In the Principia, Book 1, Proposition 1, Newton gave a geometrical proof of angular momentum conservation for central forces. A geometrical construction associated with this proof provides a very efficient graphical method to obtain approximate orbital curves for such forces that can be traced with a ruler and a pencil. Newton's geometrical construction also satisfies a discrete form of energy conservation and it is time reversal invariant. An algorithm corresponding to this construction provides an efficient and stable numerical method to integrate the equations of motion of classical mechanics. In the past, this algorithm has been rediscovered several times without recognizing its connection to Newton's geometrical construction.
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The magnitude of the displacement due to the impulse at B is cC = aδt2, where a is the acceleration due to the central force at B, but in Proposition 1, Newton does not specify the magnitude of cC.
In Proposition 1, Newton appealed to Lemma 3, Corollary 4, indicating that at each step of his geometrical construction the magnitude of the impulses are determined by a given continuous orbital curve.
The radius of curvature of orbital curves is discussed in Proposition 6, and in Lemma 11, Principia, Book 1.
As a student at Trinity College, Cambridge University, Newton became familiar with Frans van Schooten's Latin edition of Descartes Geometrie that describes geometry in an algebraic form. He neglected to study Euclidian geometry which he regarded as trivial, until Isaac Barrow, during an examination, emphasized to him its importance.
In 1685 Newton sent to the Royal Society an early draft of the Principia entitled De Motu Corporum Gyratum.
