and on Algebra as the Science of Pure Time. 375 
Thus, the power 2” and the root a increase constantly together, when both are 
positive ratios. 
The logic of this last deduction, of the Corollary (275.) from the Lemma (272.), 
must not be confounded with that erroneous form of argument which infers the truth 
of the antecedent of a true hypothetical proposition from the truth of the conse- 
quent ; that is, with the too common sophism: If a be true then B is true; but B is 
true, therefore a is true. The Lemma (272.) asserts three hypothetical propositions, 
which are tacitly supposed to be each transformed, or logically converted, according 
to this valid principle, that the falsehood of the consequent of a true hypothetical 
proposition infers the falsehood of the antecedent ; or according to this just formula : 
If a were true then B would be true ; but 2 is false, therefore a is not true. Ap- 
plying this just principle to each of the three hypothetical propositions of the Lemma, 
we are entitled to infer, by the general principles of Logic, these three converse 
hypothetical propositions : 
i ener unease as 
ify” > 2, then » > @; 
i (276.) 
if y™ ¢ v”, then y £7; 
x and y being here any positive ratios, and m any positive whole number, and the 
signs > <¢ denoting respectively “not >” and ‘‘not <” as the sign + denotes 
“not =”. And if, to the propositions (276.), we join this principle of intuition in 
Algebra, as the Science of Pure Time, that a variable moment B must either follow, 
or coincide with, or precede a given or variable moment a, but cannot do two of 
these three things at once, and therefore (by the 21st article) that a variable ratio 
must also bear one but only one of these three ordinal relations to a given or yariable 
ratio 2, which shows that 
when .y” > 2”, then:y "= 2.” andy." ¢ a", 
when 1% = 2", then 44 a" and y= > 2™, 
when y”™ < a”; then y™ } @™ and y™ + 2”, 
and that 
when y ¢ @ and y } @, then y = 2, 
when y= 2@ and y <¢ 2, then y >a, 
(278.) 
when y > @ and ya, then y < 2, 
_ we find finally that the Corollary (275.) is true. The same logic was tacitly em- 
ployed in deducing the Corollary of the Ist Lemma, in the hope that it would be 
mentally supplied by the attentive reader. It has now been stated expressly, lest any 
