272 Mh. GOODWIN, ON THE CONNEXION BETWEEN THE SCIENCES 



7. The principle of the third side of a triangle being the resultant of the other two may 

 be applied to demonstrate certain propositions in plane Geometry, which I here introduce for 

 illustration's sake. 



It may be shewn from this principle, that the lines drawn from the bisections of the sides 

 of a triangle perpendicular to the sides will pass through the same point. For suppose we 

 bisect two of the sides, and draw lines perpendicular to them, (it is of course necessary to 

 bisect the sides, because the middle point of a line is the only one which is similarly related to 

 the two extremities), then these indefinite lines determine a new point, viz. the point of inter- 

 section ; now if we perform the same operation on the third side, the result must be such that 

 no new geometrical element is determined, since everything which is functional of the third side 

 is already implicitly involved in the knowledge of the other two ; therefore this third line must 

 pass through the point of intersection of the other two, since if it did not it would determine 

 two new points, which, by what has just been said, is impossible. 



The same reasoning applies to the propositions that the lines bisecting the angles of a triangle 

 pass through the same point; and that the lines joining the angular points with the bisections 

 of the sides pass through the same point. 



And, I may remark, that we have here the explanation of the fact, that some propositions 

 in pure Geometry admit of simpler proof by referring to mechanical considerations than by the 

 ordinary geometrical methods ; as for example the last proposition of those first cited finds its 

 solution at once in the property of the center of gravity of a plane triangle. 



8. Taking the view which I have endeavoured to explain of these resultants, it will be 

 obvious how close the analogy is between this case and that of forces ; ^ 

 for if AB and BC represent two forces, then AC we know represents 

 their resultant, and in general if two sides of a triangle represent 

 two forces their resultant is given by the third, and still more generally 

 if the sides of an imperfect polygon represent forces their resultant 

 is given by the last side. Now the same thing holds in this case a~ 

 which was true in the case of Geometry, viz. that if AB, BC be given in position and 

 magnitude, the only third term determined is AC; and therefore if AB, BC represent two 

 forces, the magnitude and direction of the force AC is at once determined, but this can be 

 asserted of no other. Now I do not say that this could be considered as a proper proof of 

 the triangle of forces ; but I do think that it is a way of considering the subject which, by 

 careful thought, will lead to the intuitive perception of the truth of the proposition. It would 

 be impossible to admit this as the only proof that the force AC would balance the two AB, BC, 

 but at least it shews that AC is related to AB and BC in a manner in which no other force 

 is related, that it is at once determined by them, so that to give them is to give it, and that 

 this can be predicated in the same sense of no other force ; and from this it seems possible by 

 degrees to arrive at an intuitive perception of the truth that AC is in fact the resultant of 

 AB, BC- And after all this is the point at which we should endeavour to arrive ; the funda- 

 mental proposition in mechanics ought not to have a merely artificial basis, and to be such that 

 the mind rather concedes it because it cannot deny it, than sees it to be true ; and I cannot feel 

 a doubt but that there must be some method of viewing the subject, which if we adopt, the 

 fundamental propositions of Mechanics will gradually grow into as perfect axiomatic clearness as 

 do the simple propositions of Geometry*. 



9. To illustrate this point by contrast, let us for a moment consider the proof which is 

 frequently given in elementary treatises of the triangle of forces, I mean that which is due to 



See Note (B). 



