NATURE 



[December 20, 1894. 



It is almost needless to say that Mr. Heaviside does 

 not believe in action at a distance, that he regards energy 

 as baing continuous in space, and as moving as matter, 

 and that he treats ether as an entity, and not as a work- 

 ing hypothesis. By the way, in discussing ether, as to 

 whether it is stationary or not, stationary is generally 

 taken to mean relatively to the earth which is honoured 

 with our existence, or at least with regard to the sun 

 which is to give us light. But if motion is considered 

 with reference to infinite distances, the chances are that 

 the ether moves pjst us at a speed in comparison with 

 which '' is infinitesimal. Mr. Heaviside hopes that in 

 the future the young will be trained up to believe in ether 

 as a thing, and will therefore believe in it ; but this would 

 be a sort of religion rather than knowledge. No one 

 doubts that electrical disturbances are propagated at a 

 finite speed, and matter, with its inconvenient properties 

 removed or altered, provides a convenient working hypo- 

 thesis ; but to talk of the inconceivable as existing, is 

 using words to which no concepts belong. As most 

 people agree with Mr. Heaviside on these matters, how- 

 ever, it may be as well to say no more in a review. 

 Dealing with the medium, or rather its states, Mr. 

 Heaviside gets rid of the potential treatment. To him 

 induction and its change is of primary importance, and 

 potential is a mere derivative of it. The idea of induc- 

 tion as the essential and potential as derived is less 

 common with the academical than the practical elec- 

 trician, who also uses the notion of lines of induction. 



This treatise is remarkable, among other things, in 

 beginning almost at once with the propagation of dis- 

 turbances at the speed of light. The author hopes that 

 text-books on light will soon discuss electricity at the 

 beginning instead of at the end. He certainly 

 sets a good example by beginning a book on electro- 

 magnetism with the propagation of disturbances in time. 

 By the way, he regards chemistry as an ur.mathematical 

 science ; it is to be hoped chemistry books will soon 

 begin with thermodynamics and electricity, so as to lay 

 an engineering foundation for the study of chemical 

 action. 



Mr. Heaviside is, as is well known, a determined 

 opponent of the use of quaternions in physics, and an 

 equally strong advocate of the use of vectors; and a 

 long chapter is devoted to the " Elements of Vectorial 

 Analysis,' taking more than a third of the book. In 

 quaternions, vector products have a part at right angles 

 to both the vectors; theideasthusfittedelectromagnetism, 

 and Maxwell availed himself of the conveniences of 

 quaternion notation, and, to some extent, of quaternion 

 ideas. The relations between vectors in quaternions are 

 purely conventional, while in electricity they are physical 

 in one sense, though in another they may be due to 

 conventionalities of definition. The idea of the direction 

 of a current llowing along in a wire was derived from the 

 flow of water in a pipe, and it is possible that we might 

 have so defined electrical and magnetic quantities, and so 

 thought of them, that nothing corresponding to vector 

 products or quotients came in when passing from one to 

 the other of the electric and magnetic systems. Mr. 

 Heaviside objects altogether to quaternions in physics, 

 but does not differentiate clearly between vector and 

 NO. 131 2, VOL. 51] 



quaternion analysis, and professes that he does not or 

 cannot understand quaternions. It is not likely he can- 

 not. Perhaps he won't. One difficulty, in the way of 

 students at least, is due to writers on quaternions defin- 

 ing something that is not adding as addition, and some- 

 thing that is not multiplying as multiplication, and to 

 their removing the operand and treating the operator as 

 a quantity. This leads to S::^ being negative, to the 

 square of a vector being negative, and to the reciprocal 

 of a vector being taken in the opposite direction. When 

 an eminent scientific writer recently found, by dividing 

 the value of dy dx byj' that d dx was equal to 62S, some 

 wrongly thought he did not understand the principles of 

 the calculus ; but he was only doing in figures what is done 

 in letters in many branches of mathematics. Mr. Heavi- 

 side starts off with a definition of the " product " of two 

 vectors. The scalar part is positive, and the vector part 

 is as in quaternions, but there is no idea of the multiplier 

 rotating the multiplicand, though he gives no reason wh\- 

 the multiplicand need not be looked upon as turned 

 through a right angle. It may be asked how Mr. Heaviside 

 avoids quaternions. Using the word in one of its 

 many senses as the operator necessary to change a vector 

 into another, he avoids the difficulty, for the present, 

 by not dividing. Surely if vectors are to be multiplied 

 they must also be divided. If we have the induction 

 and current at an angle, we can find the force ; is it not 

 as reasonable to find the induction or current if the force 

 and one of them is given ? Perhaps Mr. Heaviside may 

 devise a new quotient or operator which will do this. If 

 ai3 = y we might expect that y'ii = a. This is not so in 

 quaternions, because the scalar part of aji is lost, and 

 the quaternion 7,^ gives no scalar part. To recover <i 

 there might be a term Safiji. Perhaps Mr. 

 Heaviside will give his own ideas about division in his 

 next volume. Meanwhile, though he avoids the ideas 

 of turning, every vector multiplier is just as much a 

 quaternion as any in Hamilton or Tait.as far as the versor 

 is concerned. A quaternion, though sometimes called 

 an operator, is really two operators. Mr. Heaviside 

 admits quaternions can be developed from his defi- 

 nitions. He also finds it difticult to think of energy dis- 

 appearing in one place and appearing elsewhere without 

 passing through intermediate space ; surely then he can 

 look upon a vector disappearing and reappearing in a 

 new direction, and of a new length, as having passed 

 through intermediate positions and lengths. He then 

 gets the idea of roots of quaternions, ^/ I~i, and so 

 on, without complicating vector analysis, and has a 

 system which will do all his vector algebra well, which 

 makes sense of imaginaries and exponential values of 

 sines and cosines, which does not involve the study of 

 new symbols or ideas, and which is already worked out : 

 in short, quaternions. He finds difficulty in knowing 

 when a vector is a vector, and when a quaternion. The 

 answer is: it is a vector when it is a quantity, and the 

 versor of a quaternion when it is an operator. There 

 would be no difficulty if peopledid not confuse an operator 

 with the ([uantity that specifies it. Confusion, which 

 is common to Mr. Heaviside's vector algebra, may 

 come in between the scalar and vector part of the 

 product of two vectors. He continually falls back 



