588 PHILOSOPHICAL TRANSACTIONS. [aNNO 1/22. 



hyperbola, the given parallelogram, contained by an ordinate to the asymptote 

 and the absciss from the centre, is the modulus of that system : and in other 

 systems, the modulus is generally some remarkable quantity. In the second 

 proposition he gives a concise uncommon method for calculating Briggs's Canon 

 of Logarithms; with rules for finding intermediate logarithms and numbers, 

 even beyond the limits of the canon. In the 3d proposition he constructs any 

 system of measures by a canon of logarithms ; not only when the measure of 

 some one ratio is given, but also without that datum, by seeking the modulus 

 of the system by the rule abovementioned. In the 4th, 5th, and 6th proposi- 

 tions, he squares the hyperbola, describes the logistic line and aequiangular 

 spiral by a canon of logarithms, and shows some curious uses of these proposi- 

 tions in their scholia. Take an easy example of the logometrical method, in 

 the common problem for finding the density of the atmosphere. Supposing 

 gravity uniform, every one knows, that if altitudes are taken in any arithmeti- 

 cal progression, the densities of the air in those altitudes will be in a geometrical 

 progression ; that is, the altitudes are the measures of the ratios of the densities 

 below and in those altitudes, and so the difference of any two altitudes is the 

 measure of the ratio of the densities in those altitudes. Now to determine the 

 absolute or real magnitude of these measures, the author shows, a priori, that 

 the modulus of the system, is the altitude of the atmosphere, when reduced 

 ev^ery where to the same density as below. The modulus therefore is given, as 

 bearing the same proportion to the altitude of the mercury in the barometer, as 

 the specific gravity of mercury does to the specific gravity of air, and conse- 

 quently the whole system is given. For since, in all systems, the measures of 

 the same ratios are analogous among themselves ; the logarithm of the ratio of 

 the air's density in any two altitudes, will be to the modulus of the canon, that 

 is, to the logarithm of the ratio modularis defined above, as the difference of 

 those altitudes, is to the aforesaid given altitude of the homogeneous at- 

 mosphere. 



He concludes the logometria with a general scholium, containing great variety 

 of elegant constructions both logometrical and trigonometrical ; such as give 

 the length of curves, either geometrical or mechanical ; their areas and centres 

 of gravity ; the solids generated from them, and the surfaces of these solids ; 

 with several curious problems in natural philosophy, concerning the attraction 

 of bodies, the density and resistance of fluids, and the trajectories of planets. 

 Several of these problems have two cases; one constructed by the measure of a 

 ratio, and the other by the measure of an angle. The great affinity and beauti- 

 ful harmony of the measures in these cases, have given occasion to the title of 

 the book. The measures of angles are defined, just as the measures of ratios. 



