XXI. On the Pure Science of Magnitude and Direction. By the Rev. H. Goodwin, 

 Fellow of Cuius College, and of the Cambridge Philosophical Society. 



[Read May 12, 1845.] 



In a former Memoir, I have endeavoured to point out the a priori and necessary character of the 

 fundamental proposition in Mechanics, by connecting it with the propositions of Geometry, and so 

 bringing the demonstrative character of the two sciences into one and the same point of view. I 

 there pointed out that the only elements of Force are Magnitude and Direction, and therefore that 

 the only simple ideas of which the term Force is the expression are those of Magnitude and 

 Direction, and hence, that all propositions respecting Force ought to follow demonstratively and 

 perhaps intuitively from the possession of those two ideas combined, even as the propositions 

 respecting straight lines arise necessarily from the same two ideas. In the course of that Memoir, 

 I spoke of such a Science as that of pure direction, which should include within itself the Sciences 

 of Geometry or rather of Position, of Kinematics, of Mechanics, and possibly other Sciences ; 

 it is the design of the present Memoir, to attempt to establish the fundamental Proposition of such 

 a Science, or, as perhaps it may be more properly called, the pure Science of Magnitude and 

 Direction. 



1. The fundamental problem will be, to determine the combined effect of any number of 

 causes, the magnitude and direction of each of which is given. It will be seen that this statement 

 is perfectly general ; for a line given in a certain direction may be looked upon as cause, the point 

 in space determined by its extremity as effect, or if two lines be given, having an extremity in 

 common, the line joining the other extremities which is thus determined may be regarded as effect; 

 so likewise, if a particle be animated by two simultaneous velocities, they may be looked upon as 

 causes, the resultant velocity as effect ; and if a particle be acted upon by two forces, the resultant 

 pressure will be the effect which results from the two given forces as causes ; and hence it will 

 appear, that the fundamental problem is to find the combined effect of any number of given causes. 



2. Now, if the direction in which all the causes acted were the same, it is clear that the 

 combined effect would be found by mere addition of the quantitative symbols which measure 

 their respective effects ; the only postulate here involved is, that two causes do not modify each 

 other's effects, a postulate which is of the nature of an axiom, and which merely expresses such 

 truths as these, that if a point be taken in a straight line at a distance (a) from a fixed origin, 

 and another point at a distance (6) from the former, then the distance of the point last deter- 

 mined from the origin will be a + b, or again, that if there be two forces, one of which can 

 lift a weight P, and another a weight Q, then the two together can lift a weight P + Q. 



3. Hence, when the direction of a number of causes is the same, the process of addition 

 serves to determine their combined effects, but when the directions are different, it will be necessary 

 to determine according to what law variation of direction modifies the effect of a cause ; in other 

 words, suppose we take P as the quantitative symbol of the effect of a certain cause in a given 

 direction, what will be the symbol for the effect of a cause of equal intensity whose direction 

 makes an angle G with the given direction ? 



