152 ROCHESTER ACADEMY OF SCIENCE. [Nov. 12, 



ical magnitudes, into two classes : — Psezidogi'aphs or those symbols 

 which have so far as we consider them no physical property in common 

 with the magnitudes they represent, for example, the figures i, 2, 3, 

 and the letters of the alphabet. A large 2 is no different mathematic- 

 ally from a small 2, nor an italic 2 from a Roman 2. Ideographs, those 

 symbols whose physical properties so far as we consider them may be 

 considered as identical or coextensive with the mathematical proper- 

 ties of the magnitudes they represent. 



In algebra magnitudes have two properties, size and sense of 

 opposition. Thus, -\-a and — a have the same sizes but different senses, 

 corresponding to the sense of debit and credit, or up and down, or 

 right and left, that is, they have the sense of opposition. 



The ideographs for these would be > , < which 



evidently represent two magnitudes of the same size but of opposite 

 senses, that is, of mutual opposition. 



Adopting the convention that normal or + magnitudes shall be 

 represented by strokes headed to the right, negative or opposed mag- 

 nitudes must, of course, be represented by strokes headed to the left. 



So far ideographs have no \'ery great advantage over the familiar 

 pseudographs. The question naturally arises, how about the repre- 

 sentation of ]/(-i) by ideographs? 



Before we can answer this we must ascertain clearly what is the 

 meaning of the symbol ]/. A moment's reflection will show that it 

 is the symbol of an operation to be performed upon its operand which 

 amounts to finding a mean proportion between the operand and unity, 

 such a result that its successive application twice to unity shall give 

 the original operand. Thus, \/ applied to a means the finding of 

 such a quantity (multiplier) that applied twice successively to unity it 

 shall give a as a result. Such a quantity is-j/a, for ]/a(-|/a X i) = «; 

 Y ( 16) := 4, for 4 (4. i) = 16. The test of our result is that a double 

 application of its properties to unity shall produce the original operand. 



Before going farther, too, we must ascertain clearly that multipli- 

 cation is the doing to the nudtiplicand what was done to unity to pro- 

 duce the multiplier. 



We are now prepared to ascertain what ]/(-i ) means when applied 

 to ideographs. To make it more clear, let us phrase the symbol i,/(-i )> 

 viz., find such a symbol that the performance upon it of the operation 

 which produced it from unity, shall give the result -i. 



In ideographs -}-i ^nd — i would be represented by two strokes 

 on the paper mutually opposed, thus, < > . Here the 



