A SOLUTION OF PROBLEM IX. OF THE 

 FIRST BOOK OF NEWTON'S PRINCIPIA*. 



TIIE fundamental principle of the proof I gave Dr Goodwin f 

 is, that as the time varies as the increment of area, or as the 

 square of the radius into the increment of angle, the force varies 

 as the increment of angle, so that the acceleration in an instant 



dt is equal to dd. The following application of this principle 



is perhaps more simple. Let "be the centre of force, P a 

 position of the moving body. Take PR at right angles to PO 



and equal to ~dt. Join HP', P' being the position of the body 



at the next instant. Similarly draw P'R at right angles to 

 PO and equal to PR. Join R'P", P" being a third position 

 of the body. Draw RQ parallel to RP'. Therefore the angle 

 at R being equal to the increment of the angle vector, QP is 



* Dictated to the Editor of this volume and now first published, 

 t Goodwin's Course of Mathematics. 



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