PREFACE. XV 



portions of the path, and from these the actual refractions in 

 passing over these portions are derived by making the total 

 horizontal refraction equal to the amount given by observation. 

 It should be remarked that the above calculation requires an 

 approximate knowledge of the path of the ray, whereas this 

 path is at first unknown, and cannot be accurately determined 

 without a knowledge of the refraction itself. Newton solves 

 the difficulty by an indirect method, making repeated ap- 

 proximations to the form of the path, and thus at length 

 succeeding in satisfying all the required conditions. 



The papers show that the well-known approximate formula 

 for refraction commonly known as Bradley's was really due to 

 Newton. This formula is only applicable when the object is 

 not very near to the horizon, but the method of calculation 

 employed by Newton is equally valid whatever be the apparent 

 zenith distance. 



It is well known that in the Principia Newton determines 

 the form of the solid of least resistance, thus affording the first 

 example of a class of problems which we now solve by means 

 of the Calculus of Variations. He there gives what is equivalent 

 to the differential equation to the curve by the revolution of 

 which the above-named solid is generated, without explaining 

 the method by which he has obtained it. Now among the 

 Newton papers we have found the draft of a letter to a 

 correspondent at Oxford, no doubt Professor David Gregory, 

 in which Newton gives a clear explanation of his method, which 

 is very simple and ingenious. The draft has no date, but from 

 internal evidence it was probably written about 1694. A 

 small part of the letter has perished but it is very easy to restore 

 the missing portion. The letter will be found in the Appendix 

 at the end of this preface. It may be remarked that a similar 

 method is immediately applicable to the problem of finding the 

 line of quickest descent. 



A great many of the Newton papers relate to the dis- 

 pute with Leibnitz about the discovery of Fluxions or the 

 Differential Calculus. They show that Newton's feelings were 

 greatly excited on this subject, and that he considered that 



N. b 



