Roberval / Infinitesimal Geometry
Roberval (1602-1675) studied algebra, geometry, dynamics and astronomy. With respect to differential and integral calculus, he studied cycloid that was in vogue at that point in time. Cycloid means the locus of the points on circumference of a circle rolling on a straight line.
[Tangent]
The movement of point 
is determined by the parallel motion and rotational motion. The component
of parallel motion is depicted as 
and the same for rotational motion is depicted as 
. Since their respective sizes are the same, the angle bisector 
will be the tangent of cycloid.
[Area]
Roberval used non-separability to calculate the superficial content of the area surrounded by straight lines and cycloid.
Let 
, then 
.
The next step is to equally divide semi-circle 
and segment 
by 
that leads to the following 
,
.
Draw a strait line horizontally from 
, and, in this case, 
is defined as the intersecting point with 
and 
is defined as the intersecting point on the vertical line drawn from 
. Then, the length of the segment could be expressed as 
. The locus of 
will be a 
curve.
Here, we should examine the locus of point 
. When the circle rotates and advances to point 
then point 
comes to the position of point 
. Let a point 
to the left of 
that will be equal in length to that of 
. This point 
will be defined as the position of point 
where the circle verges at point 
. The locus of 
will then becomes a cycloid.
Since the 
curve equally divides the rectangle 
, we have 
and, in 
, the length of 
is equal to length of 
, it becomes non-separable , then 
.
Therefore,
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