarcely distinct from Psilaster; | tives de la nutrition; c'est la nutrition considerée dans Sidonaster, which agrees in all points with Porcel- sa réalité, embrassée d'un coup d'œil à travers le lanaster, except that, as in other porcellanasterid temps "'? To have had this thesis worked out in a genera, the elements of the cribriform organs are methodical manner would have been great gain, but papillar instead of lamellar; and Circeaster and Lydi- the author is not strong in scientific method. He has aster, both of which are Antheneids having the abac- gleaned far and wide to illustrate“ the evolutionary tinal plates of the disk much smaller than those of aspects of nutrition," and while he has a crow to pick

with most of his authorities, who have not the It may be thought that the limits of some at least " central key of a uniform analysis," he uses them of these genera are cut too fine to last; and of the when they suit him to back up his conclusion “ that in descriptions of species it may almost be said that its silent effects nutrition is one of the most formidthey are accurate expositions of specimens rather than able factors in the shaping of individual and racial impressive definitions of nature's products; but such destinies.” The conclusion is sound, but we cannot is the way of systematic zoology nowadays.

say this of many of the arguments. The memoir is most bountifully and most beautifully illustrated by the author's own hand; the plates, which are thirteen in number, are quite above criticism.

LETTERS TO THE EDITOR. Antimony: its History, Chemistry, Mineralogy, Geo- (The Editor does not hold himself responsible for opinions logy, "Metallurgy, Uses, Preparations, Analysis, ,

expressed by his correspondents. Neither can he undertake Production, and Valuation; with complete Biblio

to return, or to correspond with the writers of, rejected graphies for Students, Manufacturers, and Users of

manuscripts intended for this or any other part of NATURE.

No notice Antimony. By Chung Yu Wang. Pp: ,x+217;

taken of anonymous communications.) illustrated. (London : C. Griffin and Co., Ltd., 1909.)

Bessel's Functions. Price 12s, 6d. net.

I Once stated that a good style of writing English is MR. WANG observes in his preface that a metallurgical

not a strong point amongst British mathematicians, and work in English by a Chinese author is unusual.

the justice of this remark is exemplified by Prof. Hill's After reading the book, the conclusion is irresistible letter on this subject (NATURE, July 8), since it contains that English metallurgists would gain if Chinese the phrases Meissel's tables, Smith's tables, Aldis' tables, authors were more numerous. Mr. Wang has treated Isherwood's tables, which are correct; and Bessel functions, his subject with the greatest respect, and has drawn British Association tables, which are wrong. It is not in up with methodical care a complete treatise which will general permissible in English to employ a proper noun be very useful to all students of the subject. The

as an adjective, for the rules of grammar require either the long and apparently exhaustive bibliography at the

use of the genitive case, or the conversion of the noun into end of each chapter would alone give the book a right

an adjective, as in the words Newtonian, Lagrangean.

The British Association is one of the most important to a place on metallurgists' shelves, but in many cases

societies in the British Empire; it long ago discarded the facts are sufficiently set forth in the present work.

the insularity of our ancestors, and has become cosmoThe author carried out some practical tests of the politan in its operations. It is therefore not too much to latest volatilisation process of extracting antimony from expect that it will conform to the rules of grammar in its its ores, which was patented last year by M. Herren- publications, and employ its influence in encouraging a schmidt, and seems to have been much impressed by good literary style. its merits. The account of these tests is, however, I do not understand what Prof. Hill means by Neumann's almost the only original matter in the book, which is functions. I believe that Neumann was the first mathemainly a compilation of previously published material, matician who studied the properties of zonal harmonics and printed without comment. Its merits lie chiefly in the allied functions of degree n+1, where n is zero or a positive logical sequence and the accuracy of the extracts.

integer; but the subject was afterwards taken up and

greatly extended by Prof. W. M. Hicks in connection with Etirage, Tréfilage, Dressage des Produits métallur-circular motion. Hicks calls these harmonics

giques. By M. Georges Soliman. Pp. 164. (Paris : toroidal functions, which is a much better phrase, since it Gauthier-Villars and Masson et Cie., n.d.) Price puts in evidence the fact that these functions are connected 3 francs.

with the potentials of anchor rings or tores.

There is also another class of functions which are zonal This interesting little work, one of the well-known

harmonics of complex degree in- . These have been “ Aide-Mémoire" series, deals with its subject from studied by Hobson (Trans. Camb. Phil. Soc., vol. xiv.. a practical point of view. It is divided into five p. 211), who calls them conal harmonics. chapters, the first considering shortly the general

A. B. BASSET. mechanical properties of metals and alloys such as Fledborough Hall, Holyport, Berks, July 9.

vortex

Musical Sands.

THE THEORY OF CROOKES'S RADIOMETER. MAY I record the existence of musical sands along the shore at the Sandbanks, Poole Harbour ?

I

HAVE noticed that the theory of this instrument Some years ago the Poole authorities erected a series is usually shirked in elementary books, even the of box groynes along this coast between Poole Head and best of them confining themselves to an account, and the Haven, and these have considerably increased the not attempting an explanation.' Indeed, if it were natural accumulations of sand, so that it is making necessary to follow Maxwell's and O. Reynolds's caleverywhere, and the growth of the marram grass on the culations, such restraint could easily be understood. dunes is in many places (independently of that recently in their mathematical work the authors named start planted) rapidly extending seawards.

from the case of ordinary gas in complete temperature The beach now, between each groyne, consists of wide and flat deposits of sand, shells, and 'flint pebbles, but equilibrium, and endeavour to determine the first about midway between the dunes and the sea, where the

effects of a small departure from that condition. So sand is comparatively free from these, musical zones are

far as regards the internal condition of the gas, their of frequent occurrence.

efforts may be considered to be, in the main, successIn walking along the shore in a westerly direction, start- ful, although (I believe) discrepancies are still outing from the first groyne, the sounding qualities of the standing. When they come to include the influence sand notably increase. Thus between the first and second of solid bodies which communicate heat to the gas groynes there are no musical patches, between the second and the reaction of the gas upon the solids, the diffiand third the sounds are very faint, and between each of culties thicken. A critical examination of these ine other groynes, until one reaches the last at the Haven memoirs, and a re-discussion of the whole question, Point, the intensity of the sound increases. In a small

would be a useful piece of work, and one that may be cove at the Point, formed by the last groyne (constructed commended to our younger mathematical physicists. of barrels of concrete and an old ship), the sand is remark

Another way of approaching the problem is to select ably musical. The increase of sound observed when walking in a

the case at the opposite extreme, regarding the gas as westerly direction is due, I think, to the fact that the

so attenuated as to lie entirely outside the field of the prevailing westerly winds, and the littoral drift, separate ordinary gaseous laws. Some suggestions tending in the finer particles from the sand and carry them eastwards,

this direction are to be found in 0. Reynolds's memoir, and a microscopic examination of samples obtained from but the idea does not appear to have been consistently distances about a mile apart on this shore confirms this. followed out. It is true that in making this supposi

This musical sand is of the Studland Bay type, and tion we may be transcending the conditions of exnear the Haven gives even better results than any I have periment, but the object is to propose the problem in found there. The occurrence of musical sands along this its simplest form, and thus to obtain an easy and particular shore through the conserving influence of the

unambiguous solution---such as may suffice for the groynes is an interesting fact, for their existence there previously was very unusual, being only once noted in very physicist will naturally wish to go further. We

purposes of elementary exposition, although the small quantity during the last twenty years. Parkstone-on-Sca, July 4. CECIL CARU'S-WILSON,

suppose, then, that the gas is so rare that the mutual

encounters of the molecules in their passage from the The Commutative Law of Addition, and Infinity.

vanes to the envelope, or from one part of the envelope REFERRING to the review of Hilbert's “ Grundlagen der

to another part, may be neglected, and, further, that Geometrie, on p. 394 of No. 2066 of NATURE (June 3),

the vanes are so small that a molecule, after impact may I point out that the commutative law of addition can

with a vane, will strike the envelope a large number be proved without the help of any axioms at all, other than

of times before hitting the vane again. those of general logic? The method, indeed, used by Peano Under ordinary conditions, if the vanes and the in 1889 (** Arithmetices Principia ...,” Turin, 1889, p. 4), envelope be all at one temperature, the included gas which is only based on axioms of a general nature (such will tend to assume the same temperature, and when as the principle of mathematical induction), and not on equilibrium is attained the forces of bombardment on such special laws as the distributive ones, appears in so far the front and back faces of a vane balance one superior to Hilbert's; and, since all Peano's axioms

another. If, as we suppose, the gas is very rare, the were proved in Mr. Russell's “ Principles of Mathematics of 1903, Hilbert's proof seems quite superseded. Further,

idea of temperature does not fully apply, but at any

rate the gas tends to a definite condition which in. the difticulties arising out of Dedekind's proof of the exist

cludes the balance of the forces of bombardment. If ence of infinite systems can be avoided without the introduction of “ metaphysical arguments about time and

the temperature be raised throughout, the velocities consciousness (see Russell, Hibbert Journal, July, 1904,

of the inolecules are increased, but the balance, of pp. 809-12), as, indeed, your reviewer seems

think course, persists. The question we have to consider possible. But the connection of the fact that the existence is what happens when one vane only, or, rather, one of an infinity of thoughts (which must be in time) with face of one vane, acquires a raised temperature. Hamilton's idea that algebra was interpretable especially The molecules arriving at the heated face have, at in the time-manifold, just as geometry is in the space any rate in the first instance, the frequencies and the manifold, is not obvious. Philip E. B. JOURDAIN. velocities appropriate to the original temperature. As The Manor House, Broadwindsor, Beaminster, Dorset, the result of the collision, the velocities are increased. July 2.

We cannot say that they are increased to the values Neither Dr. Hilbert nor the reviewer make any sug.

appropriate to the raised temperature of the surface gestion that the commutative law of addition is best proved

from which they rebound. To effect this fully would as a deduction from the laws of multiplication. But the probably require numerous collisions. Any general laws of multiplication are so often treated as deductions

increase in the velocity of rebound is sufficient to from those of addition that it is interesting to have a case

unbalanced force_tending to drive the of the converse procedure. The fact that both these opera

heated surface back, as 0. Reynolds first indicated. tions and their laws have been treated independently and

If we follow the course of the molecules after collision in a strictly logical manner by Dedekind, Peano, and others with the heated surface, we see that, in accordance is, of course, perfectly well known to all who have paid any with our suppositions, they will return by repeated attention to this part of mathematics. Whether Dedekind's collisions with the envelope to the original lower scale critics have really avoided metaphysical arguments without of velocities before there is any question of another at the same time making metaphysical assumptions is a collision with the heated face. On the whole, then, question on which a difference of opinion is permissible.

G. B. M.

I See for example Poynting and Thomson's "Heat," p. 150.

to

cause

an

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