16 Proceedings of the Royal Irish Academy. 



money may be owed by or due to a person : time may be reckoned forwards 



or backwards from a specified date ; liquid may be put into or drawn from 



a cask ; and so on. To distinguish, we measure one group in terms of a unit a, 



the other in terms of a unit /3, and for convenience a and j3 are taken to 



be of the same absolute magnitude. Thus., a mau wishing to concisely state 



the position of his affairs may do so, and make his initial approach to algebra 



by writing down, say: 



7c, 9,3, 11a. 12,3, (2A)a, 



where a is £1 owed to him, /3 £1 owed by him; and by these symbols, written 

 in a line, separated by commas, he means : — I am owed £7, but I owe £9, so 

 my position is financially the same as if I owed £2: but I am owed £11, so 

 I am owed £9 ; but I owe £12, so I owe £3: but I am owed £2 10s., so my 

 position is financially the same as if I owed 10s. A corresponding statement 

 mieht be made in which a would mean one mile walked in a forward direction 

 from an initial position on a road, and /3 a mile walked in a backward 

 direction, and the object of the statement would be, not to find the distance 

 walked, but the distance from the starting-point. Similarly, it might refer 

 to gallons of water put into or taken out of a tank, with the object of finding, 

 not the number of gallons handled, but a number of gallons which, when put 

 into or taken from the tank, as the ease may be, w r ould produce the same 

 result as the series of operations referred to by 7a, 9/3, 11a, 12/3, <25)a. 



Thus the units a, /3 combine as follows : — 7a, 9,3 = 2,3, 7a, 9a = 16a, 

 7,3, 9,3 = 1*3,3, and so on. This operation of combining the units we call 

 addition. 



Xow any calculation like the above may be altered to an equivalent one 

 in several ways. In the first place, the order may be altered in any way, 

 provided all the quantities are taken account of. The truth of this statement 

 may be derived from the assumption that the order is immaterial when we 

 are dealing with two quantities, that is to say, that the same result is obtained 

 by taking account of 7a and 9/3 in the order 7a, 9/3 or in the order 9/8, 7a, and 

 of 7o, 9a in the order 7a, 9a or 9«, 7a, and of 78, 9/3 in the order 7/3, 9/3 or 

 9,3, 7,3. Assuming this to be intuitively true, we can alter the order in which 

 a set of quantities is taken account of to be any whatsoever, by alterations in 

 each of which only two consecutive quantities are affected. For instance, to 

 show that 7a, 9,3, 11a, 12,3 = 11a, 9,3, 12/3, 7a, first alter the order of 9/3, 11a 

 to 11a, 9/9, then the order of 7a, 11a to 11a, 7a, thus bringing 11a to the first 

 position ; then bring 9/3 to the second position, and so on, thus 



7a, 9/3, 11a, 12/3 = 7a, 11a, 9,3. 12,3 = 11a, 7a, 98, 12/3 = 11a, 9/3. 7a, 12/3 

 = 11a, 9/3, 128, 7a. 



In the calculation the quantities are then said to be commutative. 



