VOL. XXXI.J PHILOSOPHICAL TRANSACTIONS. 405 



description of all geometric lines of the third, or any order whatever. But as 

 the higher kinds can be described only by means of the inferior sorts, some of 

 these must be postulated to describe those: and as straight lines are the simplest 

 and most easily described, and are always the same, that is, of one sort, there- 

 fore it was thought proper to investigate the use which they alone might be of, 

 for describing lines of all the higher orders, in the first part of this treatise; an 

 abstract of which has been published in the Transactions, N° 359. I shall 

 only add., that besides the method of describing the curves, the manner of de- 

 termining their asymptotes and species is also demonstrated ; and the more 

 simple curves of every order are particularly considered as examples of the 

 method. In the first section, the lines of the second order are considered ; in 

 the second, those of the third order, that have a punctum duplex ; in the 3d 

 section, the lines of the fourth order, and those of the third order that have no 

 punctum duplex. In the last section there are many various methods of de- 

 scribing the lines of any order. 



In the second part, the curves of the inferior orders are used for describing 

 those of the higher kinds. In the first section, the theorems published by Sir 

 Isaac Newton at the end of the enumeration of the lines of the third order 

 are demonstrated. In the second section, curves are substituted instead of 

 straight lines, in all the propositions of the first part. From one of these pro- 

 positions, lines of the 1024th order may be described by making angles move 

 on 7 conic sections; and by 3 conic sections more, lines may be described above 

 the 11,000th order. Lastly, these theorems are applied to show how the more 

 complex of the infinite order may be described from the more simple. 



In the third section, some other methods of describing curves are considered, 

 that are not so general as the preceding, but give sometimes more simple 

 methods of describing some few lines of the superior orders. Particularly the 

 epicycloids described by the motion of any curve, whether geometric or not, on 

 another equal to it, are easily constructed, and several infinite series of them 

 rectified or measured by arches of more simple curves. In this section, several 

 other descriptions of curves are treated of, that have been proposed by others. 

 In the last section, to show the use of curves in natural philosophy, two of the 

 most eminent problems in mathematical philosophy are solved. In the first, the 

 centripetal force, by which a body describes any curve, is investigated after an 

 easy manner ; and a simple construction of all those curves which a body would 

 describe, if projected with the velocity it might acquire by falling from an in- 

 finite height, in any hypothesis of gravity, is demonstrated. In the second, it 

 is found, that if any body describe a curve in a resisting medium, the resistance 

 is always as the iifioment or fluxion of a quantity, that expresses the ratio of the 



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