# Newton's calculus and the binomial theorem

I'm trying to understand the development of the calculus. Does this sound right as one of the stages?

Newton knows the binomial theorem, which gives

$$(x+y)^n={n\choose0}x^ny^0+...\;\;\;\;\;\;\;\;\;\;(1)$$

Letting $$y=δ_x$$ (I realise this is Leizbniz's notation but I find it easier to follow than Newton's $$o$$), we get

$$(x+δ_x)^n={n\choose0}x^nδ_x^0+...\;\;\;\;\;\;\;\;\;\;(2)$$

Considering $$δ_x$$ as the base of a differential triangle under a curve, the vertical of the triangle is given by $$(x+δ_x)^n-x^n$$, which gives us

$$(x+δ_x)^n-x^n={n\choose0}x^nδ_x^0+...-x^n\;\;\;\;\;\;\;\;\;\;(3)$$

But $${n\choose0}x^nδ_x^0=x^n$$, so the first part of the expansion disappears and everything else moves up one place to the left and we get

$$(x+δ_x)^n-x^n={n\choose1}x^{n-1}δ_x^1+...\;\;\;\;\;\;\;\;\;\;(4)$$

Now, the vertical $$(x+δ_x)^n-x^n$$ can be called $$δ_y$$, so now we can write

$$\frac{δ_y}{δ_x}=\frac{(x+δ_x)^n-x^n}{δ_x}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\\ = \frac{1}{δ_x}{n\choose1}x^{n-1}δ_x^1+...\;\;\;\;\;\;\;\;\;\;(5)$$

And now we can use the biniomial theorem to make a differential triangle under any polynomial; and as $$δ_x$$ gets small, the gradient approaches instantaneity.

However, that would also make $$\frac{1}{δ_x}$$ get large, so I guess I've made a mistake somewhere.

Edit: mrtaurho points out that the $$δ_x^1$$ term cancels the $$\frac{1}{δ_x}$$.

• Doesn't the $\delta_x^1$ term cancels with the $\frac1{\delta_x}$ leaving only a constant factor of $1$? However, I don't see why this is posted on History of Science and Mathematics and not on Mathematics. – mrtaurho Aug 21 '19 at 18:59
• @mrtaurho Yes, that seems like it could work. I posted it in HSM because it's a question about the history of the calculus as developed by Newton. – mjc Aug 21 '19 at 19:01
• Still this looks more like a specific issue of mathematical manipulation of equations which doesn't seem to fit within the scope of this site. See here. – mrtaurho Aug 21 '19 at 19:04
• I didn't spot any reference, positive or negative, to equation manipulation on the page you linked. If you think that's frowned upon here, you may be right (I'm fairly new here); my feeling is that the historical component is strong enough to make it out of place in general mathematics. – mjc Aug 21 '19 at 19:10
• If you would have asked: how did Newton resolved this issue (which is not really an issue, anyway) I would say it is on-topic here. Like it is stated right now I would redirect you to the particular site of the stackexchange network designed for such questions: Mahtematics. IMO it is better suited there than here. But that is only my personal opinion. – mrtaurho Aug 21 '19 at 19:16