274 Mil. GOODWIN, ON THE CONNEXION BETWEEN THE SCIENCES 



line as to magnitude only, then le^^-^ will represent the same line measured in a direction 

 making an angle 9 with some fixed line. 



Now if ABC be a right-angled triangle and BAC = 6, and AB = I in 

 magnitude, 



then AB = le^ ^~i = / cos + \/'-V I sin 6 



= AC+ \/^i .BC; / 



or, if we omit v' - 1 which is a sign of affection, i 



AB = AC-\- BC. 



We may therefore say, that regard being had to direction as well as magnitude, AB the 

 hypothenuse is the sum of the two sides AC and BC, or perhaps it would be more distinct 

 to say that the hypothenuse is the equivalent of the sides, that is to say, that considered as 

 embodiments of the ideas of direction and magnitude one is equivalent to the other ; if the 

 direction be disregarded it would be absurd to say that AB = AC + BC, and in like manner, if 

 direction only be considered, there is no equivalence between the hypothenuse and sides, but com- 

 bining the two there is an equivalence, and one may be substituted for the other in all sciences which 

 are developements of these two fundamental ideas. 



I may remark further, that we may consider the symbol le^^-^ as the type of the sciences 

 depending on these ideas, or rather one may say, that the symbol le^'^^ is the germ from which 

 may be evolved the fundamental principles of these sciences. 



12. One more observation may be made on this symbolical representation. The symbol 

 le^"^-^ is as we know equivalent to the expression /eos0 + v — 1 /sin 0, and therefore if this symbol 

 were given to a person as the representation of force, it must at once strike him that the fundamental 

 property of force was that of being made up of two other forces, which we will call as usual 

 its resolved parts. Now what I would wish to "observe, is, that this connexion supposed to be 

 suggested by the symbolical formula is precisely that which would probably be suggested to the 

 mind when it first began to engage itself with mechanical studies. 



For suppose we have a force tending to draw a particle P in any direction OP; then if we wish 

 to examine the nature of this force, and determine its laws, the obvious p p 



artifice would seem to be to constrain the particle in various ways, and 

 reason as to the result. Suppose, for instance, a plane drawn perpendi- 

 cular to OP and indefinitely near the particle P, then it is manifest that 

 the particle will not move at all, this is a point which no one will 

 question, and therefore we arrive at one property of force, namely, tiiat 

 it produces no effect in a direction perpendicular to its own. But, 

 suppose we incline the plane at some angle 9 to OP, then motion " 



will ensue if not checked, and the question is, what force acting along the plane will be just 

 sufficient to check motion ? To determine this, take any point O in the direction OP and 

 draw OP' perpendicular to the constraining plane, then it is easy to see that whatever relation 

 the line OP has to the original force, the same relation has P'P to the resolved part in the direction 

 PP' ; to make this apparent, I shall call a plane drawn through a point in the direction in which a 

 particle has a tendency to move and perpendicular to that direction the impossible plane, and 

 then the definition of OP will be, that it is the distance between the impossible planes cori-esponding 

 to Panel 0. now suppose any two other parallel planes to be drawn through P and O, and let 

 them be perpendicular to PP', then P'P is the distance between these impossible planes, as OP was 

 between tile two former. This being the case, it will be allowed that if OP represents the original 

 force, P'P will represent the resolved part in the direction PP', that is, the resolved part will 



