332 macfarlane: algebra of physics 



b 3 = b 3 . 3/3 and generally h n = b n . n/3. Again since db = 

 d (6 . |8) - db . j8 + bdp . /3 + tt/2 it follows that db 2 = 2bdb, and 

 generally db n = nb n ~ l db. Also d(ab) = adb + bda. As the 

 fundamental properties of the symbol are the same in form as 

 those of the ordinary algebraic symbol, I concluded that the 

 theorems of line algebra remained true generally without change 

 of form in plane algebra. The principle was applied to produce 

 problems in series for examination papers, and some were pro- 

 pounded in the columns of the Educational Times (e.g., Reprint, 45 : 

 28). I asked Professor Tait what he thought of the principle; 

 to which he replied that it was a pretty bold step to take. But 

 if we take conjugate powers and products, no such easy generali- 

 sation is possible. The conjugate product of a arid b has the 

 product of the lengths, but the difference of the angles, ab . — a 

 + /3, giving for the conjugate square bb = b 2 . 0. Also the con- 

 jugate of abc = abc. . a — /3 + y giving bbb = b 3 . /3. It was 

 from this point of vantage that I began to study the space-gener- 

 alisation of algebra. 



Physical arithmetic. Macmillan and Company, 1885. The 

 original title of this book was Arithmetic of Physics. In it I 

 attempted a thoro-going application of the following principle, 

 enunciated by Maxwell (Heat, p. 75). 



Every quantity is expressed by a phrase consisting of two compo- 

 nents, one of these being; the name of a number, and the other the name 

 of a thing of the same kind as the quantity to be expressed, but of a 

 certain magnitude agreed on among men as a standard or unit. 



In the book each quantity is analysed into unit, numerical value, 

 and where necessary, descriptive phrase. This descriptive phrase 

 often has reference to the direction of a line unit. A compound 

 unit is expressed in terms of its elementary units by means of 

 f 'by" and "per." Originally all the reasoning in working out 

 examples was done by means of the expression for the complex 

 unit involved; at the suggestion of Professor Tait I introduced 

 an equivalence method, which I do not now think is an improve- 

 ment. Equivalences are treacherous, especially in space-analysis. 

 The above analysis afterwards led to the idea of analysing each 

 space quantity into modulus and vector-unit, simple or compound. 



