9. _The Parallelogram of Forces is a necessary Truth._--In the series of discussions in which we are {224} here engaged, our main business is to ascertain the nature and grounds of the certainty of scientific truths. We have, therefore, to ask whether this proposition, the parallelogram of forces, be a necessary truth; and if so, on what grounds its necessity ultimately rests. We shall find that this, like the other fundamental doctrines of Statics, justly claim a demonstrative certainty. Daniel Bernoulli, in 1726, gave the first proof of this important proposition on pure statical principles; and thus, as he says[14\3], 'proved that statical theorems are not less necessarily true than geometrical are.' If we examine this proof of Bernoulli, in order to discover what are the principles on which it rests, we shall find that the reasoning employs in its progress such axioms as this;--That if from forces which are in equilibrium at a point be taken away other forces which are in equilibrium at the same point, the remainder will be in equilibrium; and generally;--That if forces can be resolved into other equivalent forces, these may be separated, grouped, and recombined, in any new manner, and the result will still be identical with what it was at first. Thus in Bernoulli's proof, the two forces to be compounded are represented by P and Q; P is resolved into two other forces, X and U; and Q into two others, Y and V, under certain conditions. It is then assumed that these forces may be grouped into the pairs X, Y, and U, V: and when it has been shown that X and Y are in equilibrium, they may, by what has been said, be removed, and the forces, P, Q, are equivalent to U, V; which, being in the same direction by the course of the construction, have a result equal to their sum. [Note 14\3: _Comm. Petrop._ vol. i.] It is clear that the principles here assumed are genuine axioms, depending upon our conception of the nature of equivalence of forces, and upon their being capable of addition and composition. If the forces, P, Q, be _equivalent_ to forces X, U, Y, V, they are equivalent to these forces added and compounded in any order; just as a geometrical figure is, by our conception of {225} space, equivalent to its parts added together in any order. The apprehension of forces as having magnitude, as made up of parts, as capable of composition, leads to such axioms in Statics, in the same manner as the like apprehension of space leads to the axioms of Geometry. And thus the truths of Statics, resting upon such foundations, are independent of experience in the same manner in which geometrical truths are so.