XX. On the Connexion between the Sciences of Mechanics and Geometry. By the 

 Rev. H. Goodwin, Fellow of Cuius College, and of the Cambridge Philo- 

 sophical Society. 



[Read February 10, 1845.] 



1. IT is well known, that the first step in proving the elementary propositions of Mechanics 

 is usually to explain that for the purposes of demonstration forces are represented by straight 

 lines, and so simple a step does this appear to be, that it has been complained that students 

 frequently do not perceive that they have passed a distinct boundary-line in their transition from 

 Geometry to Mechanics. It becomes therefore a matter of interesting inquiry, what is the ground 

 of the connexion between the two sciences ? is it merely conventional .'' or only partly so ? or not 

 at all .'' Is the substitution of lines for forces to be looked upon as a mere ingenious device, 

 or has it such a natural basis in the reality of things, as to force itself in one form or another 

 on the mind of every one capable of appreciating the subject ? This is the question which 

 I propose to examine. 



2. Let it be observed then, that an indefinite straight line is merely the expression of the 

 idea of direction : the idea of direction is a pure idea capable of no simpler expression, and, as 

 I think, obviously not acquired from experience : no child ever walked from one point to another 

 by a roundabout path, until it discovered that one path was shorter than any other ; there might 

 be a difficulty about understanding what was meant by a straight line lying evenly between its 

 two extreme points, but about the fact that you would go in one determinate direction from one 

 point if you wished to go to the other, there could be no doubt at all. I hold, therefore, that 

 the idea of direction is a pure idea, independent of all experience, and that all definitions of a 

 straight line are attempts, accompanied with more or less success, to give verbal expression to 

 .this idea*. 



And so when I draw a mark on paper which I call a straight line, this is a method of re- 

 presenting rudely to the eye a certain direction, it enables me to speak of that direction 

 intelligil)ly and to reason about it, the reasoning of course referring not to the mark on the 

 paper, but to the ideal line or direction of which that mark is the visible memorandum. 



When we speak of a Jinite straight line, we limit the idea of mere direction by introducing 

 the new one of magnitude. The idea of magnitude is merely that of comparison of one 

 quantity with another, and a straight line of certain magnitude is represented by taking two points 

 on a given indefinite straight line, such that the distance between them is so many times greater 

 than the distance between two standard points. 



Thus a finite straight line given in position is the expression of the combined pure ideas 

 of direction and magnitude ; and a mark on paper standing for such a line is the exhibition to the 

 eye of these two ideas. 



And hence, further, we may say that all propositions concerning indefinite straight lines art- 

 deductions from tlie pure idea of direction ; all propositions concerning finite straight lines not 

 given in position are deductions from the pure idea of magnitude; and all those concerning finite 

 straight lines given in position are deductions from these two pure ideas combined. 



• Sec Nute (A). 



