276 Mr. GOODWIN, ON THE CONNEXION BETWEEN THE SCIENCES 



what the writer's belief is respecting the nature of the trutlis which he is developing ; now this 

 point is entirely resolved if it is shewn that the principles of Mechanics are identical with those 

 of Geometry, that the two sciences not only have certain analogies, but are in essence identical 

 as being two developements of the selfsame ideas, and hence, if this be true, we see at once 

 the necessary character of the truths of Mechanics, or at least we shew them to stand on the 

 same ground as others which are supposed to be admitted as necessary. I will close this paper 

 by saying, that although I am well aware that what I have said in favour of the views propounded 

 may not with many appear to amount to demonstration, and indeed perhaps demonstration in 

 such a subject is not altogether possible, yet 1 am persuaded of their fundamental correctness 

 by this consideration as much as by any, viz., they do seem to point out the road to absolute 

 intuition of truth, they seem to mark out a method of thought according to which the elementary 

 truths of Mechanics will present themselves gradually with axiomatic clearness. And certainly, 

 whether this method be true or not, it cannot I think be doubted by any one who has reflected 

 on the foundations of truth, that this is the natural course, viz., that all demonstrations gradually 

 merge in intuition, and that all human knowledge converges towards that absolute intuition which is 

 the attribute of the divine mind. 



NOTES. 



Note (A), page 26;). 



The word direction appears to be the best abstract word for expressing the idea which is intended to 

 be embodied in the concrete form of a straight line; the evil of concrete terms is that they appear to 

 assign physical existence to that which can have none, and by thus leading away the mind from the 

 true idea tend to prevent the intuition of geometrical truth. If the idea intended to be embodied in the 

 terms point, straight line, and angle be conceived in their abstract form, the simple propositions respecting 

 them at once assume the character of axiomatic truth. I will here put down what appear to me to be the 

 best abstract terms for expressing these three geometrical elements — 



1 . A point = Position. 



2. A straight line = Direction. 



3. An angle = Inclination of directions. 

 I will illustrate the intuitizing force of these terms by applying them to the 



doctrine of parallels. The idea of parallelism is that of identity of direction with- 

 out identity of position ; and from this it is evident that a straight Une CD falling on 

 two parallel lines AB . A'B' makes the alternate angles equal ; for since the question 

 is one of direction only, whatever is predicated of the line AB may be predicated 

 of the line A'B', since they differ in position only and not in direction. 



Note (B), page 272. 



The proposition in pure Geometry which seems more than any other connected with Mechanics is 

 Euclid I. 32, and it will be worth while to point out the self-evidence of this proposition both for its 

 own sake, and also from the assistance it will afford in the intuition of the cognate mechanical proposition. 



