VOL. XXXII.J PHILOSOPHICAL TllANSACTIONS. 587 



An Account of a Book, entitled, Harmonia Mensurarmn^ sive Analysis et 

 Synthesis per Rationum et Angulorum mensuras pi-omotte : accedunt alia 

 Opuscula Mathematica : per Rogerum Cotesium. Edit et auxit Robertus 

 Smith. Cantab. 1722, 4to. N^ 3/2, p. ISQ. 



The book consists of three parts. In the first, called Logometria, the 

 author's chief design is to show how that sort of problems, which are usually 

 reduced to the quadrature of the hyperbola and ellipsis, may be reduced to the 

 measures of ratios and angles ; and afterwards be solved more readily by the 

 canons of logarithms and sines and tangents. He defines the measures of 

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

 magnitudes of the ratios to be measured. In this sense any canon of loga- 

 rithms is a system of numeral measures of the ratios of the absolute numbers to 

 an unit : the parts of the asymptote of the logistic line, intercepted between its 

 ordinates, are a system of linear measures of the ratios of those ordinates: the 

 areas of an hyperbola, intercepted between its ordinates to the asymptote, are a 

 system of plane measures of the ratios of those ordinates: and since there may 

 be infinite systems of measures, according as various kinds of quantities are 

 made use of, such as numbers, time, velocity, and the like ; or according as 

 the measures of any one system may be all increased or diminished in any given 

 proportion ; in such variety much confusion may possibly arise as to the kind 

 and absolute magnitudes of particular measures, which happen to fall under 

 consideration. Our author very happily removes this difficulty ; by showing 

 that the nature of the subject points out the measure of a certain immutable 

 ratio for a modulus in all systems, by which to determine the kind and absolute 

 magnitudes of all other measures in each system. 



The first proposition is to find the measure of any proposed ratio. This he 

 considers in a way so simple and general, as naturally leads to the notion and 

 definition of a modulus ; namely, that it is an invariable quantity in each 

 system, which bears the same proportion to the increment of the measure of 

 any proposed ratio, as the increasing term of the ratio bears to its own incre- 

 ment. He then shows that the measure of any given ratio, is as the modulus 

 of the system from whence it is taken : and that the modulus in every system, 

 is always equal to the measure of a certain determinate and immutable ratio, 

 which he therefore calls the ratio modularis. He shows that this ratio is ex- 

 pressed by these numbers 2.7182818 &c. to 1, or by 1 to O.3678794 &c. So 

 that in Briggs's canon, the logarithm of this ratio is the modulus of that system: 

 in the logistic line, the given subtangent is the modulus of that system : in the 



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