In his Cours de mathématiques générales Bouasse illustrates the idea of infinitely small quantities of various orders via a geometric example involving a circle.
He resorts to a property of the figure ( image below) , namely :
$(1)$ $\overline {BC} \times \overline {BD} = \overline {BA}^2$
$(2)$ $\overline {AE} \times \overline {ED} = \overline {CE}^2$
My question is a basic one and only deals with the explanation of the two above properties. In fact I have no idea of how to justify these geometric statements. Do they have any relaion with the Pythagorean theorem? Do these properties bare a name so that I could look by myself for further information?
As to infinitely small quantities of various orders, his statement is as follows :
"Let's consider AB as basic infinitely small quantity. Let's join the moving point $B$ to the fixed point $D$; and let's look for how the lines $\overline {CA}$, $\overline {CE}$, $\overline {CB}$, $\overline {AE}$ vanish ( tend to zero) , when $B$ tends toward $A$.
First, it is obvious that the arc $\overline {CA}$ and the chord $\overline {CE}$ tend to equal $\overline {AB}$. Therefore, the magnitudes [or quantities] $CA$ and $CE$ are first order quantities [ infinitely small quantities of first order] , since their ratio with the basic infinitely small quantity [namely $AB$] tends toward a finite limit ( which is $1$ in this particular example). "
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