282 Mr. GOODWIN, ON THE PURE SCIENCE 



the other in the plane perpendicular to it or the impossible plane, and this being the case, all 

 that is done by equation (B) is to assign the relative magnitudes of the two components. We 

 have, in fact, these two things known respecting the oblique cause which we denote by Pf(6), 

 first, that 



P.f(e) = Pcos0./(O) + Ps\ne.f(jj ; (B) 



and, secondly, that the oblique cause may be supposed to result from two component causes, 



for one of which 6 = and for the other = -, and putting these two things together, there 



can, I conceive, be no doubt as to the conclusion that these components are represented by P cos 6 

 and P sin 6 respectively. I am not saying that the auxiliary consideration just used is really 

 necessary for the interpretation of the equation {B), for I am inclined to believe that the generality 

 of symbolical interpretation would justify us at once in construing the equation thus: — the effect 

 of P acting at an angle = the effect of P cos acting directly, combined ivith the effect of 

 P sin acting at right angles to the original direction ; but at least the objection, if there be 

 one, seems removed by tiie consideration adduced, and that is my reason for adducing it. 



9. On the whole, I would submit that the preceding investigation not only is free from 

 solid objection, but is in fact the true mode of viewing the subject; because it rests upon the 

 leading idea of a uniform continuous passage of a cause from + to -, while its direction varies 

 continuously. And if it be objected, that physical laws cannot be conceived of as the results 

 of symbolical equations, it is to be answered that this is exactly the advantage of this mode 

 of viewina the subject, that it shews that such laws as that of the composition of forces are 

 not physical laws, in the sense of being laws known by experience or by induction from 

 observation, but are necessary laws in the most exact sense of the word : there is nothing more 

 incredible in the fact of Demoivre's formula containing the laws of Mechanics, than in that of 

 its containing the laws of Space, and it is as credible that it should be capable of proof from 

 that formula, that three forces are in equilibrium when they are each proportional to the sine 

 of the angle contained between the other two, as that the sides of a triangle are proportional 

 to the opposite sides. The fact of our making these conclusions depend on the interpretation 

 of symbols is in the present state of analysis no objection at all, and it may well be supposed 

 that some such method would be necessary in order to bring into one investigation subjects 

 at first sight apparently so distinct as the laws of space and the laws of equilibrium of forces. 



10. And I think it cannot be said that the method adopted in this paper is so artificial as those 

 which are sometimes applied to what are called fuiictional proofs of the rule for the composition 

 of forces; for although the quantity (-1)* or \/- 1 enters into the investigation, still it is to 

 be remembered, (and I wish to lay great stress upon this,) that this quantity is not introduced 

 by any principles peculiar to this paper, it enters by mathematical necessity and must be inter- 

 preted ; the only equation which it is incumbent upon me to prove is the equation f(d) = (- i)n, 

 and if this be established, the remainder necessarily follows. Indeed, so far from this method 

 being of an artificial and therefore incomplete kind, I would venture to question whether the 

 unsatisfactory character of some functional proofs of the law of composition of forces, and the 

 extremely complicated nature of all, may not arise from the oversight of the fact that a function 

 exists, which will express an oblique force in its totality and not only so far as it is effective. 

 On this subject, however, I will not enlarge, but only remark that it seems to me a point of 

 great beauty that a symbol, of such peculiar form as cos + (- l)* sin 9, which meets us at 

 every turn in analysis, should be the complete expression of a law by which all nature is 

 governed. 



