Dec. 15, 1887] 



NATURE 



147 



reproduced also by M. Balbin at pp. 136, 138, 247. 

 Dr. Gracfc, like other German authors on quaternions, 

 reproduces a great part of the " Introduction to (2:uter- 

 nions " by Kelland and Tait, and also some parts of the 

 " Elementary Treatise " by Prof. Tait ; but after having 

 once pronounced the name of Hamilton, he has done all 

 in the matter of acknowledgment, and the name of Tait is 

 not to be found in the little volume. 



We now come to the fourth class of subjects treated 

 by our Argentine author. This comprises kinematical, 

 statical, and dynamical questions. Here we meet with 

 the treatment, in good form, of questions included in 

 Hamilton's " Elements," and in the second edition (1873) 

 of Prof. Tait's " Elementary Treatise." Of this last source 

 of information our author seems to have only a second- 

 hand knowledge : he reproduces verbatim the contents of 

 § 405 of the " Elementary Treatise" (second edition), but 

 he attributes the authorship of it to M. Laisant. Evi- 

 dently, M. Laisant reproduced this § 405, which treats of 

 Foucault's pendulum, but the origin of the treatment is to 

 be found in the Proceedings of the Royal Society of 

 Edinburgh of 1869, aiictore P. G. Tait. Again, by the 

 small-print note at p. 303 we have another indication that 

 our author was unacquainted with the contents of the two 

 or three last chapters in the second edition of the 

 "Treatise." Had he known them, he could not have 

 withheld a more special acknowledgment of results worked 

 out by the immediate follower of Hamilton. 



Prof. Tait certainly can claim to have been the first to 

 make quaternions intelligible, not alone to ordinary 

 students, but to advanced mathematicians — " such as 

 liavc the [rare] gift of putting an entirely new physical 

 question into symbols." But the Edinburgh Professor 

 has particular claims to the thankfulness of students of 

 the first-named category (the writer amongst them), for, 

 under the plea of teaching the quaternion method, he has 

 given them an insight into those physico-mathematical 

 questions which are so unapproachable when obscured 

 by the apparatus of Cartesian co-ordinates. When these 

 questions are expressed and solved in quaternion lan- 

 guage, they acquire a clearness and a conciseness which 

 might well astonish their original proposers — Green, 

 let U3 say, Ampere, Poinsot, even Newton, not to name 

 living workers. We cannot be expected to enumerate 

 the list of the questions treated ; we will allude only to 

 those in which the operator v is pressed into services of 

 such marvellous fecundity, to those in which the hnear 

 vector-functions play an eminent role, and to those in 

 which the operator of conical rotation is such a powerful 

 auxiliary. 



The last chapter of the volume contains a painstaking 

 record of the history of quaternions. The English reader 

 will find much of this, and even more, in the article on 

 " quaternions" in the " Encyclopaedia Britannica." We 

 may say that the imaginaries of algebra having done 

 good service during the process of discovery, can be safely 

 now banished from the principles and practice of the qua- 

 ternion method — unless bi-quaternions are under treatment. 

 In the ordinary applications of the method the extraction 

 of the square root of the members of an equation such as 

 «- = — I (e being a unit-vector) is looked upon as imprac- 

 ticable, and the reason is clearly this : the combination 

 ef, represented by «-, is a symbol sui generis just as 



much as e itself, and cannot be decomposed or attacked 

 — to speak the language of chemistry — by the algebraical 

 operation of extracting the square root of it. To assimi- 

 late a unit-vector with J — i, the square root of negative 

 unity, is as if, in the differential calculus, one were to 



. . dy o 



assimilate a derivate, .„, with the symbol — of inde- 



termination. We cannot resist the temptation of helping 

 our author to preserve a little curiosity in the history of 

 the subject. The author records the verdict of an un- 

 named French mathematician, who says : " Quaternions 

 have no sense in them, and to try to find for them a 

 geometrical interpretation is as if one were to turn out 

 a well-rounded phrase, and were afterwards to bethink 

 oneself about the meaning to be put into the words. . . ." 

 This, after all, is rivalled by the verdict .of a German 

 mathematician, who simply declared the quaternion 

 method to be "eine Verirrung des menschlichen Geistes" 

 (an aberration of the human intellect). 



GUSTAVE PLARR. 



CABLE-LA YING. 



On a Surf- bound Coast j or, Cable-laying in the African 

 Tropics. By Archer P. Crouch, B.A. O.xon. (London : 

 Sampson Low, 1887.) 



IT is somewhat remarkable that the business of making 

 and laying submarine telegraph cables — which 

 hitherto has been a monopoly of Great Britain, and em- 

 ploys large numbers of skilled workmen of all kinds, of 

 scientific men, and of sailors — should be so little under- 

 stood by people not directly connected with it. Yet the 

 daily history of any cable-laying expedition, if faithfully 

 written, would contain matter of engrossing interest for 

 all readers. To secure a contract on advantageous terms 

 requires diplomatic talent of a high order. For, althougb. 

 the business is a British monopoly and there is no com- 

 petition with the foreigner, there is all the keener com- 

 petition between the rival British companies. Further, 

 the negotiations are almost always with Government 

 departments, either home, colonial, or foreign, and are 

 necessarily of a delicate character. In the history of any 

 particular cable the preliminary diplomatic details would 

 no doubt have by far the greatest interest for most readers, 

 but it would be obviously indiscreet and unadvisable to 

 publish them. In tendering for a cable against powerful 

 competitors it is important to have as accurate a know- 

 ledge as possible of the depth of water and the nature of 

 the bottom where the cable is to lie, in order to know 

 exactly the lengths of the different types of cable which 

 will have to be employed, and so to estimate the cost. 

 In obtaining this knowledge the cable-laying companies 

 have been the chief contributors to the science of deep- 

 sea research, or oceanography. The contract obtained, 

 the cable made, and the route determined on, the opera- 

 tion of laying has to be undertaken. When it is merely 

 a question of laying a length of cable between two 

 points over smooth ground, this is in most cases a very 

 simple affair ; although if the shore-ends of the cable have 

 to belauded on exposed beaches, as is only too often the 

 case, there is plenty of opportunity for thrilling incident 

 and hair-breadth escape. The expedition in which Mr. 



