VOL. XLII.] PHILOSOPHICAL TKANSACTIONS. 637 



In the 6th chapter, the nature and properties of logarithms are described 

 after the celebrated inventor ; and it is observed, that he made use of the very 

 terms fluxiis and fluat on this occasion. A line is said to increase or decrease 

 proportionally, when the velocity of the point, that describes it. is always as 

 its distance from a certain term of the line ; and if in the mean time another 

 point describes a line with a certain uniform motion, the space described by the 

 latter point is always the logarithm of the distance of the former from the given 

 term. Hence the fluxion of this distance is to the fluxion of its logarithm, as 

 that distance is to an invariable line ; and the fluxions of the quantities that 

 have their logarithms in an invariable ratio, are to each other in a ratio com- 

 pounded of this invariable ratio, and of the ratio of the quantities themselves. 

 Some propositions are demonstrated, that relate to the computation of lo- 

 garithms ; but this subject is prosecuted further in the second book. The lo- 

 garithmic curve is here described, with the analogy between logarithms and 

 hyperbolic ratios. 



In the 7 th chapter, after a general definition of tangents, it is demonstrated, 

 that the fluxions of the base, ordinate, and curve, are in the same proportion 

 to each other, as the sides of a triangle respectively parallel to the base, ordi- 

 nate, and tangent. When the base is supposed to flow uniformly, if the curve 

 be convex towards the base, the ordinate and curve increase with accelerated 

 motions ; but their fluxions at any term are the same as if the point which de- 

 scribes the curve had proceeded uniformly from that term in the tangent there. 

 Any further increment which the ordinate or curve acquires, is to be imputed 

 to the acceleration of tlie motions with which they flow. A ray that revolves 

 about a given centre, being supposed to meet any curve and an arc of a circle 

 described from the same centre, the fluxions of the ray, curve, and circular 

 arc, are compared together ; and several other propositions concerning tangents 

 are demonstrated from the axioms. The next chapter treats of the fluxions of 

 curve surfaces in a similar manner. 



The Qth chapter treats chiefly of the greatest and least ordinates of figures, 

 and of the points of contrary flexure and cuspids. The fluxion of the base 

 being given, when the fluxion of the ordinate vanishes, the tangent becomes 

 parallel to the base, and the ordinate most commonly is a maximum or mini- 

 mum, according to the rule given by authors on this subject. But if the second 

 fluxion of the ordinate vanish at the same time, and the third fluxion be real, 

 this rule does not hold, for the ordinate is in that case neither a maximum nor 

 minimum. If the first, second, and third fluxions vanish, and the fourth 

 fluxion be real, the ordinate is a maximum or minimum. The general rule 



