This was motivated in the early 17th century by four types of problem :
| Barrow takes another point P' on the curve, and forms PRQ and PMN , which are similar, hence the slope = .
Then when the arc PP' is sufficiently small he identifies it with the line PQ (lack of rigour again !)
y + 2ay + a = kx + ke
He subtracts y = kx to get
2ay + a = ke
The identification of PP/ with PQ is equivalent to discarding nonlinear terms in a and e
(a here), so he concludes 2ay = ke, i.e. = .
Then, since = = he has = .
Since PM is y, he has calculated MN, and hence located the position of N.
With the state of the calculus reaching this point, the stage was set for Newton and Leibniz to develop it into a form recognisable by students today.
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