Dr. Adamson on Geometrical Demonstrations. 405 



ors parallel to one another at a very small measured distance 

 asunder, with only air between them. If X be the force of 

 attraction thus measured, a the distance between the conductors, 

 S the area of each or of that portion of each which is directly 

 opposed to the other, and V the difference of the electrical poten- 

 tial kept up between them by the battery, we should have 



and therefore, if F be the electromotive force of the battery in 

 electro-magnetic measure, 



S / 



LXVI. On a proposed Test of the Necessity/ of Indirect Proof in 

 Geometrical Demonstrations, with Remarks on Methods of 

 Demonstration. By James Adamson, D.D. 

 [Concluded from p. 338.] 



AN earlier recognition of the true nature of axioms, and of the 

 real use of definitions, would have restricted considerably 

 the controversies which have arisen respecting parallelism. To 

 assume a property as a criterion of parallelism, and another as a 

 criterion of non-parallelism, except the one be proved to be a 

 deduction from the other, would not only have been disapproved 

 in Euclid, but have been avoided as contrary to reason. Con- 

 siderable oversight as to the character of defining has also 

 occurred in neglecting the principle, that the property assumed 

 as the criterion of the thing defined must be single, or be a 

 hypothesis alone, and not both a hypothesis and a conclusion. 



The relation of parallels is such, that that which is true 

 regarding two points, when a ^iven line crosses them, or that 

 which is true of two pairs of points, when there are two perpen- 

 diculars between them, determines necessarily the existence of 

 the relation, and whatever other properties belong to it. Among 

 the necessary consequences are the conclusions, that what is 

 true of two points, or of two pairs of points, is true of all others ; 

 but these properties must be made conclusions by demonstra- 

 tion. It is a remarkable characteristic of this relation, that 

 sophisms intrude so unexpectedly into the minds of inquirers, 

 and that the same failures are so often repeated. A sufficiently 

 definite idea of the real source of the difficulty has not perhaps 

 been entertained in these instances. However the properties 

 under consideration may be treated, it is evident that somewhere 

 in the series, either of the direct or of the converse statements, 



