1815.] On the Doctrine of Fluxions, 32» 



which is wanted. He will now understand that if the line F H 

 were to proceed for ever of the same length, it would generate two 

 parallelograms which would have always to one another the rate of 5 

 to 1. If, therefore, x represent the base or the perpendicular of 

 the triangle, the rate will be as x to 1. , . ,• i i 



2. With regard to the notation, if x and 1 be both multiplied by 

 any quantity whatever, their rate will not be changed; insteiid of a? 

 to I, therefore, we may employ x i to 1 x, x being any line more 

 than nothing, and less than infinite : x x, then, is the fluxion of 

 the triangle, and 1 .v the corresponding fluxion of the parallelogram ; 

 and as the triangle is the half of a square, the fluxion of a square 

 whose side is x is to the fluxion of a parallelogram whose side is I 

 as 2 X i' : 1 v. 



If now the following series of fluents be set down, the learner 

 will easily continue the fluxions. 



Fluents. Fluxions. 



1 0, for 1 being invariable has no fluxion. 



X .....1* 



XX ^xx 



XXX 3X1. T 



X* ^x*-' X 



x" n x""' .t* 



A learner who sees in the series of the fluxions above the two 

 laws of the three first terms, that of the numeral coefficients, and 

 that of the letters, will be able to continue the series to any length, 

 and to give the general expression n x""' v as the fluxion of x"; 

 because he observes that the numeral coefficients increase by unity, 

 and that tliere are as many letters in the fluxion as in the corre- 

 sponding fluent, with the last letter always dotted. 



If the result be expressed in words, we have the following rule 

 for finding the fluxion of any power of a variable quantity. 



Multiply the fluxion of the root by the exponent of the power, 

 and the product by that power of the same root whose exponent is 

 less by unity than the given exponent. 



By this rule the fluxion of x» is — x" .v; of x » is — — 



x~ • ~ ' i ; of x" y" is m if x"-' x + n x" if - ' >, when both x and 

 y vary, by considering first x as variable, and then 7/ as also 

 variable; of-;^ = x" w" " is -•? -,, • 



From all that has been said, the fluxional or dilfcrcntial calculus 

 may, in the case of one variable quantity, be defined a method for 

 finding the rate of change in a quantity, and its depciidance or 

 function. Thus the rate of change in x and its function x", is as 

 * : n x'~' .V, or as 1 : nx"~'. 



As this is not a treatise, hut a short essay, T say nothing of second 

 fluxions, which bear the same relation to fust fluxions that first 



