VOL. XXIX.] PHILOSOPHICAL TRANSACTIONS. 211 



A general Solution of a Problem concerning Curves, formerly proposed in the 

 Leipzig Acts. N° 347, p. 399. Translated from the Latin. 



In the Acta Eruditorum for Oct. 1698, p. 471, Mr. John Bernoulli writes 

 thus: " At length I have discovered the general method 1 wished for, for regu- 

 larly cutting curves that are given in position, whether algebraical or transcen- 

 dental, in an angle either right or oblique, whether invariable or varying accord- 

 ing to a given law; to which, in the opinion of M. Leibnitz, nothing can be 

 added for its further perfection, and for this reason, that it always leads to an 

 equation; in which, though the indeterminate quantities be sometimes insepa- 

 rable, the method is not the less perfect for that; for it belongs not to this, 

 but to some other method to separate them. I entreat my brother therefore to 

 try his skill in a matter of so much moment. And he will not repent of his 

 labour, if he happen to be successful. I know he will then forsake the method 

 he is now so fond of, which can only be applied on very few occasions." 



These three great men, for 4 or 5 years, had been in the habit of exercising 

 one another, in proposing and solving such kind of problems. Without the 

 spirit of divination, it would be difficult to give the very same solution as that 

 of M. Bernoulli. But it may suffice that the following solution is general, and 

 always leads to an equation. The problem is as follows. 



Problem. — Required a general method of finding a series of curves, which 

 shall cut at a given angle, or in an angle varying by a given law, curves that 

 are constituted in any other given series. 



Solution. — ^The nature of the curves to be cut gives the tangents of the same 

 at any points of intersection; and the angles of intersection give the perpendi- 

 culars of the cutting curves ; and two perpendiculars coinciding, by their last 

 concourse, give the centre of curvature of the cutting curve at the point of 

 any intersection. Let an absciss then be drawn in any convenient position, and 

 let its fluxion be unity; then the position of the perpendicular will give the first 

 fluxion of the ordinate belonging to the required curve; and the curvature of 

 this curve will give the second fluxion of the same ordinate. And thus the 

 problem will always be reduced to equations, o. e. d. 



Scholium. — It does not belong to this, but to another method, to reduce the 

 equations, and separate the indeterminate quantities, absolutely if it can be done, 

 if not, by infinite series. As this problem, however, is hardly of any use, it 

 has therefore remained neglected and unsolved for many years, in the Acta 

 Eruditorum. And for the same reason 1 shall not prosecute its solution any 

 further.* 



* These solutions seem to resemble mostly the composition of Sir I. Newton. 



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