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convention) will puzzle many a student. Prof. Joly soon gives the reason for the negative sign, though
not quite so definitely as might be desirable; but it is A MANUAL OF QUATERNIONS.
questionable if its full significance will be appreciated
until considerable progress has been made in acquiring A Manual of Quaternions. By Prof. Charles Jasper
quaternionic skill. The reader is advised to exercise Joly. F.R.S. (London: Macmillan and Co., Ltd., a strong faith, and to proceed nothing fearing. If he 1905-) Pp. xxvii + 320. Price ios, net.
persevere he will soon get out of the valley of the DROF. C. J. JOLY'S “ Manual of Quaternions" is shadow of the negative sign.
an important addition to the literature of the sub- It is possible that some critic may regard this forject. It at once takes rank with Tait's “ Treatise "assaking of Hamilton's logical basis as a confession of an eminently serviceable exposition of Hamilton's weakness. But this is not so. The weakness is in the great calculus.
average student, for whom a somewhat simple intelHamilton's own works, the “ Lectures" and the
lectual diet must be prepared in the hope that the * Elements," are in their way inimitable. Unfortu- mental digestion may be strengthened sufficiently to nately, their style is not suited to the average student assimilate the strong Hamiltonian food which Prof. eager to acquire a working knowledge of the mathe Joly serves up a little later. The truth is that very few matical method developed in them. Tait alone of the students are able to appreciate to the full an absolutely younger contemporaries of Hamilton seemed to have logical argument until they have a certain amount of been able to appreciate the “ Lectures "; but he him- practical knowledge imparted to them more or less by self used to relate how, as he laboriously read through authority. the first six, he began to despair of his own powers. So far as the principle of the method is concerned, There seemed to be such diffuse discussion, and withal Prof. Joly ranges himself at first on the side of those so little apparent progress. But the seventh lecture vector analysts who neglect the quaternion. But it came like a transformation scene. Every page dis- is only for a couple of pages at the beginning of played new beauties, every paragraph disclosed the chapter ii. On p. 8 the important formula marvellous power and variety of the method. From it Tait drew his inspiration, and proceeded to enlighten
(aß= Saß + Vab) the world as to the meaning and purpose of the is introduced as a definition of the quaternion, and the quaternion.
quaternion is never afterwards lost sight of. Its To the student who has grasped the essentials of fundamental importance and analytic value are in the method Hamilton's second volume, the “ Ele- evidence on every page. It must be admitted that by ments," will always prove a happy hunting ground; this line of approach the reader is rapidly brought but experience has shown that its very completeness into touch with the essential elements of the subject. acts as a deterrent. In the much smaller treatise There is, nevertheless, a certain arbitrariness which written by Tait, the important practical aspects of is not satisfying to the mind, nor is it clear when quaternions are more rapidly though less logically de- | all is done what is really fundamental. A critically veloped, and the chief value of Tait's work lies in his minded student might possibly be inclined to say, Why characteristic treatment of dynamical and physical not define Saß as plus the product of the lengths of the problems. It has been long felt, however, that a good vectors into the cosine of the angle between them, and working manual of quaternions was needed, by use then define the quaternion aß by the formula Vaß– Saß? of which the mathematical student could come quickly | At first sight it seems to amount to the same thing, and into touch with all that is essential in the calculus. yet, as will be found on trial, it leads to a system This is what Prof. Joly has endeavoured to supply. clothed in quaternion garments, but more like the
For reasons clearly explained in the preface, the fabulous ass in the lion's skin than the real lion. author has (reluctantly, he confesses) forsaken the Having thus established in chapter ii. the fundaHamiltonian approach. Instead of developing the cal- mental properties of the quaternion, Prof. Joly rapidly culus logically from the definition of a quaternion as runs over certain important transformations of vector the ratio of two vectors, he defines independently the products and ratios (chapters iii. and iv.), and simple quantities Saß and Vaß, and then writes the product applications to the geometry of the straight line, plane aß as equal to the sum of these two. The student and circle (chapters v. and vi.). Then follow, treated must, of course, take on trust that there is some good in separate chapters, differentiation, linear vector reason for defining Saß as minus the product of the functions, quadric surfaces, and the geometry of curves length of one vector into the length of the projection and surfaces. Here the power of the calculus asserts of the other upon it. This is, at root, the peculiarity itself strongly. Numerous examples are supplied of Hamilton's system which troubled O'Brien nearly throughout for the student to work upon and develop sixty years ago, and has not ceased to trouble occasional | his analytical skill. In subsequent chapters dynamical critics since. There is a kind of notion hovering about problems of various kinds are taken up-such as asin some minds that the positive sign in algebra is more | tatic equilibrium, screws and wrenches, strains, central natural than the negative sign, the truth being, of forces, constrained motion, motion of a rigid body, and course, that the one necessarily implies the other. It the like. A valuable and well arranged chapter on the is to be feared, however, that this apparently arbitrary operator o treats of heterogeneous strain, spherical assumption of the negative sign in translating Saß into harmonics, hydrodynamics, elasticity, electromagordinary trigonometrical notation (Clifford calls it a netic theory, and wave propagation generally. The treatment is by no means superficial, and is in many modes of treatment come and go, most of them being places highly condensed. It is all done in forty-two of little worth. But the medical practitioner is expages, a remarkable testimony to the compactness of pected to know of all these medicinal substances and quaternion notation and the brevity of quaternion vagaries of treatment, and must be prepared to employ proofs. In chapter xvii., on projective geometry, | any one of them at the suggestion of some faddist Prof. Joly gives his own interesting extension, in which who happens to consult him. From this point of view a new interpretation is assigned to the quaternion, and the first book on our list may be a useful guide, but he concludes in chapter xviii. with quaternions otherwise one would be inclined to ask, cui bono? generalised so as to be applicable to space of any Some of the substances included in the volume are number of dimensions.
by no means new, e.g. thyroid, guaiacol, and the antiThere can be no question as to the high merits of toxic sera, while others which have a greater claim the “ Manual of Quaternions.” It is a worthy com to novelty, and are, moreover, of real value, such as panion volume to the master's own great works. Like aspirin, acetozone, urotropine and cystamine, and the “ Elements" of Hamilton and the “ Elementary | purgen, are omitted. In dealing with tetanus antiTreatise ” of Tait, it is characterised by the extra- toxin, no mention is made of injection into the spinal ordinary range of mathematical subjects which comel cord or nerve trunks. As regards phosphorised prinwithin its scope. It is not merely the substitution of ciples, lecithin, glycerophosphates, &c., which have one symbol for three or one for four which makes this of late been extolled in wasting diseases and nervous condensation possible, for that, after all, is a question affections, the administration of a couple of eggs a simply of notation. But the quaternion calculus re day would probably be of far greater benefit than any joices in the possession of two remarkable operators, of the medicinal preparations of these substances. the linear vector function and the vector differen- (2) Messrs. Battle and Corner give a succinct tiator V. They operate singly and in combination account of the anatomy, pathology, symptoms, and according to laws which naturally evolve them- treatment of that common and fashionable malady selves from the fundamental laws of the calculus.appendicitis which may be safely recommended as a They can be linked together in an endless variety of guide for the medical practitioner. The origin and ways, and go far to give to Hamilton's quaternions a function of the vermiform appendix are discussed. flexibility, power, and pictorial compactness not that little blind tubular appendage of the bowel inpossessed by any other general method which is flammation of which gives rise to so much trouble. directly applicable to problems of mathematics pure The appendix has usually been regarded as a vestigial and applied. These features are exquisitely brought
structure and useless in function, but the researches out in Prof. Joly's “ Manual.”
C. G. K.
of Mr. R. Y. A. Berry, of Edinburgh, suggest that it is a specialised mass of lymphoid tissue which the
authors conceive may serve as a defensive mechanism SOME MEDICAL WORKS.
against bacterial invasion in a portion of the bowel (1) New Methods of Treatment. By Dr. Laumonier. where, for anatomical and other reasons, there is a
Translated from the second revised and enlarged delay in the passage of the intestinal contents on-
(3) This work is based on material collected in com(2) The Surgery of the Diseases of the Appendixpilation of the Erasmus Wilson lectures, 1904. The
Vermiformis and their Complications. By W. H. | author states that the main object of his lectures was Battle and E. M. Corner. Pp. xi+208. (London : to direct attention to the identity of the pathological
Constable and Co., Ltd., 1904.) Price 75. 6d. net. I changes concerned in the production of all acute per(3) Clinical and Pathological Observations on Acute forative and gangrenous processes of the alimentary
Abdominal Diseases. (The Erasmus Wilson Lec- tract. He suggests that two extremes of tissue death tures, 1904.) By E. M. Corner. Pp. 98. (London : or necrosis may be recognised, viz. that due to de
Constable and Co., Ltd., 1904.) Price 3s. 6d. net. I privation of blood and that caused by the action of (4) A Short Treatise on Anti-Typhoid Inoculation. micro-organisms. Between these two there are
By Dr. A. E. Wright. Pp. x+76. (London : Con- various grades and admixtures; the former is slow in
stable and Co., Ltd., 1904.) Price 3s. 6d. net. | action, the latter very rapid, and it is this which plays (5) The Suppression of Tuberculosis. By Prof. E. so important a part in abdominal necrosis. The work
von Behring. Authorised translation by Dr. Charles is practically a collection of notes, but is interesting Bolduan. Pp. v+85. (New York : John Wiley reading. and Sons; London : Chapman and Hall, Ltd., 1904.) (4) Prof. Wright has done well to collect into a Price 4s. 6d. net
single volume the various papers, with amplifications. (1) L VERY year' a multitude of substances, chiefly he has from time to time contributed to various
L synthetic, is introduced, every one being journals on the subject of anti-typhoid vaccination. extolled as a certain cure for this or that ailment. The method of preparation of the vaccine, theoretical By good fortune one of them is now and then found and practical considerations as to its use, and statistics to be of real value, and for a time at least finds a of its value are all considered. With regard to the place in the “ aramentaria medica," but the majority | last named, it must be mentioned that some conin a year or two pass into oblivion. Similarly new troversy has taken place in the medical Press as to