VOL. XXXIl.] PHILOSOPHICAL TKANSACTIONS. 58C) 



to be quantities of any kind, whose magnitudes are analogous to the magnitudes 

 of" the angles. Such may be the arcs or sectors of any circle, or any other 

 quantities of time, velocity, or resistance, analogous to the magnitudes of the 

 angles. Every system of these measures has likewise its modulus homogeneous 

 to the measures in that system, and may be computed by the trigonometrical 

 canon of sines and tangents, just as the measures of ratios by the canon of 

 logarithms ; for the given modulus in each system, bears the same proportion 

 to the measure of any given angle, as the radius of a circle bears to an arc 

 which subtends that angle, or the same as this constant number of degrees 

 57.2957795130 bears to the number of degrees in the said angle. 



Why the author takes his principles to be so general, will further appear by 

 an instance or two. In the problem already mentioned he measures the ratio of 

 the air's densities in any altitudes, by the altitudes themselves, making use of 

 the altitude of a uniform atmosphere for the modulus. So likewise when he 

 considers the velocities acquired, and the spaces described in given times, by a 

 body projected upwards or downwards in a resisting medium with any given 

 velocity ; he shows, that the times of descent, added to a given time, are the 

 measures of ratios, to a given modulus of time, whose terms are the sum and 

 difference of the ultimate velocity and the present velocities that are acquired : 

 that the times of ascent, taken from a given time, are the measures of angles, to 

 a given modulus of time, whose radius is to their tangents, in the ratio of the 

 ultimate velocity to the present velocities : and lastly, that the spaces described 

 in descent or ascent, are the measures of ratios to a given modulus of space, 

 whose terms are the absolute accelerating and retarding forces arising from 

 gravity and resistance, taken together at the beginning and end of those 



spaces. 



This general account may suffice to illustrate what I am going to say ; that 

 since the magnitudes of ratios, as well as their terms, may be expounded by 

 quantities of any kind, the mathematician is at liberty on all occasions to chuse 

 those which are fittest for his purpose ; and such are they without doubt, that 

 are put into his hand by the conditions of the problem. He may indeed repre- 

 sent these quantities by an hyperbola, or any other logometrical system, were 

 not his purpose answered with greater simplicity by the very system itself, which 

 occurs in each particular problem. And the same may be said for the systems 

 of angular measures, instead of recurring on all occasions to elliptical or cir- 

 cular areas. 



As to the convenience of calculating from our author*s constructions, he 

 shows that the measures of any ratios or angles, are always computed in the 

 same uniform way; by taking from the tables the logarithm of the ratio, or the 



