On Geometrical Proportion. 427 



which authors have defined the word proportion: thus, it is 

 frequently confounded with the words ratio, reason, ana- 

 logy, &cc, and sometimes two, sometimes three, and some- 

 Umzsfour quantities are said to be proportional. This mode 

 of procedure creates a good deal of confusion, and not a 

 little embarrasses the ideas of beginners ; and it is to obviate, 

 in some measure, thes'e irregularities, that I now send you 

 the following short disquisition on this interesting branch 

 of science. I must observe further, that the manner which 

 is generally practised by authors of treating the subject geo- 

 metrically, as is done in Euclid's Elements, and most of 

 the modern books of geometry, is certainly not the most 

 eli-gible, or best adapted to learners ; and, except the few 

 trifling observations which may be met with in books of 

 arithmetic, it is to books of geometry alone to which a learner 

 can have recourse for any information he may rt quire. 



To prove that what I have advanced concerning the ob- 

 scurity of the subject, when treated geometrically, is cor- 

 rect, I need only appeal to those gentlemen- who are in the 

 practice of teaching the fifth book of the Elements ; it is 

 well known that the difficulties attending it are so great, 

 that very few students ever thoroughly understand his demonr 

 strations, owing most probably to their not being able to 

 form a correct idea of his criterion of proportion. .Again, 

 "Ordinary language (as Professor Playfair observes) con- 

 veys the ideas of the different operations supposed to be per- 

 formed by these demonstrations so slowly, and breaks them 

 down into so many parts, that they make not a sufficient 

 impressiqn on the understanding ; and this generally happens 

 when the things treated of are not represented to the scn«cs 

 by diagrams, as they cannot be when we reason concerning 

 magnitudes in general, as in this part of the elements of ge- 

 ometry. It is obvious, therefore, that we ought to adopt 

 the language of arithmetic, or algebra, which by its short- 

 ness, and the rapidity with which it places objects before us, 

 makes up for its being a conventional language ; and also for 

 using symbols to denote the things we wish them to express." 



The first ideas of proportion which we generally acquire, 



are obtained by comparing natural objects with one another: 



3 thus, 



