VOL. LXVII.] PHILOSOPHICAL TRANSACTIONS. 183 



XXIII. The General Mathematical Laws ivJiick regulate and extend ProporiioJi 

 Universally \ or, a Method of Comparing Magnitudes of anij Kind together, 

 in all the possible Degrees of Increase and Decrease. By James Glenie, jI.M. 

 and Lieut, in the Artillery, p 450. 



The doctrine of proportion laid down by Euclid, and the application of it 

 given in his Elements, form the basis of all the geometrical reasoning used by 

 mathematicians, both ancient and modern. But the reasonings of geometers 

 with regard to proportional magnitudes have seldom been carried beyond the tri- 

 plicate ratio, the proportion that similar solids have to each other, when referred 

 to their homologous linear dimensions. This boundary however comprehends 

 but a very limited portion of universal comparison, and almost vanishes into no- 

 thing when referred to that endless variety of relations, which must necessarily 

 take place between geometrical magnitudes, in the infinite possible degrees of 

 increase and decrease. The first of these takes in but a very contracted field of 

 geometrical comparison; whereas the last extends it indefinitely. Within the 

 narrow compass of the first, the ancient geometers performed wonders, and their 

 labours have been pushed still further by the ingenuity and industry of the mo- 

 derns. But no author, that I have been able to meet with, gives the least iiint 

 or information with regard to any general method of expressing geometrically, 

 when any two magnitudes of the same kind are given, what degree of augmen- 

 tation or diminution any one of these magnitudes must undergo, in order to have 

 to the other any multiplicate or sub-multiplicate ratio of these magnitudes in 

 their given state; or any such ratio of them as is denoted by fractions or surds; 

 or, to speak still more generally, a ratio which has, to the ratio of the first- 

 mentioned of these magnitudes to the other, the ratio of any two magnitudes 

 whatever of the same, but of any kind. Neither have I been able to find that 

 any author has shown geometrically in a general way, when any number of 

 ratios are to be compounded or decompounded with a given ratio, how much 

 either of the magnitudes in the given ratio is to be augmented or diminished, 

 in order to have to the other a ratio, whicii is equal to tlie given ratio, com- 

 pounded or decompounded with the other ratios. To investigate all these geome. 

 trically, and to fix general laws in relation to them, is the object of this paper; 

 which, as it treats of a subject as new as it is general, I flatter myself, will not 

 prove unacceptable to this learned Society. It would be altogether superfluous to 

 mention the great advantages that must necessarily accrue to mathematics in ge- 

 neral, from an accurate investigation of this subject, since its influence extends 

 more or less to every branch of abstract science, when any data can be ascer- 

 tained for reasoning from. I shall, in a subsequent paper, take an opportunity 

 of showing how, from the theorems afterwards delivered in this, a method of 

 reasoning with finite magnitudes, geometrically, may be derived, without any 



