Explain Johann Bernoulli's postulate: "A quantity which is increased or decreased by an infinitely small quantity is neither increased nor decreased." For example Johann Bernoulli's postulate could...
Explain Johann Bernoulli's postulate: "A quantity which is increased or decreased by an infinitely small quantity is neither increased nor decreased."
For example Johann Bernoulli's postulate could be used to sanction operations such as this paraphrase of Newton's treatment as given in his Quadrature of Curves of 1704:
In the same time that x, by growing, becomes x + o, the power x^3 becomes (x + o)^3, or
and growths, or increments,
o and 3(x^2)o+3xo^2+o^3
are to each other as
1 to 3x^2+3xo+o^2.
Now let the increments vanish, and their last proportion will be 1 to 3x^2, whence the rate of change of x^3 with respect to x is 3x^2.
Explain Johann Bernoulli's postulate.
Johann Bernoulli's postulate is simple: it says that a value does not change if an infinitesimally small value is added to it or subtracted from it. An infinitesimally small value essentially takes the value zero as in comparison to the value to which it is added/subtracted it is so small it may as well be zero.
In Newton's Quadrature of Curves (1704) he uses this idea as a mathematical trick/tool to differentiate `x^3` (giving of course `3x^2)`. This is a key idea in the theory of calculus, that the gradient of a function `f(x)` at ```x` can be calculated as the ratio of the strut and base of a very tiny triangle at `(x,f(x))` of width `o` and height `f'(x) times o`. Taking a very small section of the curve at the point `x` the curve can be theoretically defined as straight and as the hypotenuse of the tiny triangle described. The area under a curve (the integral) is conceived of in a similar way in the theory of calculus. Calculus dates back to Archimedes (200BC), at least.