THURSDAY, NOVEMBER 7, 1912. MATHEMATICAL TEXT-BOOKS. (1) A Treatise on Plane Trigonometry. By Prof. E. W. Hobson, F.R.S. Third edition. Pp. Third edition. Pp. xv+383. (Cambridge University Press, 1911.) Price 125. net. (2) A Shorter Geometry. By C. Godfrey, M.V.O., and A. W. Siddons. Pp. xxii+301. (Cambridge University Press, 1912.) Price 2s. 6d. (3) A New Geometry. Books i-iii. By W. M. Baker and A. A. Bourne. Pp. xxii + 122+ iii. (London: G. Bell and Sons, Ltd., 1912.) Price IS. 6d. (4) Lessons in Geometry. Part i. By Dr. Charles McLeod. Pp. xii+212. (Aberdeen University Press, 1912.) Price 2s. 6d. net. (5) Examples in Arithmetic. Part i., with answers. Taken from "A School Arithmetic." By H. S. Hall and F. H. Stevens. Pp. ix + 115+ xxii. (London: Macmillan and Co., Ltd., 1911.) Price 1s. 6d. (6) Solutions of the Examples in Godfrey and Siddons's "Solid Geometry." By C. L. Beaven. Pp. 164. (Cambridge University Press, 1912.) Price 5s. net. (7) A B C of Hydrodynamics. By Lieut.-Col. R. de Villamil. Pp. xi+135. (London: E. and F. N. Spon, Ltd.; New York: Spon and Chamberlain, 1912.) Price 6s. net. (8) A New Algebra. By S. Barnard and J. M. Child. Volume ii. Containing Parts iv.-vi. Pp. x+301-731. (London: Macmillan and Co., Ltd., 1912.) Price 4s. (9) A Treatise on the Analytical Geometry of Three Dimensions. By Dr. George Salmon, F.R.S. Revised by Reginald A. P. Rogers. Fifth edition. In two volumes. Vol. i. Pp. xxii+ 470. (London: Longmans, Green and Co.; Dublin Hodges, Figgis and Co., 1912.) Price Porigonometry or available for English students. There is probably no lecturer at Oxford or Cambridge who does not recommend his pupils to read it. In fact, before its publication it was necessary for those who wished for a rigorous treatment of those infinite series and products which occur in higher trigonometry to have recourse to French German text-books. But recently much has been done in England to remedy this deficiency. Mr. Hardy's volume on "Pure Mathematics," Prof. Bromwich's work on "Infinite Series," and Prof. Hobson's "Theory of Functions of a Real Variable" are noteworthy examples. These, however, deal with a wider field, and students will still continue to gain their first insight into the problems of higher analysis from this volume. The need of a third edition has given the author an opportunity for a complete revision; new matter has also been inserted. The theory of the measurement of circular arcs is discussed at some length in the opening chapter. Those who are interested in mathematical history will appreciate the section on the quadrature of the circle, in which is given a modified form of Gordon's proof that is a transcendental number, thus establishing the impossibility of constructing by Euclidean methods a straight line bearing to a given straight line the ratio . But the most important changes in the work are those which relate to the theory of series and products. Many additions have been made, further examples are given to illustrate the different cases that arise, and a number of references are supplied for those who wish to make a more thorough study of the subject. We have no doubt that this volume will retain its place as the standard text-book for many years to come. (2) This volume is a carefully reasoned interpretation of the Board of Education circular on the teaching of geometry. The first stage aims at illustrating the fundamental concepts, the second leads to the discovery of the principal theorems, and the third builds up on this basis a deductive development of the subsequent propositions. The authors feel that too much time has in the past been devoted to purposeless drawing, and have therefore omitted much of the experimental work contained in their previous treatise; the number of theoretical exercises, on the other hand, has been increased. We shall be much surprised if this text-book is not widely used. (3) This is an abbreviated form of the textbook on elementary geometry by the same authors published nine years ago. The only important change is the redistribution of the propositions in been regarded as the best work on the subject | Book i., those on congruent triangles being now grouped together. The supply of riders, particularly in connection with angle properties of the circle, seems rather inadequate. We are glad to see that limit methods of proof are employed for the fundamental tangent properties. of the subject. In the hands of a skilled lecturer we think the material of this book and the lines of thought indicated would interest and stimulate class of students of ordinary ability. The character of the work leads us to believe that the measure of success by oral methods. (4) There are several novel features in this text-author's purpose would be achieved with a real book. It is divided into thirty-five sections, each of which professes to contain the material for one lesson; but we are inclined to think some of these sections will occupy four or five hours if an adequate amount of time is assigned to rider work. The theorems are not numbered, and no references to previous propositions are given in the proofs; a conversational method is employed which for beginners possesses obvious advantages, and the order of the theorems differs from that usually followed. The scope of the work includes the first three books of Euclid, the properties of similar figures, and the fundamental propositions of solid geometry. We are of opinion that the character of the book will render it more useful to the teacher than to the student. (5). In order to meet the wishes of those teachers who prefer to take all book-work orally, the authors have now issued in a separate form the exercises contained in their "School Arithmetic." The work is published in two parts, the first of which deals with fractions, decimals, factors, compound quantities, and unitary method. Both in quality and variety the collection of examples is admirable. (6) Many teachers will be glad to hear that the solutions of the exercises in Godfrey and Siddons's "Solid Geometry" have now been published. Where we have tested them we have found that they are set out very clearly. Mr. Beaven has avoided the temptation to which in such cases writers often succumb of allowing a desire for brevity to obscure lucidity of expression. The figures which illustrate the solutions of the problems on plan and elevation are drawn with great care, and the methods employed are fully explained. (8) The authors of this treatise have succeeded in producing a work wholly unlike any other textbook on the subject with which we are acquainted. They take as their motto a pregnant sentence from Tannery's "Leçons d'algèbre et d'analyse ":"J'ai horreur d'un enseignement qui n'est pas toujours sincère; le respect de la vérité est la première leçon morale, sinon la seule, qu'on puisse tirer de l'étude des sciences." Their contention is that the average schoolboy is fully capable of realising the fundamental ideas upon which the science of algebra and, in fact, all analysis is based, and that a powerful educative instrument is discarded if no attempt is made to discuss the base-principles of the subject. They regard rigour in fundamentals as important as variety in application, and the power to understand the meaning of a process as more valuable than the ability to apply it. With these ideas in the abstract few people will disagree, but the majority of teachers hold that work of this character must be reserved for specialists. They consider that the mental calibre of the ordinary schoolboy, and the limited time at his disposal, are scarcely adequate to permit of the high standard required by the authors of this work. Great things can, of course, be done by an enthusiastic and able teacher; and we have no doubt at all that Mr. Barnard obtains excellent results at Rugby by following the lines here indicated. But we question whether the ordinary teacher could be expected to meet with success. We have not the space to comment in any detail on the contents of this volume, but we would urge teachers to procure a copy and study it for themselves. They will find in it much that is highly new ideas. It is in every respect a remarkable work. (7) This is in no sense a text-book on hydro- | suggestive, and will gather from it a number of dynamics. A few formulæ are quoted from various mathematical treatises, but no proofs are given. The purpose of the author is to introduce the student to the ideas of the subject, and to point out the rather arbitrary conditions under which, in the present state of knowledge, the mathematician is compelled to work. We do not think that the novice will find the contents of this volume at all easy to understand and coordinate with his other scientific reading. The number of quotations from many different authorities and the variety of topics alluded to may well tend to confuse those who have little previous knowledg (9) By the direction of the Board of Trinity College, Dublin, a new edition of Dr. Salmon's treatise on analytic geometry of three dimensions has been prepared. The editor has retained the substance of what appeared in the fourth edition, but has brought it into line with more recent work by inserting a number of new sections; the list of references has also been supplemented. Among the additions that have been made we note some excellent plates showing models of the various species of quadrics, a paragraph on Fiedler's pro jective coordinates, an account of the parametric representation of twisted cubics and quartics, Staude's elegant thread construction for confocal ellipsoids, and considerable reference to the later results obtained in differential geometry. There are a large number of minor changes, but the numbering of articles and chapters remains unaltered. Mr. Rogers is to be congratulated on the way in which he has executed a far from easy task. (10) The author claims that the method explained in this volume abbreviates and simplifies very materially the work of surveyors and navigators. It is, of course, impossible for us to judge from a perusal of the book how far the theodolite which Mr. Fergusson has invented is successful in practical work. But Prof. Heath states that it has been tested in the engineering department of Birmingham University, and has proved extremely convenient. He also remarks that "the reduction of the results of observations can be carried on simultaneously with the field work without reference to books of tables, thus giving the surveyor full information about any part of the field observed, while he is on the spot. For all traverse surveying, especially for rapid preliminary traverse, subject to modification, the instrument possesses distinct advantages over the ordinary theodolite.” The fundamental idea in the construction of the instrument lies in the method of graduation. To explain the system adopted, we will suppose that OA, ОA100 are two radii of the circle, containing an angle of 45°. Then points A1, A2, . . . A, . are marked on the rim such that the tangent of the angle AOA is equal to r/100, where r takes all values from 1 to 100. By this means it is clear that the observer, instead of reading off the angle, obtains its tangent, which is more useful for his purpose. As a matter of fact, the new form of theodolite is also graduated in degrees so that the angle can also be obtained, if desired. The angle AOA is taken as unit, and is called the one per cent. angle. The first fifty pages explain very fully the theory, and numerous examples are given to show its application. The remaining 450 pages are occupied with tables, to the compilation of which the author has given nineteen years. The leading column gives the angle in the percentage form at intervals of o'001 per cent. below 1 per cent., and of oor per cent. above it. The other columns give logarithmic sines, cosines, tangents, secants with difference tables to seven places of decimals, and the angle in degrees to o'001 of a second. A shorter table at the end contains tangents of half angles and versines. It seems almost incredible that any one man should have been able to carry through, almost single-handed, such a laborious work as Mr. Fergusson has accomplished. It undoubtedly merits the serious consideration of those engaged in survey work. The bulky size of the volume gives it rather an alarming appearance, but the processes are in reality very simple, and a single hour's work with the instrument would probably be enough to enable any practical man to gauge its utility. (11) There is a distinct need for a treatise on statics suitable for candidates for entrance scholarships at the universities. A number of excellent introductory text-books exist, but with two possible exceptions we do not know of any work which exactly meets this demand. Many teachers will therefore welcome the publication of Prof. Loney's book, which may be regarded as a companion volume to his "Dynamics of a Particle and of Rigid Bodies," recently issued. It is assumed that the student possesses some knowledge of the methods of the calculus and the elements of analytical solid geometry. In addition to the ordinary elementary course, it contains chapters on shearing stresses, three-dimensional forces, wrenches, chains, attractions and potential, and slightly elastic beams. There is an excellent collection of examples, including some of very considerable difficulty. The author has the rare gift of writing simply, and he has chosen his material with the same skill that characterises his previous work. explaining its function in scientific method, and naturally giving prominence to J. S. Mill, though also quoting Whewell, Bain, Jevons, Alfred Sidgwick, and Welton. Book iii. consists of useful examples of scientific procedure, drawn from the investigations of White of Selborne, A. R. Wallace, Darwin, Harvey, Lord Avebury, and others; while book iv. deals with some elementary principles of science-teaching, and has some very sensible remarks on heuristic methods. The section on Bacon is particularly good, and the famous idols are lucidly explained. Indeed, the whole book is a model of clearness. If it has a fault, it is in the direction of excessive quotation; but this is difficult to avoid when an author is exceptionally well read, and it has the compensating advantage of giving the young student a wide range of actual "samples," some of which may│| lure him to the study of the authors themselves. (2) This volume contains papers by Bertrand Russell, Percy Nunn, Boyce Gibson, Dawes Hicks, W. R. Sorley, James Ward, Bernard Bosanquet, and others, on the relations of universals and particulars, animism and the doctrine of energy, the experience of power, the time difficulty in realist theories of perception, and purpose and mechanism. Perhaps the most interesting to the man of science--particularly in view of Prof. Schäfer's British Association address and the various comments thereon-is the symposium on purpose and mechanism. Prof. Sorley, instancing a workman laying the bricks in house-building, points out that the purposive process involves (1) no creation or annihilation of matter, but only rearrangement of masses; (2) no creation or annihilation of energy; (3) no violation of the law of causation; but that it is by no means established that the laws of mechanics are valid also for purposive action; e.g., the laws of motion do not explain the bricklayer. It is "ideal guidance ❞—guidance by the idea of the house or its parts, existing in the bricklayer's mind-that determines the place of each brick. Purposive action implies certain things which are inconsistent with fundamental principles of mechanics. "Energy is liberated, that is, passes from the potential to the kinetic form, as the result of a purpose, or mental idea, and the same purpose may control the direc- | will apply to every part of the human organism, while still holding that this is not a full account of the matter. The discussion was continued by Dr. Bosanquet and Prof. James Ward. (3) A paper by the late Dr. Ameghino describes. fossil remains of two human beings found on the Atlantic coast, 60 kilometres north of Necochea. The skeletons were small, almost pigmy, say about 140 metres, and of small build as to strength; skulls small, and decidedly dolichocephalic frontal part small, face prognathous. Other articles discuss intellectualism and pragmatism, the psychology of criminals, and multiple personality. J. A. H. GENERAL AND ECONOMIC GEOLOGY. (1) Earth Features and their Meaning. An Introduction to Geology for the Student and the General Reader. By Prof. W. H. Hobbs. Pp. xxxix +506. (New York: The Macmillan Co.; London: Macmillan and Co., Ltd., 1912.) Price 12s. 6d. net. (2) A Geological Excursion Handbook for the Bristol District. By Prof. S. H. Reynolds. With an Introduction by Prof. C. Lloyd Morgan, F.R.S. Pp. 224. (Bristol: J. W. Arrowsmith, Ltd.; London: Simpkin, Marshall and Co., Ltd., 1912.) Price 38. 6d. net. (3) An Introduction to British Clays, Shales, and Sands. By A. B. Searle. Pp. xi+451. (London: C. Griffin and Co., Ltd., 1912.) Price 7s. 6d. net. By (4) Graphical Solution of Fault Problems. (5) Observations on the West of England Mining tion of the movement.... If purposive action (1) Pa series of readings," the substance is a reality, then mechanism is an abstract or limited system, and cannot give an adequate account of the real process of things." Mr. A. D. Lindsay, criticising, remarks that we cannot introduce a non-mechanical into a mechanical system, however much we may insist that it is only a little one; but, at the same time, we can maintain that the mechanical explanation of a PROF ROF. HOBBS describes his new volume as "a course of lectures in expanded form. The title concisely expresses the character and scope of the work. The author discusses the figure of the earth, and lays stress upon the tetrahedral hypothesis. He touches lightly on rocks and their mode of formation; deals with earth movements, volcanic action, weathering, and the activities of water and ice, desert conditions, and the characteristics of lakes and mountains. A separate chapter is devoted to the Niagara Falls as a clock of geological time. He hints that a large amount of original and unpublished material is interwoven in the older web. This may be, but much of it has either been skilfully concealed, or it has a familiar look. The book is quite pleasant reading, and the pleasure is enhanced by the abundant and excellent illustrations. There are five appendices; two on the determination of common minerals and rocks are unsatisfactory and not necessary in a book of this kind. The third describes the author's method of explaining the meaning of contoured maps by means of apparatus; it seems an elaborate way of passing the student's time. The fourth is a short explanation of apparatus for teaching the interpretation of geological maps; it consists of different shaped blocks, representing outcrops of various kinds, to be disposed on a table ruled into squares. This might be useful in setting examination exercises. The fifth outlines several fairly lengthy geological trips in the United States, and concludes with a sketch of a geological rush across Europe. (2) A different series of geological excursions is provided for by Prof. Reynolds in his admirable little guide book. Forty excursions, all within easy distance of Bristol, are described on a uniform pian under the headings "special features," "access," "general structure," followed by, in each case, an itinerary with geological notes and a short list of references. There are numerous sketch maps and sections and an introduction to the geology of the district is written by Prof. Lloyd Morgan. (3) The literature on British clays and sands is so meagre that any serious attempt to deal with the subject is welcome. The frontispiece to Mr. Searle's book, a "map of the chief clay works in Great Britain," is fairly characteristic of much that is to follow; it presents a medley of dots conveying no information of any value whatever, but representing, no doubt, a great deal of labour. It would be incorrect to suggest that no information can be gained from the text; Mr. Searle has produced a book containing much useful and interesting matter; one who can stand the fatigue and irritation caused by the lack of arrangement and the interminable repetitions will be able to extract a great deal. Only about half the volume is devoted to the geological position, the qualities and uses of British clays, and the small amount of space allotted to sands is not sufficient to warrant any mention in the title. The remainder of the book is concerned with ordinary geology and with the properties of clays in general. Much of the geology might have been condensed or omitted; it is very bookish, and it would be easy to point out many statements that are misleading or only half true. There are chapters on the mineral and other constituents of clays; the physical and chemical properties of clays (the table of analyses is quite inadequate); materials similar to clay (including halloysite, sillimanite, kyanite, zeolites, &c.); prospecting, mining, and quarrying; the purification and preparation of clays and the legal position of clays. There is a large but not quite trustworthy index. Throughout the work there are scarcely any references to original authorities, though many are mentioned. (4) Prof. Tolman's small book of less than fifty pages deals with a subject of the greatest importance to all practical geologists and mining engineers. In most English text-books the movement on fault planes is usually treated as if it were quite simple, though everyone is aware that it has often been very complicated. The author explains very briefly, and on the whole clearly, how to represent the effects of faulting. He uses for this purpose two methods: the isometric projection, and the application of contouring. He introduces several terms in the nomenclature of fault movements which will be familiar to readers of American geological literature; his use, however, of the expression "pole" for axis is not fortunate. The book would have been more satisfactory in some respects if the author had permitted himself a little more elaboration in the treatment of the subject; it is none the less a useful pamphlet. (5) On the mining region of Cornwall and Devon Mr. Collins has produced a valuable work of reference. Perhaps the most acceptable portion will be that giving short histories of individual mines. There is also a separate list of mines arranged alphabetically. A large part of the volume is occupied with theoretical matters of the usual kind, mingled with which is more interesting local information as to the character of the ore bodies. Occasionally the author launches out into elaborate estimates of the amount of certain minerals and ores contained in the rocks; they are neither of much use nor very accurate. (6) It was an excellent idea to gather into a single volume the opinions of different mining engineers upon the type of ore formation with which each was most familiar. Separate chapters of this book are given to the Clinton iron ores, Lake Superior iron ores, the flats and pitches of the Wisconsin lead and zinc district, lead and zinc deposits of the Ozark region, native copper deposits, the Cobalt district of Ontario, the Treadwell mines, Alaska, saddle reefs, contact deposits, |