XXIX. Notices respecting New Books.

Theorie der Optischen Instrumente nach Abbe. Von Dr. SIEGFRIED CZAPSKI. Sonderdruck aus dem Handbuch der Physik von A. Winkelmann, Band II. 1893. Breslau, Ed. Trewendt; Williams and Norgate.

THE subject of Geometrical Optics might with advantage be de

fined so as to include all those parts of the theory of light and optical instruments which can be treated by geometrical and formal considerations, without detailed discussion of dynamical questions. In stricness, of course, the Theory of Light is a branch of Dynamics; but it is rather striking how completely the subject divides itself into two distinct regions. The practically important part involves but very slight recourse to dynamical treatment, while the speculative part is almost wholly concerned with dynamical questions. The title of Geometrical Optics to separate existence rests on this fact; but it may fairly be urged that, in opposition to the usual practice, the greater part of the subject of Diffraction and Interference of light should be assigned to this branch of the theory, on account both of its geometrical character and of its importance for the working of optical instruments.

As ordinarily developed, the subject is a branch of Geometry which is concerned with the reflexion and refraction of rays of light. But if it is severed too much from the practical applications that gave it birth, there is perhaps no department of mathematics that can be made so repulsive. In matters of Pure Mathematics the natural mode of progress is to adhere closely to the track of symmetry and elegance; and there is no other canon of merit that is applicable. But in the subjects that are cultivated more directly for the sake of useful applications there is no choice of topics allowed; and the aim must be to develop methods of investigation that will satisfy the demand for mathematical elegance, and will at the same time allow of an exact correlation between the mental process and the natural phenomenon.

The book before us is a treatise on Geometrical Optics, compiled from the point of view of the Theory of Optical Instruments. The position which Dr. Czapski holds, in conjunction with Professor Abbe, in relation to the famous manufacturing firm of Zeiss at Jena, is a guarantee that the more technical information contained in the work is exact and up to date. From the aspect of theory, perhaps the most valuable feature of the book is the very copious reference to the works of original writers; for to obtain a real command of any department of a subject of wide practical ramifications, the consultation of text-books, however good, is but a poor substitute for the assimilation of the ideas and points of view of its original promoters. The treatise contains a very full account of the topics which have been indicated above, with the exception of the theory of Diffraction, of which the treatment is presumably left

to another department of the general undertaking of which this book forms a part. The phrase "nach Abbe" which occurs in the title is a little puzzling to non-German readers: in extending widely the capabilities of optical construction and manufacture, a very high degree of credit is due to Prof. Abbe and his collaborators at Jena; but yet it seems strange to associate his name so markedly with the title of a theoretical treatise whose contents would appear for the most part familiar to many people who have possibly never heard of his work, and whose knowledge of Optics was acquired before his time.

From the nature of the book, as forming an article in an Encyclopedia of Physical Science, the subject is broken up somewhat into separate headings, with a view to facilitate reference. If ever the time comes for a final gathering together of the threads of this somewhat discursive subject into a compact form, it is to be hoped that much attention will be paid to the geometrical methods of discussion employed by the earlier English writers such as Robert Smith and Thomas Young, which amalgamate so easily with experimental requirements; and that the capabilities of the Hamiltonian method of Action as a basis for the analytical part of the subject will be fully utilized. J. L.

Anwendung der Quaternionen auf der Geometrie. Von

Dr. P. MOLENBROEK.

THIS is the promised sequel to the Theorie der Quaternionen by the same author, which we reviewed in November 1891. In his preface Dr. Molenbroek replies to certain of our comments at that time. He repudiates the description then given of his novel interpretation of 1 as an operator. Yet his own generalized description is in these words: "This definition shows that under the symbol-la there are included not only an infinite number of arbitrary quaternions but also a similar number of right quotients [that is, versors] whose indices are all perpendicular to a." thus, we still must believe, the operator 1 has the "singularly felicitous but hitherto unsuspected power of adjusting its axis so as to be perpendicular to any vector to which it may be applied!"

We had hoped to find in the present volume, which treats of geometrical applications, a further development of Dr. Molenbroek's pet creations-the Vektorkreis, the Vektorkegel, and the Conisch Spaltender Quaternion. From plane triangles with their "circles" and "points," to curvature and geodetics, we find many good illustrations of the power and elegance of quaternions; but of the conically spreading quaternion which transforms a vector into a conical sheet like a Japanese umbrella and then as with a fierce blast turns it inside out or in some other fashion de-axializes the original vector stem:-of this we find no mention. Our disappointment is, however, more than balanced by the real quaternion character of the book as a whole. A great many of the examples

are reproduced, with very slight changes of notation, from Professor Tait's well-known treatise, and, we are sorry to say, reproduced without acknowledgment. In his preface Dr. Molenbroek refers to Professor Tait only by way of criticism, accusing him of finding an integrating factor where none such can be. The truth is that quaternions give a solution where ordinary mathematics fail-although it would probably baffle even a Hamilton to give a geometrical interpretation of log Uq.

Occasionally Dr. Molenbroek advances along a path of his own, as, for example, in his approach to the quaternion equation of quadric surfaces. His transformation of the ordinary Cartesian equation (which is assumed) into quaternion form may well send a shiver down the back of a disciple of Hamilton.

Having neglected (as was pointed out in our former review) to discuss the operator in the Theorie, Dr. Molenbroek has to make a digression in the present volume so as to establish its elementary properties. In the preface he gives as an excuse for the former neglect the statement that Tait devotes only one line to in the theoretical part of his book. Still, Tait has it; and in the 3rd edition (1890) devotes four pages to it in the chapter on Differentation. But even in the very first edition (1867) of his treatise (in which no hard-and-fast line was drawn between the theoretical part and the applications) Tait gives ample evidence of the importance of ; and we still think that a more recent writer, wise in the accumulated experience of his predecessors, should have had something to say concerning the theory of this remarkable differentiating operator. Dr. Molenbroek himself employs it with effect in his discussion of partial differential equations. It plays a conspicuous role in the theory of orthogonal surfaces. It is preeminently the force-function operator. It is the heart of Green's theorem, the key-note of spherical harmonics. But we now learn on Dr. Molenbroek's authority that its value in physical applications is much over-rated (sehr überschätzt). We shall wait with interest his promised demonstration of this statement, which will probably be a unique feature of a succeeding volume. Meanwhile it is interesting to note that the last chapter of the present volume, which discusses with considerable elegance Hamilton's theory of rectilinear rays, ends with the equation, S. 77=0.

It is satisfactory to find in Dr. Molenbroek an enthusiastic admirer of the quaternion and all that it involves. Adhering more or less closely to the methods and notation of Hamilton, he has made an honest endeavour to familiarize German reading students with the beauty and power of the system. We trust that his efforts will be rewarded and that in Leyden, at any rate, will grow up a school of Hamiltonian workers who will effectively help in developing the infinite resources of quaternions,

Physikalisches Praktikum, mit besonderer Berücksichtigung der physikalisch-chemischen Methoden. (Practical Physics, with special reference to physico-chemical methods.) By E. WIEDEMANN and H. EBERT. Second Edition, 1893. Vieweg und Sohn, Brunswick.

DURING the past year the subject of practical physics seems to have received an unusual amount of attention both in Germany and England. Some morths ago we had occasion to notice a new work by Traube dealing with the methods of practical chemical physics, and the present volume bears a very similar title. Until quite recently Kohlrausch's classical text-book was almost universally adopted in German laboratories, but alterations in design of apparatus, changes in methods of teaching, and a necessity for the measurement of constants the importance of which was formerly unrecognized, have combined to make the work appear antiquated in a modern laboratory. The authors have treated the subject in a comprehensive manner as far as description of experiments is concerned; each section is introduced by an epitome of the theory of the experiments to be performed; this is followed by accounts of the experiments, apparatus and requisite manipulation, and when necessary the method of calculating results is indicated. The data of actual experiments are given as examples, and these show the degree of accuracy attainable. It would be an improvement if these data were stated in the way in which a student would be expected to write them down in his note-book, the method of entering results being one of the greatest difficulties experienced by the younger students in a laboratory. As they stand the quantities are merely designated by letters, of which the meanings can only be found by reference to the context.

As the work was written more especially for chemical students, the sections devoted to Electricity and Magnetism are not so complete as the rest of the book; for example the methods of measuring induction-coefficients and the capacity of a condenser are omitted. The rough determination of the specific inductive capacity of a dielectric, and of the resistance of a badly conducting liquid might, however, have been included with advantage.

The lists of apparatus required for each experiment form an excellent feature of the book, and will be of great service to the laboratory assistant. The illustrations are likewise good, especially those in colours, representing the solar spectrum, chromatic aberration, and grating spectra respectively. Indeed, the volume quite maintains the high standard which physicists have previously ascribed to the scientific work of its two authors.

JAMES L, HOWARD,

La Chaleur (Heat). By PIERRE DE HEEN, Professor of Experimental Physics in the University of Liége. Liége: Nierstrasz,

1894.

IN the somewhat remarkable preface to this work the author discusses the two methods of studying science by mere experiment and deduction therefrom, and by assumption of a theory which is afterwards put to the test of crucial experiment, respectively. It is only by the latter method that we succeed in acquiring a knowledge of the causes of things, which is the true aim of physical science. In the ordinary text-books of heat (and indeed of all branches of physics, except perhaps sound) the former method has been generally adopted; a miscellaneous collection of facts is given concerning expansion, the laws of gases, calorimetry, &c., and these lead to a higher analytical or mechanical development of the subject, but no theory is given to unite them into a connected system. Such a connecting link is first met with in the kinetic or molecular theory of gases, by which the behaviour of gases under various conditions of temperature and pressure is completely explained. By a suitable choice of temperature and pressure, however, a gas may be changed to liquid, or a liquid to solid, so that a similar theory must apply to all three states of matter. But as the molecules become crowded together the increasing complexity of their motions defies all calculation, although Van der Waals and others of the same school, to whose influence the present work is doubtless due, have achieved a certain amount of success in the case of liquids near their critical temperatures. Again, if we grant that the properties of liquids and solids are capable of explanation by molecular motion, it follows that such properties will be functions of the molecular weights of the substances. This conclusion has been verified in many cases in which a complete explanation is yet wanting.

In the volume before us heat is regarded as a molecular motion, and its chief phenomena are explained from this point of view as far as is possible in the present state of knowledge. The author assumes a knowledge of elementary phenomena, and occasionally borrows formulæ from thermodynamics, so that his work must be considered as an exposition of molecular physics rather than a text-book of heat. We have found several misprints in the book, notably in connexion with names of authors, for example Clairant, Peluj, Quinke, all of which might be corrected in a future edition. JAMES L. HOWARD,