An Example from Newton’s
‘Method of Fluxions’

(slightly abbreviated)

I The Relation of Flowing Quantities (Fluents) to one another being given, to determine the Relation of their Fluxions.

If the Relation of Flowing Quantities x, y be

x3 – ax2 + axy – y3 = 0

it will be found that

:   : :  3y2 – ax : 3x2 – 2ax + ay.

In moment ‘o’:                            x x + o , y y + o.

Hence

x3 – ax2 + axy – y3

x3 + 3x2 o + 32 o ox + 3o3 – ax2 – 2aox – a2oo +                                      

                                                             axy + aoy + aox + aoo – y3– 3y2o – 3ooy – 3 o = 0.

Expunge   x3 – ax2 + axy – y3 ( = 0 ) and divide through by o:

3x2 + 32 ox + 3o2 – 2aox – a2o + ay + ax + ao – 3y2 – 3oy – 3 = 0.

But whereas o is supposed to be infinitely little, the terms that are multiplied by it will be nothing in respect of the rest. Therefore I reject them:

3x2 – 2aox + ay + ax – 3y2 – 3 = 0

(3x2 – 2ax + ay) + (ax – 3y2) = 0,

whence the Result.

 

Home Archimedes Calculus Takes Form Newton Example Calculus Today