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student may find something to interest him, and, with such limitations as noticed above, it is extraordinarily

full and complete. Altogether a book which should be THE THEORY OF CONTINUOUS GROUPS. widely read. Introductory Treatise on Lie's Theory of Finite Con

So much so that it is both difficult and uncongenial tinuous Transformation Groups. By John Edward

to offer any criticisms, were only a review complete

without some. To us it seems that some account of Campbell, M.A., Fellow and Tutor of Hertford College, Oxford, and Mathematical Lecturer at systems of equations which in the aggregate define a University College, Oxford. Pp. xx+416. (Oxford :

finite continuous group forms the most natural introClarendon Press, 1903.) Price 14s. net.

duction to the theory; though Lie's account of them

comes near the end of his third volume he is there THE HE theory of continuous groups should appeal to revising his fundamental principles, and the ideas inall who are interested in mathematics; it is based

volved are very simple. Reference to Schlesinger's on the fundamental ideas involved in cases of change “ Treatise on Linear Differential Equations ” (Bd. ii., of the algebraic notation, and as such is an illumin- | Teil i., p. 23) shows how this suggestion works out ating synthesis of a large number of our elementary in detail. It seems right that the student should early operations; and the principal notions of the theory, learn, for instance, how far the linear transformations once laid bare, are so simple and admit of so many which leave x2 + y2 unaltered fall under Lie's terminfamiliar applications that these should form

an ology. Perhaps, again, fuller references to anticipaintegral part of elementary teaching, particularly in tions of the ideas which Lie has coordinated into one analytical geometry and differential equations. As to

system would have helped the student. Such may be its philosophical import, the theory is of the greatest found widely scattered in all the early masters; two value in the analysis of our geometrical conceptions, that are handy to us are in Sylvester's writings. In being an indispensable part of that algebraic scheme

1852, when Lie was ten years old, Sylvester (“ Collected which, at present running parallel with these, may Works,” vol. i., pp. 326, 353), while ascribing the modify them still more than hitherto before the notion partly to others, writes of continuous parallelism is recognised again as an identity.

infinitesimal variation, and that “concomitance canLie himself, though directing attention to the fact not exist for infinitesimal variations without, by that he heard as a student, in 1863, lectures from

necessary implication, existing for finite variations Sylow on Galois's theory of discontinuous groups, and also.” Or, again, the deduction, so interesting when acknowledging his indebtedness to several writers on

we first came across it, of the equations for the inpartial differential equations, would seem to have been finitesimal motion of a rigid body, from the invariance interested, above all other things, in the transform- of the expression dx? + dy? + dza, is in a paper of ations of analytical geometry; and while the precise Sylvester's of 1839 (ib., p.

34). Again, it propositions of his theory of groups must be primarily appears to us, though recognising the value of Mr. attributed to his study of systems of linear partial Campbell's proofs of the fundamental theorems, that differential equations, his bias was at first, and largely much would have been gained in directness, withthroughout, to arrive at his conclusions by the help out appreciable increase of the necessarily analytical of geometrical intuition. Thus, though he has character of much of the subject, by a frank recognition succeeded so extraordinarily in what he tells us was of Schur's forms for the first parameter group in terms one of his objects, drawing again into organic union of the constants of structure; of this we are, perhaps, branches of mathematics which threatened to pursue not impartial judges (see Proc. Lond. Math. Soc., solitary developments, there is, some may think, a vol. xxxiv. p. 91), as equally not of Mr. Campbell's certain underlying vagueness of definition as to the use of the word united in his exposition of Lie's decharacter of the functions to which his theories apply. | finition of an integral of a partial differential equation, This even has, perhaps, some advantages.

having ventured elsewhere to introduce the words Of these various points of view the book now under connected and connectivity, which latter seems better notice gives the English student an excellent means than the mere symbol M, which Mr. Campbell adopts of judging. With roughly the same purpose as the from Lie (see “ Encyc. Brit.,” vol. xxvii. p. 452). But simplified German account of Lie's theory (Scheffers, we have a more serious contention with Mr. Campbell 1893, 800 pages), it is briefer, and yet quite clear in about a matter in which opinions will be widely divided; statement; it contains more of the application of Lie's no doubt it is proper that a beginner's course in the theory to the solution of partial differential equations, theory of groups should insist primarily on the group and it offers alternative proofs, due to its writer, of the property, and not confuse this by complicated considerfundamental theorems of the subject. Like the ations in regard to the properties of functions; but in German book, it largely leaves aside the developments our opinion no account can be regarded as modern subsequent to Lie, such as the intricate theory of the which does not face the difficulties; it seems to us misstructure of groups, and the application to the trans- | leading, without careful explanations, to use language formation group of systems of differential equations, about functions in general which applies in the first initiated by Picard, and leaves wholly aside Lie's instance only to the simplest algebraic functions. On criticism of the axioms of geometry, while it accepts . 11 we read : bk can in general be expressed ... Lie's function theory throughout; but it abounds in in order that (2) may remain an analytic function of apt examples, chosen mainly from differential equa- its arguments.” In what way is the student to tions and geometry, so that almost any mathematical imagine the function defined after it has ceased to be an analytic function of its arguments? or does the modern practice. Dr. Smith's treatment resulted in word analytic mean regular? and what is the meaning a varnish or enamel ” of linseed oil and coal-tar of expressed? Again, on p. 98 : “ This transformation pitch being baked on to the cleaned surface of the pipe, of the variables has only involved algebraic processes.' the oil oxidising more or less completely during the The processes in question consist in reverting certain operation. The modern substitute for this is, too power series; now a power series is an entirely symbolic often, a mere dipping of the pipe in crude tar, or in thing unless we have very simple rules for the law of tar diluted with “ dead oil.” From the wording of the its coefficients; how can the reversion of a power original patent this process may, on a technicality, pass series in general be regarded as a practicable process under Angus Smith's name; but our author has no likely to aid the effective determination of the integrals doubt that if the inventor were living he would conof a differential equation ? and at any rate it does not demn the whole thing from beginning to end. It is seem fair to describe it as an algebraic process. More- “adulterating his invention and stealing his reputaover, apart from such indefiniteness, and passing over tion." such phrases as (p. 24)“ where t is a constant so small Mr. Sabin describes a process of his own, which has, that its square may be neglected,” there is the question, he tells us, been successfully applied to large pipeapparently unconsidered in this book, of how far Lie's lines in America, and is in use in the United States propositions can be proved for functions which are not Navy for the protection of heavy copper mains. It is analytic, in regard to which various investigations are evidently based upon a study of the Angus Smith already forthcoming.

process. It consists in applying to the pipes a thin But we gladly turn from such criticisms to remark coating of a mixture of linseed oil and asphaltum, and again on the merits of the book, choosing two random afterwards heating the pipe to 400° F. until the oil examples, one of the practical spirit in which it is is completely oxidised. The product is said to be a written, the other of the author's eye for a neat result. hard, elastic enamel. One result is that, whereas the On p. 256 the author frankly uses the known theorems aforesaid copper mains had formerly an average“ life" as to forces in three dimensions to abbreviate the re- of about six months, they have now lasted three or duction of the equation of a linear complex. On four years, and their ultimate durability is not yet p. 243 the author arrives at the theorem that Ampère's determined. partial differential equation of the second order is re- There is some curious lore in the author's historical ducible by a contact transformation either to s=o or to summary. The connection between electricity and rt-sa=0, according as it possesses two distinct systems “ Berenice with the golden hair," between varnish and of intermediary integrals or only two coincident the Queen of Cyrene, is a good example of etymological systems. In conclusion, we would express our admir- ramifications. One quaint recipe of 1520 is worth ation for the form and printing of the volume.

quoting : H. F. B. A most excellent varnish for varnishing arque

buses, crossbows, and iron armour: Take of linseed

oil two pounds, sandarac one pound, Greek pitch two TECHNOLOGICAL CHEMISTRY.

ounces. Boil the oil, then dissolve in it the other in

gredients, and strain through a much-worn linen The Industrial and Artistic Technology of Paint and cloth; and when you wish to use the varnish, scrape

Varnish. By A. H. Sabin, M.S. Pp. vi+372. and polish the work and heat it in a hot oven, because (New York: John Wiley and Sons; London : Chap- that is the best place to heat it .... then lay it on man and Hall, Ltd., 1904.) Price 125. 6d. net.

thinly with an instrument of wood, so that you may Food Inspection and Analysis. By Albert E. Leach, changing colour.

not burn your fingers, and it will make a beautiful S.B., Analyst of the Massachusetts State Board of

“And if you supplied the place of Greek pitch with Health. Pp. xiv +787. (New York : John Wiley naval pitch, I think it would make the work black and Sons; London : Chapman and Hall, Ltd., 1904.) when you varnished it." Price 31s. 6d. net.

The treatise can be read with profit either by the 1 HIS is a gossipy, pleasantly volume,

the style of which will be indicated when we chemist who wishes to know something of paint and remark that the book is prefaced by an extract from varnish technology. Quintilian, and closes with a poetical quotation. It (2) There is a Madras story of a native woman, who, treats, generally in untechnical and even colloquial charged with possessing illicit salt, would offer no language, of varnishes and paints, their history, defence; wherefore she was about to be mulcted in fabrication, and uses. Principles, not formulæ, are the sum of one rupee. Before closing the case, howusually given by the author; the book is in no sense ever, the magistrate thought he might just as well a collection of recipes.

satisfy himself that the substance really was salt, and If there is not much of strictly scientific value in the forthwith proceeded to taste it. Thereupon the lady treatise, there is a good deal which is of practical raised her voice in a very effective interjection : interest. The chapter upon the protection of metals only,” said she, “ not only does the sahib fine me one against corrosion, for instance, may be recommended rupee, but lo! he eats the ashes of my dead husband." to the notice of engineers, and also that on the coating Fortunately for magistrates, such appeals to the of water-pipes. As regards this latter question, the palate are rarely either necessary or sufficient, nowaauthor points out that the essential feature of the days, for disposing of legal cases relating to the “ Angus Smith process ” has been misapprehended in identity and purity of foodstuffs. Much more cum

" Not


brous machinery has had to be devised. To summarise is well known to have attended to the subject for many and explain this machinery is the aim of the work years. under notice. In the main it is intended for the food The amount of contradictory evidence is remarkable. analyst, and the author's idea has been to give this In the case of the earlier experimenters, with more or official some information, not only on the subject of less faulty methods, this is not surprising; but the food-analysis, but also on various collateral matters

ing strikes one in many modern instances. with which he is brought into contact. Thus there are The question of the amount of transpiration in moist sections discussing the equipment of the laboratory, tropical regions, as compared with Europe, is a case the storage of samples, legal precautions, the duties in point. Another instance is what the author deof the food inspector, and certain processes of food scribes as a “ seven years' war (1884–1891) between manufacture.

Wille and Lundström as to the absorption of water All the ordinary foodstuffs are dealt with, a chapter by the aërial parts of plants. Other disputed points being allotted to each group of allied products, such are the effect of salt solutions supplied to the tranas cereals, spices, alcoholic beverages, and so on. The spiring plants, and the influence of varying amounts descriptions are written clearly; an excellent selection of CO, in the atmosphere; and many other cases might of the salient facts and the best methods of examin- / be cited. ation has been made; and to each division an extensive The relation of plants to water, though a subject of bibliography is appended. Microscope work is a primary importance, is still to a great extent in the special feature, and the volume is enriched by a series elementary stage of inquiry. A large number of the of forty plates, containing about four times as many statements quoted by Burgerstein are little more than photomicrographs of the principal vegetable and disconnected facts, and, in spite of the interesting book animal structures met with in the examination of he has made of them, they still seem to us to await foods.

a somewhat different treatment. The chief criticism to offer on the book is that the The subject-matter of the book falls into two treatment of so much material in one volume-even classes :-(1) the loss of water-vapour considered as one of eight hundred pages-must necessarily be in physical phenomenon; (2) the biological inquiry into the nature of a summary. Hence in many instances the adaptation of plants to the distribution of water the information, though sufficient for routine work, considered as environment. From both points of view is not full enough to be of much value when cases of transpiration should be considered side by side with real difficulty arise.

assimilation and respiration, and this manner of lookOne notes several examples of careless transcription ing at the subject has not, in our judgment, been kept in looking through the work. On p. 441 the so-called sufficiently in mind by the author. The point is that “ Koettstorfer's equivalent” for butter-fat is given a the same organs—the stomata—serve for gaseous exmaximum value of 241 and a minimum of 253. It change and for the evaporation of water. Burgerstein might be guessed that these two numbers have been discusses at the end of his book the question whether, transposed; but on the next page the value of the as some have supposed, transpiration is a necessary constant in question is given as 224. The author has, evil. This might have been discussed from a broader in fact, failed to distinguish between the “ equivalent” standpoint, and would have been in place in an earlier and the "value” of the saponification experiment. chapter. It does not seem necessary to treat the view In the table on p. 441 the values of the insoluble acids referred to as entirely false. Plants undoubtedly have for oleomargarine are transposed; the specific gravity to strike a balance between the possession of a free has no temperature of reference; and a faulty arrange- stomatal connection with the atmosphere and the conment of the table makes it appear that butter-fat and sequent danger of evaporating more water than they margarine possess, somehow, a maximum and a can take up from the soil. This compromise includes minimum temperature; whilst in the data for edible also the value of the transpiration-stream in supplyoils and fats on p. 380 the limiting values are again ing minerals to the aërial parts, on which Burgerstein transposed.

rightly lays stress. All we suggest is that the whole Nevertheless, it would be unfair to judge the book problem, being of a fundamental character, might well by these slips. It contains a large amount of inform- have been dealt with more liberally, and been given a ation and, though written more particularly from the place preliminary to the details of transpiration. American point of view, will be found a useful con- A fault in Burgerstein's treatment of transpiration, spectus of the whole field of food control.

though a fault difficult to avoid, is that he does not C. SIMMONDS. keep before the reader the fact that the condition of

the stomata—whether open, half open, or shut-is far

and away more important than all the other internal THE TRANSPIRATION OF PLANTS.

conditions put together. Like the rest of the world, Die Transpiration der Pflanzen. Eine Physiologische he is well aware of this, but we doubt whether the

Monographie von Dr. Alfred Burgerstein, A. 0. uninstructed reader would here learn to think of the Universitätsprofessor in Wien. Pp. x+283. (Jena : problem in this way. To take an example, he deGustav Fischer, 1904.) Price 7.50 marks.

scribes (p. 62) how, when part of the foliage is reTHIS 'HIS book is a classified analysis of the published moved, the remaining leaves transpire more actively

work on transpiration from the time of Hales than before. Here we want a discussion of the possible onward, with a running criticism by the author, who effects, direct or indirect, of the operation on the


stomata of the remaining leaves. The same thing is seem to expect a series of ready-made lessons on a true of the discussion (p. 81) on the transpiration of

variety of nature subjects, basing their demand on the flowers as compared with leaves, where the reader is

ground that they have no time (or is it that they have

no inclination?) to make the necessary studies for left in ignorance of how far the facts are explicable

themselves. If this course were adopted, it would lead by reference to the stomata.

to two evils. First, all the observations (if they could But it is not merely in relation to isolated problems be so called) would come from the teacher and not that we feel the want of more information with regard from the pupils; and, secondly, knowledge thus to the stomata. We should expect to find a full acquired by the teacher could not possibly raise the

delights of genuine nature-study in the minds of his general discussion of their importance in regard to

scholars. Prof. Miall has therefore preferred to make transpiration. This would have included a reference

an effort to instil and encourage the habit of observto Horace Brown's work on the static diffusion of gas ation and inquiry in a few teachers (who will necesthrough these openings, and a consideration of the sarily be the best of their kind) by showing them what question how far evaporation can be checked by the may be learnt by careful observation of the common

natural objects to be met with among their daily closure of the stomata. Again, we should have liked

surroundings, rather than by pandering to the popular a discussion of the trustworthiness and general value

clamour for cut and dried lessons—which are really of the microscopic measurements of the stomata in

not nature-study at all. How he has succeeded reliving plants. Burgerstein gives interesting

mains to be seen. If we may venture to predict, it will account of the methods depending on the yield of be the clever and inquiring teachers who will praise

and take advantage of his efforts, and the dullards and water-vapour, such as Stahl's cobalt test, &c., by which

plodders who will condemn them and say that they are it can be roughly determined that the stomata are

unsuited to their purpose. “ widely open or “ nearly shut.” But if we are to

Although the author modestly says that he gives distinguish the stomatal factor from other factors in

only a few lessons, his articles or essays are no less experiments on transpiration, numerical statements as than fifty-four in number, and cover a very wide range to the condition of the stomata are wanted, and the of subjects, including cheese-grubs,, glow-worms,

water-lilies, London pride, the human face and hand, question whether such data are available might well

and museums and their teachings. As an example of have been discussed. With regard to method, Burger

the large amount of information Prof. Miall manages stein seems to us a little hard on the various “ poto- to give in a very small compass, we may refer to the meter " methods, by which a general idea of the tran- exceedingly interesting account of the ancestry and spiration curve is obtained by measuring the intake of evolution of insects in the chapter on the “ cheese

hopper.An excellent work which should be in the water. He is justified in saying that these methods

hands of all teachers is our verdict.

R. L. do not estimate transpiration but absorption; but we think he undervalues the fact that, with cut branches Ideals of Science and Faith. Essays by Various and for not too extended periods of time, the intake Authors, edited by the Rev. J. E. Hand. Pp. xix+

333. (London : George Allen, 1904.) Price 5s. net. so closely corresponds to transpiration that the method

“On all sides” (to quote the preface)" is a growing cannot be neglected, and is certainly of great value for

recognition that the ideals common to both Religion purposes of demonstration.

and Science are not only numerous but are indeed the Though we have criticised “ Die Transpiration der very ideals for which the nobler spirits on both sides Pflanzen,” we are far from meaning to condemn it; care most." Necessarily the treatment is varied, we have, indeed, read it with interest and profit. Any perhaps too varied, but the editor gently deprecates

criticism of this feature. Prof. Patrick Geddes has one intending to make a study of the subject cannot do better than read it with care. He will thus be made

room to discourse on the excellence of teaching boys

to make boxes; and the theologians, under "A aware of many pitfalls, and will have a guide to the Presbyterian Approach," "A Church of England chief points which need fresh investigation.

Approach," and the like, hardly give one a definite F. D. view of “ A Christian Approach.”

In the papers of the men of science and philosophers

the general position is that science does not deal with OUR BOOK SHELF.

the whole of life, and that it can no longer meet the

claims of faith with a House, Garden, and Field; a Collection of Short Lodge defends the idea of continuous guidance on the

certainly not." Sir Oliver Nature Studies. By L. C. Miall. Pp. x+316; illus

part of the Deity, seeks to reconcile Pantheism and the trated. (London : E. Arnold, 1904.) Price 6s.

belief in a personal God, and complains that religious This admirable little work appears to be by far the people seem to be losing some of their faith in prayer. best aid to the proper teaching of nature-study that has Prof. J. Arthur Thomson and Prof. Patrick Geddes lay hitherto come under our notice, the author having very stress on the altruistic side of the struggle for existwisely refrained from furnishing the teacher with a Prof. Muirhead maintains that we must limit manual which would do away with all necessity for causation and the conservation of energy to the original study and observation on his part, and enable material world, and must look for some other concephim to read the various lessons to his pupils without tion when we come to the action of the mind itself. effort or thought. The object of the writer is, indeed, “We use a saw to make a fiddle; we throw it (sic) as much to educate the teacher as to enable the latter aside when we come to play upon it (sic).” The Hon. to teach his pupils. For example, in the article on Bertrand Russell's paper-“ An Ethical Approach "bananas, Prof. Miall, when he asks the reason for the is the most eloquent; much of it is Lucretius, Book iii., peculiar shape of that popular fruit, under the guise rewritten (could one be more complimentary?), with of leaving the reply to the pupil is really testing the the difference that Mr. Russell recognises more depowers of observation and reasoning possessed by the finitely the need for religion and worship, albeit the teacher himself.

worship of a God who is not Force but " created by As the author observes in his introduction, teachers our own love of the good.”


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