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VIEWS OF LIMITS AND NUMBERS 77 

CHAPTER VI 

MODERN VIEWS OF LIMITS AND NUMBERS 

LET us try to form a clear idea of the conception which showed 
itself to be fundamental in the principles of the infinitesimal 
calculus, the conception of a limit. 

Notice that the limit of a sequence is a number which is 
already defined. We cannot prove that there is a limit to a 
sequence unless the limit sought is among the numbers already 
defined. Thus, in the system of " numbers " here we must 
refer back to the second chapter consisting of all fractions 
(or ratios), we can say that the sequence (where 1 and 2 are 
written for the ratios ^ and f) 1, l + , ! + + , . . ., has 
a limit (2), but that the sequence 

1, 1+ T V, 1 + T V + T ^,1 +^+1-^+1^, . . .,or 1-4142. . . . 

got by extracting the square root of 2 by the known process 
of decimal arithmetic, has not. In fact, it can be proved that 
there is no ratio such that it is a limit for the above sequence. 
If there were, and it were denoted by "x," we would have 
a; 2 = 2. Here we come again to the question of incommen- 
surables and "irrational numbers." The Greeks were quite 
right in distinguishing so sharply between numbers and 
magnitudes, and it was the tacit, natural, and unjustified 
not, as it happens, incorrect presupposition that the series of 
numbers, completed into the series of what are called "real 
numbers" by "irrational numbers," exactly corresponds to 
the series of points on a straight line. The series of points 
which represents the sequence last named seems undoubtedly 
to possess a limit ; this limiting point was assumed to repre- 
sent some number, and, since it could not represent an integer 
or a ratio, it was said to represent an " irrational number," 
J'2. Another irrational number is that which is represented 
by the incommensurable ratio of the circumference of a circle 
to its diameter. This number is denoted by the Greek letter 
"IT" and its value is nearly 3-1416. . . Of course, the process 
of approximation by decimals never comes to an end. 

The subject of limits forced itself into a very conspicuous 
place in the seventeenth and eighteenth centuries owing to the