Barrow / Calculus
Barrow (1630-1677) had made the great achievements in optics/mathematics by publishing Lectiones Opticae in 1669 and Lectiones Geometriae in 1670. Lectiones Geometriae contained a section explaining that the calculations of tangential method and quadrature method are reversed.
[The Definition of Tangent]
There is a point on a straight line shared with a curved line and all the points on a straight line close enough to that point are on the same side of the curved line.
[Finding Tangents]
Draw a line of tangency at the point 
on the curved line 
and call the point of intersection with a straight line 
a 
, and find the length of the straight line 
.
Let the arc 
shall be infinitely small, draw 
parallel to 
, draw 
parallel to 
. Let 
.
Then
When 
approaches 0, then the position of point 
could be calculated.
This calculation should comply with the stated rules below:
(1)Exclude the term that contains the power of 
or 
, and its product.
(2)Exclude the term that does not contain 
or 
.
(3)Substitute 
for 
, and 
for 
.
(Example)
Draw a line of tangency like: 
[Calculating the area]
A curved line 
(function 
)bearing away from axis 
is given.
(Note)
Pay strict attention because it is different from the present system of coordinates. It is not minus (negative) just because function is subjacent to axis 
.
Constant number 
is introduced in order to adjust the dimension.
Draw curved lines that satisfy the conditions below:
The area of 
The area of 
The area of 
And then plot a point 
on 
that makes the area of 
=
.
Due to these conditions, the straight line 
is now a tangent at point 
. (Proof is omitted)
Let 
,
then, the area of 
----- (1)
Due to the property of tangent,
In modern expression, it becomes:
----- (2)
From the above (1) and (2), it becomes obvious that the calculations to find the tangent and area are reversed.
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