The theory of statics

In a science that has existed from the time of Archimedes, and occupied the attention of the most celebrated Mathematicians of many centuries, much new matter cannot be expected. Little more now remains to be done than to collect, arrange, and simplify the results already obtained; this therefore is the task I proposed to myself, in undertaking the preparation of the following treatise on Statics.

That a treatise was necessary, which should present the subject under one continuous chain of reasoning, with a uniform system of notation, and having its several parts dependent on the same first principles and definitions, will not I think be denied. In what degree, however, the present will supply the want of such a, treatise, must now be left to the public to determine.

The first chapter commences with general notions of mechanical force. I have purposely avoided setting forth any abstract ideas of its nature  in as much as such considerations, so far from throwing light on the subject, do generally involve us in the greater darkness the further we pursue them.

It appeared to me, that while we are ignorant of motion, we must of necessity remain also ignorant of the existence of force; and as the idea of matter does not include in it the idea of motion, we infer that motion is not an essential, but a consequential state of matter; for it can scarcely be imagined how matter can pass into a certain state, which is not essential to its existence, without the agency of an independent cause. Wherefore, without further inquiry into the nature of this cause, I have denominated it Force; and as motion is the most simple, natural, and general evidence our senses can receive of the existence of force; and seeing, moreover, that we have originally (that is to say, independently of subsequent deductions of reason) no other test of its action, I have not scrupled to take motion as the measure of its intensity in Statics, as well as in Dynamics.

I am aware that the introduction of motion as a necessary fundamental idea in a treatise on Statics, (the science of Rest), does, in a certain degree, appear absurd, and has accordingly been reprobated by some writers; but, not with standing all that has been said upon the distinction between Statical and Dynamical force, it does not appear that any real difference in the nature of the two forces has ever been clearly made out; the only difference established having reference rather to the state of the body acted upon, or of that in which the force resides, than to the forces themselves. No one, I believe, will question the correctness of the two following fundamental principles :

1st. We know nothing of force but what we learn from its effects.
2ndly. Force is that which produces, or tends to produce motion.

Now the latter virtually confesses that motion is the only natural and uninfluenced effect of force; and, therefore, we infer from the former, that motion is the only natural and uninfluenced means we can have of discovering the properties of force. Upon these grounds, as well as the considerations before mentioned, that there is no difference in the nature of Dynamical and Statical force; I have taken the direction in which a free particle begins to move when acted on by a force as the direction in which the force acts; and have used the quantity of motion produced in a given time as the measure of its intensity.

It this measure be found inconvenient in practice a more convenient method of measuring forces must be investigated, but we cannot assume weight as such a measure, because such an assumption supposes us to be acquainted with the nature of the force of gravity, when, at the same time, we are supposed altogether ignorant of every force.

Many writers on Statics commence with the consideration of the lever, and take certain experimental results respecting its equilibrium as the fundamental principles of the science. This method, though allowable, on account of its great simplicity, when the science was in its infancy, ought not now to be adopted, when, as is agreed upon by all, the science has attained a very considerable degree of maturity. In such a system of Statics, the composition of forces acting at a point is usually deduced from the conditions of equilibrium of two forces acting on the bent lever, which have been previously deduced from the straight lever, by introducing the principle of the transmissibility of force. Now it is manifest, that when forces act at a point, no transmission can take place; and, consequently, the composition and resolution of forces would remain unaltered, if the principle of transmission were not true; wherefore, in establishing the composition of force by the lever, or by any other method involving the transmission of force, we introduce a superfluous principle, and thereby make it depend upon experiments foreign to the subject, when in fact it ought to follow from the definition of force, and be independent of all experiments, whose sole object is not limited to the illustration of the nature of force itself: and, therefore, every demonstration of it, which professes to be grounded on philosophic principles, and the rules of strict and logical reasoning, ought not to involve the transmission of force either directly or by implication.

In the second chapter the law of the transmission of force is proved by a simple and obvious experiment, and its effect in reducing two very extensive classes of forces to the formulae of Chapter I, is pointed out in Arts. 44, 45. I have thought it necessary to insist upon the impossibility of establishing this principle by theory, because several authors have attempted to do so. Of attempts of this kind, those which depend upon removing successively different portions of the rigid body, until it is reduced to a straight line, are evidently not at all to the purpose, since they take for granted that these portions may be removed without prejudicing the effect, which is precisely the property to be proved.

It will not, I think, be said, that I have laid too great stress upon this point, when it is considered that there is scarcely a single property of forces acting on a rigid body which can be demonstrated without introducing the principle in question; and, indeed, it is of such importance, that were it not for the single exception of a very remarkable class of forces denominated couples, all forces whatever acting on a rigid body might at once be reduced by its means to others acting at its point. With respect to the last mentioned class of forces, considerable care has been taken clearly to point out their origin, and to show that their separate treatment is not so much a matter of convenience as of necessity. In the enunciations and demonstrations of their properties I have, for the preservation of uniformity, frequently deviated considerably from their admirable inventor Poinsot.

Though it was no part of my design, in compiling the present treatise, to enter into the lengthened details of practical Mechanics, yet I have been anxious so far to give it a practical bias as should make the student acquainted with the imperfections and deficiencies of theory, and the difficulties to be expected in applying it to practice; and in several instances (which may serve as precedents for other cases), I have shown how, by proper experiments, such an extension of the theory is effected as enables us to overcome several of the most important of these impediments.