Jakob Bernoulli’s solution I'm reading this.
It says,
from $ds^2 = dx^2 + dy^2$, we can get
$2dsd^2s =2dxd^2x $
I cannot understand this process. 
Please give me some references.(simpler is better)

If one feels uncomfortable with differentials it is possible to explain with ordinary derivatives.
Let's divide $ds^2 = dx^2 + dy^2$ by $dy^2$ to get
$$
\left(\frac{ds}{dy}\right)^2=\left(\frac{dx}{dy}\right)^2+1\qquad\Leftrightarrow\qquad (s'(y))^2=(x'(y))^2+1.
$$
Now differentiate both sides w.r.t. $y$ using the chain rule
$$
2s'\cdot s''=2x'\cdot x''\qquad\Leftrightarrow\qquad 2\frac{ds}{dy}\frac{d^2s}{dy^2}=2\frac{dx}{dy}\frac{d^2x}{dy^2}.
$$
Multiplying by the denominator we obtain $2ds\cdot d^2s=2dx\cdot d^2x$
