Read The Nature of Mathematics Online

62 THE NATURE OF MATHEMATICS 

And now let us calculate s approximately. Starting from O, 
in the first small interval OH the rectilinear triangle OHK, 
where HK is the ordinate at H, represents approximately the 
space described. In the next small interval HL, where the 
length of HL is equal to that of OH, the space described is 
represented by the rectilinear figure KHLM. The rectangle 
KL is the space passed over the uniform velocity HK in time 
KL; and the triangle KNM is the space passed over by a 
motion in which the velocity increases from zero to MN. 
And so on for other intervals beyond HL. Thus s is ultimately 
given (approximately) as the number of square units in a 
polygon which closely approximates to the figure AOB 

We must now say a few words about the meaning of the 
letters in geometrical and mechanical equations which, following 
Descartes, we use instead of the proportions used by Galileo 
and many of his contemporaries and followers. It seems 
better, when beginning mechanics, to think in proportions, but 
afterwards, for convenience in dealing with the symbolism of 
mathematical data, it is better to think in equations. 

A typical proportion is : Final velocities are to one another 
as the times ; or, in symbols, " V : V : : T : T'." Here "V" 
(for example) is just short for " the velocity attained at the 
end of the period of time " (reckoned from some fixed instant) 
denoted by " T" and V : V, and T : T', are just numbers 
(real numbers) ; and the proportion states the equality of 
these numbers. Hence the proportion is sometimes written 
" V : V' = T : T'" If, now, v is the numerical measure, 

v t 

merely, of V, v' that of V', and so on, we have = ~, or 

v t 

vt' = v't. 

In the last equation, the letters v and t have a mnemonic 
significance, as reminding us that we started from velocities and 
times, but we must carefully avoid the idea that we are 
" multiplying " (or can do so) velocities by times ; what we are 
doing is multiplying the numerical measures of them. People 
who write on geometry and mechanics often say inaccurately, 
simply for shortness, " let s denote the distance, t the time," 
and so on; whereas, by a tacit convention, small italics are 
usually employed to denote numbers. However, in future,