4 A Discourse on the Theory of Fluxions. 
accelerated velocity, and let the increments Gs Pm be gen- 
erated in the same time; let also: Pv be the increment that 
would have 
F G 8 
A P 
described in the same time that Pv and Ps are described, as 
before mentioned; then it is manifest, that this uniform ve- 
locity must be between the velocities at P and m, that is, V 
-++w is greater than V and less than V-+-r, or w is greater 
than o and less than r. Also, since the spaces described in 
the same time are as the velocities, V : Vt-w::Pv : Pm. 
Now in every state of these increments,V ; V--w::Pv : Pm; 
and by contiually diminishing the time, and consequently 
the increments, we diminish r and w, but V remains con- 
stant; it is manifest therefore that the ratio of V : V--w, 
and consequently that of Pv : Pm, continually approaches 
towards a ratio of equality, and when ime, and con- 
sequently the increments, become actually =O, then r=O; 
consequently w=Q; therefore the limit of the ratio of P 
v } Pm becomes that of V; V, a ratio of equality. Hence 
of the ratio of Gs: Pv, or it is Gs: Pv, that ratio being 
constant.” 
_ _ From the foregoing reasoning it is manifest, that the limi- 
ting ratio of the increments expresses accurately the rate of 
increase in the fluent at any assigned point in its generation. 
An example in geometrical quantities of two dimensions may 
be derived from the square, in which the two generating lines 
constitute the limit, The ultimate ratio, then, expressed by 
its usual representatives will be 2x : 1, or, combining the 
