894 Dr. Adamson on Geometrical Demonstrations, 



Nothing purely geometric can enter among them. Axioms in 

 fact may be either corollaries or theorems. In the former case, 

 they may need no demonstration, or may be termed self-evident ; 

 in the latter case, they may, in some instances, to some minds, 

 and in some instances, to all minds, require demonstration j but 

 this does not affect their essential nature, nor the relations they 

 bear to the subjects of specific sciences. 



Everything, except their application, ought to be eliminated 

 from the reasonings of specific sciences; and if requiring de- 

 monstration, they shoiJd be treated symbolically, or algebraic- 

 ally. Geometiy therefore has not assumed its proper logical 

 form until we have relieved its demonstrations from all those 

 portions of them which constitute reasonings applicable not 

 purely to geometric, but abstractly to all modes of magnitude. 

 These steps in reasoning should be provided for by axiomic theo- 

 rems, just as they would be in sciences relating to force, money, 

 sound, &c. This would contribute greatly to clearness and ele- 

 gance in demonstration. No finer instance of such effects can 

 perhaps be found than in the applications of Euclid's criterion of 

 proportionality, which, under a general law, makes relations of 

 number or of quotient, in regard to all kinds of quantity, to be 

 deducible from relations of excess or difference. Thus the same 

 rule is applicable to every mode of geometrical magnitude. An 

 interesting example may perhaps be found also, in the use of the 

 following theorem and its converse. If A:B = C:D; then 

 A2:A2--B2=A2-C2:(A±D)2-(B±C)^ whence (A + D)^- 

 (B + C)2=(A~D)2 — (B — C)2. If we employ opposite terms to 

 indicate the extremes and means of a proportion, then by using 

 that word along with the others adjacent and alternate, we can 

 exhibit these results in various very beautifully symmetrical forms 

 of expression. From the relation between the ordinates to the axis 

 and the segments of the axis, in conic sections, we deduce in- 

 stantly by the above theorem or its results, the conclusions, that 

 the ratio of the distances from a point in the curve to the focus 

 and to the directrix is constant ; and that the sum or the dif- 

 ference of the lines to the foci is constant. The complex pro- 

 portions generally used for these purposes, may be considered as 

 replaced by the steps of reasoning in the theorem ; which are 

 obvious. 



Since axioms ought not to introduce hypotheses, we see at 

 once what is the real place and office of geometrical definitions. 

 In fact, as men do in all other sciences, and as they generally 

 endeavour, though sometimes unconsciously, to do in geometry, 

 definitions must be made the foundation of all conclusions. Pro- 

 perties of objects are necessarily derived from their nature, either 

 when assumed for consideration individually, or in combination. 



