IX. On the Substitution of Methods founded on Ordinary Geometry for Methods 

 based on the General Doctrine of Proportions, in the Treatment of some Geo- 

 metrical Problems. By G. B. Airy, Esq. Astronomer Royal. 



[Read Bee. 7, 1857-] 



The doctrine of Proportions, laid down in the Fifth Book of Euclid's Elements, is, so far 

 as I know, the only one which is applicable to every case without exception. It is subject 

 only to the condition, that the quantities compared, in each ratio, shall be of the same kind 

 (without requiring generally that the quantities in the different ratios shall be of the same 

 kind); a condition which appears essential to the idea of ratio. 



This generality, however, as in other instances, is not without its inconvenience. The 

 methods of demonstration which are applied by Euclid are very cumbrous and exceedingly 

 difficult to retain in the tnemory, and I know but one instance (that of the proposition ex cequali 

 in ordine perturbata, as amended by Professor De Morgan) in which it has been found prac- 

 ticable to simplify them. It is therefore natural that attempts should be made, in special 

 applications of the doctrine of proportions, to introduce the facilities which are special to each 

 case. 



In the special application in which numbers are the subject of proportion, methods have 

 long since been introduced, departing widely in form from Euclid's, yet demonstrably leading 

 to the same results, and possessing all desirable facility of application. 



No attempt, I think, has been made to avoid the necessity for employing Euclid's gene- 

 ralities, when geometrical lines alone are the subject of consideration. Yet there are cases in 

 which these generalities have always been openly or tacitly employed, but in which the nature 

 of the investigation seems to indicate that there is no need to introduce proportions at all. 

 I was led to this train of thought by considering the well-known theorem, " If pairs of tan- 

 gents be drawn externally to each couple of three unequal circles, the three intersections of the 

 tangents of each pair will be in one straight line." This, I believe, has always been proved 

 by the use of certain propositions of proportion. Yet the theorem starts from data without 

 proportions, and leads to a conclusion without proportions; and it seems wrong that it should 

 be conducted by intermediate steps of proportions, the theorems of which have been proved by 

 methods based fundamentally on considerations of arbitrary equimultiples. 



It appeared to me, on examination, that this and similar investigations, of which lines only 

 are the subject, might be put in a simple and satisfactory form, referring to nothing more 

 advanced than the geometry of Euclid's Second Book, by a new treatment of a theorem equi- 

 valent to Euclid's simple ex cequali, and of the doctrine of similar triangles. I beg leave to 



