stand that these letters represent numbers, and that the numbers are the numbers of pounds which the forces are respectively capable of just lifting. 8. He will understand also that it is for convenience, and not of necessity, that reference is made to weight, as a standard by which to measure force. Instead of taking as the unit of force that force which will just lift a weight of 1 lb., we might take the force which would just break a given beam, or just bend a given spring, but it is manifest that this would not be so convenient a method. 9. In order to describe a force something else must be given beside the number of pounds which it will just lift. Two horses of equal strength move a carriage because they are harnessed in such manner as to pull in parallel directions; if they were harnessed on opposite sides of the carriage, so as to pull in opposite ways, the carriage would not move; and yet they might be exerting precisely the same amount of force as before. Hence to know the real effect of a force, we must know not only its magnitude, but also its direction; not only the intensity with which it pulls, but the direction in which it tends to make the particle on which it acts move, and in which the particle would begin to move if not hindered. And not only must we know the magnitude and direction of a force, in order to know its real effect, but it is easy to satisfy ourselves that there is nothing else connected with a force that we can know. If we know the magnitude and direction, then we know the force completely. 10. Hence a very convenient mode of representing forces suggests itself, which will be used continually throughout this treatise. In order to determine or describe a finite straight line, it is necessary that we should know concerning it just these two things and no more, which have been described as necessary in the case of a force. We must know the magnitude of the straight line and its direction, and we then know everything. It is clear therefore that we may take a finite straight line to represent a force: the direction in which the straight line is drawn will shew the direction of the force, and with regard to its magnitude it will only be necessary to agree that a certain length of line shall represent a certain amount of force; as for instance, that 1 inch shall stand for 1 lb.; and then by measuring the line wo shall know how many pounds it is intended to represent. 11. This method of representing forces, which is of infinite use in mechanics, is called representing forces geometrically. I will endeavour to describe it more precisely. Let AB represent in magnitude and direction a certain force P. Then P A stands for P lbs., and if we agree to re B present a pound by an inch, AB must contain P inches. Moreover the position of AB on the paper represents P's direction, or (as it is sometimes called) the line of P's action. There is nothing however in this representation to shew whether the force is acting from A towards B, or from B towards A. This deficiency in the representation may be supplied by the introduction of an arrowhead, as in the annexed figure, in which the force is supposed to act upon a particle A B at A, and to tend to make it move from left to right across the paper. Sometimes the symbol which denotes the magnitude of the force is also introduced; thus the annexed figure would represent a force P, tending to make the particle A at A move in the direction of the arrow-head. P 12. This method may be further applied to represent two or more forces acting upon the same particle. Thus the particle A may be under the action of two forces P and Q, acting in different directions, and we should represent this state of things by such a figure as is here given. And so on for any number of forces. A 13. The representation here given of two forces acting on the same particle introduces us to that which is the fundamental problem of Statics. It is manifest that two forces acting on a particle as in the last figure cannot possibly counteract each other, cannot produce equilibrium. Motion would ensue; and such motion cannot be prevented, except by the action of some third force. Of course two equal forces, if they act in opposite directions, can keep a particle at rest, or in equilibrium; but in no other case is this possible; in general there must be three forces at least, in order that a particle may be kept by their united action in equilibrium. Then suppose we have two forces P and Q acting upon a particle A, and suppose that the particle would N R P X begin to move in the direc tion AX; produce AX backwards to N, and suppose AN to be a string made fast at N; then it is evident that the particle cannot move at all; and therefore it will be possible to find a force R, which acting in the direction AN will counteract the combined effect of the forces P and Q. Thus it is clear that three forces may be found to keep a particle at rest; and that if two forces be given, a third may always be found which shall counteract the effect of the other two. The discovery of the magnitude and direction of this third force, which will counteract the effect of two given forces, is the fundamental problem in Statics. 14. There are two methods of solving this problem. The one by reference to experiment; the other by mathematical demonstration depending upon axioms after the manner of geometry. This latter is the true scientific method; but there is considerable advantage to be derived from treating the subject in the first instance experimentally; the reader will find himself familiarized by the experimental treatment with the new conceptions belonging to the subject, before he has to deal with those conceptions in their more abstract form; he will know in fact distinctly what it is that he is required to prove mathematically, before he is called to the effort of following and understanding the proof. 15. I say the reader; but I cannot refrain from taking this opportunity of reminding the student, that he will find writing a much more effective mode of study than reading. Let him write out from the book several times any difficult proposition, and he will find that he has gained more knowledge of the proposition than he could have gained in a much longer time spent in merely reading it. The method of writing, which appears slow and laborious, is in reality to all, except a very few, an important economy of time and trouble. 16. The next two Chapters deal with statical principles treated experimentally. CHAPTER II. EXPERIMENTAL MECHANICS. COMPOSITION AND RESOLUTION OF FORCES WHICH ACT AT ONE POINT, OR ON THE SAME PARTICLE. 1. THE HE simplest case of forces acting at the same time upon the same particle, is that of two forces acting in such a manner as to keep it at rest. It is evident that a particle under the action of two forces cannot be at rest unless the two forces be exactly equal in magnitude and opposite in direction. Let us denote by P and Q two forces acting upon a particle in opposite directions, then for equilibrium we must have 2. If the forces P and Q be not equal, the greater will preponderate; and the particle will be under exactly the same circumstances as if it were acted upon by the excess of the greater over the smaller force. For instance, if a particle be acted upon by a force of 5 lbs., tending to draw it from left to right across the leaf of this book, and by a force of 3 lbs., tending to draw it from right to left, the particle will be under exactly the same circumstances as if it were acted upon by a force of 2 lbs., tending to draw it from left to right. In this case the force of 2 lbs. is said to be the resultant of the two opposite forces 5 lbs. and 3 lbs. More generally, if the two forces P and Q act in opposite directions, and P be the greater, P-Q will be the result ant of P and Q, and will tend in the same direction as rý The resultant of these two forces may be denoted by R, . we shall then have be etion UU deter same thing, It is evident that if P act upon the particle and tend to draw it in one direction, and Q together with R tend to draw it in the opposite direction, the particle will be at rest. If P and Q tend to draw the particle in the same direction, and we call R their resultant, we shall have in like manner P+Q R.................. (4). 3 And generally, if a particle be acted upon by any number of forces P1 P2 P ̧... tending to draw it in one direction, and by any number Q1 Q2 Q3 ..... tending to draw it in the opposite direction, and we call R the resultant, we shall have P1+P2+P+... - Q1 Q2-Q3 2 - = R...... (5). 3. It is frequently convenient to use a symbol for a force, which shall indicate not only the magnitude of the force, but also the direction in which it acts. Now in Trigonometry we make use of the signs + and to indicate the directions in which straight lines are drawn, and very great advantage is derived therefrom. If we take a fixed point A, and draw a B' straight line AB of length a in a B given direction, and then draw a straight line AB' of the same length (a) in the exactly opposite direction, we distinguish AB from AB' by calling AB + a, and AB' — a. It is quite unnecessary to explain to any person, who is acquainted with Trigonometry, the remarkable simplicity and generality which is given to formulæ by this method. Suppose then we adopt a similar convention respecting forces; that is, if A be a particle acted upon by a force P tending to move it from A towards B, let us denote the force by P, and then we can denote by -P an equal force tending to move the particle from A towards B'. of 4. We can by this convention enunciate, in a very neat nd simple form, a proposition which expresses the rule for fding the resultant of any number of forces, acting upon all, vrticle along the same straight line, but some tending to it in one direction along the straight line, and others 16. it in the exactly opposite direction; for we may treated expe troub |