270 Mr. GOODWIN, ON THE CONNEXION BETWEEN THE SCIENCES 



We may put what has been said in other words by asserting that all properties of straight 

 lines are functions either of their direction, or their magnitude, or both ; a straight line has no 

 other elements than these, and therefore every thing which is predicated of a straight line is 

 predicated simply in consequence of that straight line having a certain direction and a certain 

 magnitude. 



3. Now from this point, I think we can see a simple road into Mechanical Science ; for if 

 there be anything physical which depends on no other elements, than those of direction and 

 magnitude, there is no reason why a mark on a piece of paper should not stand for this physical 

 embodiment of the two ideas as well as for the geometrical: and further, if there be anything 

 physical, of which it can be predicated that it has no other elements than direction and magni- 

 tude, then all propositions which have been proved for straight lines will have their corresponding 

 propositions, in fact will be true with a change of phraseology, in physics. 



In devisinof a method therefore for representing to the eye the forces on which we reason in 

 Statics, the question is not whether a force can be conveniently represented by an ideal straight 

 line, but whether a force has such qualities that the same representation which serves for demon- 

 strations respecting straight lines, will also serve for demonstrations respecting forces. 



4. Now when we come to examine a Statical force, we find that it does involve, or rather it 

 is a physical expression of, those two ideas of direction and magnitude, and of no others. For 

 we measure a Statical force by the pressure which will counteract it ; and what are the questions 

 as to the counteracting force.' these two — in what direction it must be applied, and with what 

 intensity ; it is clear that neither of these is sufficient without the other ; for a particle left to 

 itself under the action of a force will move off in a certain determinate direction, and it is a truth 

 which requires no proof, but is purely axiomatic, that a force, however great, applied in any 

 other but the exactly reverse direction will not prevent motion ; and so likewise it is a self-evident 

 fact, that the counteracting pressure must be of a certain determinate magnitude and no other. 

 Thus, to a person who understands what I mean by the term Force, it will be apparent that the 

 only ideas involved, are those of direction and magnitude ; any cause tending to produce motion 

 which involves any other element for its complete determination is not a Force, it may be called 

 so popularly, but jt is not included in the mathematical definition. 



And it may be observed here, that as in Euclid, the definition given of a straight line, viz. 

 " that it lies evenly between its extreme points," is virtually superseded by the axiom, that " two 

 straio-ht lines cannot inclose a space," so in elementary books of Mechanics, although the definition 

 is o-iven of a force that it is " any cause which produces or tends to produce motion," yet the 

 fundamental proposition is usually made to depend on the axiom or fact (or whatever it is to be 

 called) that a force may be supposed to act at any point in its direction, which is the same thing 

 as saying, that if the magnitude be given the force depends on direction only. 



When the science of Mechanics was first studied, the simple view of force which I have given 

 would, of course, not be immediately taken ; the effect of force would probably be supposed to 

 depend on other circumstances ; but this is a matter of no consequence : the question is merely, 

 what we mean by force now, and what it is supposed to mean in all mechanical treatises ; and 

 it signifies not whether we start with the idea of a cause of tendency to motion involving the ideas 

 of direction and magnitude only, and call the embodiment of that idea force by definition, or 

 whether we examine the world we live in, and shew that such are the elements and the only 

 elements of force. 



5. Let it be granted then that the only ideas involved in that of force, are those of direction 

 and magnitude, and we come to the case (already spoken of by anticipation) of a thing physical, 

 involving exactly the same ideas as the straight line in Geometry ; and we therefore lay down 

 this proposition, that every theorem regarding straight lines will have its fellow in Mechanics, that 



