triangle of forces? To which I reply, that when two straight lines A B, B C are given in position and magnitude, the straight line joining

the points A and C will be as strictly their geometrical resultant as the force represented by AC will be the resultant of the forces repre

A.

FIG. 1.

C

B

sented by A B, B C. For by speaking of the resultant of two straight lines we necessarily imply that the two lines are given to determine some third object; and that object must be a straight line, since the resultant of two things of the same kind must obviously be of the same kind with those which produce it; and if there be any line which is to be considered as the resultant of A B, BC, it must be A C, since this is the only new line whose position and magnitude are in any way whatever determined by the positions and magnitudes of A B and BC. If, therefore, we mean by the resultant of two straight lines given in position the straight line which is determined in magnitude and position by these straight lines- and this seems to be the obvious or only meaning to be assigned to the term resultant-then AC is the resultant of A B, BC.

The problem of finding the resultant of two straight lines given in position may be generalised into that of finding the resultant of any number of straight lines forming an imperfect polygon. For if all the sides of a polygon be given except one, then

that one will be the resultant of all the rest; it is the only new straight line whose position and magnitude become determinate in virtue of the other sides being given. It may be said that the extremity of one of the last sides may be joined with one of the angular points, and that thus some other line will be determined but the obvious answer is, that this will not employ all the data, and that the line so determined I will be the resultant of all those which have been really used. In fact, a straight line may be given just as really, though not so directly, by giving in position all the other sides of a polygon of which this straight line forms the last: to give these other sides is precisely the same thing as to give the line itself.

Conversely, a straight line may be considered as the resultant of any system of straight lines which with it will form a polygon; and also in such polygon any one side may be called the resultant of all the rest. If two be missing, they cannot be replaced; if one only, then is that missing one just as fixed and determinate as if it were represented by an ink-mark as part of the polygon.

The principle of the third side of a triangle being the resultant of the other two may be applied to the demonstration of certain propositions in plane geometry, which I here introduce for illustration's sake.

It may be shown from this principle that the straight lines drawn from the bisections of the sides of a triangle perpendicular to the sides will pass

through the same point. For suppose we bisect two of the sides, and draw straight lines perpendicular to them (it is of course necessary to bisect the sides, because the middle point of a line is the only one which is similarly related to the two extremities), then these indefinite lines determine a new point, namely, the point of intersection. Now, if we perform the same operation on the third side, the result must be such that no new geometrical element is determined, since everything determinable by the third side is already implicitly involved in the knowledge of the other two. Therefore this third line must pass through the intersection of the other two, since if it did not it would determine two new points, which, by what has just been said, is impossible.

The same reasoning applies to the theorems, that the lines bisecting the angles of a triangle pass through the same point, and that the lines joining the angular points with the bisections of the sides pass through the same point.

And I may remark that we have probably here the explanation of the fact, that propositions in pure geometry sometimes admit of simpler proof by reference to mechanical considerations than by the ordinary geometrical methods; for example, the last proposition of those just cited finds its solution at once in the property of the centre of gravity of a plane triangle.

Taking the view which I have thus endeavoured

to expound of the resultants of straight lines, it wi be obvious how close is the analogy between this cas and that of forces. For if A B, BC (see diagram, p. 7 represent two forces, then A C, as we know, represent their resultant; and in general, if two sides of triangle represent two forces, their resultant is give by the third; and still more generally, if the sides o an imperfect polygon represent forces, their resultan is given by the last side. Now the same thing holds true in this case which was true in the case of geometry-namely, that if A B, BC be given in position and magnitude, the only third thing determined is AC; and therefore, if A B, A C represent two forces, the magnitude and direction of the force AC is at once determined, but this can be asserted of no other force. Now I do not say that this could be accepted as a full and proper proof of the triangle of forces; but I do think that it is a mode of considering the subject which, by careful thought, may lead to the intuitive perception of the truth of the proposition. It might be impossible to admit this as the only proof that the force AC would balance the two A B, BC; but at least it shows that AC is related to A B and B C in a manner in which no other force is related, and that it is determined by them, so that to give them is to give it, and that this can be predicated in the same sense of no other force; and from this it seems possible to arrive at an intuitive perception of the truth that A C is in reality the resultant of A B and B C. And this is

the point at which we should endeavour to arrive; the fundamental proposition in mechanics ought not to have a merely artificial basis, ought not to be so proved that the mind rather concedes its truth because it cannot deny it, than sees it to be true; and I cannot feel a doubt but that there must be some method of viewing the subject, by adopting which the fundamental propositions of mechanics will gradually grow into as perfect axiomatic clearness as do the simple propositions of geometry.

To illustrate this point by contrast, let us consider for a moment the proof which is frequently given, or used to be given, in elementary treatises, of the triangle or parallelogram of forces; I mean that which is due to and goes by the name of Duchayla.

Now, this proof is certainly convincing; that is to say, it is not possible to point out any flaw in the steps of demonstration; but for persuading the intellect it appears to have no fitness. The proof is essentially artificial, and is based on a simple case of composition of forces, which seems very insufficient to suggest, as it is pretended that it does, the result sought. The character of the proof seems, if I may so express myself, to be that of cunning rather than that of open-faced argument; and yet I think that, however unsatisfactory the proof may appear when viewed in this light, we feel convinced that there must be a meaning and a principle in it, and that these are only smothered and obscured by the artificial contriv