506 HUGH MACCOLL : 4. Similarly we may define the class of infinitesimal ratios, a class which we will here denote by h. Just as every infinite number or ratio H is too large to be expressible, either exactly or approximately, solely in terms of any finite number F, or finite numbers F lf F 2 , etc., so every infinites- imal ratio h is too small to be expressible, either exactly or approximately, solely in terms of any finite number or ratio F, or finite numbers F lf F 2 , etc. Thus, just as M M , though inconceivably large, is still not infinite, so its reciprocal 1 -i- M M , though inconceivably small, is still not infinitesimal. 5. Using the symbol A* to assert that the number or ratio A belongs to the class x, where x may stand for F or H or h, these conventions or definitions give us several evident for- mulae, of which the following are a few : (1) (FH) H ; (2) (F*)*; (3) ; (4) ; (5) (6) (A)*; (7) (F^) F ; (8) (HF) H ; (9) H, + F - H 3 ; (10) 5 -H,; (H)(f) H ; (12) (g)*. What leads to much confusion is the fact that mathe- maticians also use the word infinity and the symbol x to denote 1 2 such expressions as -, -, etc., which represent no real ratios at all but pure non-existences, such as in my two preceding ar- ticles I have denoted by the symbol 0. But as these pseudo- 1 2 ratios -, -, etc., form a different class of non-existences from the pseudo-ratios -, -, etc., it would be convenient in mathe- 1 !_ matical reasoning to restrict the symbol to the latter class, and the symbol x to the former. Thus, the symbol repre- sents, as it were, the death of a real infinitesimal ratio h in passing from the positive to the negative state, or vice versa ; while the other non-existence symbol x (which may be called pseudo-infinity} similarly represents, as it were, the death of a real infinite ratio H, in making the same transition. We shall then get the following self-evident formulae, in which (to prevent ambiguity) the symbol :: will be used (instead of =) to assert equivalence of propositions, not equivalence of ratios. (17) (Fx )- ; (18) (F0); (19) (x F) 00 ; (20) (?)*; (21) (J)!
