138 



NA TURE 



[Dec. 25, 1873 



but the unsatisfactory nature of this mode of treatment is 

 shown by the fact that in all modern books on trigono- 

 metry an angle is represented as generated by motion 

 round an axis in a definite direction. 



There are thus three geometrical quantities having di- 

 rection, and the more than magical power of the method 

 of Quaternions resides in the spell by which these three 

 orders of quantities are brought under the sway of the 

 same system of operators. 



The secret of this spell is twofold, and is symbolised 

 by the vine-tendril and the mason's rule and square. The 

 tendril of the vine teaches us the relation which must be 

 maintained between the positive direction of translation 

 along a line and the positive direction of rotation about 

 that line. When we have not a vine-tendril to guide us, 

 a corkscrew will do as well, or we may use a hop-tendril, 

 provided we look at it not directly, but by reflexion in a 

 mirror. 



The mason's rule teaches us that the symbol, as 

 written on paper, is not a real line, but a mere injunc- 

 tion, commanding us to measure out in a certain direction a 

 vector of a length so many times that of the rule. With- 

 out the rule the symbol would have no definite meaning. 

 Thus the rule is the unit of the Quaternion system, while 

 the square reminds us that the right angle is the unit 

 versor. 



The doctrine of the unit is a necessary part of every 

 exact science, but in Quaternions the application of the 

 same operators to versors, vectors, and areas is utterly un- 

 intelligible without a clear understanding of the function 

 of the unit in the science of measurement. 



Whether, however, it is better to insinuate the true doc- 

 trine into the mind of the student by a graduated series 

 of exercises, or to inculcate it upon him at once by 

 dogmatic statements, is a question which can only be 

 determined by the experience of a new generation, who 

 shall have been born with the extraspatial unit ever 

 present to their consciousness, and whose thoughts, 

 guided by the vine-tendril along the Quaternion path, 

 shall turn always to the right hand, and never to the left. 

 Prof. Kelland tells us in the preface to the work to which 

 we have alluded that, whereas Sir W. R. Hamilton 

 and Prof. Tait have written treatises on Quaternions 

 for mathematicians, the time has come when it 

 behoves some one to write for those who desire 

 to become mathematicians. Whatever, therefore, ad- 

 vanced mathematicians may think of this book, they 

 ought to reserve their judgment as to its difficulty till 

 they have ascertained how it is assimilated by those for 

 whom it is written — those in whom the desire to be- 

 come mathematicians has not yet become alloyed with 

 the consciousness that they are mathematicians. For 

 while Prof. Kelland — as he has elsewhere told us — finds 

 but little difficulty in teaching the elements of the doc- 

 trine of Vectors to his junior classes, Hamilton himself, 

 the great master of the spell, when addressing mathe- 

 maticians of established reputation, found, for his Quater- 

 nions, but few to praise and fewer still to love. 



Prof Kelland, by the clearness and orderliness of his 

 statements, and by boldly getting rid of everything which 

 is unnecessarily abstruse, has done more than any other 

 man towards rendering the subject easy to the student, 

 and reconciling even the case-hardened mathematician to 



the new method, as applied to geometrical questions of 

 old-established truth. 



The other aspect of Quaternions, as a method which 

 every mathematician must learn in order to deal with the 

 questions which the progress of physics brings every day 

 into greater prominence, is hinted at by Prof Tait in the 

 last chapter of the book. He there introduces us to the 

 linear and vector function of the first degree under its 

 kinematical aspect of a homogeneous strain. The im- 

 portance of functions of this kind may be gathered from 

 the fact that a knowledge of their properties supplies the 

 key to the theory of the stresses as well as the strains in 

 solid bodies, and to that of the conduction of heat and elec- 

 tricity in bodies whose properties are different in different 

 directions, to the phenomena exhibited by crystals in the 

 magnetic field, to the thermo-electric properties of crystals, 

 and to other sets of natural phenomena, one or more of 

 which the scientific progress of every year brings before us. 



But as we believe that Prof. Tait is about to bring out 

 a new edition of his treatise on Quaternions, in which 

 this higher aspect of the subject will be brought more pro- 

 minently forward, we reserve our remarks on Quaternions 

 as an instrument of physical research till we have the 

 subject presented to us by Prof Tait in a form which 

 adequately represents its latest developments. 



MARKHAM'S " UNKNOWN REGION" 



7 he Threshold of the Unknown Region. By Clements 

 R. Markham, C.B., F.R.S., Secretary of the Royal 

 Geographical Society, formerly of H.M. Arctic ship 

 Assistance. (London : Sampson Low and Co., i 873). 



HE must be a sorry story-teller who manages to make 

 a traveller's tale uninteresting, especially if the 

 traveller be a voyager, and still more if his voyages have 

 led him into unknown regions. Of all forms of narrative 

 we think it will be generally acknowledged that narratives 

 of discovery are by far the most popular, as is testified by 

 the abundance of this kind of literature, historical and 

 fictitious, provided for the delectation of the young. No 

 doubt this may be largely accounted for by the fact that 

 a discoverer of new lands is continually unveiling the un- 

 known to those who listen to his tale, thereby appealing 

 to one of the strongest and most fruitful characteristics of 

 the human mind, that of curiosity. Every step taken by 

 a discoverer, every knot sailed by his "good ship," we 

 know will lead him among fresh wonders. Once 

 upon a time the Unknown Region — that is, the region 

 unknown to those peoples who have had a thirst for 

 knowledge to any fruitful extent — was in sooth wide 

 enough, when first our Aryan forefathers left their 

 eastern home, and had " all the world before them where 

 to choose." Even four centuries ago the greater part of 

 the earth waited the coming of the European descendants 

 of those primitive discoverers who first turned their faces 

 eagerly and inquisitively to the unknown west. But ever 

 since then the boundary of the Unknown Region has 

 been gradually pushed farther and farther back, until now 

 there remains comparatively little to be found out in order 

 to enable geographers to complete the configuration of the 

 lands of the globe. The extent of our dwelling-place is 

 now pretty well known, though there is yet abundance of 



