■I 
 
 -« • >«.. 
 
 • ill'!' III!
 
 THE LIBRARY 
 
 OF 
 
 THE UNIVERSITY 
 
 OF CALIFORNIA 
 
 LOS ANGELES 
 
 GIFT OF 
 
 John S.Prell
 
 s-^ 
 
 \v
 
 THE PRINCIPLES 
 
 OF 
 
 MECHANICS 
 
 DESIGNED 
 
 FOR THE USE OF STUDENTS 
 
 UNIVERSITY. 
 
 BY JAMES WOOD, D.D. 
 
 DEAN OF ELY, AND 
 MASTER OF ST. JOHN's COLLEGE, CAMBRIDGE. 
 
 EIGHTH EDITION. 
 
 CAMBRIDGE: 
 
 Printed by J. Smith, Printer to the University ; 
 
 SOLD BY 
 
 J. & J. J. DEIGHTON, T. STEVENSON, AND R. NEWBY, CAMBRIDGE ; 
 AND B. FELLOWES, LUDGATE-«TREET, LONDON. 
 
 JOHfV iSri^ELL 
 
 CM & Mechanical Engineer. 
 
 8AN FltAAClSLU, VAL.
 
 Engineering 
 Library 
 
 M53 
 
 t 
 
 CONTENTS. 
 
 Page 
 On Matter and Motion 2 
 
 On the Laws of Motion IG 
 
 On the Composition and Resolution of Motion 28 
 
 On the Mechanical Powers 42 
 
 On the Center of Gravity 84 
 
 On the Collision of Bodies 101 
 
 On accelerated and retarded Motion 121 
 
 On the Oscillation of Bodies 142 
 
 On Projectiles 165 
 
 Appendix. Action of Bodies on Machines in Motion 180 
 
 733257
 
 THE PRINCIPLES 
 
 OF 
 
 MECHANICS 
 
 The term Mechanics has at different times, and by 
 different writers, been applied to branches of science essen- 
 tially distinct from each other. It was originally confined 
 to the doctrine of equilibrium, or the investigation of the 
 proportion of powers when they balance each other. 
 
 Later writers, adapting the term to their discoveries, 
 have used it to denote that science which treats of the 
 nature, production, and alteration of motion ; giving to 
 the former branch, by way of contra-distinction, the name 
 of Statics. 
 
 Others, giving the term a still more comprehensive 
 meaning, have applied it to both these sciences. 
 
 None of these definitions will exactly suit our present 
 purpose; the first being too contracted, and the others 
 much too extensive, for a treatise which is intended to 
 be an introduction only, to the higher branches of philo- 
 sophy. Our system of Mechanics will comprise the doctrine 
 of equilibrium, and so much of the science of motion as is 
 necessary to explain the effects of impact and gravity. 
 
 A .
 
 SECTION I 
 
 ON MATTER AND MOTION. 
 
 DEFINITIONS. 
 
 Art. (1.) Matter is a substance, the object of our 
 senses, in which are always united the following properties ; 
 extension, figure, solidity, mobility, divisibility, gravity, 
 and inactivity. 
 
 (2.) Extension may be considered in three points of 
 view: 1st. As simply denoting the part of space which lies 
 between two points, in which case it is called distance. 
 2d. As implying both length and breadth, wheil it is 
 denominated surface or area. 3d. As comprising three 
 dimensions, length, breadth, and thickness; in which case 
 it may be called bulk, capacity, or content. It is used 
 in the last of these senses when it is said to be a property 
 of matter. 
 
 (3.), Figure is the boundary of extension. The por- 
 tions of matter, from which we receive our ideas of this 
 substance, are bounded ; that is, they have figure. 
 
 (4.) Solidity is that property of matter by which it 
 fills space ; or, by which any portion of matter excludes 
 every other portion from that part of space which it oc- 
 cupies; and tluis it is capable of resistance and protrusion.
 
 DEFINITIONS. 3 
 
 " There is no idea which we receive more constantly from 
 sensation than solidity. Whether we move or rest, in what 
 posture soever we are, we always feel something under us 
 that supports us, and hinders our farther sinking down- 
 wards; and the bodies which we daily handle make us 
 perceive that, whilst they remain between them, they do 
 by an insurmountable force hinder the approach of the 
 parts of our hands that press them*.^' 
 
 (5.) Mobility, or a capacity of being transferred from 
 one place to another, is a quality found to belong to all 
 bodies upon which we can make suitable experiments ; and 
 hence we conclude that it belongs to all matter. 
 
 (6.) Divisibility signifies a capacity of being separated 
 into parts. That matter is thus divisible, our daily ex- 
 perience assures us. How far the division can actually be 
 carried, is not so easily seen. We know that many bodies 
 may be reduced to a very fine powder by trituration ; by 
 chemical solution, the parts of a body may be so far di- 
 vided as not to be sensible to the sight ; and by the help 
 of the microscope we discover myriads of organized bodies, 
 totally unknown before such instruments were invented. 
 These and similar considerations, lead us to conclude, that 
 the division of matter is carried to a degree of minuteness 
 far exceeding the bounds of our faculties ; and it seems not 
 unreasonable to suppose, that this capacity of division is 
 without limit; especially, as we can prove, theoretically, 
 that any portion of extension is divisible into parts less 
 and less without end-f*. 
 
 * Locke's Essay, Book II. Chap. iv. 
 
 t Porro corporum partes divisas et sibi rautuo contiguas ab invicem 
 separari posse ex phaenomenis novimus, et partes indivisas in partes 
 minores ratione distingui posse ex mathematica certum est. Utrum vera 
 partes illae distinctae et nonduin divisas per vires nature dividi et ab invi- 
 cem separari possint, incertum est. Newt. Prineip. Lib. III. Reg. 3.
 
 DEFINITIONS. 
 
 From the extremities of the line AB, draw AC, BDy 
 parallel to each other, and in opposite directions; in ^C 
 take any number of points E, F, G, &c. and join DE, 
 
 DF, DG, &c. these lines will cut AB in different points ; 
 and since, in the indefinite line AC, an unlimited number 
 of points may be taken^ the number of parts into which 
 AB is divisible, is indefinite. 
 
 This property of extension may also be proved ex 
 
 ahsurdo. If possible, let AB he the least portion of a 
 circular arc; take C the center, join CA, CB, and with 
 the center C, and any radius C«, less than CA, describe 
 a circle cutting CA and CB in the points a and b ; then, 
 because AB and ab are similar arcs, they are as their radii; 
 therefore «6 is less than AB; or a portion of extension less 
 than the least possible has been found, which is absurd. 
 Hence, any portion of extension is divisible into parts less 
 and less, without ever coming to a limit.
 
 DEFINITIONS. &■ 
 
 It has been supposed by some writers that there are 
 certain indivisible particles of matter, of the same figure 
 and dimensions, by the different modifications of which 
 different bodies are formed. As no arguments are adduced 
 in favour of this hypothesis, and as experiment seems to 
 lead us to a contrary conclusion, we cannot allow it a place 
 amongst the principles upon which our theory is to rest. 
 
 (7-) Gravity is the tendency which all bodies have to 
 the center of the earth. 
 
 We are convinced of the existence of this tendency by 
 observing that, whenever a body is sustained, it's pressure 
 is exerted in a direction perpendicular to the horizon ; and 
 that, when every impediment is removed, it always de- 
 scends in that direction. 
 
 The weight of a body is it's tendency to the earth, 
 compared with the like tendency of some other body, which 
 is considered as a standard. Thus, if a body with a 
 certain degree of gravity be called one pound, any other 
 body which has the same degree of gravity, or which by 
 it's gravity will produce the same effect, under the same 
 circumstances, is also called a pound; and these two 
 together, two pounds, &c. 
 
 Gravity is not an accidental property of matter, arising 
 from the figure or disposition of the parts of a body ; for 
 then, by changing it's shape, or altering the arrangement 
 of the particles which compose it, the gravitation of the 
 mass would be altered. But we find that no separation of 
 the particles, no change of the structure, which human 
 power can effect, produces any alteration in the weight. 
 
 As gravity is a property belonging to every particle 
 of a body, independent of its situation with respect to other 
 particles, the gravity of the whole is the aggregate of the 
 gravities of all it's parts. Thus, though the weight of the 
 whole is not altered by any division, or new arrangement
 
 6 DEFINITIONS. 
 
 of the particles, yet every increase or diminution of their 
 number produces a corresponding increase or diminution 
 of the weight. 
 
 Our present subject does not lead us to consider gravi- 
 tation in any other point of view than simply as a tendency 
 in bodies to the center of the earth, or to attend to it''s 
 effects at any considerable distance from the surface; it 
 may not, however, be improper to observe that the opera- 
 tion of this principle is much more extensive. Every 
 portion of matter gravitates towards every other portion, 
 in that part of the system of nature which falls under our 
 observation. The gravitation indeed of small particles 
 towards each other is not perceived, on account of the 
 superior action of the earth*; yet it has been found, by 
 the accurate observations of Dr. Maskelyne, in Scotland, 
 that the attraction of a mountain is sufficient to draw the 
 plumb-line sensibly from the perpendicular. 
 
 Sir I. Newton has discovered that the moon is re- 
 tained in her orbit by the agency of a cause similar to that 
 by which a body falls to the ground, differing from it only 
 in degree ; and this in consequence of the greater distance 
 of the moon from the earth's center. The same author 
 has demonstrated that the planets are retained in their 
 respective orbits by a principle of the same kind; and 
 later writers have shewn, that the minutest irregularities 
 in their motions may be satisfactorily deduced from the 
 known laws of it's operation. 
 
 (8.) Inactivity may be considered in two lights: 
 1st. As an inability in matter to change it's state of rest 
 or uniform rectilinear motion: 2d. As that quality by 
 
 * This attraction is not sufficient to explain the common experiment 
 of two particles of the same kind, as quicksilver, &c. when placed upon 
 a smooth horizontal plane, running together. If the effect were not 
 produced by some power different from gravitation, a drop of oil would 
 run, in the same manner, towards a drop of water ; which is not found 
 to be the case.
 
 DEFINITIONS. / 
 
 which it resists any such change *. In this latter sense it 
 is usually called the force of inactivity, the inertia, or the 
 vis inertice. 
 
 The inactivity of matter, according to the former ex- 
 planation, is laid down as a law of motion ; the truth of 
 which we shall endeavour to establish in the next section. 
 
 That a body resists any change in it's state of rest, or 
 uniform rectilinear motion, is known from constant ex- 
 perience. We cannot move the least particle of matter 
 without some exertion; nor can we destroy any motion 
 without perceiving some resistance -[-. Thus, we see, in 
 general, that inertia is a property inherent in all bodies 
 with which we are concerned ; different indeed in different 
 cases, but existing, in a greater or less degree, in all. The 
 quantity we are not at present considering ; the existence 
 of the property, every one, from his own observation, 
 will readily allow. We know indeed from experience |, 
 that the inertia of a body is not altered by altering the 
 arrangement of it's parts; but if one portion of matter 
 be added to another, the inertia of the whole is increased ; 
 and if any part be removed, the inertia is lessened. 
 This clearly shews that it exists, independently, in every 
 particle, and that the whole inertia is the aggregate of 
 all if s parts. 
 
 Hence it follows, from our notion of quantity, that 
 if to a body with a certain quantity of inertia, another, 
 
 * That is, a change from rest to uniform rectilinear motion, or a 
 change in its uniform rectilinear motion. The term is sometimes ap- 
 plied to the resistance which a body makes to the production, or altera- 
 tion of motion, when this resistance acts at a mechanical advantage, 
 or disadvantage. 
 
 t It must he observed, that this resistance is distinct from and 
 independent of gravity ; because it is perceived where gravity produces 
 no effect: as, when a wheel is turned round it's axis, or a body moved 
 along an horizontal plane. 
 
 \ See Art. 25.
 
 "8 DEFINITIONS. 
 
 which has an equal quantity, be added, the whole inertia 
 Avill be doubled; and that by the repeated addition of 
 equal quantities, the whole inertia will be increased in 
 the same proportion with the number of parts. 
 
 These properties, which are always found to exist 
 together in the same substance, have sometimes been said 
 to be essential to matter. Whether they are, or are not 
 necessarily united in the same substance, it is impossible 
 to decide, nor does it concern us to inquire. The business 
 of natural philosophy is not to find out what might have 
 been the constitution of nature, but to examine what it 
 is in fact : and to account for the phaenomena, which fall 
 under our observation, from those properties of matter 
 which we know by experience that it possesses. 
 
 (9.) By the quantity of matter in a body, we under- 
 stand the aggregate of ifs particles, each of which has 
 a certain degree of inertia. Or, in other words, if we 
 suppose bodies made up of particles, each of which has the 
 same inertia, the quantity of matter in one body, is to the 
 quantity of matter in another, as the number of such 
 particles in the former, to the number in the latter*. 
 
 When we consider bodies as made up of parts, and 
 compare them in this respect, it becomes necessary to 
 give a definite and precise description of those parts ; 
 otherwise our notion of the quantity will be vague and 
 inaccurate. Now the only properties of matter which 
 admit of exact comparison, and which depend upon the 
 number, and not upon the arrangement of the particles, 
 
 * Quantitas materist' est inensura ejusdem orta ex illius densitate 
 et magnitudine conjunctim. Newt. Princip. Def. 1. 
 
 Ejusdem esse densitatis dico, quarum vires inertise sunt ut magni- 
 tudines. Lib. III. Prop. 6. Cor. 4. 
 
 Attendi enim oportet ad punctorum numerum, ex quibus corpus 
 movendum est conflatura. Puncta vero ea int^ se aequalia censeri 
 debent, non quaB acque sunt parva, sed in quae eadem potentia aequales 
 exerit effectus. Ei'i,. Mech. 139.
 
 DEFINITIONS. 9 
 
 are weight and inertia ; either of which may properly be 
 made use of as a measure of the quantity of matter ; and 
 since, at a given place, they are proportional to each 
 other, as we shall shew hereafter (Art. 25), it is of little 
 consequence which measure we adopt. The inertia has 
 been fixed upon, because the gravity of a body, though 
 invariable at the same place, is different at different dis- 
 tances from the center of the earth ; w^hereas, the inertia 
 is always, and under all circumstances, the same. 
 
 The density of a body is measured by the quantity 
 of matter in a given hulk; and it is said to be uniform 
 when equal quantities of matter are always contained in 
 equal bulks. 
 
 (10.) By motion we understand the act of a body's 
 changing place ; and it is of two kinds, absolute and 
 relative. 
 
 A body is said to be in absolute motion when it is 
 actually transferred from one point in fixed space to 
 another ; and to be relatively in motion^ when it's situation 
 is changed with respect to the surrounding bodies. 
 
 These two kinds of motion evidently coincide when 
 the bodies, to which the reference is made, happen to 
 be fixed. In other cases, a body relatively in motion, 
 or relatively at rest, may or may not be absolutely in 
 motion. Thus, a spectator standing still on the shore, 
 if his place be referred to a ship which sails by, is rela- 
 tively in motion ; and the several parts of the vessel are 
 at rest, with respect to each other, though the whole is 
 transferred from one part of space to another. 
 
 The motion of a body is swifter or slower^ according 
 as the space passed over, in a given time, is greater or 
 less. 
 
 When a body always passes over equal parts of space 
 in equal successive portions of time, its motion is said
 
 10 DEFINITIONS. 
 
 to be uniform. When the successive portions of space, 
 described in equal times, continually increase, the motion 
 is said to be accelerated ; and to be retarded, when those 
 spaces continually decrease. Also the motion is said to 
 be uniformly accelerated or retarded, when the increments 
 or decrements of the spaces, described in equal successive 
 portions of time, are always equal. 
 
 (11.) The degree of swiftness or slowness of a body's 
 motion is called it's velocity ; and it is measured by the 
 space, uniformly described, in a given time. 
 
 The given time, taken as a standard, is usually one 
 second; and the space described is measured in feet. 
 Thus, when v represents a body's velocity, v is the number 
 of feet which the body would uniformly describe in one 
 second. 
 
 If a body's motion be accelerated or retarded, the 
 velocity at any point is not measured by the space actually 
 described in a given time, but by the space which would 
 have been described in the given time, if the motion had 
 continued uniform, from that point ; or had, at that point, 
 ceased to increase or decrease. 
 
 (12.) CoR. 1. If two bodies move uniformly on the 
 same line, in opposite directions, their relative velocity 
 is equal to the sum of their absolute velocities, since the 
 space by which they uniformly approach to, or recede 
 from, each other, in any time, i§ equal to the sum of the 
 spaces which they respectively describe in that time. 
 
 When the bodies move in the same direction, their 
 relative velocity is equal to the difference of their absolute 
 velocities. 
 
 (13.) CoK. 2. When a body moves with an uniform 
 velocity, the space described is proportional to the time 
 of it';s motion.
 
 DEFINITIONS. 11 
 
 Let the body describe the space a in the time 1 ; then 
 since the motion is uniform, it will describe the space ta 
 in the time ^; that is, the space described is proportional 
 to the time. 
 
 (14.) CoR. 3. When bodies have different uniform 
 motions, the spaces described are proportional to the times 
 and velocities jointly *. 
 
 Let V and v be the velocities of two bodies A and B ; 
 T and t the times of their motions; S and 5 the spaces 
 described. Also let S' be the space described by B in 
 the time T: 
 
 Then S : S' :: V ; V (Art. 11), 
 S' : s :: T '. t (Art. 18). 
 
 Comp. S : s :: TV: tv ; 
 that is, Soc TV {Alg. Art. 195). 
 
 Ex. Let the times be to each other as 6 : 5, and the 
 velocities as 2 : 3 ; then 
 
 »S' : * :: 2 X 6 : 3 X 5 :: 4 : 5. 
 (15.) Cor. 4. Since S oc TV, we have Foc~, and 
 
 S 
 7' oc -- , {Alg. Art. 205). 
 
 Ex. 1. Let A move uniformly through 5 feet in s'\ 
 and B through f) feet in 7''; required the ratio of tlie 
 velocities. 
 
 * Since the times and velocities may, in each case, be represented 
 by numbers, there is no impropriety in speaking of their products. 
 The truth of this observation will be evident, if the proposition be 
 expressed in different words. When the uniform velocities of two bodies 
 are in the ratio of the numbers V and v, and the times of their motions 
 in the ratio of the numbers T and t, the spaces described are in the ratio 
 of the numbers TV and tv.
 
 12 DEFINITIONS. 
 
 5 9 
 
 r :«::-:-:: 35 : 27. 
 
 3 7 
 
 Ex. 2. Let ^'s velocity be to ^'s velocity as 5 to 4 ; 
 to compare the times in which they will describe 9 and 
 7 feet respectively. 
 
 9 7 
 T : ^ ;:-:-:: 36 : 35. 
 
 5 4 
 
 (16.) CoR. 5. Since the areas of rectangles are in the 
 ratio compounded of the ratios of their sides, if the bases 
 of two rectangles represent the velocities of two motions, 
 and altitudes the times, the areas will represent the spaces 
 described. 
 
 (I7.) The quantity of motion, or momentum of a 
 body, is measured by the velocity and quantity of matter 
 jointly. 
 
 Thus, if the quantities of matter in two bodies be 
 represented by 6 and 7? and their velocities by 9 and 8, 
 the ratio of 6 x 9 to 7 X 8, or 27 to 28, is called the ratio 
 of their momenta. 
 
 (18.) CoR. r. If M be the momentum of a body, 
 Q it''s quantity of matter, and V iVs velocity, then 
 
 M M 
 
 since J/ oc Q T, we have Q ex: -— ; and F oc — . 
 
 V Q 
 
 Ex. If the quantities of motion be as 6 to 5, and the 
 velocities as 7 to 8, what is the ratio of the quantities of 
 matter ? 
 
 M , 6 5 
 
 Since Q oc — , we have Q : 9 ::-:-:: 48 : 35. 
 
 (I9.) CoR. 2. If M be given, Qoc— ; and con- 
 versely if Q -^ — , il/ is invariable. {Algebra, Art. 206.)
 
 DEFTNITIOXS. 13 
 
 (20.) Whatever changes, or tends to change, the state 
 of rest or uniform rectilinear motion of a body, is called 
 force. 
 
 Thus, impact, gravity, pressure, &c. are called forces. 
 
 When a force produces it's effect instantaneously, it 
 is said to be impulsive^. When it acts incessantly, it 
 is called a constant^ or continued force. 
 
 Constant forces are of two kinds, uniform and variable. 
 A force is said to be uniform, when it always produces 
 equal effects in equal successive portions of time; and 
 variable., when the effects produced in equal times are 
 unequal. 
 
 Forces, which are known to us only by their effects, 
 must be compared by estimating those effects under the 
 same circumstances. Thus, impulsive forces must be 
 measured by the whole effects produced ; uniform forces, 
 by the effects produced in equal times ; and variable forces, 
 by the effects which would be produced in equal times, 
 were the forces to become and continue uniform during 
 those times. 
 
 The effects produced by the actions of forces are of 
 two kinds, velocity and momentum; and thus we have 
 two methods of comparing them, according as we con- 
 ceive them to be the causes of velocity or momentum. 
 
 (21.) The accelerating force is measured by the 
 velocity uniformly generated in a given time, no regard 
 being had to the quantity of matter moved. 
 
 Thus, if the velocities uniformly generated, in two 
 cases, in equal times, be as 6 to 7, the accelerating forces 
 are said to be in that ratio. 
 
 * Though we cannot conceive finite effects to be produced otherwise 
 than by degrees, and consequently in successive portions of time; yet 
 when these portions are so small as' not to be distinguishable by our 
 faculties, the effects may be said to be instantaneous.
 
 14 DEFINITIONS. 
 
 The accelerating force of gravity, at the same place, 
 is invariable ; for all bodies falling freely, in an exhausted 
 receiver, acquire equal velocities in any given time. 
 
 (22.) The moving force is measured by the vwmen- 
 tum uniformly generated in a given time. 
 
 If the momenta thus generated, in two cases, be as 14 
 to 1 5, the moving forces are said to be in that ratio. 
 
 (23.) CoR. 1. Since the momentum is proportional to 
 the velocity and quantity of matter, the moving force varies 
 as the accelerating force and quantity of matter jointly. 
 
 The moving force of gravity varies as the quantity 
 of matter moved, because the accelerating force is given 
 (Art. 21). 
 
 (24.) Cor. 2. Hence it follows that the accelerating 
 force varies as the moving force directly, and the quantity 
 of matter inversely. 
 
 Prop. I. 
 
 (25.) The vis inerticB of a body is proportional to ifs 
 weight. 
 
 The inertia, as was observed on a former occasion, is 
 the resistance which a body makes to any change in it's 
 state of rest or uniform rectilinear motion (Art. 8.) ; and 
 this resistance is manifestly the same in two bodies, if the 
 same force, applied in the same manner, and for the same 
 time, communicate to each of them the same velocity. 
 
 Let two bodies, A and B, equal in weight, be placed 
 in two similar and equal boxes, which are connected by 
 a string passing over a fixed pulley ; then these will ex- 
 actly balance each other; and if the whole be put into 
 motion, the gravity can neither accelerate nor retard that 
 motion; the whole resistance therefore to the communica- 
 tion of motion in the system, arises from the inertia of
 
 MEASURE OF THE INERTIA. 15 
 
 the weights, the inertia of the string and pulley*, the 
 friction upon the axis, and the resistance of the air-f*. 
 
 Now let a weight C be added on one side, and let the 
 velocity generated in any given time, in the whole system, 
 by this additional weight, be observed. 
 
 Then in the place of J, or B^ substitute any other mass 
 of the same weight, and it will be found that C will, in the 
 same time, generate the same velocity in this system as in 
 the former; and, therefore, the whole resistance to the 
 communication of motion must be the same. Also the 
 inertia of the string and pulley, the friction of the axis, 
 and the air^s resistance, are the same in the two expe- 
 riments ; consequently, the resistance arising from the 
 inertia of the weights is the same: That is, so long as 
 the weight remains unaltered, whatever be the form or 
 constitution of the body, the inertia is the same. 
 
 Also, since the whole quantity of inertia is the aggre- 
 gate inertia of all the parts, if the weight be doubled, an 
 equal quantity of inertia is added to the former quantity, 
 or the whole inertia is doubled ; and in the same manner, 
 if the weight be increased in any proportion, by the re- 
 peated addition of equal weights^ the inertia is increased 
 in the same proportion. 
 
 It may be observed, that the velocity generated in a 
 given time, is the same, whether the system begins to 
 move from rest or not ; therefore the inertia is the same, 
 whether the system be at rest or in motion. 
 
 (26.) Cor. Since the quantity of matter is measured 
 by the inertia (Art. 9.), it is also proportional to the 
 weight. 
 
 * See Note, page 8. 
 
 t This experiment may be made with great accuracy by means of 
 a machine, invented by Mr. Atwood, for the purpose of examining the 
 motions of bodies when acted upon by constant forces. This machine 
 is described in his well-known treatise on the Redilinmr Motion and 
 Rotaimn of Bodies, (p. 299.)
 
 SECTION H. 
 
 ON THE LAWS OF MOTION. 
 
 THE FIRST LAW. 
 
 (27.) If a body he at rest, it will continue at rest, 
 and if ifi motion, it will continue to move uniformly 
 forward in a right line, till it is acted upon by some 
 external force. 
 
 That a body at rest cannot put itself in motion, we 
 know from constant and universal experience. 
 
 That a body in motion will continue to move uniformly 
 forward in a right line till it is acted upon by some exter- 
 nal force, though equally certain, is not, it must be allowed, 
 equally apparent ; since all the motions which fall under 
 our immediate observation, and rectilinear motions in par- 
 ticular, are soon destroyed. If however we can point out 
 the causes which tend to destroy the motions of bodies, and 
 shew, experimentally, that, by removing some of them and 
 diminishing others, the motions continually become more 
 uniform and rectilinear, we may justly conclude that any 
 deviation from the first direction, and first velocity, must 
 be attributed to the agency of external causes ; and that 
 there is no tendency in matter itself, either to increase or 
 diminish any motion impressed upon it. 
 
 Now the causes which retard a body's motion, besides 
 collision, or the evident obstruction which it meets with
 
 THE LAWS OF MOTION. . 17 
 
 from sensible masses of matter, are gravity, friction, and 
 the resistance of the air ; and it will appear, by the follow- 
 ing experiments, that when these are removed, or due 
 allowance is made for their known effects, we are necessarily 
 led to infer the truth of the law above laid down. 
 
 1st. If a ball be thrown along a rough pavement, it's 
 motion, on account of the many obstacles it meets with, 
 will be very irregular, and soon cease ; but if it be bowled 
 upon a smooth bowling-green, it's motion will continue 
 longer, and be more rectilinear ; and if it be thrown along 
 a smooth sheet of ice, it will preserve both it's direction 
 and it's motion for a still longer time. 
 
 In these cases, the gravity, which acts in a direction 
 perpendicular to the plane of the horizon, neither accelerates 
 nor retards the motion ; the causes which produce the latter 
 effect are collision, friction, and the air's resistance; and 
 in proportion as the two former of these are lessened, the 
 motion becomes more nearly uniform and rectilinear. 
 
 2d. When a wheel is accurately constructed, and a 
 rotatory motion about it's axis communicated to it, if the 
 axis, and the grooves in which it rests, be well polished, 
 the motion will continue a considerable time; if the axis 
 be placed upon friction wheels, the motion will continue 
 longer ; and if the apparatus be placed under the receiver 
 of an air pump, and the air be exhausted, the motion will 
 continue, without visible diminution, for a very long time. 
 
 In these instances, gravity, which acts equally on op- 
 posite points of the wheel, neither accelerates nor retards 
 the motion ; and the more care we take to remove the 
 friction, and the resistance of the air, the less is the first 
 motion diminished in a given time. 
 
 3d. If a body be projected in any direction inclined to 
 the horizon, it describes a curve, which is nearly the com- 
 mon parabola. This effect is produced by the joint action 
 
 B
 
 18 THE LAWS OF MOTION. 
 
 of gravity, and the motion of projection; and since the 
 effect produced by the former is known, the effect produced 
 by the latter may be determined. This, it is found, would 
 carry the body uniformly forward in the line in which it 
 was projected ; as will fully appear when we come to the 
 doctrine of projectiles. The deviation of the curve de- 
 scribed from the parabolic form is sufficiently accounted 
 for by the resistance of the air. 
 
 From these, and similar experiments, we are led to 
 conclude that all bodies in motion would uniformly per- 
 severe in that motion, were they not prevented by external 
 impediments; and that every increase or diminution of 
 velocity, every deviation from the line of direction, is to 
 be attributed to the agency of such causes. 
 
 (28.) It may not be improper to observe, that this law 
 suggests two methods of distinguishing between absolute 
 motions, and such as are only apparent ; one, by considering 
 the causes which produce the motions; and the other, by 
 attendincr to the effects with which the motions are accom- 
 panied *. 
 
 1st. We may sometimes distinguish absolute motion, 
 or change of absolute motion, from that which is merely 
 apparent, by considering the causes which produce them. 
 
 When two bodies are absolutely at rest, they are re- 
 latively so; and the appearance is the same, when they 
 are moving in the same direction, at the same rate; a 
 relative motion therefore can only arise from an absolute 
 motion, or change of absolute motion, in one or both of 
 the bodies. We have seen also, in the last article, that 
 motion, or change of motion, cannot be produced but by 
 force impressed ; and therefore, if we know that such a 
 cause exists, and acts upon one of the bodies, and not 
 
 Newt. Princip. Schol. ad Def.
 
 THE LAWS OF MOT [ON. 19 
 
 upon the other, we conclude that the relative motion 
 arises from a change in the state of rest, or absolute motion 
 of the former; and that with respect to the latter, the 
 effect is merely apparent. Thus, when a person on ship- 
 board observes the coast receding from him, he is convinced 
 that the appearance arises from a motion, or change of 
 motion, in the ship ; upon which a cause, sufficient to pro- 
 duce this effect, acts, namely, the force of the wind or tide. 
 
 The precession of the equinoxes arises from a real 
 motion in the earth, and not from any motion in the 
 heavenly bodies; because we know that there is a force 
 impressed upon the earth, which is sufficient to account 
 for the appearance, 
 
 2d. Absolute motion may sometimes be distinguished 
 from apparent motion, by the effects produced. 
 
 If a body be absolutely in motion, it endeavours by 
 it's inactivity to proceed in the line of it's direction ; if the 
 motion be only apparent, there is no such tendency. 
 
 It is in consequence of the tendency to persevere in 
 rectilinear motion that a body revolving in a circle con- 
 stantly endeavours to recede from the center. The effort 
 thus produced is called a centrifugal force ; and as it arises 
 from absolute motion only, whenever it is observed, we 
 are convinced that the motion is real. 
 
 In order to see the nature and origin of this force, 
 A D 
 
 suppose a body to describe the circle ABC; then at any 
 
 b2
 
 20 THE LAWS OF MOTTOX. 
 
 point A, it is moving in the direction of the tangent AD, 
 and in this direction, by the first law of motion, it en- 
 deavours to proceed; also, since every point D in the 
 tangent is without the circle, this tendency, to move on 
 in the direction of the tangent, is a tendency to recede 
 from the center of motion; and the body will actually 
 fly off, unless it is prevented by an adequate force. 
 
 The following experiment is given by Sir I. Newton 
 to shew the effect of the centrifugal force, and to prove 
 that it always accompanies an absolute circular motion. 
 
 Let a bucket, partly filled with water, be suspended 
 by a string, and turned round till the string is considerably 
 twisted; then let the string be suffered to untwist itself, 
 and thus communicate a circular motion to the vessel. 
 At first the water remains at rest, and ifs surface is smooth 
 and undisturbed ; but as it gradually acquires the motion 
 of the bucket, the surface grows concave towards the 
 center, and the water ascends up the sides, thus en- 
 deavouring to recede from the axis of motion ; and this 
 effect is observed gradually to increase with the absolute 
 velocity of the water, till at length the water and the 
 bucket are relatively at rest. When this is the case, let 
 the bucket be suddenly stopped, and the absolute motion 
 of the water will be gradually diminished by the friction 
 of the vessel ; the concavity of the surface is also diminished 
 by degrees, and at length, when the water is again at rest, 
 the surface becomes plane. Thus we find that the cen- 
 trifugal force does not depend upon the relative, but 
 upon the absolute motion, with which it always begins, 
 increases, decreases, and disappears. 
 
 A single instance will be sufficient to shew the great 
 utility of this conclusion in natural philosophy. 
 
 The diurnal rotation of the heavenly bodies may, as 
 far as the appearance is concerned, be accounted for, either 
 by supposing the heavens to revolve from ' east to west,
 
 THE LAWS OF MOTION. 21 
 
 and complete a revolution in twenty-four hours ; or, the 
 earth to revolve from west to east, in the same time : 
 but the sensible diminution of gravity as we proceed 
 towards the equator, and the oblate figure of the earth, 
 which are the effects of a centrifugal force, prove that the 
 appearance is to be ascribed to a real motion in the earth. 
 
 THE SECOND LAW OF MOTION. 
 
 (29.) Motion, or change of motion, produced in a 
 body, is proportional to the force impressed, and takes 
 place in the direction in which the force acts. 
 
 It has been seen in the preceding articles, that no 
 motion or change of motion is ever produced in a body 
 without some force impressed ; we now assert that it cannot 
 be produced without an adequate force; that when bodies 
 act upon each other, the effects are not variable and 
 accidental, but subject to general laws. Thus, whatever 
 happens in one instance, will, under the same circum- 
 stances, happen again; and when any alteration takes 
 place in the cause, there will be a corresponding and 
 proportional alteration in the effect produced. Were not 
 cause and effect thus connected with, and related to, each 
 other, we could not pretend to lay down any general rules 
 respecting the mutual actions of bodies ; experiment could 
 only furnish us with detached and isolated facts, wholly 
 inapplicable on other occasions; and that harmony, which 
 we cannot but observe and admire in the material world, 
 would be lost. 
 
 In order to understand the meaning and extent of 
 this law of motion, it will be convenient to distinguish
 
 22 THE LAWS OF MOTION. 
 
 it into two cases ; and to point out such facts, under each 
 head, as tend to establish it's truth. 
 
 1st. The same force, acting freely for a given time, 
 will always produce the same effect, in the direction in 
 which it acts. 
 
 Ex. 1. If a body, in one instance, fall perpendicularly 
 through 16^ feet in a second, and thus acquire a velocity 
 which would carry it, uniformly, through SSifeet in that 
 time, it will always, under the same circumstances, acquire 
 the same velocity. 
 
 The effect produced is the same, whether the body 
 begins to move from rest or not. 
 
 Ex. 2. If a body be projected perpendicularly down- 
 wards, the velocity of projection, measured in feet (Art. 11.), 
 will, in one second, be increased by 32^ ; and if it be pro- 
 jected perpendicularly upwards, it will, in one second, be 
 diminished by that quantity. 
 
 Ex. 3. If a body be projected obliquely, gravity will 
 still produce if s effect in a direction perpendicular to the 
 horizon; and the body, which by it's inactivity would 
 have moved uniformly forward in the line of it's first 
 motion, will, at the end of one second, be found 16^ feet 
 below that line ; having thus acquired a velocity of 32^ feet 
 per second, in the direction of gravity. 
 
 2d. If the force impressed be increased or diminished 
 in any proportion, the motion communicated will be in- 
 creased or diminished in the same proportion. 
 
 Ex. If a body descend along an inclined plane, the 
 length of which is twice as great as it's height, the force 
 which accelerates it's motion is half as great as the force 
 of gravity ; and, allowing for the effect of friction, and 
 the resistance of the air, the velocity generated in any
 
 THE LAWS OF MOTION. 23 
 
 time is half as great as it would have been, had the body 
 fallen, for the same time, by the whole force of gravity *. 
 
 (30.) In estimating the effect of any force, two cir- 
 cumstances are to be attended to : First, we must consider 
 what force is actually impressed; for this alone can pro- 
 duce a change in the state of motion or quiescence of a 
 body. Thus, the effect of a stream upon the floats of 
 a water-wheel is not prodviced by the whole force of the 
 stream, but by that part of it which arises from the excess 
 of the velocity of the water above that of the wheel ; and 
 it is nothing, if they move with equal velocities. — Secondly, 
 we must consider in what direction the force acts; and 
 take that part of it, only, which lies in the direction in 
 which we are estimating the effect. Thus, the force of 
 the wind actually impressed upon the sails of a windmill, 
 is not wholly employed in producing the circular motion ; 
 and therefore in calculating it's effect, in this respect, we 
 must determine what part of the whole force acts in the 
 direction of the motion. 
 
 In the following pages, we shall see a great variety 
 of instances in which this method of estimating the effects 
 of forces is applied; and the conclusions thus deduced, 
 
 * The experiments which most satisfactorily prove the truth of this 
 law of motion, are made with Mr. Atwood's machine, mentioned on 
 a former occasion. 
 
 Let two weights, each of which is represented by 9 m, balance each 
 other on this machine; and observe what velocity is generated in one 
 second, when a weight 2m is added to either of them. Again, let 
 the weights 8w, Sm, be sustained, as before, and add 4m to one of them, 
 then the velocity generated in one second is twice as great as in the 
 former instance. Since, therefore, the mass to be moved is the same 
 in both cases, viz. 20m together with the inertia of the machine, it 
 is manifest that when the moving force is doubled (Art. 23.), the 
 momentum generated is also doubled, and, by altering the ratio of 
 the weights, it may be shewn, in any other case, that the momentum 
 communicated is proportional to the moving force impressed.
 
 24 THE LAWS OF MOTION. 
 
 being found, without exception, to agree with experiment, 
 we cannot but admit the truth of the principle. 
 
 (31.) Cor. Since the effect produced upon each other 
 by two bodies, depends upon their relative velocity, it 
 will always be the same whilst this remains unaltered, 
 whatever be their absolute motions. 
 
 THE THIRD LAW OF MOTION. 
 
 (32.) Action and reaction are eqtial, and in opposite 
 directio7is. 
 
 Matter not only perseveres in it's state of rest or 
 uniform rectilinear motion, but also by it's inertia resists 
 any change. Our experience with respect to this reaction, 
 or opposition to the force impressed, is so constant and 
 universal, that the very supposition of ifs non-existence 
 appears to be absurd. For who can conceive a pressure 
 without some support of that pressure ? Who can sup- 
 pose a weight to be raised without force or exertion? 
 Thus far then we are assured by our senses, that whenever 
 one body acts upon another, there is some reaction: The 
 law farther asserts, that the reaction is equal in quantity 
 to the action. 
 
 By action, we here understand moving force, which, 
 according to the definition (Art. 22.), is measured by the 
 momentum which is, or would be generated, in a given 
 time ; and to determine whether action and reaction, in 
 this sense of the words, are equal or not, recourse must 
 be had to experiment. 
 
 Take two similar and equal cylindrical pieces of wood, 
 from one of which projects a small steel point; suspend 
 them by equal strings, and let one of them descend through
 
 THE LAWS OF MOTION. 25 
 
 any arc and impinge upon the other at rest; then, by 
 means of the steel point, the two bodies will move on 
 together as one mass, and with a velocity equal to half 
 the velocity of the impinging body. Thus the momentum, 
 which is measured by the quantity of matter and velocity 
 taken jointly, remains unaltered ; or, as much momentum 
 as is gained by the body struck, so much is taken from 
 the momentum of the striking body, or communicated to 
 it in the opposite direction. 
 
 If the striking body be loaded with lead, and thus made 
 twice as heavy as the other, the common velocity after impact 
 is found to be to the velocity of the impinging body :: 2 : 3 ; 
 and because the joint mass after impact : quantity of matter 
 in the striking body :: 3 : 2, the momentum after impact : 
 momentum before :: 3 x 2 : 2 x 3, or in a ratio of equality, 
 as in the former case. 
 
 In making experiments to establish the third law of 
 motion, allowance must be made for the air's resistance; 
 and care must be taken to obtain a proper measure of the 
 velocity before and after impact. See Sir I. Newton's 
 Scholium to the Laws of Motion. 
 
 (33.) The third law of motion is not confined to cases 
 of actual impact ; the effects of pressures and attractions, 
 in opposite directions, ax'e also equal. 
 
 When two bodies sustain each other, the pressures in 
 opposite directions must be equal, otherwise motion would 
 ensue ; and if motion be produced by the excess of pressure 
 on one side, the case coincides with that of impact *. 
 
 * The effects of pressure and impact are manifestly of the same 
 kind, and produced in the same way ; excess of pressure, on one side, 
 produces momentum, and equal and opposite momenta support each 
 other by opposite pressures. 
 
 Thus also pressures may be compared, either by comparing the 
 weights which they sustain, or the momenta which they would generate 
 under the same circumstances.
 
 26 THE LAWS OF MOTION. 
 
 When one body attracts another, it is itself also equally 
 attracted ; and as much momentum as is thus communicated 
 to one body, will also be communicated to the other in 
 the opposite direction. 
 
 A loadstone and a piece of iron, equal in weight, and 
 floating upon similar and equal pieces of cork, approacli 
 each other with equal velocities, and therefore with equal 
 momenta; and when they meet, or are kept asunder by 
 any obstacle, they sustain each other by equal and opposite 
 pressures. 
 
 (34.) CoR. Since the action and reaction are equal 
 at every instant of time, the whole effect of the action in 
 a finite time, however it may vary, is equal to the effect 
 of the reaction; because the whole effects are made up 
 of the effects produced in every instant. 
 
 SCHOLIUM. '. 
 
 (35.) These laws are the simplest principles to which 
 motion can be reduced, and upon them the whole theory 
 depends. They are not indeed self-evident, nor do they 
 admit of accurate proof by experiment, on account of the 
 great nicety required in adjusting the instruments, and 
 making the experiments; and on account of the effects 
 of friction, and the air's resistance, which cannot entirely 
 be removed. They are however constantly, and invariably, 
 suggested to our senses, and they agree with experiment 
 as far as experiment can go ; and the more accurately 
 the experiments are made, and the greater care we take 
 to remove all those impediments which tend to render the 
 conclusions erroneous, the more nearly do the experiments 
 coincide with these laws. 
 
 Their truth is also established upon a different ground : 
 from these general principles innumerable jmrticular con- 
 clusion.'i have been deduced ; sometimes the deductions are
 
 THE LAWS OF MOTION. 27 
 
 simple and immediate, sometimes they are made by tedious 
 and intricate operations ; yet they are all, without excep- 
 tion, consistent with each other and with experiment: it 
 follows therefore that the principles, upon which the cal- 
 culations are founded, are true*. 
 
 (36.) It will be necessary to remember, that the laws 
 of motion relate, immediately^ to the actions of particles 
 of matter upon each other, or to those cases in which 
 the whole mass may be conceived to be collected in a 
 point; not to all the effects that may eventually be pro- 
 duced in the several particles of a system. 
 
 A body may have a rectilinear and rotatory motion 
 given it at the same time, and it will retain both. The 
 action also, or reaction, may be applied at a mechanical 
 advantage or disadvantage, and thus they may produce, 
 upon the whole, very different momenta: these effects 
 depend upon principles which are not here considered, 
 but which must be attended to in computing such effects. 
 
 * Atwood on the Motions of Bodies, p. 358.
 
 SECTION III 
 
 ON THE COMPOSITION AND RESOLUTION 
 OF MOTION. 
 
 Prop. II. 
 
 (37-) Two lines, which represent the momenta com- 
 municated to the same or equal bodies, will represent the 
 spaces uniformly described by them in equal times; and 
 conversely, the lines which represent the spaces uniformly 
 described by them in equal times, will represent their 
 momenta. 
 
 The momenta of bodies may be represented by num- 
 bers, as was seen Art. 17; but in many cases it will be 
 much more convenient to represent them by lines, because 
 lines will express not only the quantities of the momenta, 
 but also the directions in which they are communicated. 
 
 Any line drawn in the proper direction may be taken 
 to represent one momentum ; but to represent a second, 
 a line, in the direction of the latter motion, must be taken, 
 having the same proportion to the former line, that the 
 second momentum has to the first. 
 
 Let two lines, thus taken, represent the momenta 
 communicated to the same, or equal bodies ; then since 
 Moc V X Q (Art. 17.), and Q is here given, Moc V; there- 
 fore the lines, which represent the momenta, will also
 
 THE COMPOSITION, &C. OF MOTION. 29 
 
 represent the velocities, or the spaces uniformly described 
 in equal times. Again, if the lines represent the spaces 
 uniformly described in equal times, they represent the 
 velocities, and since Q is given, VocQVocM; therefore 
 the lines represent the momenta. 
 
 Prop. III. 
 
 (38.) Two uniform motions^ which, when communi- 
 cated separately to a body, would cause it to describe the 
 adjacent sides of a parallelogram in a given time, will, 
 when they are communicated at the same instant, cause 
 it to describe the diagonal in that time; and the motion 
 in the diagonal will be uniform. 
 
 Let a motion be communicated to a body at A, which 
 would cause it to move uniformly from A to B in T" , and 
 
 at the same instant, another motion which alone would 
 cause it to move uniformly from J to C in T" ; complete 
 the parallelogram BC, and draw the diagonal AD', then 
 the body will arrive at the point D, in T", having de- 
 scribed AD with an uniform motion. 
 
 For the motion in the direction AC can neither ac- 
 celerate nor retard the approach of the body to the line 
 BD which is parallel to AC, (Art. 29. Ex. S.) ; hence, 
 the body will arrive at BD, in the same time that it would 
 have done, had no motion been communicated to it in the 
 direction AC, that is in T''. In the same manner, the 
 motion in the direction AB can neither make the body
 
 30 
 
 THE COMPOSITION AND 
 
 approach to, nor recede from, CD; therefore, in conse- 
 quence of the motion in the direction AC, it will arrive 
 at CD in the same time that it would have done, had 
 no motion been communicated in the direction AB, that is 
 in T". Hence it follows that, in consequence of the two 
 motions, the body will be found both in BD and CD 
 at the end of T", and will therefore be found in D, the 
 point of their intersection. 
 
 Also, since a body in motion continues to move uni- 
 formly forward in a right line, till it is acted upon by 
 some external force (Art. 27-)? the body A must have 
 described the right line AD, with an uniform motion. 
 
 (39.) To illustrate this proposition, suppose a plane 
 ABDC, as the deck of a ship, to be carried uniformly 
 forward, and let the point A describe the line AC in T"; 
 also, let a body move uniformly in this plane from A to 
 B, in the same time. Complete the parallelogram BC, 
 and draw the diagonal AD. Then at the end of T" the 
 body, by its own motion, will arrive at B; also by the 
 motion of the plane, AB will be brought into the situation. 
 CD, and the point B will coincide with D ; therefore the 
 body will upon the whole, at the end of T", be found 
 in D. In any other time t" let the point A be carried 
 from A to M hy the motion of the plane, and the body 
 
 from A to L hy it's own motion ; complete the parallelo- 
 gram ALNM, and join AN% then, as in the preceding 
 case, the body will, at the end of t", be found in A^; 
 and since the motions in the directions AC, AB are 
 uniform, AC : AM :: T - t :: AB : AL (Art. 13.);
 
 RESOLUTION OF MOTION. 31 
 
 that is, the sides of the parallelograms, about the common 
 angle LAM, are proportional, and consequently the paral- 
 lelograms are about the same diagonal JD (Euc. 26. vi.) ; 
 therefore the body at the end of any time t" will be found 
 in the diagonal AD. It will also move uniformly in the 
 diagonal; for, from the similar triangles AMN, ACD, 
 we have AD : AN :: AC : AM :: T : t, or the spaces 
 described are proportional to the times. (See Art. 10.) 
 
 (40.) CoK. 1. The reasoning in the last article is 
 applicable to the motion of a point. 
 
 (41.) Cor. 2. If two sides of a triangle, AB, BD, 
 taken in order, represent the spaces over which two uniform 
 motions would, separately, carry a body in a given time ; 
 when these motions are communicated at the same instant 
 to the body at A, it will describe the third side AD, 
 uniformly, in that time. 
 
 For, if the parallelogram BC be completed, the same 
 motion, which would carry a body uniformly from B to D, 
 would, if communicated at A, carry it in the same manner 
 from A to C; and in consequence of this motion, and of 
 the motion in the direction AB, the body would uniformly 
 describe the diagonal AD, which is the third side of the 
 triangle ABD. 
 
 (42.) CoR. 3. In the same manner, if the lines AB, 
 B 
 
 E 
 
 BC, CD, DE, taken in order, represent the spaces over
 
 32 THE COMrOSITIOX AND 
 
 which any uniform motions would, separately, carry a body, 
 in a given time, these motions, when communicated at the 
 same instant, will cause the body to describe the line AE 
 which completes the figure, in that time ; and the motion 
 in this line will be uniform. 
 
 (43.) Cor. 4. If AD represent the uniform velocity 
 of a body, and any parallelogram ABDC (Art. 38.), be 
 described about it, the velocity AD may be supposed to 
 arise from the two uniform velocities AB, AC, or AB, 
 BD; and if one of them, AB, be by any means taken 
 away, the velocity remaining will be represented by JC 
 or BD. (See Art. 11.) 
 
 (44.) Def. a force is said to be equivalent to any 
 number of forces, when it will, smgly, produce the same 
 effect that the others produce jointly, in any given time. 
 
 Prop. IV. 
 
 (45.) If the adjacent sides of a parallelogram repre- 
 sent the quantities and directions of two forces, acting at 
 the same time upon a body, the diagonal ivill represent one 
 equivalent to them both. 
 
 Let AB, AC represent two forces acting upon a body 
 at A, then they represent the momenta communicated to 
 it in those directions (Art. 22.), and consequently the 
 
 spaces which it would uniformly describe in equal times 
 (Art. 37). Complete the parallelogram CB, and draw the 
 diagonal JZ); then, by the last proposition, AD is the 
 space uniformly described in the same time, when the two
 
 RESOLUTION OF MOTION. 33 
 
 motions are communicated to the body at the same instant; 
 and since AB, AC, and AD, represent the spaces uniformly 
 described by the same body, in equal times, they represent 
 the momenta, and therefore the forces acting in those di- 
 rections ; that is, the forces J5, AC*, acting at the same 
 time, produce a force which is represented, in quantity and 
 direction, by AD. 
 
 Def. The force represented by AD is said to be com- 
 pounded of the two, AB, AC. 
 
 (46.) Cor. l. If two sides of a triangle, taken in 
 order, represent the quantities and directions of two forces, 
 the third side will represent a force equivalent to them 
 both. 
 
 For a force represented by BD, acting at A, will pro- 
 duce the same effect that the force AC, which is equal to it 
 and in the same direction, will produce ; and AB, AC, are 
 equivalent to AD ; therefore AB, BD are also equivalent 
 to AD. 
 
 (47.) Cor. 2. If any lines AB, BC, CD, DE, taken 
 
 B 
 
 in order, represent the quantities and directions of forces 
 
 * In this, and many other cases, where forces are represented by 
 lines, the lines are used, for the sake of conciseness, to express the 
 forces which they represent. 
 
 c
 
 34 
 
 THE COMPOSITION AND 
 
 communicated at tlie same time to a body at A, the line 
 AE, which completes the figure, will represent a force 
 equivalent to them all. 
 
 For the two AB, BC are equivalent to AC % also, AC^ 
 CD, that is, AB, BC, CD, are equivalent to AD ; in the 
 same manner AD, DE, that is, AB, BC, CD, and DE, 
 are equivalent to AE. 
 
 (48.) Cor. 3. Let AB and AC represent the quantities 
 and directions of two forces, join BC and draw AE bisecting 
 
 it in E, then will 2 AE represent a force equivalent to them 
 both. 
 
 For, if the parallelogram be completed, since the di- 
 agonals bisect each other, AD, which represents a force 
 equivalent to AB and AC, is equal to 2AE. 
 
 (49.) Cor. 4. If the angle at which two given forces 
 act be diminished, the compound force is increased. 
 
 Let AB, AC be the two given forces; complete the 
 
 A,^r ^ 
 
 parallelogram ABDC and draw the diagonal AD, this
 
 RESOLUTION OF MOTION. 35 
 
 represents the compound force. In the same manner, if 
 AE be taken equal to AB, and AE, AC, represent the 
 two forces, then AF the diagonal of the parallelogram 
 AEFC, represents the compound force ; and since the 
 angle BAC is greater than the angle EAC, ACD which 
 is the supplement of the former, is less than ACF the 
 supplement of the latter; also, CF = AE = AB =,CD ; 
 therefore in the two triangles ACD, ACF, the sides AC, 
 CD are respectively equal to AC, CF, and the Z ACD 
 is less than the z ACF; consequently AD is less than 
 AF (Euc. 24. i.). 
 
 (50.) Cor. 5. Two given forces produce the greatest 
 effect when they act in the same direction, and the least 
 when they act in opposite directions ; for, in the former 
 case, the diagonal AF becomes equal to the sum of the 
 sides AC, CF; and in the latter, to their difference. 
 
 (51.) Coil. 6. Two forces cannot keep a body at rest, 
 unless they are equal and in opposite directions. 
 
 For this is the only case in which the diagonal, repre- 
 senting the compound force, vanishes. 
 
 (52.) Cor. 7. In the composition of forces, force is 
 lost ; for the forces represented by the two sides AB, BD 
 (Art. 45.), by composition produce the force represented 
 by AD; and the two sides AB, BD, of a triangle, are 
 greater than the third side AD. 
 
 Prop. V. 
 
 (53.) If a body, at rest, be acted upon at the same 
 time by three forces which are represented in quantity and 
 direction by the three sides of a triangle, taken in order, 
 it will remain at rest. 
 
 c 2
 
 36 THE COMPOSITION AND 
 
 Let AB, BC, and CA, represent the quantities and 
 
 c 
 
 A 
 
 directions of three forces acting at the same time upon 
 a body at A ; then since AB and BC are equivalent to 
 AC (Art. 46.) ; AB, BC and CA are equivalent to AC 
 and CA ; but AC and CA, which are equal and in opposite 
 directions, keep the body at rest ; therefore AB, BC, and 
 CA, will also keep the body at rest. 
 
 Prop. VI. 
 
 (54.) If a body be kept at rest by three forces, and 
 two of them be represented in quantity and direction by 
 two sides AB, BC*, of a triangle, the third side, taken 
 in order, will represent the quantity and direction of the 
 other force. 
 
 Since AB, BC represent the quantities and directions 
 of two of the forces, and AB, BC are equivalent to AC, 
 the third force must be sustained by JC; therefore CA 
 must represent the quantity and direction of the third 
 force (Art. 51.). 
 
 (55.) Cor. If three forces keep a body at rest, they 
 act in the same plane ; because the three sides of a triangle 
 are in the same plane (Euc. 2. xi.). 
 
 Prop. VII. 
 
 (56.) If a body be kept at rest by three forces, acting 
 upon it at the same time, any three lines, which are in the 
 directions of these forces, and form a tria^igle, will repre- 
 sent them. 
 
 Fig. Art. 53.
 
 HESOLUTION OF MOTION. 37 
 
 I^et three forces, acting in the directions AB, AC, AD, 
 keep the body A at rest; then AB, AC, AD are in the 
 same plane (Art. 55.). In AB take any pointy B, and 
 through B draw BI parallel to AC, meeting DA produced 
 in /; then will AB, Bl, and I A represent the three forces. 
 
 For AB being taken to represent the force in that 
 direction, if BI do not represent the force in the direction 
 AC or BI, let BF be taken to represent it; join AF; 
 
 then since three forces keep the body at rest, and AB, BF 
 represent the quantities and directions of two of them, FA 
 will represent the third (Art. 54.), that is, FA is in the 
 direction AD, which is impossible (Euc. 11. i. Cor.); 
 therefore BI represents the force in the direction AC', 
 and consequently I A represents the third force (Art. 54.). 
 
 Any three lines, respectively parallel to AB, BI, lA, 
 and forming a triangle, will be proportional to the sides 
 of the triangle ABI, and therefore proportional to the 
 three forces. 
 
 (57.) CoR. 1. If a body be kept at rest by three 
 forces, any two of them are to each other inversely as the 
 sines of the angles which the lines of their directions make 
 with the direction of the third force. 
 
 Let ABI be a triangle whose sides are in the directions 
 of the forces; then these sides represent the forces; and 
 AB : BI :: sin. BIA : sin. BAI :: sin. lAC : sin. BAI 
 :: sin. CAD : sin. BAD.
 
 38 
 
 THE COMPOSITION AND 
 
 (58.) Cor. 2. If a body, at rest, be acted upon at the 
 same time by three forces, in the directions of the sides of 
 a triangle taken in order, and any two of them be to each 
 other inversely as the sines of the angles which their direc- 
 tions make with the direction of the third, the body will 
 remain at rest. 
 
 For, in this case, the forces will be proportional to the 
 three sides of the triangle, and consequently they will 
 sustain each other (Art. 53,). 
 
 Prop. VIII. 
 
 (59.) If a body be kept at rest by three forces, and 
 lines be drawn at right angles to the directions in which 
 they act, forming a triangle, the sides of this triangle will 
 represent the quantities of the forces. 
 
 Let AB, BC, CA be the directions in which the forces 
 act ; and let them form the triangle ABC ; then the lines 
 AB, BC, CA, will represent the forces (Art. 5Q.). Draw 
 
 the perpendiculars DH, EI, FG, forming a triangle GHI ; 
 then since the four angles of the quadrilateral figure ADHF 
 are equal to four right angles, and the angles at D and F 
 are right angles, the remaining angles DHF, DAF are 
 equal to two right angles, or to the two angles DHF, 
 DHG ; consequently, the angle DAF is equal to the angle 
 IHG. In the same manner, it may be shewn, that the 
 angles ABC, BCA are respectively equal to GIH, HGI ',
 
 RESOLUTION OF MOTION. 39 
 
 therefore the triangles ABC and GHI are equiangular; 
 hence, the sides about their equal angles being proportional, 
 the forces which are proportional to the lines AB, BC, CA, 
 are proportional to the lines HI, IG, GH. 
 
 Cor. If the lines DH, EI, FG be equally inclined 
 to the lines DB, EC, FA, and form a triangle GHI, the 
 sides of this triangle will represent the quantities of the 
 forces. 
 
 Prop. IX. 
 
 (60.) If any number of forces, represented in quan- 
 tity and direction by the sides of a polygon, taken in order, 
 act at the same time upon a body at rest, they will keep it 
 at rest. 
 
 Let AB, BC, CD, DE, and EA (Fig. Art. 4?.), re- 
 present the forces ; then since AB, BC, CD and DE are 
 equivalent to AE (Art. 47.) ; AB, BC, CD, DE, and EA, 
 are equivalent to AE and EA ; that is, they will keep the 
 body at rest. 
 
 Prop. X. 
 
 (61.) If any number of lines, taken in order, repre- 
 sent the quantities and directions of forces tvhich keep a 
 body at rest, these lines will form a polygon. 
 
 Let AB, BC, CD and DE represent forces which keep 
 a body at rest (Fig. Art. 47.); then the point E coincides 
 with A. If not, join AE, then AB, BC, CD, and DE, are 
 equivalent to AE', and the body will be put in motion by 
 a single force AE, which is contrary to the supposition ; 
 therefore the point E coincides Avith A, and the lines form 
 a polygon. 
 
 This and the last proposition are true when the forces 
 act in different planes.
 
 40 THE COMPOSITION AND 
 
 Prop. XI. 
 
 (62.) A single force may be resolved into any number 
 of forces. 
 
 Since the single force AD is equivalent to the two, AB, 
 BD, it may be conceived to be made up of, or resolved 
 into, the two, AB, BD. The force AD may therefore 
 
 be resolved into as many pairs of forces as there can be 
 triangles described upon AD, or parallelograms about it. 
 Also AB, or BD, may be resolved into two; and, by 
 proceeding in the same manner, the original force may 
 be resolved into any number of others. 
 
 (63.) Cor. l. If two forces are together equivalent 
 to AD, and AB he one of them, BD is the other. 
 
 (64.) Cor. 2. If the force AD be resolved into the 
 two, AB, BD, and AB be wholly lost, or destroyed, the 
 effective part of AD is represented in quantity and direction 
 hy BD. 
 
 (65.) Cor. 3. In the resolution of forces, the whole 
 quantity of force is increased. For the force represented 
 by AD is resolved into the two AB, BD which are together 
 greater than AD (Euc. 20. i.).
 
 RESOLUTION OF MOTION. 
 
 41 
 
 Prop. XII. 
 
 (66.) The effects of forces, when estimated in given 
 directions, are not altered by composition or resolution. 
 
 Let two forces AB, BC, and the force AC which is 
 equivalent to them both, be estimated in the directions AP, 
 
 AQ. Draw BD, CP parallel to AQ ; and CE parallel to 
 AP. Then the force AB is equivalent to the two AD, 
 DB ; of which AD is in the direction AP, and DB in the 
 direction AQ; m the same manner, BC is equivalent to the 
 two BE, EC ; the former of which is in the direction BD 
 or QA, and the latter in the direction EC or AP ; therefore 
 the forces AB, BC, when estimated in the directions AP, 
 AQ, are equivalent to AD, EC, DB, and BE; or, AD, 
 DP, DB and BE, because EC is equal to DP ; and since 
 DB and BE are in opposite directions, the part EB of the 
 force DB is destroyed by BE ; consequently, the forces are 
 equivalent to AP, DE, or AP, PC Also AC, when es- 
 timated in the proposed directions, is equivalent to AP, 
 PC; therefore the effective forces in the directions AP, 
 AQ are the same, whether we estimate AB and BC, in 
 those directions, or AC, which is equivalent to them. 
 
 (67.) Cor. When AP coincides with AC, EC also 
 coincides with it, and D coincides with E. In this case 
 the forces DB, BE wholly destroy each other ; and thus, 
 in the composition of forces, force is lost.
 
 SECTION IV. 
 
 ON THE MECHANICAL POWERS. 
 
 (68.) The mechanical powers are the most simple 
 instruments used for the purpose of supporting weights, 
 or communicating motion to bodies ; and by the combi- 
 nation of which, all machines, however complicated, are 
 constructed. 
 
 These powers are six in number, viz. the lever; the 
 ivheel and axle; the pulley; the inclined plane; the 
 wedge ; and the screw. 
 
 Before we enter upon a particular description of these 
 instruments, and the calculation of their effects, it is neces- 
 sary to premise, that when any forces are applied to them, 
 they are themselves supposed to be at rest ; and conse- 
 quently, that they are either without weight, or that 
 the parts are so adjusted as to sustain each other. They 
 are also supposed to be perfectly smooth ; no allowance 
 being made for the effects of adhesion. 
 
 When two forces act upon each other by means of 
 any machine, one of them is, for the sake of distinction, 
 called the potver, and the other the weight. 
 
 ON THE LEVER. 
 
 (69.) Def. The Lever is an inflexible rod, moveable 
 in one plane, upon a point whicli is called the fulcrum, 
 or center of motion.
 
 ON THE LEVER. 
 
 43 
 
 The power and weight are supposed to act in the 
 plane in which the lever is moveable round the fulcrum, 
 ana tend to turn it in opposite directions. 
 
 (70.) The properties of the lever cannot be deduced 
 immediately from the propositions laid down in the last 
 section, because the forces acting upon the lever are not 
 applied at a point, which is always supposed to be the 
 case in the composition and resolution of forces; they 
 may however be derived from the following principles, 
 the truth of which will readily be admitted. 
 
 Ax. 1. If two weights balance each other upon a 
 straight lever, the pressure upon the fulcrum is equal 
 to the sum of the weights, tvhafever be the length of the 
 lever *. 
 
 Ax. 2. If a weight be supported upon a lever tvhich 
 rests on two fulcrums, the pressure upon the fulcrums 
 is equal to the whole weight. 
 
 Ax. 3. Equal forces, acting perpendicularly at the 
 edotremities of equal arms of a lever, ewert the same effort 
 to turn the lever round. 
 
 PRor. XIII. 
 
 (71 .) If two equal weights act perpendicularly upon 
 a straight lever, the effort to put it in motion, round 
 a7iy fulcrum, tvill be the same as if they acted together 
 at the middle point between them. 
 
 Let A and B be two equal weights, acting perpen- 
 dicularly upon the lever FB, whose fulcrum is F. Bisect 
 AB in C; make CE=CF; and at E suppose another ful- 
 crum to be placed. 
 
 * The effect produced by the gravity of the lever is not taken into 
 consideration, unless it be expressly mentioned.
 
 44 ON THE LEVER. 
 
 Then since the two weights A and B are supported 
 by E and F, and these fulcrums are similarly situated 
 
 E 
 
 B 
 
 c 
 
 A 
 
 F 
 
 A 
 
 B 
 
 E 
 
 
 E 
 
 A 
 
 A 
 
 
 A 
 
 u 
 
 A 
 
 o 
 
 with respect to the weights, each sustains an equal pres- 
 sure ; and therefore the weight sustained by E is equal 
 to half the sum of the weights. Now let the weights A 
 and B be placed at C, the middle point between A and B, 
 and consequently the middle point between E and F; 
 then since E and F support the whole weight C, and 
 are similarly situated with respect to it, the fulcrum E 
 supports half the weight ; that is, the pressure upon E 
 is the same, whether the weights are placed at A and B, 
 or collected in C, the middle point between them; and 
 therefore, the effort to put the lever in motion round F, 
 is the same on either supposition. 
 
 (72.) CoR. If a weight be formed into a cylinder 
 AB (Fig. Art. 73.) which is every where of the same 
 density, and placed parallel to the horizon, the effort of 
 any part AD, to put the whole in motion round C, is 
 the same as if this part were collected at E, the middle 
 point of AD. 
 
 For the weight AD may be supposed to consist of 
 pairs of equal weights, equally distant from the middle 
 point. 
 
 What is here affirmed of weights, is true of any forces 
 which are proportional to the weights, and act in the same 
 directions. 
 
 Prop. XIV. 
 
 (73.) Two iceights, or two forces, acting perpen- 
 dicularly upon a straight lever, will balance each other, 
 ivhen they are reciprocally proportional to their distances 
 from the fulcrum.
 
 ON THE LEVER. 45 
 
 Case l. When the weights act on contrary sides of 
 the fulcrum. 
 
 Let X and y be the two weights, and let them be formed 
 into the cylinder AB^ which is every where of the same 
 density. Bisect AB in C ; then this cylinder will balance 
 
 itself upon the fulcrum C (Art. 72.). Divide AB into two 
 parts in D, so that AD : DB :: oo : y, and the weights 
 of AD and DB will be respectively x and ?/; bisect AD 
 in E and DB in F\ then since AD and DB keep the 
 lever at rest, they will keep it at rest when they are col- 
 lected at E and F (Art. 72.) ; that is, c^, when placed 
 at Ey will balance y^ when placed at i^; and x : y :: AD : 
 BD :: AB-BD : AB - AD :: 2CB-2BF : 2AC-2AE 
 :: 2CF : 2CE :: CF : CE. 
 
 Case 2. When the two forces act on the same side 
 of the center of motion. 
 
 Let AB be a lever whose fulcrum is C; ^ and B 
 two weights acting perpendicularly upon it ; and let 
 A : B :: BC : AC; then these weights will balance each 
 other, as appears by the former Case. Now suppose a 
 power sufficient to sustain a weight equal to the sum of 
 the weights A and B, to be applied at C, in a direction 
 
 o ± O 
 
 ^ B 
 
 opposite to that in which the weights act; then will this 
 power supply the place of the fulcrum (Art. 70. Ax. 1.); 
 also, a fulcrum placed at A^ or B, and sustaining a weight 
 A, or B, will supply the place of the body there, and 
 the equilibrium will remain. Let B be the center of
 
 1 (J 
 
 46 ON THE LEVER. 
 
 motion; then we have a straight lever whose center of 
 motion is B, and the two forces A and A + B, acting 
 perpendicularly upon it at the points A and C, sustain 
 each other; also, A : B :: BC : AC; 
 
 therefore^ : A + B :: BC : BA. 
 
 (74.) Cor. 1. If two weights, or two forces, acting 
 perpendicularly on the arms of a straight lever, keep each 
 other in equilibrio, they are inversely as their distances 
 from the center of motion. 
 
 For the weights will balance when they are in that 
 proportion, and if the proportion be altered by increasing 
 or diminishing one of the weights, it's effort to turn the 
 lever round will be altered, or the equilibrium will be 
 destroyed. 
 
 (75.) CoR. 2. Since A : B :: BC : AC when there 
 is an equilibrium upon the lever AB, whose fulcrum is 
 C, by multiplying extremes and means, A x AC = BxBC. 
 
 (76.) Cor. 3. When the power and weight act on 
 the same side of the fulcrum, and keep each other in 
 equilibrio, the weight sustained by the fulcrum is equal 
 to the difference between the power and the weight. 
 
 (77) Cor. 4. In the common balance, the arms of 
 the lever are equal; consequently, the power and weight, 
 or two weights, which sustain each other, are equal. In 
 the false balance, one arm is longer than the other ; there- 
 fore the weight, which is suspended at this arm, is pro- 
 portionally less than the weight which it sustains at the 
 other. 
 
 (78.) CoR. 5. If the same body be weighed at the 
 two ends of a false balance, it"'s true weight is a mean 
 proportional between the apparent weights.
 
 ON THE LEVER. 
 
 47 
 
 Call the true weight x, and the apparent weights, 
 when it is suspended at A and B, a and h respectively ; 
 
 then a 
 
 00 : 
 
 : AC : 
 
 BC, 
 
 and OD 
 
 b : 
 
 : AC : 
 
 BC; 
 
 therefore a 
 
 : OS : 
 
 : w : 
 
 b. 
 
 (79.) Cor. 6. If a weight C be placed upon a lever 
 which is supported upon two props A and B in an hori- 
 
 A 
 
 c 
 
 A 
 
 zon 
 B 
 
 ital position, the pressure upon A : the pressure upon 
 :: BC : AC. 
 
 For if B be conceived to be the fulcrum, we have 
 this proportion, the weight sustained by A : the weight 
 C :: BC : AB; in the same manner, if A be considered 
 as the fulcrum, then the weight C : the weight sustained 
 by jB :: AB : CA ; therefore, eoj ijequo, the weight sustained 
 by A : the weight sustained hy B :: BC : AC 
 
 (80.) CoR. 7. If a given weight P be moved along 
 the graduated arm of a straight lever, the weight W, 
 
 ^ , 
 
 m 
 
 which it will balance at A, is proportional to CD, the 
 distance at which the given weight acts. 
 
 When there is an equilibrium, W x AC = P x DC 
 (Art. 75.) ; and AC and P are invariable ; therefore Woe DC 
 {Alg. Art. 19p.).
 
 48 
 
 ON THE LF.VER. 
 
 Prop. XV. 
 
 (81.) If two forces, acting upon the arms of any 
 lever, keep it at rest, they are to each other inversely 
 as the perpendiculars drawn from the center of motion 
 to the directions in which the forces act. 
 
 Case 1. Let two forces, A and B, act perpendicu- 
 larly upon the arms CA, CB, of the lever ACB whose 
 fulcrum is C, and keep each other at rest. Produce BC 
 
 D 
 
 r 
 
 to D, and make CD = CA ; then the effort of A to move 
 the lever round C, will be the same, whether it be sup- 
 posed to act perpendicularly at the extremity of the arm 
 CA, or CD (Art. 70. Ax. 3.) ; and on the latter sup- 
 position, since there is an equilibrium, A : B :: CB : CD 
 (Art. 74.) ; therefore A : B :: CB : CA, 
 
 Case 2. When the directions AD, BH, in which 
 the forces act, are not perpendicular to the arms. Take 
 
 N 
 
 AD and BH, to represent the forces ; draw CM and CN 
 at right angles to those directions; also draw AF per- 
 pendicular, and DF parallel to AC, and complete the
 
 ON THE LEVER. 
 
 # 
 
 parallelogram GF'^ then the force AD is equivalent to 
 the two AF, AGy of which, AG acts in the direction of 
 the arm, and therefore can have no effect in causing, or 
 preventing any angular motion in the lever about C. Let 
 BH be resolved, in the same manner, into the two Bt, 
 BK of which BI is perpendicular to, and BK in the 
 direction of the arm CB% then BK will have no effect 
 in causing, or preventing any angular motion in the lever 
 about C ; and since the lever is kept at rest, AF and BI^ 
 which produce this effect, and act perpendicularly upon 
 the arms, are to each other, by the 1st case, inversely as 
 
 the arms; that is, AF : BI :: CB : CA, 
 or AFxCA = BIx CB. 
 Also in the similar triangles ADF, ACM, 
 AF : AD :: CM : CA, 
 and AFxCA = AD X CM; 
 in the same manner, BI x CB = BH x CN; 
 therefore ADxCM=BHx CN, 
 and AD : BH :: CN : CM, 
 
 (82.) Cor. 1. Let a body IK be moveable about 
 the center C, and two forces act upon it at A and B, in 
 
 the directions AD, BH, which coincide with the plane 
 ACB; join AC, CB; then this body may be considered 
 as a lever ACB, and drawing the perpendiculars CM, CN, 
 
 D
 
 50 
 
 OK THE LEVER. 
 
 there will be an equilibrium, when the force acting at 
 A: the force acting at B :: CN : CM^. 
 
 (83.) CoE. 2. The effort of the force A, to turn 
 the lever round, is the same, at whatever point in the 
 direction MD it is applied; because the perpendicular 
 CM remains the same. 
 
 (84.) Cor. 3. Since CA : CM :: rad. : sin. CAM, 
 
 _,__ CA X sin. CAM _ . 
 
 CM = ; ; and m the same manner. 
 
 CN 
 
 rad. 
 
 CB X sin. CBN 
 rad. 
 
 equilibrium, 
 
 power at A : weight at B :: 
 
 ; therefore, when there is an 
 CBx sin. CBN CAx sin. CAM 
 
 rad. rad. 
 
 :: CB X sin. CBN : CA x sin. CAM. 
 
 (85.) CoR. 4. If the lever ACB be straight, and the 
 U 
 
 tB 
 
 H 
 
 directions AD, BH, parallel, A : B :: BC i AC; because, 
 in this case, sin. C^Ji'=sin. CBH. 
 
 Hence also, AxAC—B/ BC. 
 
 See Art. 74.
 
 ON THE LEVER. 
 
 51 
 
 (86.) Cor. 5. If two weights balance each other 
 upon a straight lever in any one position, they will balance 
 each other in any other position of the lever ; for the weights 
 act in parallel directions, and the arms of the lever are 
 invariable. 
 
 (870 ^^^' ^- ^^ ^ ^^"' balanced in a common pair 
 of scales, press upwards by means of a rod, against any 
 point in the beam, except that from which the scale is 
 suspended, he will preponderate. 
 
 Let the action upwards take place at D, then the 
 scale, by the reaction downwards, will be brought into the 
 situation iE; and the effect will be the same as if DA, AE, DE^ 
 constituted one mass ; that is, drawing EF perpendicular to 
 
 CA produced, as if the scale were applied at F (Art. 83.); 
 consequently the weight, necessary to maintain the equi- 
 librium, is greater than if the scale were suffered to hang- 
 freely from J, in the proportion of CF : CA (Art. 80.) 
 
 (88.) CoR. 7. Let AD represent a wheel, bearing 
 
 c 
 
 G 
 
 E 
 
 \^ F 
 
 
 
 "^xJA 
 
 
 J 
 
 ) B 
 
 
 a weight at it''s center C; AB an obstacle over which it 
 
 D2
 
 52 ON THE LEVER. 
 
 is to be moved by a force acting in the direction CE ; join 
 CA, draw CD perpendicular to the horizon, and from A 
 draw AG, AF, at right angles to CE, CD. Then CA 
 may be considered as a lever whose center of motion is 
 A, CD the direction in which the weight acts, and CE the 
 direction in which the power is applied ; and there is an 
 equilibrium on this lever when the power : the weight :: 
 AF : AG. 
 
 Supposing the wheel, the weight, and the obstacle 
 given, the power is the least when AG is the greatest; 
 that is, when CE is perpendicular to CA, or parallel to 
 the tangent at A. 
 
 (89.) Cor. 8. Let two forces acting in the directions 
 ADy BH, upon the arms of the lever ACB, keep each other 
 in equilibrio; produce DA and HB till they meet in P; 
 join CP, and draw CL parallel to PB ; then will PL, LC 
 
 represent the two forces, and PC the pressure upon the 
 fulcrum. 
 
 For, if PC be made the radius, CM and CN are the 
 sines of the angles CPM, CPN, or CPL, PCL ; 
 
 and PL : LC :: sin. PCL : sin. LPC :: CN : CM; 
 
 therefore PL, LC, represent the quantities and directions 
 of the two forces, which may be supposed to be applied at 
 P (Art. 83.), and which are sustained by the reaction of 
 the fulcrum ; consequently, CP represents the quantity 
 and direction of that reaction (Art. 54.), or PC represents 
 the pressure upon the fulcrum.
 
 ON THE LEVER 53 
 
 Prop. XVI. 
 
 (90.) In a co7nhination of straight levers, AB, CD, 
 whose centers of motion are E and F, */ they act perpen- 
 dicularly upon each other, and the directions in which the 
 power and weight are applied he also perpendicular to the 
 armsy there is an equilibrium when P : W :: EB x FD : 
 EA X FC. 
 
 For, the power at A : the weight at B, or C :: EB 
 
 PI 
 
 A ^ h 5_ 
 
 E B 
 
 C 
 
 L 
 
 wd 
 
 : EA ; and the weight at C : the weight at D :: FD : FC ; 
 therefore, P : W :: EB X FD : EA x FC 
 
 By the same method we may find the proportion between 
 the power and the weight, when there is an equilibrium, in 
 any other combination of levers. 
 
 (91.) CoR. If E and F be considered as the power 
 and weight, A and D the centers of motion, we have, as 
 before, E : F :: FD x BA : AE x CD. Hence the pressure 
 upon E : the pressure upon F :: FD x BA : AE x CD.
 
 5#! ON THE LEVER. 
 
 Prop. XVII. 
 
 (92.) Any weights will keep each other in equilibrio 
 on the arms of a straight lever ^ when the products^ which 
 arise from multiplying each weight by ifs distance from 
 the fulcrum^ are equals on each side of the fulcrum. 
 
 The weights Ay B, D, and E, F, will balance each 
 other upon the lever AF whose fulcrum is C, if A x AC 
 -^BxBC + DxDC = ExEC + FxFC. 
 
 In CF take any point JC, and let the weights r, s, t, 
 placed at X, balance respectively, A, B^ D; then A x AC 
 = rxXC; BxBC = sxXC; DxDC = txXC, (Art. 85.); 
 or, AxAC + BxBC + DxDC = (r ^rs + t) x XC. In 
 
 A- 
 
 B 
 
 Y 
 
 D 
 
 c 
 
 E 
 
 X 
 
 Y 
 
 
 
 
 
 A 
 
 
 
 ^r 
 t 
 
 
 the same manner, let p and q, placed at F, balance respec- 
 tively, iJ and F; then (p-\-q)x YC = Ex EC -]- F x FC ; 
 but by the supposition A x AC + Bx BC -\-Dx DC = E x 
 EC^FxFC; therefore {r ^ s -\-t)x XC z=z{p + q) xYC, 
 and the weights r, s, t^ placed at X, balance the weights 
 p^ q, placed at F; also A, B, D, balance the former 
 weights, and E, F, the latter; consequently A^ B, Z>, 
 will balance E and F. 
 
 (93.) CoR. 1. If the weights do not act in parallel 
 directions, instead of the distances we must substitute the 
 perpendiculars, drawn from the center of motion, upon 
 the directions. (See Art. 81.) 
 
 (94.) Cor. 2. In Art. 80. the lever is supposed to 
 be without weight, or the arms AC, CD to balance each 
 other: In the formation of the common steelyard the 
 longer arm CB ib heavier than CJ^ and allowance must
 
 ON THE WHEEL AND AXLE. 
 
 55 
 
 be made for this excess. Let the moveable weight P, 
 
 A 
 
 E C 
 
 W 
 
 B 
 
 ^ 
 
 when placed at E, keep the lever at rest ; then w^ien W 
 and P are suspended upon the lever, and the whole remains 
 at rest, W sustains P, and also a weight which would sup- 
 port P when placed at E ; 
 
 therefore W x AC = PxDC + P x EC = Px DE; 
 
 and since AC and P are invariable. Woe ED; the gradua- 
 tion must therefore begin from E; and if P, when placed 
 at P, support a weight of one pound at A, take FG, GD, &c. 
 equal to each other, and to EF, and when P is placed 
 at G it will support two pounds ; when at D it will sup- 
 port three pounds, &c. 
 
 ON THE WHEEL AND AXLE. 
 
 (95.) The wheel and axle consists of two parts, a 
 cylinder AB moveable about it's axis CD, and a circle 
 
 EF so attached to the cylinder that the axis CD passes 
 through it's center, and is perpendicular to it's plane.
 
 56 
 
 ON THE WHEEL AND AXLE. 
 
 The power is applied at the circumference of the wheel, 
 usually in the direction of a tangent to it, and the weight is 
 raised by a rope which winds rounds the axle in a plane at 
 right angles to the axis. 
 
 Prop. XVIII. 
 
 (96.) There is an equilibrium upon the wheel and 
 axle, when the power is to the weight, as the radius of 
 the axle to the radius of the wheel. 
 
 The effort of the power to turn the machine round the 
 axis, must be the same at whatever point in the axle the 
 wheel is fixed; suppose it to be removed, and placed in 
 such a situation that the power and weight may act in the 
 
 same plane, and let CA, CB, be the radii of the wheel and 
 axle, at the extremities of which the power and weight acts; 
 then the machine becomes a lever ACS, whose center of 
 motion is C ; and since the radii CA, CB, are at right angles 
 to AP and BW, we have P: W :: CB : CA (Art. 82.). 
 
 (97) Cor. l. If the power act in the direction Ap, 
 draw CE perpendicular to Ap, and there will be an equi- 
 librium when P : W :. CB I CE (Art. 82.).
 
 ON THE WHEEL AND AXLE. 5^ 
 
 The same conclusion may also be obtained by resolving 
 the power into two, one perpendicular to AC, and the other 
 parallel to it. 
 
 (98.) Cor. 2. If 2R be the thickness of the ropes 
 by which the power and weight act, there will be an equi- 
 librium when P : W :: CB + R : CA-V R, since the power 
 and weight must be supposed to be applied in the axes of 
 the ropes. 
 
 The ratio of the power to the weight is greater in this 
 case than the former ; for if any quantity be added to the 
 terms of a ratio of less inequality, that ratio is increased 
 {Alg. Art. 162.). 
 
 (99.) Cor. 3. If the plane of the wheel be inclined to 
 the* axle at the angle EOD, draw ED perpendicular to 
 
 CD; and considering the wheel and axle as one mass, 
 there is an equilibrium when P : W :: the radius of the 
 axle : ED. 
 
 (100.) Cor. 4. In a combination of wheels and axles, 
 where the circumference of the first axle is applied to the 
 circumference of the second wheel, by means of a string, 
 or by tooth and pinion, and the second axle to the third 
 wheel, &c. there is an equilibrium when P : W :: the 
 product of the radii of all the axles : the product of the 
 radii of all the wheels. (See Art. 90.). 
 
 (101.) Cor. 5. When the power and weight act in 
 parallel directions, and on opposite sides of the axis, the 
 pressure upon the axis is equal to their sum; and when 
 they act on the same side, to their difference. In other 
 cases the pressure may be estimated by Art. 89.
 
 58 
 
 ON THE PULLEY. 
 
 ON THE PULLEY. 
 
 (102.) Def. a Pulley is a small wheel moveable 
 about it's center, in the circumference of which a groove 
 is formed to admit a rope or flexible chain. 
 
 The pulley is said to be Jtxed^ or moveable, according 
 as the center of motion is fixed or moveable. 
 
 Prop. XIX. 
 
 (103.) In the single fixed pulley^ there is an equi- 
 iibriumy when the power and weight are equal. 
 
 Let a power and weight P, IF, equal to each other, 
 act by means of a perfectly flexible rope PDW, which 
 passes over the fixed pulley ADB ; then, whatever force 
 
 B 
 
 is exerted at D in the direction DAP, by the power, an 
 equal force is exerted by the weight in the direction 
 DBW\ these forces will therefore keep each other in 
 equilibrio. 
 
 CoR. 1. Conversely, when there is an equilibrium, the 
 power and weight, are equal*. 
 
 CoR. 2. The proposition is true in whatever direction 
 the power is applied ; the only alteration made, by changing 
 
 * Sec Art. 7i.
 
 ON THE PULLEY. 
 
 59 
 
 it"'s direction, is in the pressure upon the center of motion. 
 (See Art. 106.). 
 
 Prop. XX. 
 
 (104.) In the single moveable pulley, whose strings 
 are parallel^ the power is to the weight as 1 to 2^. 
 
 A string fixed at E, passes under the moveable pulley 
 J, and over the fixed pulley B; the weight is annexed 
 
 to the center of the pulley A, and the power is applied 
 at P. Then since the strings EA, BA are in the direction 
 in which the weight acts, they exactly sustain it; and 
 they are equally stretched in every point, therefore they 
 sustain it equally between them ; or each sustains half 
 the weight. Also, whatever weight AB sustains, P sus- 
 tains (Prop. xix. Cor. 1.), therefore P \ W :: 1 : 2. 
 
 Prop. XXI. 
 
 (105.) In general^ in the single moveable pulley^ the 
 power is to the weighty as radius to twice the cosine of 
 the angle which either string makes with the direction 
 in which the weight acts. 
 
 * In this and the following propositions, the power and weight 
 arc supposed to be in equilibrio.
 
 60 
 
 ON THE PULLEY. 
 
 Let AW he the direction in which the weight acts; 
 produce BD till it meets AW in C, from A draw AD at 
 right angles to AC, meeting BC in D; then if CD be 
 taken to represent the power at P, or the power which acts 
 in the direction DB, CA will represent that part of it 
 which is effective in sustaining the weight, and AD will 
 be counteracted by an equal and opposite force, arising 
 
 from the tension of the string CE; also, the two strings 
 are equally effective in sustaining the weight; therefore 
 2 AC will represent the whole weight sustained; conse- 
 quently, P : W :: CD : 2AC :: rad. : 2 cos. DC A. 
 
 (106.) Cor. l. If the figure be inverted, and E and 
 B be considered as a power and weight which sustain each 
 other upon the fixed pulley A, W is the pressure upon 
 the center of motion; consequently the power : the pres- 
 sure :: radius : 2 cos. DC A. 
 
 (IO7.) CoR. 2. When the strings are parallel, the 
 angle DC A vanishes, and it's cosine becomes the radius; 
 in this case, the power : the pressure :: 1 : 2.
 
 ON THE PULLEY. 
 
 61 
 
 Prop. XXII. 
 
 (108.) In a system where the same string passes 
 round any number of pulleys^ and the parts of it between 
 the pulleys are parallel, P : W :: 1 : the number of 
 strings at the lower block. 
 
 Since the parallel parts, or strings at the lower block, 
 are in the direction in which the weight acts, they exactly 
 support the whole weight ; also, the tension in every point 
 of these strings is the same, otherwise the system would 
 not be at rest, and consequently each of them sustains 
 
 an equal weight; whence it follows that, if there be n
 
 62 
 
 ON THE PULLEY. 
 
 strings, each sustains - th part of the weight ; therefore, P 
 
 sustains --th part of the weight, ov P : W :: ^ i \ .: \ : n. 
 n n 
 
 (109.) CoR. If two systems of this kind be com- 
 bined, in which there are m and n strings, respectively, 
 at the lower blocks, P : W w \ -. mn. 
 
 Prop. XXIII. 
 
 (110.) In a system where each pulley hangs by a 
 separate string, and the strings are parallel, P : W :: 1 : 
 that power of 2 whose index is the number of moveable 
 pulleys. 
 
 In this system, a string passes over the fixed pulley 
 
 EF a 
 
 A, and under the moveable pulley J?, and is fixed at E;
 
 ON THE PULLEY. 63 
 
 another string is fixed at 5, passes under the moveable 
 pulley C, and is fixed at F% &c. in such a manner that 
 the strings are parallel. 
 
 Then, by Art. 104, when there is an equilibrium, 
 
 P : 
 
 the weight at B 
 
 :: 1 
 
 : 2 
 
 the weight at B : 
 
 the weight at C 
 
 :: 1 
 
 : 2 
 
 the weight at C 
 
 the weight at D 
 
 :: 1 
 
 : 2 
 
 &c. 
 
 
 
 
 Comp. P : W :: 1 : 2 X 2 X 2 X &c. continued to as many 
 factors as there are moveable pulleys ; that is, when there 
 are 7i such pulleys, P : W :: I : 2^\ 
 
 (111.) CoR. 1. The power and weight are wholly 
 sustained at A, E, F, G, &c. which points sustain respec- 
 tively, 2P, P, 2P, 4P, &c. 
 
 (112.) Cor. 2. When the strings are not parallel, 
 P : W :: rad. : 2 cos. of the angle which the string makes 
 with the direction in which the weight acts, in each case 
 (Art. 105.). 
 
 Prop. XXIV. 
 
 (113.) In a system of n pulleys each hanging by a 
 separate string, where the strings are attached to the 
 weight as is represented in the anneoced figure, P : W :: 
 1 : 2"— 1. 
 
 A string, fixed to the weight at P, passes over the 
 pulley C, and is again fixed to the pulley P; another 
 string, fixed at E, passes over the pulley P, and is fixed 
 to the pulley A ; &c. in such a manner that the strings 
 are parallel:
 
 64 
 
 ON THE PUI.LET. 
 
 Then, if P be the power, the weight sustained by 
 the string DA is P; also the pressure downwards upon 
 A^ or the weight which the string AB sustains, is SiP 
 (Art. 107.); therefore the string EB sustains 2P; &c. 
 and the whole weight sustained is PH-2P + 4P+&C. 
 
 Hence, P : W :: 1 : 1 -f 2 -f 4 -f- &c. io n terms :: 1 : 2'^ — 1 
 {Alg, Art. 222.). 
 
 (114.) Cor. 1. Both the power and the weight are 
 sustained at H. 
 
 (115.) Cor. 2. When the strings are not parallel, 
 the power in each case, is to the corresponding pressure 
 upon the center of the pulley :: rad. : 2 cos. of the angle 
 made by the string with the direction in which the weight 
 acts (Art. 106.). Also, by the resolution of forces, the 
 power in each case, or pressure upon the former pulley,
 
 ON THE TXn.TNED PLANE. 
 
 is to the weight it sustains :: rad. : cos. of the angle made 
 by the string with the direction in which the weight 
 acts. 
 
 ON THE INCLINED PLANE. 
 
 Prop. XXV. 
 
 (116.) //' a body act upon a perfectly hard and 
 ^smooth plane, the effect jjroduced upon the plane is in 
 a direction per pendic alar to ifs surface. 
 
 Case 1. When the body acts perpendicularly upon 
 the plane, it's force is wholly effective in that direction ; 
 since there is no cause to prevent the effect, or to alter 
 it's direction. 
 
 Case 2. When the direction in which the body acts 
 is oblique to the plane, resolve it's force into two, one 
 parallel, and the other perpendicular, to the plane; the 
 former of these can produce no effect upon the plane, 
 because there is nothing to oppose it in the direction in 
 which it acts (See Art. 29.) ; and the latter is wholly 
 effective (by the first case) ; that is, the effect produced 
 by the force is in a direction perpendicular to the plane. 
 
 (11 7.) Cor. The reaction of the plane is in a di- 
 rection perpendicular to it's surface (Art. 32.). 
 
 Prop. XXVI. 
 
 (118.) When a body is sustained 7(pon a plane which 
 is inclined to the horizon, P : W :: the sine of the plane'' s 
 inclination : the sine of the angle which the direction of 
 the power makes ivith a perpendicular to the plane. 
 
 Let BC be parallel to the horizon, BA a plane inclined 
 to it; P a body, sustained at any point upon the plane 
 
 E
 
 66 
 
 ox THE INCLINED PLANE. 
 
 by a power acting in the direction PV. From P draw 
 PC perpendicular to BA, meeting BC in C; and from C 
 draw CV perpendicular to BC, meeting PF in F*. Then 
 the body P is kept at rest by three forces which act upon 
 
 it at the same time: the power, in the direction PV; 
 gravity, in the direction VC ; and the reaction of the 
 plane, in the direction CP (Art. 117.); these three forces 
 are therefore properly represented by the three lines PV, 
 VC, and CP (Art. 56.) ; oiP : W :: PV : VC :: sin. PCV : 
 sin. VPC; and in the similar triangles APC, ABC 
 (Euc. 8. vi.), the angles ACP, and CBA are equal; there- 
 fore P : W:: sin. ABC : sin. VPC. 
 
 (119.) CoR. 1. When PV coincides with PA, or 
 the power acts parallel to the plane, P : W :: PA : AC :: 
 AC : AB. 
 
 (120.) Cor. 2. When PV coincides with Pv, or the 
 power acts parallel to the base, P : W :: Pv : vC :: AC 
 : CB ; because the triangles Pv C, ABC are similar. 
 
 (121.) CoR. 3. When PV is parallel to CV, the 
 power sustains the whole weight. 
 
 (122.) CoR. 4. Since P : W :: sin. ABC : sin. VPC, 
 we have Px sin. VPC = Wx sin. ABC; and if W, and 
 
 * That PV, CV, are in the same plane, appears from Art. 55.
 
 ON THE TNCLTNED Pt.ANEi 
 
 the sine of the z ABC be invariable, P oc 
 
 sin. VPC 
 
 1 
 
 (Jig- Art. 206.) ; therefore P is the least, when . 
 
 is the least, or sin. VPC the greatest; that is, when 
 sin. VPC becomes the radius, or PF coincides with PA. 
 Also, P is indefinitely great when sin. VPC vanishes ; that 
 is, when the power acts perpendicularly to the plane. 
 
 (123.) CoE. 5. If P and the /iABC be given, W 
 oc sin. VPC; therefore W will be the greatest when 
 sin. VPC is the greatest, that is, when PV coincides with 
 PA. Also, W vanishes when the sin. VPC vanishes, or 
 PV coincides with PC. 
 
 (124.) CoR. 6. The power : the pressure :: PV : PC 
 :: sin. PCV : sin. PVC :: sin. ABC : sin. PVC 
 
 (125.) CoR. 7. When the power acts parallel to the 
 plane, the power : the pressure :: PA : PC :: AC : BC 
 
 (126.) Cor. 8. When the power acts parallel to the 
 base, the power : the pressure :: Pv : PC :: AC : AB. 
 
 (I27.) Cor. 9. P X sin. PFC = the press. X sin. ^5C; 
 and when P and the z ABC are given, the pressure 
 ocsin, PVC; therefore the pressure will be the greatest 
 when PV is parallel to the base. 
 
 (128.) Cor. 10. When two sides of a triangle, taken 
 in order, represent the quantities and directions of two 
 forces which are sustained by a third, the remaining side, 
 taken in the same order, will represent the quantity and 
 direction of the third force (Art. 54.). Hence, if we sup- 
 pose PF to revolve round P, when it falls between Pj?, 
 which is parallel to VC, and PE, the direction of gravity 
 remaining unaltered, the direction of the reaction must 
 be changed, or the body must be supposed to be sustained 
 against the under surface of the plane. When it falls 
 between PE and ,vP produced, the direction of the power 
 
 E 2
 
 G8 ON THE WEDGE. 
 
 must be changed. And when it falls between vP produced, 
 and PC, the directions of both the power and reaction 
 must Be different from what they were supposed to be 
 in the proof of the proposition; that is, the body must 
 be sustained against the under surface of the plane, by 
 a force which acts in the direction VP. 
 
 (129.) Cor. 11. If the weights P, IF, sustain each 
 other upon the planes AC, CB, which have a common 
 
 altitude CD, by means of a string PC IV which passes over 
 the pulley C, and is parallel to the planes, then 
 
 P : W :: AC : BC 
 
 For, since the tension of the string is every where the 
 same, the sustaining power, in each case, is the same ; 
 and calling this power a, 
 
 P : X :: AC : CD (Art. II9.); 
 x : W:: CD : CB. 
 comp. P : W:: AC : CB. 
 
 ON THE WEDGE. 
 
 (130.) Def. a Wedge is a triangular prism; or a 
 solid generated by the motion of a plane triangle parallel 
 to itself, upon a straight line which passes through one 
 of it's angular points *. 
 
 * See also Euc. B. XL Def. 13.
 
 ON THE WEDGE. 
 
 69 
 
 Knives, swords, coulters, nails, &c. are instruments of 
 this kind. 
 
 The wedge is called isosceles or scalene, according as 
 the section of it, made by a plane perpendicular to it's 
 sides, is an isosceles or scalene triangle. 
 
 Prop. XXVII. 
 
 (131.) If two equal forces act upon the sides of an 
 isosceles wedge at equal angles of inclination^ and a force 
 act perpendicularly upon the hack, they will keep the 
 wedge at rest, when the force upon the hack is to the 
 sum of the forces upon the sides, as the product of the 
 sine of half the vertical angle of the wedge and the sine 
 of the angle at which the directions of the forces are 
 inclined to the sides, to the square of radius. 
 
 Let AVB represent a section of the wedge, made by 
 a plane perpendicular to it's sides ; draw VC perpendicular 
 
 to AB', DC, dC, in the directions of the forces upon the 
 sides; and CE, Ce, at right angles to ^F, 57; join Ee, 
 meeting CV m F. 
 
 Then, in the triangles VCA, VCB., since the angles 
 VCA, CAV, are respectively equal to VCB, VBC, and 
 VC is common to both, AC=CB, and the z CVA= z CVB. 
 Again, in the triangles. ^Ci>, BCd, the angles DAC, CD A,
 
 70 ON THE WEDGE. 
 
 are equal to the angles CBd, BdC, and AC = BC; there- 
 fore. DC = dC. In the same manner it may be shewn 
 that CE=Ce, and AE = Be; hence the sides AV, BV, 
 of the triangle A VB, are cut proportionally in E and e ; 
 therefore Ee is parallel to AB (Euc. 2. vi.), or perpen- 
 ^ dicular to CF; also, since CE = Ce, and CF is common 
 
 ' Z. to the right-angled triangles CEF, CeF, we have EF=eF 
 
 \/ (Euc. 47. i.) 
 
 I\ Now since DC and c/ C are equal, and in the directions 
 
 V of the forces upon the sides, they will represent them ; 
 
 1^ resolve DC into two, DE, ECy of which DE produces 
 no effect upon the wedge, and EC, Avhich is effective 
 (Art. 116.), does not wholly oppose the power, or force 
 upon the back ; resolve EC therefore into two, EF^ 
 parallel to the back, and FC perpendicular to it, the 
 latter of which is the only force which opposes the power. 
 In the same manner it appears that eF, FC, are the only 
 effective parts of dC, of which FC opposes the power, 
 and ejP is counteracted by the equal and opposite force 
 EF ., hence if 9,CF represent the power, the wedge will be 
 kept at pest * ; that is, when the force upon the back : the 
 sum of the resistances upon the sides :: 2C-F : DC-\-dC 
 :: 9.CF : 2DC :: CF : DC; ami 
 
 CF : CE :: sin. CEF : rad. :: sin. CVE : rad. 
 CE : DC :: sin. CDE : rad. 
 Comp. CF : DC :: sin. CF^xsin. CDE : (rad.)^. 
 
 (132.) Coil. 1. The forces do not sustain each other 
 because the parts DE, de, are not counteracted. 
 
 (133.) Cor. 2. If the resistances act perpendicularly 
 upon the sides of the wedge, CDE becomes a right angle, and 
 P : the sum of the resistances :: sin. CVE x rad. : (rad.)- 
 :: sin. CVE : rad. :: AC : AV. 
 
 * The directions of the three forces must meet in a pointy otherwise 
 a rotatory motion will be given to the wedge.
 
 ON THE WEDGE. 
 
 71 
 
 (134.) Cor. 3. If the directions of the resistances be 
 perpendicular to the back, the angle CDE= /.CVE, and 
 P : the sum of the resistances :: (sin. CVE)^ : (rad.)- :: 
 
 (135.) Cor. 4. When the resistances act parallel to 
 the back, sin. CDJ = sin. CJV, and P : the sum of the re- 
 sistances :: sin. CVA xsiu. CAV : (rad.)" :: CAxCV : AV^ 
 :: CEx AV* : AV :: CE : AV. 
 
 (136.) Cor. 5. In the demonstration of the propo- 
 sition it has been supposed that the sides of the wedge 
 are perfectly smooth; if on account of the friction, or 
 by any other means, the resistances are wholly effective, 
 join Dd, which will cut CF at right angles in y, and 
 
 resolve DC, dC into Dy, yC, dy, yC, of which Dy and 
 dy destroy each other, and 2yC sustains the power. 
 Hence, the power : the sum of the resistances 
 
 :: 2yC : 2DC ::yC : DC :: sin. CDy or DC A : rad. 
 
 (I37.) Cor. 6. If Ee cut DC and dC in x and ^, 
 the forces, wC, %C, when wholly effective, and the forces 
 DC, dC, acting upon smooth surfaces, will sustain the 
 same power 2CF. 
 
 By similar triangles, CE : CA :; CV : AV; 
 therefore CEXAV=^CAX CF>
 
 72 
 
 OX THE WEDGE. 
 
 (138.) Cor. 7- If from any point P in the side AV, 
 PC be drawn, and the resistance upon the side be repre- 
 sented by it, the effect upon the wedge will be the same 
 as before ; the only difference will be in the part PE which 
 is ineffective. 
 
 (139.) Cor. 8. If DC be taken to represent the 
 resistance on one side, and pC, greater or less than dC, 
 represent the resistance on the other, the wedge cannot 
 be kept at rest by a power acting upon the back ; because, 
 on this supposition, the forces which are parallel to the 
 back are unequal. 
 
 This Proposition and it's Corollaries have been deduced 
 from the actual resolution of the forces, for the purpose of 
 shewing what parts are lost, or destroyed by their opposition 
 to each other ; the same conclusions may, however, be very 
 concisely and easily obtained from Art. 142. 
 
 Prop. XXVIII. 
 
 (140.) When three forces, acting perpendicularly 
 upon the sides of a scalene tvedge, keep each other in 
 equilibrio, they are proportional to those sides. 
 
 Let GI, HI, DI, the directions of the forces, meet 
 in /; then since the forces keep each other at rest, they are 
 proportional to the three sides of a triangle which are re- 
 spectively perpendicular to those directions (Art. 59.); that 
 is, to the three sides of the wedge.
 
 ON THE SCREW. ^3 
 
 (141.) Cor. l. If the lines of direction, passing- 
 through the points of impact, do not meet in a point, the 
 wedo-e will have a rotatory motion communicated to it; 
 and this motion will be round the center of gravity of the 
 wedge. (See Art. 184.) 
 
 (142.) CoR. 2. When the directions of the forces are 
 not perpendicular to the sides, the effective parts must be 
 found, and there will be an equilibrium when those parts 
 are to each other as the sides of the wedge. 
 
 ON THE SCREW, 
 
 (143.) Def. The Screw is a mechanical power, 
 which may be conceived to be generated in the following 
 manner : 
 
 Let a solid and a hollow cylinder of equal diameters be 
 taken, and let ABC be a right-angled plane triangle whose 
 base BC is equal to the circumference of the solid cylinder; 
 apply the triangle to the convex surface of this cylinder, in 
 such a manner, that the base BC may coincide, with the 
 circumference of the base of the cylinder, and BA will 
 form a spiral thread on it's surface. By applying to the 
 cylinder, triangles, in succession, sinjilar and equal to 
 ABC, in such a manner, that their bases may be parallel 
 to BC, the spiral thread may be continued ; and supposing 
 this thread to have thickness, or the cylinder to be pro- 
 tuberant where it falls, the external screw will be formed, 
 
 ,S c 
 
 in which the distance between two contiguou>s threadt^.
 
 74 
 
 ON THE SCREW. 
 
 measured in a direction parallel to the axis of the cylinder, 
 is AC. Again, let the triangles be applied in the same 
 manner to the concave surface of the hollow cylinder, and 
 where the thread falls let a groove be made, and the in- 
 ternal screw will be formed. The two screws being thus 
 exactly adapted to each other, the solid or hollow cylinder, 
 as the case requires, may be moved round the common axis. 
 
 by a lever perpendicular to that axis; and a motion will 
 be produced in the direction of the axis, by means of the 
 spiral thread. 
 
 Prop. XXIX. 
 
 (144.) When there is an equilibrium upon the screw, 
 P : W :: the distance between two contiguous threads, 
 measured in a direction parallel to the aocis : the circum- 
 ference of the circle which the power describes. 
 
 Let BCD represent a section of the screw made by 
 a plane perpendicular to it's axis, CE a part of the spiral 
 thread upon which the weight is sustained ; then CE is 
 a portion of an inclined plane, whose height is the distance
 
 ON THE SCREW. 
 
 75 
 
 between two threads, and base equal to the circumference 
 BCD. Call F the power which acting at C in the plane 
 BCD, and in the direction CI perpendicular to AC, will 
 sustain the weight W, or prevent the motion of the screw 
 round the axis; then since the weight is sustained upon 
 
 the inclined plane CE by a power F acting parallel to it's 
 base, F : W :: the height : the base (Art. 120.) :: the 
 distance between two threads : the circumference BCD. 
 Now, instead of supposing the power F to act at C, let 
 a power P act perpendicularly at G, on the straight lever 
 GCA, whose center of motion is A, and let this power 
 produce the same effect at C that F does; then, by the 
 property of the lever, P : F :: CA : GA :: the circum- 
 ference BCD : the circumference FGH. We have there- 
 fore these two proportions. 
 
 F : W 
 
 P : F 
 
 comp. P : W 
 
 : distance between two threads : BCD 
 
 BCD : FGH 
 
 : distance between two threads : FGH. 
 
 (145.) Coil. 1. In the proof of this Proposition the 
 whole weight is supposed to be sustained at one point C 
 of the spiral thread ; if we suppose it to be dispersed over 
 the whole thread, then, by the Proposition, the power at 
 G necessary to sustain any part of the weight : that part 
 :: the distance between two threads : the circumference 
 of the circle FGH; therefore the sum of all these powers, 
 or the whole power : the sum of all the corresponding
 
 76 ON THE SCREW. 
 
 weights, or the whole weight, :: the distance between 
 two threads : the circumference of the circle FGH {Alg. 
 Art. 183.). 
 
 (146.) Cor. 2, Since the power, necessary to sustain 
 a given weight, depends upon the distance between two 
 threads and the circumference FGH^ if these remain un- 
 altered, the power is the same, whether the weight is 
 supposed to be sustained at C, or at a point upon the 
 thread nearer to, or farther from, the axis of the cylinder. 
 
 (I47.) Some Authors have deduced the properties of 
 the mechanical powers immediately from the Third Law 
 of Motion; contending that if the power and weight be 
 such as would sustain each other, and the machine be put 
 into motion, the momenta of the power and weight are 
 equal ; and consequently, that the power x the velocity of 
 the power = the weight x the velocity of the weight ; or 
 the power's velocity : the weight's velocity :: the weight : 
 the power. 
 
 Though this conclusion be just, the reasoning by which 
 it is attempted to be proved is inadmissible, because the 
 Third Law of Motion relates to the action of one body 
 immediately upon another (Art. 36.). It may however be 
 deduced from the foregoing Propositions ; and as it is, in 
 many cases, the simplest method of estimating the power 
 of a machine, it may not be improper to estabhsh it's truth. 
 
 In the application of the rule, two things must be at- 
 tended to : 1st, We must estimate the velocity of the power 
 or weight in the direction in which it acts. 2dly, We must 
 estimate that part only of the power or weight which is 
 effective. 
 
 These two considerations are suggested by the Second 
 Law of Motion, according to which motion is communicated 
 in the direction of the force impressed, and is proportional 
 to that force.
 
 RATIO OF THE VELOCITIES, &C. 77 
 
 Prop. XXX. 
 
 (148.) The velocity of a body in any one direction 
 AB being given, to estimate ifs velocity in any other 
 direction BP. 
 
 Suppose the motion of A to be produced by a force 
 acting in the direction BP, by means of a string which 
 passes over a pulley at P ; produce PB to 0, making PO 
 = PA; join AO ; then OB is the space which measures 
 the approach of A to P. ^ow let the pulley be removed 
 to such a distance that the angle at P may be considered 
 as evanescent, and the power will always act in the same 
 direction BP ; also, the angles at A and are equal ; and 
 
 they are right angles, because the three angles of the 
 triangle APO are equal to two right angles, and the 
 angle at P vanishes ; therefore, the space described in the 
 direction OP, or BP, is determined by drawing AO per- 
 pendicular to OP. If the space described in the direction 
 xy, which is parallel to OP, be required, produce AO to 
 X, and from B draw By at right angles to cvy; then the 
 figure OByoB is a parallelogram, and OB = aiy the space 
 required. Also, if the motion in the direction AB be 
 uniform, the motion in the direction BP, or xy, is uniform; 
 since AB : OB :: rad. : cos. ABO. Hence, the velocity 
 in the direction AB : the velocity in the direction BP :: 
 AB : OB (Art. 11.).
 
 78 
 
 RATIO OF THE VELOCITIES 
 
 Prop. XXXI. 
 
 (149.) If a 'power and weight sustain each other on 
 any machine, and the whole he put in motion, the velocity 
 of the power : the velocity of the weight :: the weight : the 
 power. 
 
 Case 1. In the lever ACB, let a power and weight, 
 acting in the directions AD, BH, sustain each other, and 
 let the machine be moved uniformly round the center C, 
 
 through a very small angle AC a,, Join Aa^ Bb; draw 
 CM, ax, at right angles to MD; and CN, by, at right 
 angles to NB ; then J's velocity : B^s velocity :: Aj? : By 
 (Art. 148.). Now the triangles Axa, MCA, are similar; 
 because z xAC = z AMC + z MCA (Euc. 32. i.) and 
 A aAC= jL AMC', therefore, AwAa^ z MCA ; and the 
 angles at M and x are right angles; consequently, the 
 remaining angles are equal; and 
 
 Ax 
 also, in the sim. A^ AC a, BCb, A a 
 and in the sim. A^ Bhy, BCN, Bb 
 by composition, Ax 
 or the power's vel. : the weight's vel. :: the weight : the 
 power (Art. 81.). 
 
 Aa : 
 
 : CM : 
 
 CA 
 
 Bb : 
 
 : CA : 
 
 CB 
 
 By : 
 
 : CB : 
 
 CN 
 
 By : 
 
 : CM : 
 
 CN
 
 OF THF. POWER AND WEIGHT, 
 
 79 
 
 Case 2* In the wheel and axle^ if the power be made 
 to describe a space equal to the circumference of the wheel 
 with an uniform motion, the weight will be uniformly 
 raised through a space equal to the circumference of the 
 axle; hence, the power's velocity : the weight's velocity :: 
 the circumference of the wheel : the circumference of the 
 axle :: the radius of the wheel : the radius of the axle :: 
 the weight : the power (Art. 96.)- 
 
 Case 3. In the single fixed pulley^ if the weight be 
 uniformly raised 1 inch, the power will uniformly describe 
 
 1 inch in the direction of it's action ; therefore the power's 
 velocity : the weight's velocity :: the weight : the power. 
 
 Case 4. In the single moveable pulley where the 
 strings are parallel, if the weight be raised 1 inch, each 
 of the strings is shortened 1 inch, and the power describes 
 
 2 inches; therefore, P's velocity : PF's velocity :: W : P 
 (Art. 104.). 
 
 Case 5. In the system of pulleys described in Art, 108,
 
 80 RATIO OF THE VELOCITIES 
 
 if the weight be raised 1 inch, each of the strings at the 
 lower block is shortened 1 inch, and the power describes 
 n inches; therefore, P's velocity : W\ velocity :: W : P. 
 
 In this system of pulleys, whilst 1 inch of the string 
 passes over the pulley A^ 2 inches pass over the pulley B^ 
 
 3 over C, 4 over D, &c. 
 
 Hence it follows, that if in the solid block A, the 
 grooves A, C, E, &c. be cut, whose radii are 1,3, 5, &c. 
 and in the block B, the grooves B, D, F, &c. whose radii 
 are 2, 4, 6, &c. and a string be passed round these grooves 
 as in the annexed figure ; the grooves will answer the pur- 
 pose of so many distinct pulleys, and every point in each, 
 moving with the velocity of the string in contact with it, 
 the whole friction will be removed to the two centers of 
 motion in the blocks A and B. 
 
 Case 6. In the system of pulleys described in Art. 110, 
 each succeeding pulley moves twice as fast as the preceding ; 
 
 therefore, Ws velocity : CTs velocity :: 1 : 2 
 
 Cs velocity : ^'s velocity :: 1 : 2 
 
 B''s velocity : P''s velocity :: 1 : 2 
 
 &c. 
 
 comp. W'*s velocity : i^s velocity :: 1 : 2x2x2x&c. 
 :: P : W. 
 
 Case 7- In the system. Art. 113, if the weight be 
 raised 1 inch, the pulley B will descend 1 inch, and the 
 pulley A will descend 2 -f 1 inches ; in the same manner, 
 the next pulley will descend 2x(2-|-l)+l inches, or 
 
 4 + 2 + 1 inches ; &c. therefore P's velocity : Ws velocity 
 :: 1 + 2 + 4 + &c. : 1 :: W : P. 
 
 Case 8. Let a body be uniformly raised along the 
 inclined plane BA from B to P, by a power acting parallel 
 to PV; upon BP describe a semi-circle BOP, cutting BC
 
 OF THE POWER AND WEIGHT. 
 
 81 
 
 in M; produce VP to 0, join BO, PM, MO. Then since 
 the angles BOP, BMP, in the semi-circle, are right 
 
 angles, OP and MP are spaces uniformly described in the 
 same time, by the power and weight in their respective 
 directions (Art. 148.); also, because L POM — L PBM 
 = z PCV, and z 0PM = I PVC (Euc. 29. i.), the tri- 
 angles POM, PVC are similar, and OP : MP :: VC : PV, 
 or the power's velocity : the weight's velocity :: the weight 
 : the power, in the case of an equilibrium (Art. 118.). 
 
 Case 9. In the isosceles wedge, wC is the only effec- 
 tive part of the resistance DC (see Art. 13?.); ^^^^ ^^ 
 perpendicular to CD produced; then if the wedge be 
 moved uniformly from C to V, CO is the space uniformly 
 
 described by the resisting force (Art. 148.) ; hence, the 
 power's velocity : the velocity of the resisting force :: CV 
 : CO :: Cx : CF :: the resistance : the power. 
 
 F
 
 82^ RATIO OF THE VELOCITIES, &C. 
 
 Case 10. hi the screw, whilst the power uniformly 
 describes the circumference of the circle FGH (Art. 144.), 
 the weight is uniformly raised through the distance be- 
 tween two contiguous threads; therefore P's velocity : 
 Ws velocity :: the circumference of the circle FGH : the 
 distance between two threads :: W : P. 
 
 Case 11. In any combination of the mechanical 
 powers, lei P : W, W : R, R : S, &c. be the ratios of the 
 power and weight in each case, when there is an equili- 
 brium ; then, 
 
 P^s velocity : Ws velocity :: W : P 
 
 W\ velocity : i?'s velocity :: R : W 
 
 R\ velocity : *S"s velocity :: S : R 
 &c. 
 
 comp. P\ velocity : *S"s velocity :: S : P. 
 
 SCHOLIUM. 
 
 (150.) It has been usual to distinguish Levers into 
 three kinds, according to the different situations of the 
 power, weight, and center of motion; there are however 
 only two kinds which essentially differ; those in which 
 the forces act on contrary sides of the center of motion, 
 as the common balance, steel-yard, &c. and those in which 
 they act on the same side, as the stock-knife, shears 
 which act by a spring, oars, &c. The proportion between 
 the forces, when there is an equilibrium, is expressed in 
 the same terms in each case ; but the levers differ in this 
 respect, that the pressure upon the fulcrum depends upon 
 the sum of the forces in the former case, and upon their 
 difference in the latter; and consequently, the friction 
 upon the center of motion, cceteris paribus, is greater in 
 the former case than the latter.
 
 SCHOLIUM. 83 
 
 (151.) The pulley has, by some Writers, been re- 
 ferred to the lever, and they have justly deduced it's 
 properties from the proportions which are found to obtain 
 in that mechanical power ; for, the adhesion of the pulley 
 and the rope, which takes place at the circumference 
 of the pulley, will overcome the friction at the center 
 of motion ; both because it acts at a mechanical advantage, 
 and because the surface in contact is greater in the former 
 case than in the latter; and the friction depends, not 
 only upon the weight sustained, but also upon the quantity 
 of surface in contact: Thus, in practice, the rope and 
 pulley move on together, and the pulley becomes a lever. 
 
 (152.) The Wedge has hitherto chiefly been applied 
 to the purposes of separating the parts of bodies, and it's 
 power, notwithstanding the friction, is much greater than 
 the theory leads us to expect; the reason is, the effect 
 is produced by impact, and the momentum thus generated 
 is incomparably greater than the effect of pressure, in 
 the same time. Mr. Eckhard, a very ingenious mechanic, 
 by combining it with the wheel and axle, has constructed 
 a machine, the power of which, considering it's simplicity, 
 is much greater than that of any machine before inventedc 
 
 F2
 
 SECTION V, 
 
 ON THE CENTER OF GRAVITY. 
 
 (153.) Def. The Center of Gravity of any body^ 
 or system of bodies, is that point upon which the body 
 or system, acted upon only by the force of gravity, will 
 balance itself in all positions*. 
 
 (154.) Hence it follows, that if a line or plane, which 
 passes through the center of gravity, be supported, the 
 body, or system, will be supported in all positions. 
 
 (155.) Conversely, if a body, or system, balance 
 itself upon a line or plane, in all positions, the center of 
 gravity is in that line or plane. 
 
 If not, let the line or plane be moved parallel to itself 
 till it passes through the center of gravity, then we have 
 increased both the quantity of matter on one side of the 
 line or plane, and if s distance from the line or plane, 
 and diminished both, on the other side; hence, if the 
 body balanced itself in all positions in the former case, 
 it cannot, from the nature of the lever, balance itself in 
 all positions^ in the latter; consequently, the center of 
 gravity is not in this line, or plane (Art. 154.); which is 
 contrary to the supposition. 
 
 * That there is such a point in every body, or system of bodies, will 
 be shewn hereafter.
 
 ON THE CENTER OF GRAVITY. 85 
 
 (156.) Cor. By reasoning in the same manner, it 
 appears that a body, or system of bodies, cannot have 
 more than one center of gravity. 
 
 Prop. XXXIL 
 
 (I57.) To find the center of gravity of any number 
 of particles of matter. 
 
 Let A^ B, C, D, &c. be the particles; and suppose 
 J, B, connected by the inflexible line AB without weight *; 
 
 ^ D 
 
 divide AB into two parts in jG, so that A : B :: BE : EA, 
 or comp. A+B : B :: AB : EA; then will A and B 
 balance each other upon E, or if E be supported, A and B 
 will be supported in all positions (Art. S6.) ; let E be 
 supported on the line CE, then are A and B supported 
 in all positions ; also the pressure upon the point E is 
 equal to the sum of the weights A and B (Art. 70. Ax. 1,). 
 Join EC, and take J + 5 : C :: CF : FE, or A-^B-\-C 
 : C :: EC : FE ; then if F be supported, E and C will 
 be supported, that is, A, j&, and C, will be supported, in 
 all positions of the system ; and the pressure upon F will 
 be the sum of the weights. A, B, and C In the same 
 manner, join FD, and divide it into two parts in G, so 
 that A + B + C : D :: DG : FG, or A-\-B + C + D : D 
 :: FD : FG, and the system will balance iteelf in all po- 
 sitions upon G ; that is, G is the center of gravity of the 
 .system. 
 
 * The particles must be supposed to be connected, otherwise they 
 could not act upon each other, so as to balance upon any point.
 
 86 ON THE CENTER 
 
 (158.) CoK. 1. From this Proposition it appears that 
 every body, or system of bodies, has a center of gravity. 
 
 (159.) Cor. 2. If the particles be supposed to be 
 connected in any other manner, the same point G will be 
 found to be their center of gravity (Art. 156.). 
 
 (160.) CoR. 3. The effect of any number of particle}^ 
 in a system, to produce or destroy an equilibrium, is the 
 same, whether they are dispersed, or collected in their 
 common center of gravity. 
 
 (161.) CoR. 4. If A, B, C, &c. be bodies of finite 
 magnitudes, G, the center of gravity of the system, may 
 be found by supposing the bodies collected in their re- 
 spective centers of gravity. 
 
 (162.) CoR. 5. If the bodies A, B, C, &c. be in- 
 creased or diminished in a given ratio, the same point G 
 will be the center of gravity of the system. For the 
 points E, F, G, depend upon the relative, and not upon 
 the absolute weights of the bodies. 
 
 (163.) CoR. 6. If any forces, which are proportional 
 to the weights, act in parallel directions at A, B, C, Z>, 
 they will sustain each other upon the point G; and this 
 point is still called the center of gravity, though the 
 particles are not acted upon by the force of gravity. 
 
 (164.) CoR. 7. A force applied at the center of gravity 
 of a body cannot produce a rotatory motion in it. For 
 every particle resists, by it's inertia, the communication 
 of motion, and in a direction opposite to that in which 
 the force applied tends to communicate the motion ; these 
 resistances, therefore, of the particles, act in parallel di- 
 rections, and they are proportional to the weights (Art. 25.); 
 consequently, they will balance each other upon the center 
 of gravity.
 
 OF GRAVITY. 8? 
 
 Prop. XXXIII. 
 
 (165.) To find the center of gravity of a right line^. 
 
 The center of gravity of a right line, composed of 
 particles of matter which are equal to each other and uni- 
 formly dispersed, is it's middle point. For, there are equal 
 weights on each side, equally distant from the middle 
 point, which will sustain each other, in all positions, upon 
 that point <Art. S6.). 
 
 Prop. XXXIV. 
 
 (166.) To find the center of gravity of a parallelo- 
 gram. 
 
 Let AB be an uniform lamina of matter in the form 
 of a parallelogram; bisect AO, AT, in K and D; draw 
 
 A_ D 
 
 E 3 
 
 KL, DE, respectively, parallel to AI, AO, cutting each 
 other in C; this point C is the center of gravity of the 
 figure. For if the parallelogram be supposed to be made 
 up of lines parallel to AI, any one of these, as KL, is 
 bisected by the line DE (since AC, CI, are parallelograms, 
 and therefore, KC=^AD — DI^CL); consequently, each 
 line will balance itself upon DE (Art. l65.), or the whole 
 figure will balance itself upon DE, in all positions ; there- 
 
 * When we speak of a line or plane as having a center of gravity, 
 we suppose it to be matle up of particles of matter, uniformly diffused 
 over it.
 
 88 ON THE CENTER 
 
 fore, the center of gravity is in that line (Art. 155.). In 
 the same manner it may be shewn that the center of gravity 
 of the figure is in the line KL, consequently C, the inter- 
 section of the two lines, is the center of gravity required. 
 
 Prop. XXXV. 
 (167) ^^ ^^^ ^^^ center of gravity of a triangle. 
 
 Let ABC be an uniform lamina of matter in the form 
 of a triangle; bisect AB, AC^ in 2), £^; join CD^ BE, 
 
 cutting each other in G, this point is the center of gravity 
 of the triangle. 
 
 Suppose the triangle to be made up of lines parallel 
 to CA, of which let cea be one; then since the triangles 
 BEC, Bee, are similar, 
 
 BE : EC :: Be : ec; also, in the triangles 
 BEA, Bea, EA : BE :: ea : Be; 
 by composition, EA : EC :: ea : ec; 
 
 and EA = EC, therefore ea = ec; and consequently, the 
 line ac will balance itself in all positions upon BE. For 
 the same reason, every other line parallel to AC will 
 balance itself, in all positions, upon BE, or the whole 
 triangle will balance itself in all positions upon BE ; there- 
 fore the center of gravity of the triangle is in that line. 
 In the same manner it may be proved that the center
 
 OF GRAVITY. 89 
 
 of gravity is in the line CD ; therefore it is in G, the 
 intersection of the two lines BE, CD. 
 
 (168.) Cor. The distance of G from B is two-thirds 
 of the line BE. Join ED-., then since AD = DB, and 
 AE = EC, ED is parallel to BC (Euc. 2. xi.) ; therefore, 
 the triangles AED, ACB, are similar, 
 
 and CB : CA :: ED : EA ; 
 hence, CB : £/> :: CA : EA :: 2 : 1. 
 
 Also, the triangles CGB, EGD, are similar, 
 
 therefore, BG : Ci5 
 
 alternately, BG : GjE 
 
 hence, ^G : BE 
 
 : GE : ED; 
 : CB : jE:/) 
 
 : 2 : 3. 
 
 Prop. XXXVI. 
 
 (169) To Jlnd the center of gravity of any recti- 
 linear figure. 
 
 Let ABCDE be an uniform lamina of matter of the 
 proposed figure. Divide it into the triangles ABC, ACD, 
 ADE, whose centers of gravity a, b, c, may be found by 
 the last Proposition ; then if the triangles be collected 
 in their respective centers of gravity (Art. iGo.), their 
 common center of gravity may be found as in Prop. 32. ; 
 
 D C 
 
 that is, join ab and take db : ad :: the triangle ABC : the
 
 90 ON THE CENTER 
 
 triangle ADC, and d is the center of gravity of the two 
 triangles ABC, ACD. Join dc, and take ce : ed :: the 
 sum of the triangles ABC, ACD : the triangle AED, and 
 € is the center of gravity of the figure. 
 
 Prop. XXXVII. 
 
 (I70.) Tojind the center of gravity of any number 
 of bodies placed in a straight line. 
 
 Let A, B, C, D, be the bodies, collected in their re- 
 spective centers of gravity ; S any point in the straight 
 line SAD ; O the center of gravity of all the bodies. 
 Then since the bodies balance each other upon O, 
 
 § A B_0 C p 
 
 A X AO + B X BO = C X CO + D x DO (Art. 92.) ; 
 that is, A X (SO-SA) -\- B x {SO -SB) 
 = Cx (SC-SO) +Dx (SD - SO) ; 
 hence, by mult, and transposition, 
 
 A X SO-\-Bx SO + Cx SO-\-DxSO 
 = Ax SA + BxSB + CxSC-\-DxSD; therefore, 
 AxSA + BxSB + Cx SC + D x SD 
 
 SO = 
 
 A+B+C+D 
 
 (I7I) CoR. If any of the bodies lie the other way 
 from S, their distances must be reckoned negative ; and 
 if SO be negative, the distance SO must be measured from 
 S in that direction which, in the calculation, was supposed 
 to be negative. (Sec Alg- Art. 472.)
 
 OF (IKAVIT^. 
 
 91 
 
 Prop. XXXVIII. 
 
 (I72.) If perpendiculars he drawn from any number 
 of bodies to a given plane, the sum of the products of each 
 body, multiplied by ifs perpendicular distance from the 
 plane, is equal to the product of the sum of all the bodies 
 multiplied by the perpendicular distance of their common 
 center of gravity from the plane. 
 
 Let A, B, C, &c. be the bodies, collected in their 
 respective centers of gravity ; PQ the given plane ; draw 
 A a, Bb, Cc, at right angles to PQ, and consequently, 
 parallel to each other (Euc. 6. xi.); join AB, and take 
 
 - t> 
 
 a ^ 
 
 
 '•■e 
 
 J%^ 
 
 ''H) 
 
 y\ 
 
 
 ^V 
 
 S 
 
 c 
 
 L ' 
 
 AE : EB :: B : A, then E is the center of gravity of 
 A and B; through E draw Ee perpendicular to PQ, or 
 parallel to Aa, and wEy perpendicular to A a or Bb; 
 then in the similar triangles AEx, EBy, 
 
 Aaj : AE :: By : BE, 
 
 alternately, Aa; : By :: AE : BE :: B : A ; 
 
 therefore A x Ax = B x By, 
 
 that is, A X (.%a — Aa) = Bx (Bb — yb), 
 
 or since Ea, Eb, are parallelograms, 
 
 A X iEe-Aa)=^B x {Bb--Ec);
 
 if^ ON THE CENTER 
 
 and by multiplication and transposition, 
 
 J xEe-\- B xEe = A xAa + BxBh, 
 that is, {A + B)xEe = AxAa + BxBh. 
 
 Again, join EC, and take CG : GE :: A i- B : C, 
 then G is the center of gravity of the bodies A, B, C; 
 draw Gg perpendicular to PQ ; and it may be shewn, 
 as before, that 
 
 (A + B)xEe-^ CxCc = (A + B + C)x Gg, 
 
 or Ax Aa-^B X Bb+C X Cc = (A-i-B-hC) X Gg. 
 
 The same demonstration may be extended to any number 
 of bodies. 
 
 /-i^ox r^ TT r. AxAa-\-BxBh-\-CxCc 
 (1/3.) Cor. 1. Hence G^= . , „ , ^ ; 
 
 and if a plane be drawn parallel to PQ, and at the distance 
 Gg from it, the center of gravity of the system lies some- 
 where in this plane. In the same manner two other planes 
 may be found, in each of which the center of gravity lies, 
 and the point where the three planes cut each other, is the 
 center of gravity of the system. 
 
 (174.) Cor. 2. If any of the bodies lie on the other 
 side of the plane, their distances must be reckoned negative. 
 
 (I75.) CoR. 3. Wherever the bodies are situated, if 
 their respective perpendicular distances from the plane 
 remain the same, the distance of their common center of 
 gravity from the plane w^ill remain the same. 
 
 (I76.) CoR. 4. Let the bodies lie in the same plane, 
 and let perpendiculars, A a, Bb, Cc, Gg, be drawn to any 
 given line in that plane, then 
 
 A x Aa-\- B x Bb-\- C X Cc 
 ^'^ = J + B + C • 
 
 (177-) Con. ,). If A and B be on one side of the 
 plane, and C on the other, and the plane pass through
 
 OF GRAVITY. 93 
 
 the center of gravity, then A x Aa + B x Bh = C x Cc. 
 For Ggx{A-^B^'C) = AxAa-^BxBh-CxCc, and 
 Gg = 0, therefore A x Aa + B x Bh - C x Cc = 0', or 
 AxAa + BxBh = CxCc. 
 
 Prop. XXXIX. 
 
 (I78.) If any momenta he communicated to the parts 
 of a system, ifs center of gravity will move in the satne 
 manner that a body, equal to the sum of the hodies in the 
 system, would move, were it placed in that center, and 
 the same momenta, in the same directions, communicated 
 to it. 
 
 Let A, B, C, be the bodies in the system, and the 
 points A, B, C, their respective centers of gravity ; join 
 BC, and take BT : TC :: C : B; join AT, and take 
 
 TE : EA :: A : B + C, 
 or TE : TA :: A : A+B^C, 
 
 then will E be the center of gravity of the system (Art. 161.). 
 Suppose the momentum communicated to A would 
 
 A 
 
 > 
 
 / 
 
 cause it to move from A to w in GT", and at x let the 
 body be stopped ; join Tx ; 
 
 take TF : Tx :: A : A-{-B-\-C,
 
 9^ ON THE CENTER 
 
 then F is the center of gravity of the bodies when they 
 are at .v, B, C; join EF; 
 
 since TE : TJ :: A : A -\- B -h C :: TF : Tx, 
 
 EF is parallel to Ax (Euc. 2. vi.), and consequently the 
 triangles TEF, TA x, are similar ; 
 
 therefore J^F : Ax :: A : A + B + C. 
 
 Hence if one body A in the system be moved from A 
 to tT, the center of gravity is moved from E to F \ which 
 point may be thus determined; draw EF parallel to Ax^ 
 and take EF : Ax :: A : A-\B^C. 
 
 Next let a momentum be communicated to B, which 
 would cause it to move from B io y va. T" \ at y let the 
 body be stopped ; then, according to the rule above laid 
 down, draw FG parallel to By^ and take 
 
 FG : By V. B '. J+^ + C, 
 
 and G will be the center of gravity of the bodies when 
 they are at x^ y^ C. In the same manner, let a momen- 
 tum be communicated to C, which would cause it to 
 move from C to ;$? in T'\ and at !s let the body be stop- 
 ped ; draw GH parallel to C^, and take 
 
 GH : C:^ :: C : A -{- B -b C, 
 
 then H is the center of gravity of the bodies when they 
 are at .j?, y, ^. If now the momenta, instead of being 
 communicated separately, be communicated at the same 
 instant to the bodies, at the end of T'' they will be 
 found in x, y, %, respectively ; therefore, at the end of T'\ 
 their common center of gravity will be in H. 
 
 Now let £ be a body equal to A +B -t Cy and let the 
 same momentum be communicated to it that was before 
 communicated to A, and in the same direction ; then since 
 EF is parallel to Ax, EF is in the direction in which 
 the body E will move ; also, since the quantities of motion
 
 OF GRAVITY. 95 
 
 communicated to A and E are equal, their velocities are 
 reciprocally proportional to their quantities of matter 
 (Art. 19.), or E's velocity : J's velocity :: A : A + B + C 
 :: EF : Ax; therefore, EF and Aoe are spaces described 
 by E and A in equal times (Art. 11.), or E will describe 
 the space EF in T". In the same manner FG is the space 
 which the body E will describe in T", if the momentum, 
 before communicated to B, be communicated to it ; and 
 GH the space it will describe in T", if the momentum 
 before communicated to C, be communicated to it ; join 
 EH; and when the motions are communicated at the same 
 instant to E, it will describe EH in T" (Art. 42.). Hence 
 it follows, that when the same momenta are communicated 
 to the parts of a system, and to a body, equal to the sum 
 of the bodies, placed in the common center of gravity, this 
 body and the center of gravity are in the same point at the 
 end of T" ; and T may represent any time ; therefore, they 
 are always in the same point. 
 
 The same demonstration may be applied, whatever be 
 the number of bodies in the system. 
 
 (I79.) CoR. 1. If the parts of a system move uni- 
 formly in right lines, the center of gravity will either 
 remain at rest, or move uniformly in a right line. For 
 if the momenta communicated to the several parts of the 
 system be communicated to a body, equal to the sum of 
 the bodies, placed in the center of gravity of the system, 
 it will either remain at rest or move uniformly in a right 
 line (Art. 27.)- 
 
 (180.) Cor. 2. If two weights support each other 
 upon any machine, and it be put in motion, the center of 
 gravity of the weights will neither ascend nor descend. 
 For the momenta of the weights, in a direction perpen- 
 dicular to the horizon, are equal and opposite (Art. 149.); 
 therefore, if they were communicated to a body equal to 
 the sum of the bodies, placed in the common center of 
 gravity, they would neither cause it to ascend nor descend.
 
 96 ON THE CENTER 
 
 (181.) Cor. 3. The motion or quiescence of the 
 center of gravity is not affected by the mutual action of 
 the parts of a system upon each other. For action and 
 reaction are equal and in opposite directions, and equal 
 and opposite momenta communicated to a body, equal to 
 the sum of the bodies in the system, will not disturb it's 
 motion or quiescence. 
 
 (182.) CoR. 4. The effect of any force to commu- 
 nicate motion to the common center of gravity, is the same, 
 upon whatever body in the system it acts. 
 
 (183.) CoR. 5. If G be the center of gravity of the 
 
 particles of matter. A, B, C, Z), which are acted upon only 
 by their mutual attractions, they will meet at G. For they 
 must meet, and their common center of gravity will remain 
 at rest (Art. 181.); therefore, they must meet at that 
 center. 
 
 (184.) Cor. 6. If a rotatory motion be communicated 
 to a body, and it be then left to move freely, the axis of 
 rotation will pass through the center of gravity. For the 
 center of gravity itself, either remaining at rest or moving 
 uniformly forward in a right line, has no rotation. 
 
 Prop. XL. 
 
 (185.) If a body be placed upon an horizontal plane, 
 and a liiie drawn from ifs center of gravity perpendicular 
 to that plane, the body will be sustained, or 7iot, according 
 as the perpendicular falls ivithifi or without ifs base.
 
 OF GRAVITY 
 
 97 
 
 Let ABDC represent the body, G it's center of gravity; 
 draw GE perpendicular to the horizon ; join CG^ and with 
 the radius CG describe the circular arc HGF ; then the 
 body cannot fall over at C, unless the center of gravity 
 
 A _B 
 
 describes the circular arc GF. Suppose the whole force 
 of gravity applied at G (Art. 178.), and take GE to re- 
 present it ; draw E.v perpendicular to CG ; then the 
 force GE is equivalent to the two G.37, a^E, of which Gw 
 cannot move the body either in the direction GF or GH ; 
 and when E falls within the base, cvE acts at G in the 
 direction GH; therefore the center of gravity cannot de- 
 scribe the arc GF, that is, the body cannot fall over at C 
 In the same manner it may be shewn that it cannot fall 
 over at D. 
 
 When the perpendicular GE falls without the base, 
 ,vE acts in the direction GF, and since there is no force 
 
 to counteract this, the center of gravity will move in that 
 direction, or the body will fall. 
 
 (186.) CoR. 1. In the same manner it may be shewn, 
 that if a body be placed upon an inclined plane, and the 
 lateral motion be prevented by friction, the body will be 
 sustained or not, according as the perpendicular to the 
 horizon, drawn through it's center of gravity, falls within 
 or without the base. 
 
 G
 
 98 
 
 ON rHK CENTER 
 
 Ex. Let ABCD represent a cube of imilorm density, 
 placed upon the inclined plane RS., G it's center of gravity ; 
 draw GE perpendicular to CD, and GFH perpendicular to 
 the horizon ; then this body will not be sustained upon the 
 inclined plane, if the angle of the plane's inclination SR T, 
 exceed half a right angle. For if the Z FRH be greater 
 than half a right angle, the z RFH, or GFE, is less than 
 
 half a right angle, and the Z FGE is greater than half a 
 right angle; therefore, EF is greater than EG, or EC, 
 and the body will roll. 
 
 (187) CoR. 2. The higher the center of gravity of a 
 body is, ccsteris paribus, the more easily it is overturned. 
 
 The same construction being made as in the Proposition, 
 the whole weight of the body : that part of the weight which 
 keeps it steady upon it's base, or opposes any power em- 
 ployed to overturn it :: GE : xE :: GC : CE; and when 
 CE and the whole weight of the body are given, the force 
 
 which keeps the body steady oc -— {Jig. Art. 206.); 
 
 therefore as GC increases, that is, as GE increases, the 
 force which keeps the body steady decreases, or the more 
 easily will the body be overturned. 
 
 (188.) CoR. S. When CE vanishes with respect to 
 GC, the force which keeps the body steady vanishes, and 
 the bodv may be overturned by a very small force. Thus
 
 OF GRAVITY. 
 
 99 
 
 it is extremely difficult to balance a body upon a point 
 placed under the center of gravity. 
 
 Prop. XLI. 
 
 (189.) If a body he suspended by any point, it will 
 not remain at rest till the ceyiter of gravity is in the line 
 which is drawn through that point, perpendicular to the 
 horizon. 
 
 Let S be the point of suspension of the body ABC; 
 
 R 
 
 G it's center of gravity ; join SG and produce it ; through 
 *S', and G, draw RST, and GH, perpendiculars to the 
 horizon; then the effort of gravity, to put the body in 
 motion, is the same that it would be, were all the particles 
 collected at G; take GH to represent the force in that 
 direction, and draw HI perpendicular to GI; then the 
 force GH is equivalent to the two GI, IH, of which GI 
 is sustained by the reaction at the point of suspension ^S*, 
 and IH is employed in moving the center of gravity in 
 a direction perpendicular to SG ; therefore the center of 
 gravity cannot remain at rest till IH vanishes ; that is, 
 till the angle IGH, or GST, vanishes, or SG coincides 
 with RT. 
 
 G 2
 
 100 ON THE CENTER OF GRAVITY. 
 
 (190.) Cor. Hence it follows, that if a body be sus- 
 pended successively by different points, and perpendiculars 
 to the horizon be drawn through the point of suspension, 
 and passing through the body, the center of gravity will 
 lie in each of these perpendiculars, and consequently, in 
 the point of their intersection.
 
 SECTION VI. 
 
 ON THE COLLISION OF BODIES. 
 
 (191.) Def. Hardness, which is found in different 
 bodies in different degrees, consists in a firm cohesion of the 
 component particles ; and that body is said to be harder 
 than another, whose particles require a greater force to 
 separate them. By a perfectly hard body we mean one 
 whose parts cannot be separated, or moved one amongst 
 another by any finite force. 
 
 (192.) Def. The tendency in a b'ody to recover it's 
 former figure, after having been compressed, is called elas- 
 ticity. That body is said to be more elastic than another, 
 which recovers it's figure with the greater force, supposing 
 the compressing force the same. By a perfectly elastic 
 body we mean one which recovers it's figure with a force 
 equal to that which was employed in compressing it. 
 
 That such a tendency exists in bodies is evident from 
 a variety of experiments. If an ivory ball, stained with 
 ink, be brought gently into contact with an unstained ball, 
 the spot received by the latter will be very small, since two 
 spheres touch each other only in a single point ; but if one 
 of the balls be made to impinge upon the other, the spot 
 will be enlarged ; and the greater the force of impact, the 
 greater will be the surface stained ; hence it is manifest, 
 that one, or both of the balls, has been compressed, and 
 afterwards recovered it's spherical figure. Almost all bodies 
 with which we are acquainted are elastic in a greater or 
 less degree; but none perfectly so. In steel balls, the 
 force of elasticity is to the compressing force as 5 to 9 ; in
 
 102 ON THE COLLISION 
 
 glass, as 15 to l6; though in all cases, the force of elas- 
 ticity seems to depend, in some measure, upon the diameter 
 of the ball. 
 
 (193.) Def. The impact of two bodies is said to be 
 direct, when their centers of gravity move in the right line 
 which passes through the point of impact. 
 
 In considering the effects of collision, the bodies are 
 usually supposed to be spheres of uniform density ; and in 
 their actions upon each other, not to be affected by gravity, 
 or any other force but that of inertia. 
 
 Prop. XLII. 
 
 (194.) If the impact of two jJ^rfectly hard bodies be 
 direct, after impact they tcill either remain at rest, or 
 ?nove on, uniformly, together. 
 
 Since there is no force to turn either body out of the 
 line of direction, they will continue in that line after im- 
 pact*. Let A and B be the two bodies, moving in the 
 same direction, and let A overtake B; then will A continue 
 to accelerate B's motion, and B will continue to retard ^'s, 
 till their velocities are equal, at which time they will cease 
 to act upon each other ; and since there is no force to 
 separate them, they will move on together, and their com- 
 mon velocity, by the First Law of Motion, will be uniform. 
 When they move in opposite directions, if their forces be 
 equal, they will rest after impact; if ^'s force be greater than 
 B''s, the whole velocity of B will be destroyed, and ^'s not 
 being destroyed, A will communicate velocity to B, and B 
 by it's reaction will retard A, till they move on together, 
 as in the former case. 
 
 * The momenta of the particles in each body are proportional to 
 their weights, since their velocities are equal ; these momenta, therefore, 
 will not turn the body to either side of the line passing through the 
 center of gravity (Art. 163.).
 
 OF BODIES. 
 
 Prop. XLIIL 
 
 103 
 
 (195.) If the impact of two perfectly hard bodies he 
 direct, their common velocity may he found by dividing 
 the whole momentum before impact, estimated in the di- 
 rection of either motion, by the sum of the quantities of 
 matter. 
 
 Let A and B be the quantities of matter contained 
 in the bodies, a and h their velocities; then, when they 
 move in the same direction, Aa + Bb is the whole mo- 
 mentum in that direction, before impact. When they 
 move in opposite directions, Aa — Bb is the whole mo- 
 mentum estimated in the direction in which A moves. 
 
 In the former case, as much as A a, the momentum 
 of A, is diminished, so much is Bb, the momentum of B, 
 increased by the impact (Art. 32.); therefore Aa + Bb 
 is equal to the whole momentum after impact. 
 
 In the latter case, it A a be greater than Bb, before 
 the bodies can begin to move together, Bb, the momentum 
 of B, must be destroyed ; and therefore ^'s momentum 
 must be diminished by the quantity Bb (Art. 32.). Thus, 
 when the bodies begin to move in the same direction, 
 Aa — Bb is their whole momentum; and as much mo- 
 mentum as is afterwards communicated to B, so much 
 is lost by A; therefore Aa — Bb is equal to the whole 
 momentum after impact. 
 
 If A a be less than Bb, the momentum after impact, 
 in the direction of ^'s motion, will be Bb — Aa; or, 
 in the direction of A\ motion, Aa — Bb. 
 
 Let V be the common velocity after impact; 
 then (A -\- B) X ii is the whole momentum ; consequently, 
 
 {A^B)x v=: Aa -f Bb. and v. = ^- ^ . 
 ~ A-Y B
 
 104 ON THE COLLISION 
 
 In which expression, the positive sign is to be used when 
 the bodies move in the same direction before impact, and 
 the negative sign, when they move in opposite directions. 
 
 (196.) Cor. l. When the bodies move in opposite 
 directions with equal momenta, they will remain at rest 
 after impact. In this case Ja — Bb = 0; therefore v = 0. 
 
 (197-) Cor. 2. If J56 be greater than A a, v is 
 negative. This shews that the bodies will move in the 
 direction of B^s motion, which was supposed, in the pro- 
 position, to be negative. 
 
 Prop. XLIV. 
 
 (198.) In the direct impact of two perfectly hard 
 bodies A and B, estimating the effects in the direction 
 of A's motion^ A + B : A :: the relative velocity of the 
 two bodies : the velocity gained by B. And A 4- B : B :: 
 their relative velocity : the velocity lost by A. 
 
 The same notation being retained; when the bodies 
 move in the same direction, a — b is their relative velocity 
 (Art. 12.) ; and u, their common velocity after impact, 
 
 is (Art. 195.); therefore, the velocity ^«mec^ by J?, 
 
 A -{- B 
 
 Aa + Bb , Aa—Ab . 
 
 or V — b= — 6 = -— -— ; hence, 
 
 A-h B A+B 
 
 A^B : A :: a — b : the velocity gained by B. 
 
 Aa-^Bb Ba-Bb , 1 •, / , 1 t 
 
 Also, a -— , or , . ^- is the velocity lost by A ; 
 
 A-\'B A + B 
 
 hence A -\- B : B :: a ~ b : the velocity lost by A. 
 
 When the bodies move in opposite directions, a -{■ b 
 
 Aa-Bb 
 is their relative velocity (Art. J2.); and v = j , p
 
 OF BODIES. 105 
 
 (Art. 195.); also, the velocity communicated to B upon 
 the whole, in the direction of ^'s motion, is 
 
 Aa-Bh , , . Aa-\-Ab 
 ^ + ^' Q^* ^ . p +^^ that IS, ; 
 
 therefore, A + B:A::a-\-b: the velocity gained by B. 
 
 111.- Aa-Bh Ba + Bb 
 
 The velocity lost by ^ is a -— , or — — — ; 
 
 ^ ^ A+B A+B 
 
 therefore, A -r B : B :: a -\- b : the velocity lost hy A. 
 
 Ex. Let the weights of A and B be 10 and 6*; their 
 velocities 12 and 8, respectively; then, when they move 
 in the same direction, 10 + 6" : 10 :: 12- 8 : j^ = 2^, the 
 velocity gained by B; and 10 + 6' : 6 :: 12 - 8 : ^ = 1^, 
 the velocity lost by A. 
 
 When they move in opposite directions, 12 + 8 is their 
 relative velocity ; and 10 + 6 : 10 :: 12 + 8 : ^ = 12i, the 
 velocity gained by B in the direction of ^'s motion. Also, 
 since it had a velocity 8 in the opposite direction before 
 impact, ifs velocity after impact is 4^ in the direction 
 of A's motion. Again, 10 + 6 : 6 :: 12 + 8 : ^^ = 7!, the 
 velocity lost by ^. 
 
 (I99.) CoR. 1. Whilst the relative velocity remains 
 the same, the velocity gained by B, and the velocity lost 
 by A, are unaltered. 
 
 (200.) CoR. 2. Hence it also follows that the velo- 
 cities, gained by B, and lost by A, are the same, whether 
 both bodies are in motion, or A impinges upon B at rest, 
 with a velocity equal to their relative velocity in the former 
 case. 
 
 * Sec Art. 26.
 
 106 ON THE COLLISION 
 
 (201.) Cor. 3. If the relative velocity be the same, 
 the momentum communicated is the same, whether A 
 impinges upon Bj or B upon J. 
 
 Call r the relative velocity ; then when A impinges 
 
 Ar 
 upon B, A -r B : A :: r : — — — , the velocity gained by 
 
 ABr 
 
 B ; therefore — is the momentum gained by B. When 
 
 Br 
 B impinges upon A, A -{- B : B :: r : , the velocity 
 
 ABr 
 
 gained by A ; therefore — is the momentum gained 
 
 by A ; which is also the momentum gained by B on the 
 former supposition. 
 
 Prop. XLV. 
 
 (202.) When the bodies are perfectly elastic, the 
 velocity gained by the body struck., and the velocity lost 
 by the striking body, will be twice as great as if the 
 bodies were perfectly hard. 
 
 Let A and B be the bodies ; then, as in Art. 194, A 
 will accelerate 5's motion, and B will retard ^'s, till their 
 velocities are equal ; and if they were perfectly hard thev 
 would then cease to act upon each other, and move on 
 together; thus, during the first part of the collision, the 
 same effect is produced, that is, the same velocity is gained 
 and lost, as if the bodies were perfectly hard. But, during 
 this period, the bodies are compressed by the stroke, 
 and since they are, by the supposition, perfectly elastic, 
 the force with which each will recover it's former shape 
 is equal to that with which it was compressed ; therefore, 
 each body will receive another impulse from the elasticity- 
 equal to the former ; or B will gain, and A lose, upon 
 the whole, twice as great a velocity as if both bodies had 
 been pei'fectly hard.
 
 OF BODIES, 107 
 
 (203.) The same demonstration may be applied to 
 the case where one body is perfectly hard, and the other 
 perfectly elastic. 
 
 Prop. XLVI. 
 
 (204.) In the direct impact of two perfectly elastic 
 bodies A and B, A + B : 2 A :: their relative velocity 
 before impact : the velocity gaified by B in the direction 
 of A's motion ; and A + B : 2 B : : their relative velocity : 
 the velocity lost by A, in that direction. 
 
 Call r the relative velocity of the bodies ; ,v the velocity 
 gained by B, and y the velocity lost by Ay when both 
 bodies are perfectly hard; then 2w is the velocity gained 
 by B, and 2t/ the velocity lost by A, when they are 
 perfectly elastic ; and 
 
 A^B '. A :. r : m (Art. 198.) ; therefore, 
 
 A-^-B : 2A :: r .: 2a: (Alg. Art. 185.), the velocity gained 
 
 hj B. 
 
 Again, A -[- B : B :: r : y (Art. 198.); therefore, 
 A-\-B : 2B :: r : 2y, the velocity lost by A. 
 
 Ex. Let the weights of the bodies be 5 and 4, their 
 velocities 7 and 5; then, when they move in the same 
 direction, 5+4 : 10 :: 7 — 5 : — = 2-, the velocity gained 
 by B ; therefore 5+2-, or 7 g is ^'s velocity after impact. 
 Also, 5 + 4 : 8 :: 7 — 5 : ^ = 1^, the velocity lost by A; 
 
 7 2 
 
 §,or5^^ 
 
 120 
 
 therefore 7 — 1 5, or 5^, is ^'s velocity after impact. When 
 they move in opposite directions, 5 + 4 : 10 :: 7 + 5 : ~ 
 z= 13^, the velocity gained by B. Also, since it had a 
 velocity 5 in the opposite direction, it's velocity after 
 impact, in the direction of ^'s motion, is 13^—5, or 8^. 
 Again, 5 + 4 : 8 :: 7 + 5 : ^ = lo| J's velocity lost; and 
 since it had a velocity 7 before impact, after impact it 
 will move in the opposite direction with a velocity 3^.
 
 108 ON THE COLLISIOX 
 
 (205.) CoK. 1. When A = B, the bodies interchange 
 velocities. For, in this case, A -i- B = 2A = 2B ; therefore, 
 the velocity gained by B, and the velocity lost by A, are 
 respectively equal to their relative velocity before impact. 
 Let a and b be their velocities before impact ; then, when 
 they move in the same direction, a — b is the velocity 
 gained by B, or lost by A ; therefore a — b ^ b, or «, is 
 5's velocity after impact ; and a — (a — b), or b, is ^""s 
 velocity. If b be negative, or the bodies move in opposite 
 directions, a + b — b, or a, is B''s velocity, and « — (a + 6), 
 or — 6, is A'^s velocity after impact. 
 
 (206.) CoR. 2. If the bodies move in opposite di- 
 rections with equal quantities of motion, the whole mo- 
 mentum of each will be destroyed during the compression, 
 and an equal one generated by elasticity in the opposite 
 direction; each body will therefore be reflected with a 
 velocity equal to that which it had before impact. 
 
 (207.) CoR. 3. The relative velocity of the bodies 
 after impact is equal to their relative velocity before 
 impact. 
 
 Let a and b be the velocities of the bodies before 
 impact; p and q their velocities after; then a — b = q—p. 
 
 ^ A T^ A ^ 2A X (a — b) , , . 
 
 For, A + B : 2A :: a-b : _1___I the velocity 
 
 A + B ^ 
 
 11 ^ 1 p 7 2Ax(a—b) 
 
 gamed by B ; therefore q = b-\- 
 
 Also A + B : 2B :: a- b 
 
 A + B 
 
 2B X (a-b) 
 
 A + B 
 
 the velocity lost by A ; 
 
 2Bx{a-b) 
 
 therefore jj = a — 
 
 and q — p = b - a •{- 
 
 A + B 
 
 (2A + 2B) x(a-b) 
 A + B 
 = b — a ^ 2a — 2b = a — b.
 
 OF BODIES. 109 
 
 When the bodies A and B move in opposite directions, 
 the sign of h is negative; in other respects the demon- 
 stration is the same. 
 
 (208.) Cor. 4. The sum of the products of each 
 body, multiplied by the square of it's velocity, is the same 
 before and after impact. 
 
 The notation in the last Article being retained ; 
 
 Aa + Bh = Aj) + Bq {Art. S4>.); 
 
 hence Aa — Ap = Bq—Bb 
 
 or Ax {a ^p) — B X {q - b). 
 
 Also a — h = q — p (Art. 20?.) ; 
 
 or a + p = q -\-h; 
 
 therefore A x (a —p) x (a -\- p) z=: B x {q — h) x (q -\- b) ; 
 
 or Aa'-Ap^^Bq-'-Bh'; 
 
 therefore A a" -f- Bb" = Ap~ -\- B(f. 
 
 If any of the quantities, 6, ja, q^ be negative, it's square will 
 be positive, and therefore the conclusion will not be altered. 
 
 (209.) CoR. 5. . If there be a row of equal elastic 
 bodies, A^ 5, C, Z), &c. at rest, and a motion be com- 
 municated to A^ and thence to jB, C, -D, &c. they will all 
 remain at rest after the impact, except the last, which will 
 move off with a velocity equal to that with which the first 
 moved. 
 
 For A and B will interchange velocities (Art. 205) ; 
 that is, A will remain at rest, and B move on with ^'s 
 velocity. In the same manner it may be shewn that all 
 the others will remain at rest after impact, except the 
 last; which will move off with the velocity communicated 
 to ^. 
 
 (210.) Cor. 6. If the bodies decrease in magnitude, 
 they will all move in the direction of the first motion, 
 and the velocity communicated to each succeeding body 
 will be greater than that which was communicated to the 
 precedhig.
 
 110 ON I'HK roLLISTOX 
 
 For, A -\- B : 9.B :: A\ velocity before impact : the 
 velocity lost by A ; and since 2 jB is less than A-\- B^ A 
 does not lose it's whole velocity ; therefore it will move 
 on after impact in the direction of the first motion. Also, 
 A + B : 2 A :: A^ velocity before impact : the velocity 
 gained by B ; and since 2 A is greater than A + B^ the 
 velocity gained by 5 is greater than ^'s velocity before 
 impact. In the same manner it may be shewn that 5, 
 
 C, D, &c. will move on in the direction of the first motion ; 
 and that the velocity communicated to each will be greater 
 than that which was communicated to the preceding body. 
 
 (211.) Coil. 7. If the bodies increase in magnitude, 
 they will all be reflected back, except the last; and the 
 velocity communicated to each succeeding body will be 
 less than that which was communicated to the preceding. 
 
 For, in this case, 2 B is greater than A -\- B \ therefore, 
 A loses more than it^s whole velocity, or it will move in the 
 contrary direction. Also, 2 J is less than A-\-B; therefore, 
 the velocity gained by B is less than A^s velocity before 
 impact. In the same manner it may be shewn that B, C, 
 
 D, &c. will be reflected ; and that the velocity commu- 
 nicated to each will be less than that which was com- 
 municated to the preceding body. 
 
 (212.) Cor. 8. The velocity thus communicated from 
 A through B to C, when B is greater than one of the two 
 A, C, and less than the other, exceeds the velocity wbich 
 would be communicated immediately from A to C 
 
 Let a represent A''s velocity ; then 
 
 2 Ad 
 
 A -\- B : 2A :: a : ~ , the velocity of B; 
 
 A -\- B 
 
 . _ ^ _ 2Aa 2Aa 2B 
 
 and B + C:,B::j^:j^x-^^, 
 
 the velocity communicated from B to C. 
 
 2Aa 
 Again, A -\- C : 2A :: a : — — - , 
 
 A -h C
 
 OF BODIES. Ill 
 
 the velocity communicated immediately from A to C. 
 Hence it follows, that the velocity communicated to C, 
 by means of B, is greater than that which would be com- 
 municated to it immediately, if 
 
 2Aa 2B ^ ^ 2Aa 
 
 X — — - be greater than 
 
 A + B B+C ^ A + C 
 
 that is, if ^ + C be greater than ( -^ + ^) >^ (B + C) ^ 
 
 9.B 
 
 AC 
 
 or 2 v4 + 2 r greater than A -\- C + B -\ ; 
 
 B 
 
 AC 
 or A-\-C greater than B -\ — — . 
 
 B 
 
 Suppose A = B + ,r, C — B + y\ 
 
 then A-{-C = 9.B-\-x-\-y, and 
 
 „ AC ^ B~ + BoG-{-By -\- xy ^ wy 
 
 B+~=B+ ^^ l=2B + .v + y+^i 
 
 therefore, the velocity communicated to C by means of J5, 
 is greater than the velocity communicated to it without B, 
 
 if 9.B -\- X -\-y be greater than 2B-\-x-\-y ^ — ^; which 
 
 B 
 
 will always be the case when ocy is negative, or when x 
 
 and y have different signs ; that is, when B is less than one 
 
 of the bodies, A^ C, and greater than the other*. 
 
 (213.) CoR. g. If the bodies be in geometrical pro- 
 gression, the velocities communicated to them will be in 
 geometrical progression ; and when there are n such bodies, 
 whose common ratio is r, the velocity of the first : the 
 velocity of the last :: (l-\-ry-^ : 2"'^. 
 
 * The velocity comnmnicated from A through B to C, is a maximum 
 when A, B, and C, are in j^eoraetrical progression. (Flux. Art. 21. 
 Ex. 11.)
 
 112 ON THE (Or.LISTOX 
 
 Let A, Jr, Ar-, Ar', &c. be the bodies; «, h, c, d, &c. 
 the velocities successively communicated to them ; then 
 
 A -\- Ar : 9.A :: a : b, or 
 
 1 + r : 2 :: a : b; and in the same manner, 
 
 1 + r : 2 :: b : c 
 
 1 + r : 2 :: c : d, kc. 
 
 therefore a : b :: b : c :: c : d, &c. Also, by compo- 
 sition, (l+r)^"^ : 2""^ :: a : the velocity of the last. 
 
 (214.) Con. 10. If the number of mean proportionals, 
 interposed between two given bodies A and X, be increased 
 without limit, the ratio of A''s velocity to the velocity thus 
 communicated to X will approximate to the ratio of 
 A^ X : 1^ A as ifs limit. 
 
 Let A^ 5, C, Z), JT be the bodies ; «, 6, c, d, x 
 
 the velocities communicated to them. Then since the 
 number of bodies interposed between A and X is increased 
 without limit, their differences will be diminished without 
 limit ; let J + ^ = jB ; then 
 
 2^ + sr : 2^ ..a : b 
 
 or A -\ — : A :: a : b 
 
 2 
 
 * and J + ? : ^ :: .J1lV% : ^/~A :: ^^ : J~A'. 
 
 * Since (^ + |)%\4^ :: A' + Az + ^ : A^ :: J^^+^ .- A 
 
 '-'- B-\- —-2 : A, the ratio of (^+5) " ^^ when z is continually 
 diminished, approximates to the ratio of B : A, and consequently, 
 the ratio of ^ + - : A approximates to the ratio of \/B : \/A as it's 
 limit.
 
 OF BODIES. 113 
 
 therefore, ^J B : fj^ '■'■ « : h 
 
 inthesamemanncr, ^C : y/ B :: h : c 
 
 ^'D : ^~C :: c : d 
 
 &c. 
 
 comp. /y/jf : ^y J :: a : w. 
 
 Cor. The conclusion is the same when the inter- 
 mediate bodies vary according to any other law, if the 
 difference of the succeeding bodies, in every part of the 
 series, be evanescent. 
 
 Prop. XLVII. 
 
 (215.) in the direct impact of two imperfectly elastic 
 bodies A and B, if the compressing force he to the force 
 of elasticity :: 1 : m, then A + B : (l+m)xA :: their 
 relative velocity before impact : the velocity gained by B 
 in the direction of A^s motion. And A + B : (l + m) x B 
 :: their relative velocity before impact : the velocity lost 
 by A, in that direction. 
 
 By reasoning, as in Art. 202, it appears that the velocity 
 gained by B and the velocity lost by J, during the com- 
 pression, are the same as if the bodies were perfectly 
 hard; and the velocity communicated by the elasticity is 
 to the velocity communicated by the compression :: m : 1. 
 Call r the relative velocity before impact, w the velocity 
 gained by B^ and y the velocity lost by A, during the 
 compression ; then (l -^m) x oe is the velocity gained by 
 By and {l -\-m) xy the velocity lost by A, upon the whole. 
 Now 
 
 A + B : A :: r : .T (Art. IpS.), 
 
 and A -\- B : B :: r : y ; 
 
 H
 
 114 ON THE COLLISION 
 
 therefore, A + B : (l-\-m)xA :: r : (\-^m)x w, 
 the velocity gained hjB; 
 
 and J + B : (l+w)x5 :: r : (l+m)xy, 
 the velocity lost by A. 
 
 (216.) CoR. 1. The relative velocity before impact 
 : the relative velocity after impact :: 1 : m. 
 
 Let a and b be the velocities of the two bodies before 
 impact, p and q their velocities after ; then 
 
 A-^B : (l+m)x^ :: a-b : ^^ j^^ ' 
 
 the velocity gained by B ; 
 
 (\ -^m)x A X (a — b) 
 
 therefore, q = b-\- 
 in the same manner, p = a^ 
 hence, q — p = b^a-\- 
 
 A + B 
 
 (1+ m) X jB X {a - b) 
 
 (1 + m) X ( J + -S) X (a — b) 
 
 A^B 
 
 or 6 — a + a - fe -j- m X (a — 6), i. e. m x (a — 6), 
 
 is the relative velocity after impact ; 
 
 and a — b : mx(a — b) :: 1 : m. 
 
 When the bodies move in opposite directions, the sign 
 of b is negative. 
 
 (217.) Cor. 2. Hence it appears that if the velocities 
 of the bodies before and after impact be known, the elastic 
 force is known.
 
 OF BODIES. 115 
 
 (218.) Cor. 3. If A impinge upon B at rest, A will 
 remain at rest after impact when A : B :: m : I. 
 
 In this case A loses it's whole velocity, 
 and A-\-B : (l ^m)xB :: a : the velocity lost by A; 
 therefore A -{- B = (l + m) x B, and A—mB; 
 consequently, A : B :: m : I. 
 
 (219.) Cor. 4. The momentum communicated is the 
 same, whether A impinges upon B, or B upon A, if the 
 relative velocity be the same. This is the case when the 
 bodies are perfectly hard (Art. 201.); and the effect pro- 
 duced in elastic bodies is in a given ratio to that which 
 is produced when the bodies are perfectly hard. 
 
 Prop. XLVIII. 
 
 (220.) When a perfectly hard body impinges obliquely 
 on a perfectly hard and immoveable plane AB, in the 
 direction CD, after impact it will move along the plane, 
 and the velocity before impact : the velocity after :: radius 
 : the cosine of the angle CDA. 
 
 Take CD to represent the motion of the body before 
 
 impact ; draw CE parallel, and DE perpendicular to AB. 
 Then CD may be resolved into the two CE, ED, (Art. 43.), 
 of which ED is wholly employed in carrying the body in a 
 direction perpendicular to the plane; and since the plane 
 is immoveable, this motion will be wholly destroyed, (See 
 Art. Il6.). The other motion CE, which is employed in 
 carrying the body parallel to the plane, will not be affected 
 
 h2
 
 116 
 
 ON THE COLLISION 
 
 by the impact ; and consequently, there being no force to 
 separate the body and the plane, the body will move along 
 the plane ; and it will describe DB = CE in the same time 
 that it described CD before impact ; also, these spaces are 
 uniformly described (Art. 27-); consequently, 
 
 the velocity before impact : the velocity after :: CD : CE 
 
 :: radius : sin. z CDE :: radius : cos. z CD A. 
 
 (221.) Cor. The velocity before impact : the differ- 
 ence between the velocity before and the velocity after, that 
 is, the velocity lost :: radius : rad. —cos. z CDA :: rad. : 
 the versed sine of the angle CDA. 
 
 Prop. XLIX. 
 
 (222.) If a perfectly elastic body impinge upon a 
 perfectly hard and immoveable plane AB, in the direction 
 CD, it will be reflected from it in the direction DF, which 
 makes, with DB, the angle BDF equal to the angle ADC. 
 
 Let CD represent the motion of the impinging body ; 
 A 2 B 
 
 draw CF parallel, and DE perpendicular to AB ; make 
 EF=CE, and join DF. Then the whole motion may be 
 resolved into the two CE, ED, of which CE is employed 
 in carrying the body parallel to the plane, and must there- 
 fore remain after the impact ; and ED carries the body in 
 the direction ED, perpendicular to the plane; and since 
 the plane is immoveable, this motion will be destroyed 
 during the compression, and an equal motion will be gene- 
 rated in the opposite direction by the force of elasticity.
 
 OF BODIES. 
 
 117 
 
 Hence it appears, that the body at the point D has two 
 motions, one of which would carry it uniformly from D 
 to Ey and the other from E to F, in the same time, viz. 
 in the time in which it described CD before the impact ; 
 it will, therefore, describe DF in that time (Art. 38.). 
 Also, in the triangles CDE, EDF, CE is equal to EF, 
 the side ED is common, and the z CED is equal to the 
 
 z DEE; therefore, the z CDE = the Z EDF; hence, the 
 
 zCDA = thezFDB. 
 
 (223.) CoR. 1. Since CD = DF, and these are spaces 
 uniformly described in equal times, before and after the 
 impact, the velocity of the body after reflection is equal 
 to its velocity before incidence. 
 
 (224.) CoR. 2. If the body and plane be imperfectly 
 elastic, take DE : Doo :: the force of compression : the 
 force of elasticity; draw xf parallel and equal to EF, 
 join F/, Df\ then the two motions which the body has 
 at D are represented by Dx, ^'^Z*? and the body will 
 describe Z>/, after reflection, in the same time that it 
 described CD before incidence; therefore, the velocity 
 
 before incidence : the velocity after reflection :: CD : Df 
 :: DF : Df :: sin. DfF, or sin. of it's supplement EDf : 
 sin. DFf, or sin. FDE :: sin. EDf: sin. EDC. 
 
 * Here we suppose the coirmon surface of the body and plane, 
 during the impact, to remain parallel to AB, in which case there is 
 no cause to accelerate or retard the motion CE (See Art, 116.)
 
 118 
 
 ox THE COLLISION 
 
 Prop. L. 
 
 (225.) Having given the radii of two spherical bodies 
 moving in the same plane, their velocities, and the direc- 
 tions in which they move, to find the plane which touches 
 them both at the point of impact. 
 
 Let AE, BE, meeting in E, be the directions in which 
 the bodies A and B move ; and let AE and BD be spaces 
 uniformly described by them in the same time; complete 
 the parallelogram ABKE; join KD, and with the center 
 E and radius equal to the sum of the radii of the two 
 bodies, describe a circular arc cutting KD in H; join 
 
 EH, and complete the parallelogram EHMR. Then R 
 and M will be the places of the centers of the two spheres 
 when they meet ; and if RC be taken equal to the radius 
 of the sphere A, the plane CL, which is drawn through 
 C perpendicular to MR, will be the plane required. 
 
 - Since MH is parallel to AE or BK, the triangles 
 DMH, DBK, are similar, and BK : BD :: MH : MD ; 
 or AE : BD :: RE : MD; therefore AE : BD :: AR : BM 
 (Euc. 19. V.) ; and since AE and BD are spaces described 
 in the same time by the uniform motions of A and B, AR 
 and BM, which are proportional to them, will be described 
 in the same time ; when, therefore, the center of the body
 
 OF BODIES. 
 
 119 
 
 A is in R, the center of the body B is in M, and the 
 distance MR = HE = the sum of the radii of the bodies ; 
 hence, they will be in contact when they arrive at those 
 points. Also, MR which joins their centers will pass 
 through the point of contact; and LC will be a tangent 
 to them both. 
 
 Pkop. LI. 
 
 (226.) Having given the motions, the quantities of 
 matter, and the radii of two spherical bodies which im- 
 pinge obliquely upon each other, to find their motions 
 after impact. 
 
 Let LN be the plane which touches the bodies at the 
 point of impact ; produce AB, which joins the centers of 
 the bodies, indefinitely both ways; through the centers 
 A and B, draw EAF, GBH, parallel to LN, let CA, DB, 
 represent the velocities of the bodies before impact ; resolve 
 CA into the two CI, I A *, of which CI is parallel, and 
 I A perpendicular to LN', also resolve DB into two, DK 
 parallel to LN, and KB perpendicular to it. Then CA 
 and the angle CAI, which the direction of A\ motion 
 
 c 
 
 
 ^''x. K 
 
 
 
 E ^^ 
 
 ^.f^^'"^^ 
 
 p 
 
 F 
 
 jj V 
 
 
 
 \ 
 
 y S 
 
 
 a 
 
 d/ 
 
 IC 
 
 <i 
 
 makes with AI perpendicular to LN, being known, CI 
 
 * Sec Art. 43.
 
 120 ON THE COLLISION OF BODIES. 
 
 and lA are known; in the same manner, DK and KB 
 are known. Now CI, DK, which are parallel to the plane 
 jLiV, will not be affected by the impact; and lA, KBy 
 which are perpendicular to it, are the velocities with which 
 the bodies impinge directly upon each other, and their 
 effects may be calculated by Prop. 44, when the bodies 
 are perfectly hard ; and by Prop. 47, when they are elastic. 
 Let AR and BS be the velocities of the bodies after im- 
 pact, thus determined; take AF=CI, and BH=iDK\ 
 complete the parallelograms RF, SH and draw the dia- 
 gonals APy BQ; then the bodies will describe the lines 
 AP, BQ, after impact, and in the same time that they 
 describe CA, DB, before impact.
 
 SECTION VII 
 
 ON THE 
 
 RECTILINEAR MOTIONS OF BODIES 
 
 ACCELERATED or RETARDED 
 
 BY UNIFORM FORCES. 
 
 Prop. LII. 
 
 (227.) If a body be impelled in a right line by an 
 uniform force^ the velocity communicated to it is pro-, 
 portional to the time of ifs motion*. 
 
 The accelerating force is measured by the velocity 
 uniformly generated in a given time (Art. 21.), and in 
 this case, the force i« invariable, by the supposition ; 
 therefore, equal increments of velocity are always generated 
 in equal times (Art. 20.) ; and since a body, by the first 
 law of motion, retains the increments of velocity thus 
 communicated to it, if, in the time t, the velocity a be 
 generated, in the time mt the velocity ma is generated; 
 that is, the velocity generated is proportional to the time 
 {Alg. Art. 193.). 
 
 * By force, in this and the following Propositions, we understand 
 the accelerating force, no regard being paid to the quantity of matter 
 moved, unless it be expressly mentioned. Also, the direction in which 
 the force acts, to generate or destroy velocity, is supposed to coincide 
 with the direction of the motion.
 
 122 ON THE RECTILINEAR 
 
 Prop. LIII. 
 
 (228.) If bodies he impelled in right lines by different 
 uniform forces, the velocities generated in any times are 
 proportional to the forces and times jointly. 
 
 Let F and / be the forces, T and t the times of their 
 action, V and v the velocities generated; also, let oo be 
 the velocity generated by the force / in the time T ; then, 
 
 V . X :: F : f (Art. 21.); 
 
 X : V :: T : t (Art. 227-); 
 
 comp. V : V :: FT : ft; that is, the velocities 
 generated are proportional to the forces and times jointly 
 (Alg. Art. 195.). 
 
 Ex. If a force, represented by unity, generate a 
 velocity represented by 2 m, in one second of time, what 
 velocity will the force F generate in T seconds ? 
 
 Since Voc FT, we have 1x1 : FT :: 2m : 2mFT,- 
 the velocity required. 
 
 V 
 (229.) CoR. Since Foe FT, Toe- (^/^. Art. 205.). 
 
 F 
 
 Prop. LIV. 
 
 (230.) If a body'^s motion be retarded by an uniform 
 force, the velocity destroyed in any time is equal to that 
 which would be ge^ierated in the same time, were the 
 motion accelerated by the same force. 
 
 The force impressed is the same, by the supposition, 
 whether the body move in the direction of the force, or 
 in the opposite direction ; therefore, the velocity generated
 
 MOTIONS OF BODIES. 123 
 
 in the former case, is equal to the velocity destroyed, in 
 the same time, in the latter (Art. 29.). 
 
 (231.) Cor. l. Hence, the velocity destroyed by an 
 uniform force is proportional to the time of it's action. 
 
 For, the velocity generated by the action of the force 
 is in that ratio (Art. 227.)- 
 
 (232.) CoR. 2. The velocities destroyed by different 
 uniform forces, are proportional to the forces and times 
 jointly (Art. 228.). 
 
 Prop. LV. 
 
 (233.) If a body he moved through any space, from 
 a state of rest, by the action of an uniform force, and 
 then be projected in the opposite direction with the velocity 
 acquired, and move till that velocity is destroyed, the 
 whole spaces described in the two cases are equal. 
 
 The velocity generated in any time, is equal to the 
 velocity destroyed in the same time by the action of the 
 same force (Art. 230.) ; hence, the whole times of motion, 
 in the two cases, are equal ; also, if equal times be taken, 
 from the beginning of the motion in the former case, and 
 from the end of the motion in the latter, the velocities 
 at those instants are equal. Since then the whole times 
 of motion are equal, and also the velocities at all cor- 
 responding points of time, the whole spaces described are 
 equal. 
 
 Prop. LVI. 
 
 (234.) If a body be moved from a state of rest by 
 the action of an uniform force, the space described, 
 reckoning from the beginning of the motion, varies as 
 the square of the time, or as the square of the last ac- 
 quired velocity.
 
 124 
 
 ON THE RECTILINEAR 
 
 Take AB to represent the time of the body's motion ; 
 draw BC at right angles to AB, and let BC represent 
 
 I) 
 
 \v 
 
 H r 
 
 M 
 
 N 
 
 the last acquired velocity; join AC; divide the time AB 
 into ' small equal portions AD, DE, EF, FG, &c. ; and 
 from the points D, E, F, G, &c. draw DK, EL, FM, 
 GN, &c. parallel to BC, meeting AC in the points K, 
 L, M, N, &c. ; complete the parallelograms DX, EW, 
 FV, GT, &c. 
 
 Then, in the similar triangles ABC, ADK, we have 
 AB : AD :: BC : DK; and since BC represents the velo- 
 city acquired in the time AB, DK represents the velo- 
 city acquired in the time AD (Art. 227.) ; in the same 
 manner it appears, that EL, FM, GN, &c. represent the 
 velocities generated in the times AE, AF, AG, &c. Now, 
 if the body move with the uniform velocity DK, during 
 the time AD, and with the uniform velocities EL, FM, 
 GN, &c. during the times DE, EF, FG, &c. respectively, 
 the spaces described may properly be represented by the 
 rectangles DX, EW, FV, GT, &c. (Art. l6.) ; therefore, 
 the whole space described, on this supposition, will be 
 represented by the sum of the rectangles, or by the triangle 
 ABC, together with the sum of the triangles AXK, KWL, 
 LVM, MTN, &c. that is, because the bases of these small 
 triangles are respectively equal to IB, and the sum of 
 their altitudes is equal to BC, by the triangle ABC, 
 together with half the rectangle BQ. Let the intervals 
 AD, DE, EF, FG, &c. be diminished without limit with 
 respect to AB, and the rectangle BQ, is diminished without 
 limit with respect to the triangle ABC; or ABC -\-^BQ
 
 MOTIONS OF BODIES. 
 
 125 
 
 approaches to ABC as it's limit ; tlierefore, when the 
 motion of the body is constantly accelerated, the space 
 described is represented by the area of the triangle ABC. 
 The space described in any other time AG, reckoning 
 from the beginning of the motion, is represented, on the 
 same scale, by the area of the triangle AGN; and these 
 triangles being similar, the space described in the time 
 AB : the space described in the time AG :: AB'^ : AG'. 
 
 Also, BC, GN, represent the velocities generated in 
 the times AB, AG; and from the same similar triangles, 
 the space described in the time AB : the space described 
 in the time AG :: BC' : GN'. 
 
 (235.) Ex. If a body be accelerated from a state 
 of rest by an uniform force, and describe m feet in the 
 
 first second of time, it will describe 4>m, Qm, l6m, 
 
 mT^feet, in the 2,3,4/, T first seconds. 
 
 (236.) Cor. 1. The space described, reckoning from 
 the beginning of the motion, is half that which would 
 be described in the same time with the last acquired 
 velocity continued uniform. 
 
 Complete the parallelogram BD; then, it appears from 
 the Proposition, that the space described in the time AB, 
 reckoning from the beginning of the motion : the space 
 described in the time IB with the uniform velocity BC 
 
 :: the triangle ABC : BQ. Also, the space described in 
 the time IB, with the uniform velocity BC : the space
 
 126 
 
 ON THE RECTILINEAll 
 
 described in the time AB, with the same uniform velocity 
 :: IB : AB (Art. 13.) :: BQ : BD; and by compounding 
 these two proportions, we have the space described in 
 the time AB, when the body^s motion is accelerated from 
 a state of rest : the space described in the same time 
 with the last acquired velocity continued uniform :: the 
 triangle ABC : the rectangle BD :: 1 : 2*. 
 
 (237.) CoR. 2. The space described in the time GB 
 is represented by the area GBCN; or, if NM be drawn 
 parallel to GB, by the rectangle GM together with the 
 triangle NMC Now, GM represents the space which a 
 body would describe in the time GB, with the uniform 
 velocity GN; and the triangle NMC, which is similar 
 to the triangle ABC, represents the space through which 
 the body would be moved from a state of rest, by the 
 action of the force, in the time GB; thus, the space 
 described in any time, when a body is projected in the 
 direction of the force, is equal to the space which it would 
 have described, in that time, with the first velocity con- 
 tinued uniform, together with the space through which 
 it would have been moved from a state of rest, in the 
 same time, by the action of the force. 
 
 (238.) CoR. 3. If a body be projected in a direction 
 opposite to that in which the uniform force acts, with 
 the velocity BC, and move till that velocity is destroyed. 
 
 • This proof has been misunderstood ; it amounts to this : 
 The rectangle BQ, represents the space uniformly described, with 
 the velocity BC, in the time BI, on the same scale that the triangle 
 ABC represents the space through which the body is drawn, by the 
 action of the uniform force, in the time AB; and also, on the same 
 scale that DB represents the space uniformly described in the time 
 AB, with the velocity BC; consequently, the spaces described, when 
 the body's motion is accelerated from rest for the time AB, and when 
 the velocity BC remains uniform for the same time, are represented, 
 on the same scale, by the triangle ABC and the rectangle BD.
 
 MOTIONS OF BODIES. 127 
 
 the whole time of its motion is represented by BA, 
 (Art. 230.), and the space described by the area ABC 
 (Art. 233.). 
 
 Also, the space described in the time BG is repre- 
 sented, on the same scale, by the area BGNC that is, 
 by the rectangle BL diminished by the triangle CLN, 
 or CNM. Thus it appears, that the space described 
 in the time BG, is equal to that which would have been 
 described with the first velocity continued uniform during 
 that time, diminished by the space through which the 
 body would have been moved from a state of rest, in 
 the same time, by the action of the uniform force. 
 
 Prop. LVII. 
 
 (239.) When bodies are put in motion by uniform 
 forces, the spaces described in any times, reckoning from 
 the beginning of the motion in each case, are proportional 
 to the times and last acquired velocities jointly. 
 
 Let S and s be the spaces described in the times T 
 and #; V and v the velocities acquired; then 2S and 9.S 
 are the spaces which would be described in the times 
 T and t, with the uniform velocities V and v (Art. 236.) ; 
 and the spaces described with uniform velocities are pro- 
 portional to the times and velocities jointly (Art. 14.) ; 
 hence, 
 
 2S : 2s :: TV : tv, 
 
 or A^ : s :: TV : tv (Alg. Art. 184.); 
 
 that is, SocTV (Alg. Art. 195.). 
 
 (240.) Cor. Hence, the times vary as the spaces 
 directly, and the last acquired velocities inversely.
 
 128 ON THE RECTILINEAR 
 
 Prop. LVIII. 
 
 (241.) The spaces described, reckoning from the be- 
 ginning of the motions, vary also as the forces and squares 
 of the times ; or as the squares of the velocities directly ^ 
 and the forces inversely. 
 
 In general, So^TV (Art. 239.); and VocFT (Art. 
 228.) ; hence, TV oc FT^ {Alg. Art. 203.) ; therefore, 
 
 S<x.FT\ Also, Toe-; therefore, TFoc4\ and 
 
 i' F 
 
 V- 
 consequently, S oc -— . 
 F 
 
 (242.) CoR. 1. If T be given, SgcF; that is, the 
 spaces described in equal times, by bodies which are put 
 in motion by uniform forces, are proportional to those 
 forces. 
 
 jr- 
 (243.) Cor. 2. Since ^oc—, we have V" oc FS 
 
 F 
 
 (Alg. Art. 203.) ; that is, the squares of the velocities 
 
 communicated are proportional to the forces and spaces 
 
 described jointly. 
 
 (244.) CoR. 3. If V be given, aS'oc^;- 
 
 F 
 
 (245.) CoR. 4. When bodies in motion are retarded 
 by uniform forces, and move till their whole velocities 
 are destroyed, the spaces described vary as the forces 
 and squares of the times; or, as the squares of the first 
 velocities directly and the forces inversely. 
 
 For, the time in which any velocity is destroyed, is 
 equal to the time in which it would be generated by the 
 same force ; also, the spaces described, on supposition 
 that the body in the latter case is moved from a state
 
 MOTIONS OF BODIES, 129 
 
 ©f rest, are equal (Art. 233.) ; therefore, the same expres- 
 sions which represent the relations of the forces, spaces, 
 times, and velocities, in accelerated motions, represent 
 them also when the motions are retarded, and the bodies 
 move till their whole velocities are destroyed. 
 
 Thus, when equal bodies are made to impinge upon 
 banks of earth, sand, &c. where the retarding forces are 
 invariable, the depths to which they sink, or the whole 
 spaces described, are as the squares of the first velocities 
 directly and the forces inversely; and the resisting forces 
 are as the squares of the first velocities directly and the 
 spaces inversely. 
 
 Prop. LIX. 
 
 (246.) If a body be moved from, a state of rest by 
 the action of an uniform force, the spaces described in 
 equal successive portions of time, reckoned from the be- 
 ginning of the motion, are as the odd numbers 1, 3, 5, 7, 
 9, &c. 
 
 If m be the space described in the first portion of 
 time, 4 m will be the space described in the first two 
 portions (Art. 235.); therefore, 4m — m, or 3m, will be 
 the space described in the second portion alone. Also, 
 9 m will be the space described in the first three portions 
 of time, and consequently, 9 m --4 m, or 5 m, will be the 
 space described, in the third portion, &c. Thus the spaces 
 described in the equal successive portions of time, are 
 m, 3m, 5 m, 7m, 9m, &c. which are as the odd numbers 
 1, 3, 5, 7, 9, &c. 
 
 (247.) CoR. When a. body is retarded by an in- 
 variable force, the spaces described in equal portions of 
 time, reckoning from the end of the motion, are as the 
 odd numbers 1, 3, 5, 7, 9? &c. 
 
 I
 
 130 ON THE llECTILINEAR 
 
 For, when a body moves till it's whole motion is de- 
 stroyed by an uniform force, the space described in any 
 time is equal to that which would be described in the 
 corresponding time, were the body moved from a state of 
 rest bv the action of the same force (See Art. 233.). 
 
 Prop. LX. 
 
 (248.) The force of gravity, at any given place, is 
 an uniform force, which ahvays acts in a direction 
 perpendicular to the horizon, and accelerates all bodies 
 equally. 
 
 The same body will, by it's gravity, always produce 
 the same effect under the same circumstances; thus, it will, 
 at the same place, bend the same spring in the same degree; 
 it will also fall through the same space in the same time, if 
 the resistance of the air be removed ; therefore, the force 
 of gravity is uniform. Also, all bodies which fall freely 
 by this force, descend in lines perpendicular to the horizon ; 
 and, in an exhausted receiver, they all fall through the 
 same space in the same time; consequently, gravity acts 
 in a direction perpendicular to the horizon (Art. 29.), and 
 accelerates all bodies equally (Art. 242.). 
 
 It is found by experiments made on the descent of 
 heavy bodies, and on the oscillations of bodies in small 
 circular arcs (Art. 302.), that every body which falls freely 
 in vacuo by the force of gravity, descends from rest through 
 l6^ feet in one second of time. 
 
 This fact being established, every thing relative to the 
 descent of bodies when they are accelerated by the force 
 of gravity, and to their ascent when they are retarded by 
 that force, supposing the motions to be in vacuo, may he 
 deduced from the foregoing Propositions.
 
 MOTIONS OF BODIES. 
 
 131 
 
 1st. When a body falls by the force of gravity, the 
 velocity acquired in any time, as T'\ is such as would carry 
 it uniformly over 2m T feet in l'' ; where m^=lQ^. 
 
 Since a body falls l6^ feet in l", it acquires a velocity 
 which would carry it uniformly through 32| feet in l" 
 (Art. 236.); and when a body is accelerated by a given 
 invariable force, the velocity generated is proportional to 
 the time (Art. 226.); therefore, l" : T" :: 32^ : 32| T, the 
 velocity acquired in T" ; that is, the velocity acquired is 
 such as would carry the body uniformly over 32^ T feet 
 in l". Let V be the velocity acquired, and m=l6^, then 
 F=2mT. 
 
 2d. The space fallen through in T\ reckoned from 
 the beginning of the motion, is m T^ feet. 
 
 For ^oc T' (Art. 234.) ; therefore, 1" : T^ :: m : mT~, 
 the space described in T". That is, S — m T". 
 
 Ex. 1. In S" a body falls 9w^, or 9 x l6^ = 144|: feet. 
 Ex. 2. In ^' a body falls - , or l6^ x J = 4,^ feet. 
 
 3d. The space fallen through to acquire the velocity F, 
 
 is — feet. 
 4m 
 
 For, aS'ocT' (Art. 234.); therefore, (2 m)'' : V- v. m '. S, 
 
 m\d S = - — feet. 
 
 4 m 
 
 Ex. If a body fall from rest till it has acquired a 
 velocity of 20 feet per second, the space fallen through 
 
 .20x20 
 
 as :; — , or 6.21 feet, nearly. 
 
 64 ' 
 
 I 2
 
 132 ox THE RECTILINEAR 
 
 From the three preceding expressions, V = 2mT'^ 
 
 V" 
 S = 7nT-; and S = — ; any one of the quantities S, 
 4<m 
 
 T, F, being given, the other two may be found. 
 
 Ex. To find the time in which a body will fall 90 feet ; 
 and the velocity acquired. 
 
 Since S = mT% T-=-, and T=\/ -; in this 
 m m 
 
 T= y — p =2.36 seconds, nearly. 
 
 case _ 
 
 16 
 
 12 
 
 V- , 
 
 Also, *S'= — ; therefore, V=2^mS; in this case, 
 4>m ^ 
 
 V=2^ l6i X 90 = 76 feet per second, nearly. 
 
 4th. If a body fall from rest by the force of gravity, 
 the spaces described in any equal successive portions of 
 time, reckoning from the beginning of the motion, are 
 as the numbers 1, 3, 5, 7, &c. Thus, the spaces fallen 
 through in the l'^, 2^, 3^ 4*'' seconds, are l6^^, 3 x l6^, 
 5 X l6j^, 7 X l6^ feet, respectively. Also, if a body, 
 projected upwards, move till it''s whole velocity is de- 
 stroyed, the spaces described in equal successive portions 
 of time are as the numbers 1, 3, 5, 7, &c. taken in an 
 inverted order. Thus, if the velocity be wholly destroyed 
 in 4", the spaces described in the 1^^ 2*^, 3^, 4*^ seconds, are 
 7 X l6^, 5 X 16^, 3 X l6j^, 16^ feet, respectively. 
 
 5th. If a body begin to move in the direction of gravity 
 with any velocity, the whole space described in any time is 
 equal to the space through which the first velocity would 
 carry the body, together with the space through which it 
 would fall by the force of gravity in that time (Art. 237.)- 
 
 Ex. If a body be projected perpendicularly downwards, 
 with a velocity of 20 feet per second, to find the space 
 described in 4".
 
 MOTIONS OF BODIES. 133 
 
 The space described in 4", with the first velocity, is 
 4 X 20, or 80 feet ; and the space fallen through in 4", 
 by the action of gravity, is l6^ x l6, or 257^ feet; there- 
 fore, the whole space described is 80 + 257y, or 337 j feet. 
 
 6th. If a body be projected perpendicularly upwards, 
 the height to which it will ascend in any time is equal to 
 the space through which it would move with the first ve- 
 locity continued uniform, diminished by the space through 
 which it would fall by the action of gravity in that time 
 (Art. 238.). 
 
 Ex. I. To what height will a body rise in 3'\ if 
 projected perpendicularly upwards with a velocity of 100 
 feet per second ? 
 
 The space which the body would describe in s", with 
 the first velocity, is 300 feet ; and the space through which 
 the body would fall by the force of gravity in 3" is 16^x9, 
 or 144|: feet; therefore the height required is 300— 144;|, 
 or 155^ feet. 
 
 Ex. 2. If a body be projected perpendicularly upwards 
 with a velocity of 80 feet per second, to find it's place at 
 the end of 6'\ 
 
 The space which would be described in 6", with the 
 first velocity, is 480 feet, and the space fallen through in 
 the same time is 16^ x 36, or 579 feet ; therefore the dis- 
 tance of the body from the point of projection, at the end 
 of 6", is 480 — 579, or — 99 feet. The negative sign 
 shews that the l^ody will be below the point of projection 
 (See Alg. Art. 472.). 
 
 Prop. LXI. 
 
 (249.) The force which accelerates or retards a body's 
 motion upon an inclined plane, is to the force of gravity, 
 as the height of the plane is to ifs length.
 
 134 
 
 ON THE EECTILINEAR 
 
 Let JC be the plane, BC it's base, parallel to the 
 horizon, AB if s perpendicular height, D the place of a 
 body upon it. From the point D draw DE parallel to AB, 
 and take DE to represent the force of gravity; from E 
 draw EF perpendicular to AC Then the whole force 
 DE is equivalent to the two DF, FE, of which FE is 
 
 perpendicular to the plane, and, consequently, is supported 
 by the plane's reaction (Art. Il6.); the other force DF, 
 not being affected by the plane, is wholly employed in 
 accelerating or retarding the motion of the body in the 
 direction of the plane; therefore, the accelerating force 
 : the force of gravity :: DF : DE :: (from the similar 
 triangles DEF, ABC) AB : AC. 
 
 (250.) Con. 1. Since the accelerating force, on the 
 same plane, is in a given ratio to the force of gravity, it 
 is an uniform force. 
 
 (251.) CoE. 2. If H be the height of an inclined 
 plane, L it's length, and the force of gravity be represented 
 by unity, the accelerating force on the inclined plane is 
 
 represented by — . 
 
 For, the accelerating force : the force of gravity (1) 
 
 IT 
 
 :: H : L; therefore the accelerating force = — . 
 
 (252.) CoR. 3. Since H : L :: the sine of the plane's 
 jr 
 
 inclination : the radius, — , or the accelerating force, varies 
 as the sine of the plane's inclination to the horizon.
 
 MOTIONS OF BODIES. 135 
 
 (253.) Cor. 4. If a body fall down an inclined plane, 
 the velocity V, generated in T'\ is such as would carry it 
 
 TT 
 
 uniformly over — x2mT feet in l"; where m= l6^. 
 
 In general, Foe/' T (Art. 228.); therefore, the velocity 
 acquired when a body falls by the force of gravity : the 
 velocity acquired on the inclined plane :: the product of 
 the numbers which represent the force and time in the 
 former case : the product of the numbers which represent 
 them in the latter*; also, the force of gravity being repre- 
 sented by unity, the accelerating force upon the plane 
 
 TT 
 
 is — , and the velocity generated by the force of gravity 
 
 TT 
 
 in \" is 2m; therefore, 2m : V :: 1 x 1 : ~r x T \ and 
 
 /. 
 
 TT 
 
 — x2mTt. 
 
 Ex. Thus, if the length of an inclined plane be twice 
 as great as it"'s height, a body which falls down this plane 
 will, in 3", acquire a velocity of ^ x 32^ x 3, or 48^; feet 
 per second. 
 
 (254.) Cor. 5. The space fallen through in T'\ from 
 
 TT 
 
 a state of rest, is — x mT^^ feet. 
 
 x> 
 
 In general, SocFT" (Art. 241.); therefore, the space 
 through which a body falls by the action of gravity in 
 l" : the space through which it falls down the inclined 
 plane in T" :: the product of the numbers which represent 
 the force and square of the time' in the former case : the 
 
 * See Note, page 11. 
 
 t In this, and the following Articles, the planes are supposed to be 
 perfectly smooth, and the resistance of the air inconsiderable.
 
 136 ON THE RECTILINEAR 
 
 product of the numbers which represent them in the latter ; 
 or, if S be the space described upon the plane, 
 
 m ' S :: ixl'' : ^x T\ and ^ = ^ x mT\ 
 
 Li Li 
 
 Ex. 1. If L = 2jff, the space through which a body 
 falls in 3" is ^ x l6^ x 9, or 72§ feet. 
 
 Ex. 2. To find the time in which a body will descend 
 12 feet down this plane. 
 
 H L X S 
 
 Since ^ = - x mT^ T^ = = (in this case) 
 
 L H X m 
 
 I X 12 X --r = 1.49; and T = 1.2, nearly. 
 l6— 
 
 (255.) Cor. 6. The space through which a body 
 must fall, from a state of rest, to acquire the velocity V, 
 
 L V ^ 
 IS — X — feet. 
 H 4W 
 
 In general, S oc —- (Art. 241.); therefore, the space 
 
 through which the body falls by the force of gravity : 
 
 V" . 
 the space through which it falls down the plane :: — in 
 
 72 
 
 the former case : — in the latter; and if m (l6^) be the 
 F^ 
 
 space fallen through by the action of gravity, 2 m is the 
 velocity acquired ; hence, 
 
 1 H H 4^m 
 
 Ex. 1. If Z = 2/f, and a body fall from a state of 
 
 rest till it has acquired a velocity of 30 feet per second, 
 
 2 900 
 the space described is - x — r = 27-97 feet, nearly. 
 1 64i
 
 MOTIONS OF BODIES. 137 
 
 Ex.2. If a body fall 12 feet from a state of rest 
 down this plane, to find the velocity acquired. 
 
 T V~ J-f 
 
 Since 6^ = — x , we have V^^^mS x — = (in 
 
 this case) 64^ x 12 x ^ = 386 ; hence, V = 19.6 feet per 
 second, nearly. 
 
 CoR. 7. In the same manner, if a body be acted upon 
 by any uniform force, which is to the force of gravity 
 as F : 1, and V represent the velocity generated, T the 
 time in seconds, *S' the space described, in feet, reckoned 
 from the beginning of the motion, then 
 
 V=2mFT; S = mFT^; and V^ = 4.mFS. 
 
 Prop. LXII. 
 
 (256.) The velocity which a body acquires in falling 
 down the whole length of an inclined plane, varies as the 
 square root of the perpendicular height of the plane *. 
 
 In general, when the force is uniform, V^oc FS (Art, 
 
 IT 
 
 243.); in this case. Foe—, and S = L, by the sup- 
 position; therefore, 
 
 V^oc^ xLocH; and Foe ^Jl (Jig. Art. 202 f.). 
 
 (257.) CoR. 1. When the heights of two inclined 
 planes are equal, the velocities acquired in falling down 
 their whole lengths are equal. 
 
 (258.) CoR. 2. The velocity which a body acquires 
 in falling down the length of an inclined plane is equal 
 
 * Bodies, in this, and the subsequent Propositions, are supposed to 
 fall from a state of rest. 
 
 t See also Cor. 6. Ex. 2. of the last Proposition.
 
 138 ON THE RECTILIKKAIi 
 
 to the velocity which it would acquire in falling down 
 it's perpendicular height. 
 
 Pkop. LXIII. 
 
 (259.) The time of a body's descent down the whole 
 length of an inclined plane, varies as the length directly, 
 and as the square root of the perpendicular height in- 
 versely. 
 
 S 
 In general, *S'oc TF (Art. 239.) ; therefore, Toe-; 
 
 and in this case, Foe a^ H (Art. 25^.) ; consequently, 
 
 S L ... 
 
 Toe 
 
 ^h""^ 
 
 (260.) Cor. l. If the height, or the last acquired 
 velocity, be given, TocL. 
 
 (261.) Cor. 2. If the inclination be given, or HocL, 
 
 then T^oe— ocL, and T oc ^ L. That is, the times 
 
 of descent, down planes equally inclined to the horizon, 
 vary as the square roots of their lengths. 
 
 (262.) Cor. 3. The time of descent down an inclined 
 plane, is to the time of falling down it's perpendicular 
 height, as the length of the plane, to it's height. 
 
 Prop. LXIV. 
 
 (263.) If chords be drawn in a circle from the ex- 
 tremity of that diameter which is perpendicular to the 
 horizon, the velocities which bodies acquire by falling 
 down them are proportional to their lengths ; and the 
 times of descent are equal. 
 
 * Sec also Art. 2o\. Ex. 2.
 
 MOTIONS OF BODIES. 
 
 139 
 
 Let ACB be the circle, AB it's diameter perpendicular 
 to the horizon; BC a chord drawn from the extremity B 
 of the diameter; join AC, and draw CD perpendicular 
 
 to AB, or parallel to the horizon. Then CB may be 
 considered as an inclined plane whose perpendicular height 
 is DB, and the velocity acquired in falling down it varies 
 as sj DB (Art. 256.). Now, from the similar triangles 
 DBC, ABC, DB : CB V. CB : AB', therefore, 
 
 DB = 
 
 AB 
 
 CB 
 
 ^»^Y^^ = 7^^ 
 
 CB 
 
 consequently, Foe — ;=; and AB is invariable, there- 
 sjAB 
 
 fore Foe CB. 
 
 S 
 Again, Tec- (Art. 240.), and in this case, CB, which 
 
 is the space described, has been proved to be proportional 
 
 CB 
 
 to the velocity acquired ; therefore T oc , or the time 
 
 CB 
 
 of descent is invariable. 
 
 (264.) Con. 1. The time of descent down any chord 
 CB, is equal to the time of descent down the diameter AB. 
 
 (265.) Cor. 2. In the same manner, the time of 
 descent down ^C is equal to the time of descent down
 
 140 ON THE RECTILINEAR 
 
 AB; therefore the time of descent down AC is equal to the 
 time of descent down CB. 
 
 (266.) CoR. 3. The times of descent down the chords 
 thus drawn, in different circles, are proportional to the 
 square roots of the diameters. 
 
 For, the times of descent down the chords are equal to 
 the times of descent down the diameters which are perpen- 
 dicular to the horizon ; and these times vary as the square 
 roots of the diameters. (See Art. 234.). 
 
 (2670 When a body falls freely by the force of gra- 
 vity, every particle in it is equally accelerated ; that is, 
 every particle descends towards the horizon with the same 
 velocity ; in this descent, therefore, no rotation will be 
 given to the body. The same may be said when a body 
 descends along a perfectly smooth inclined plane, if that 
 part of the force which acts in a direction perpendicular to 
 the plane (Art. 249.), be supported; that is, if a perpendicular 
 to the plane, drawn from the center of gravity of the body, 
 cut the plane in a point which is in contact with the body. If 
 this part of the force be not sustained by the plane, the body 
 will partly roll and partly slide, till this force is sustained ; 
 and afterwards the body will wholly slide. When the late- 
 ral motion is entirely prevented by the adhesion of the body 
 to the plane, we have before seen on what supposition the 
 body will roll (Art. 186.); if the adhesion be not sufficient 
 to prevent all lateral motion, this body will partly slide 
 and partly roll; and to estimate the space described, the 
 time of it's motion, or the velocity acquired, we must have 
 recourse to other principles than those above laid down. 
 On this subject the Reader may consult Professor Vince's 
 Plan of a Course of Lectures, p. 39. 
 
 (268.) When a body falls freely by the force of gravity^ 
 or descends along a perfectly smooth inclined plane, the ac- 
 celerating force is the same, whatever be the weight of the 
 body (Arts. 248, 249.); consequently, the moving force, on
 
 MOTIONS OF BODIES. 141 
 
 either supposition, is proportional to the quantity of matter 
 moved. In all cases, the accelerating force varies as the 
 moving force directly and the quantity of matter inversely 
 (Art. 24.) ; and when the moving force and quantity of 
 matter moved are invariable, the accelerating force is uni- 
 form, and it's effects may be estimated by the rules laid 
 down in the first part of this section. 
 
 Ex. If two bodies, whose weights are P and Q, be 
 connected by a string, and hung over a fixed pulley, to 
 find how far the heavier P will descend in T" . 
 
 The moving force of gravity is proportional to the 
 weight ; if therefore P be taken to represent the moving 
 force of the former body when it descends freely, Q will 
 represent the moving force of the latter, and P—Q will 
 represent the moving force when the bodies are connected 
 and oppose each other's motion ; hence, neglecting the 
 inertia of the string and pulley, the accelerating force of 
 gravity : the accelerating force in this case 
 
 • 16^ X ^""^ X T^ 
 
 
 
 P 
 
 P-Q 
 
 
 
 '''' P '' 
 
 P+Q " 
 
 and. 
 
 , since 
 
 FT' oc S, 
 
 
 
 1X1^ 
 
 P-Q 
 
 r :: I6i 
 
 the 
 
 space 
 
 required. 

 
 SECTION VIII. 
 
 ON THE 
 
 OSCILLATIONS OF BODIES IN CYCLOIDS 
 
 AND IN 
 
 SMALL CIRCULAR ARCS. 
 
 Prop. LXV. 
 
 (269.) ^F a body descend down a system of inclined 
 fplanes, the velocity acquired, on the supposition that no 
 motion is lost in passing from one plane to another , is 
 equal to that which would be acquired in falling through 
 the perpendicular height of the system. 
 
 Let ABCD be the system of planes ; draw AE, DF, 
 
 AG E 
 
 parallel to the horizon ; produce CB, DC, till they meet
 
 ON THE OSCILLATIONS OF BODIES. 143 
 
 AE in G and E; and draw EF perpendicular to DF. 
 Then the velocity acquired by a body in falling from A 
 to B, is equal to that which it would acquire in falling 
 from G to B, because the planes AB, GB, have the same 
 perpendicular height (Art. 257.) ; and since, by the sup- 
 position, no velocity is lost in passing from one plane to 
 another, the body will begin to descend down BC with the 
 same velocity, whether it fall down AB or GB ; conse- 
 quently, the velocity acquired at C will be the same on 
 either supposition. Also, the velocity acquired at C is 
 equal to that which would be acquired in falling down 
 EC (Art. 257.); and no velocity being lost at C, the body 
 will begin to descend down CD with the same velocity, 
 whether it fall from A through B and C to D, or from E 
 to D; and the velocity acquired in falling down ED is 
 equal to the velocity acquired in falling through the per- 
 pendicular height EF (Art. 258.); therefore, the velocity 
 acquired in falling down the whole system, is equal to the 
 velocity acquired in falling through the perpendicular 
 height of the system. 
 
 Prop. LXVI. 
 
 (270.) If a body fall from a state of rest down a 
 curve surface which is perfectly smooth, the velocity ac- 
 quired is equal to that which would he acquired in falling 
 from rest through the same perpendicular height. 
 
 When a body passes from one plane AB to another 
 5C, the whole velocity : the quantity by which the velocity 
 is diminished :: radius : the versed sine of the Z ABG 
 (Art. 221.); when, therefore, the angle ABG is diminished 
 without limit, the velocity lost is diminished without limit ; 
 and if the lengths of the planes, as well as their angles of 
 inclination ABG, BCE, be continually diminished, the 
 system approximates to a curve, as it's limit, in which no 
 velocity is lost ; consequently, the whole velocity acquired
 
 144 ON THE OSCILLATIONS 
 
 is equal to that which a body would acquire in falling 
 through the same perpendicular altitude (Art. 269*-). 
 
 (271.) Cor. 1. If a body be projected up a curve, 
 the perpendicular height to which it will rise is equal to 
 that through which it must fall to acquire the velocity of 
 projection. 
 
 For the body in if s ascent will be retarded by the same 
 degrees that it was accelerated in ifs descent. 
 
 (272.) Cor; 2. If BAb be a curve in which the 
 
 A 
 
 lowest point is A, and the parts AB, Ab, are similar and 
 
 * When the chord of an arc is diminished without limit with respect 
 to the diameter, the versed sine is diminished without limit with respect 
 to the chord ; because, the diameter : the chord :: the chord : the versed 
 sine; hence, the ratio of the diameter to the versed sine, and consequently, 
 the ratio of the radius to the versed sine, is, in this case, indefinitely greater 
 than the ratio of the diameter to the chord. Let BC be one of the evan- 
 escent planes, F the velocity of the descending body at B, V+v it's 
 velocity at C ; produce CB to G, and let GB be the space through which 
 the body must descend to acquire the velocity V; then, 
 
 V : F+v :: \/GB : \/gbTBC ; 
 and when GB : BC :: the radius : an evanescent chord, 
 
 V: F^v :: GB : G5 + ^ (see Note, p. 112.); 
 therefore, V : v :: GB : ^ :: 2GB : BC. 
 
 Also, F : the velocity lost at 5 :: radius : the versed sine of the angle 
 ABG. Hence it follows, that the ratio of F to the velocity lost at B, is 
 indefinitely greater than the ratio F to the velocity acquired in the descent 
 from B to C; and consequently, the velocity lost at B is indefinitely less 
 than the velocity acquired in the descent from B to C; in the same man- 
 ner, the velocity lost at any other plane is indefinitely less than the 
 velocity acquired in the descent down that plane ; therefore, the velocity 
 lost in the whole descent is indefinitely less than the whole velocity 
 acquired.
 
 OF BODIES. 
 
 145 
 
 equal, a body in falling down BA will acquire a velocity 
 which will carry it to 6; and since the velocities at all 
 equal altitudes in the ascent and descent are equal, the 
 whole time of the ascent will be equal to the time of 
 descent. 
 
 (273.) CoR. 3. The same proposition is true, if the 
 body be retained in the curve by a string which is in every 
 point perpendicular to it. For the string will now sustain 
 that part of the weight which was before sustained by the 
 curve (Art. 117.). 
 
 Prop. LXVII. 
 
 (274.) The times of descent down similar systems of 
 inclined plajies, similarly situated, are as the square roots 
 of their lengths, on the supposition that no velocity is lost 
 in passing from one plane to another. 
 
 Let ABCD, abed, be two similar systems of inclined 
 planes, similarly situated; that is, let AB : ah :: BC : bo 
 :: CD : cd; the angles ABC, BCD, respectively equal to 
 the angles abc, bed; and the planes AB, ah, equally in- 
 clined to the horizon. Complete the figures as in the last 
 
 Proposition; then, since AB : ah :: BC : he :: CD : cd, 
 AB : ah :: AB -\- BC + CD : ah + he -h cd (A lg. Art. 183.); 
 and >/^ : ^^ :: ^ABi-BC + CD : ^ab + bc-^-cd. 
 
 K
 
 146 ON THE OSCILL AXIOMS 
 
 Also, since the angles ABC, abc, are equal, their sup- 
 plements, the angles ABG, cibg, are equal; and the 
 angles of inclination to the horizon BAG, hag, are 
 equal; therefore, the triangles ABG, ahg^ are similar, and 
 AB : BG :: ab : bg; alter. AB : ah :: BG : bg :: BC : be; 
 consequently, BG : bg :: BG + BC (GC) : bg^bc {gc) 
 :: AB : ab. In the same manner, ED : ed :: AB : ab. 
 Then, because the planes AB, ab, are equally inclined to 
 the horizon, the time of descent down AB : the time down 
 ab :: ^ AB : ^ ab (Art. 26l.); and if the bodies fall 
 down GC, gc, the time down GC : the time down gc :: 
 /sj GC : sj gc :: sj AB : sj^h', also, the time down 
 GB : the time down gb :: ^^B : J^ :: >/j^ : ^ab; 
 hence the whole time down GC : the whole time down gc 
 :: the time down GB : the time down gb; therefore, the 
 remainder, the time down BC : the remainder, the time 
 down be, in the same ratio, or as sj AB : yj ab (Euc. 19- v.) ; 
 and since, by the supposition, no motion is lost is passing 
 from one plane to another, the times of descent down BC 
 and be are the same, whether the bodies descend from A 
 and a, or from G and g; consequently, when the bodies 
 descend down the systems, the time down BC : the time 
 down be :: sj AB : a^ ab. In the same manner it may be 
 shewn that the time down CD : the time down cd:: /y/^ZS : 
 »^ ah. Hence, the time down AB : the time down ab :: 
 the time down BC : the time down be :: the time down 
 CD : the time down cd; therefore, the time down AB + 
 BC + CD : the time down ab-^bc + cd :: the time down AB 
 : the time down ab :: tj AB -. ;^ ah (Algebra, Art. 183.) :: 
 ^AB-hBC-hCD: Jab^he + cd. 
 
 (275.) CoR. 1. If the lengths of the planes, and their 
 angles of inclination ABG, ACE, he be continually di- 
 minished, the limits, to which these systems approximate, 
 are similar curves, similarly situated, in which no velocity 
 is lost (Art. 270.) ; hence, the whole times of descent down 
 these curves vill be as the square roots of their lengths.
 
 OF BODIES. 
 
 147 
 
 (276.) Cor. 2. The times of descent down similar 
 circular arcs, similarly situated, are as the square roots of 
 the arcs, or as the square roots of their radii. 
 
 (277-) Def. If a circle be made to roll in a given 
 
 
 H 
 
 
 L 
 
 
 
 ^ 
 
 -^ 
 
 ^^ 
 
 3t ^N/"^^^^-^.^^^ 
 
 \ 
 
 cY 
 
 
 
 { Y 
 
 ' / 
 
 ^\ 
 
 A 
 
 I 
 
 
 GW 
 
 \ 
 
 / \ 
 
 ^--^_ 
 
 
 y\^ 
 
 y 
 
 \ 
 
 \ 
 
 
 r^ 
 
 
 ^ 
 
 r 
 
 plane upon a straight line AB, the point C in the circum- 
 ference, which was in contact with AB at the beginning of 
 the motion, will, in a revolution of the circle, describe a 
 curve ACEB called a cycloid. 
 
 The line AB is called the base of the cycloid. 
 
 The circle HCD is called the generating circle. 
 
 The line FE., which is drawn bisecting AB at right 
 angles, and produced till it meets the curve in E, is called 
 the aoois^ and the point E, the vertex, of the cycloid. 
 
 (278.) CoR. 1. The base AB is equal to the circum- 
 ference of the generating circle; and AF to half the 
 circumference. 
 
 (279.) Cor. 2. The axis FE is equal to the diameter 
 of the generating circle. 
 
 When the generating circle comes to F, draw the di- 
 ameter Fob\ which will be perpendicular to AB (Euc. 18. iii.); 
 and because the circle has completed half a revolution, x is 
 the generating point ; that is, w is a point in the cycloid, or 
 X coincides with E. 
 
 K 2
 
 U8 
 
 ox THE OSCILLATION'S 
 
 Prop LXVIIL 
 
 (280.) If a line CGK, drawn from a point C in the 
 
 cycloid^ parallel to the base AB, meet the generaimg circle, 
 described upon the axis, in G, the circular arc 'EG is equal 
 to the right line CG, 
 
 Let the generating circle HCD touch the base in D 
 when the generating point is at C ; draw DH perpendicular 
 to AB, and it will be the diameter of the circle HCD 
 (Euc. 19. iii.), and therefore equal to FE ; join CH, GE; 
 and since DH = FE, and DI = FK (Euc. 34. i.), the 
 remainders IH and KE are equal; consequently, CI, 
 which is a mean proportional between HI and ID (Euc. 
 Cor. 8. vi.), is equal to KG, which is a mean proportional 
 
 
 tL 
 
 
 
 
 / 
 
 
 ^ 
 
 ~^>C'''/^^^/ 
 
 ■"v^ 
 
 \ 
 
 ( / 
 
 
 
 w 
 
 'J 
 
 N. 
 
 A ^ 
 
 
 \ 
 
 ^v 
 
 .1 
 
 
 ] 
 
 ? 
 
 B 
 
 between EK and .ff^F; to each of these equals add IG, 
 and CG = IK. Also, C/f, which is a mean proportional 
 between IH and HD, is equal to GE, which is a mean 
 proportional between EK and -EF; therefore, the arc 
 C-H'=the arc GE (Euc. 28. iii.); and since every point 
 in CD has been successively in contact with AD, CD = 
 AD, and HCD = AF (Art. 278.) ; hence, the arc CH 
 = DF; therefore, the arc EG = DF=IK=CG. 
 
 CoR. If the line KGC be always drawn perpendicular 
 to EF the diameter of the circle EGF, and GC be taken 
 equal to the arc EG, the locus of the point C is a cycloid, 
 whose axis is EF.
 
 OF BODIES, 
 
 149 
 
 Proi', LXIX. 
 
 (281.) If a line LM, drawn from L parallel to the 
 base AB, meet the generating circle described upon the 
 axis in M, and EM />e joined, the tangent to the cycloid 
 at the point L is parallel to the chord EM. 
 
 Draw SR parallel and indefinitely near to ZJ/; join 
 EM, RM, SL ; produce EM till it meets SR in P; draw 
 EN, MN, touching the circle in E and M, and meeting 
 each other in N, 
 
 Then, since RM is ultimately in the direction of the 
 
 X W 
 
 tangent MN (Newt. Lem. 6.), the angles RMP, EMN, are 
 equal ; and because EN is parallel to RS (Euc. 18. iii.) the 
 angles MPR^ MEN, are equal ; hence the triangles EMN, 
 RMP, are equi-angular, and EN : MN :: RP : RM ; 
 and since EN=NM, RP = RM=ihe arc RM (Newt. 
 Lem. 7.). Again, since the arc EMR = RS (Art. 280.), 
 and RM = RP, the remainders, the arc EM and the 
 right line PS, are equal; also, ML— the arc EM:, there- 
 fore, PS = ML; consequently, SL is equal, and parallel 
 to PM (Euc, 33, i.*); and since SL is ultimately in the 
 direction of the tangent at L (Newt, Lem, 6,), MP^ or 
 EM, is parallel to the tangent at L. 
 
 (282.) CoR. The tangent to the cycloid at B or A, 
 is perpendicular to AB, 
 
 * The Proposition may be justly applied, because the fliffcrcnce 
 between LM and SP h cvancycent with respect to Mi% or LS.
 
 15Q OK THE OSCILLATIONS 
 
 Prop. LXX. 
 
 (283.) The same construction being made, the cycloidal 
 arc EL zs double 0/ EM the corresponding chord of the 
 generating circle described upon the ams. 
 
 Join ER, and in EP take Eo=zER% join Ro. Then, 
 when the arc MR, and consequently the angle MER, is 
 diminished without limit, the sum of the angles ERo, 
 EoR, approximates to two right angles as it's limit; and 
 these angles are equal to each other ; therefore, each of 
 them is a right angle; and since the angles RMo, RoM, 
 are respectively equal to RPo, RoP, and i?o is common to 
 the triangles RoM, RPo, Mo = oP, and MP = 2Mo; 
 also. Mo (ER — EM) is the quantity by which the chord 
 EM increases, whilst the cycloidal arc EL increases by 
 LS; and it appears from the demonstration of the last 
 Proposition that MP = LS = arc LS (Newt. Lem. 7.)? 
 therefore, the arc LS = 2Mo; or, the cycloidal arc EL 
 increases twice as fast as the corresponding chord EM; 
 and they begin together at E ; consequently, the cycloidal 
 arc EL is double of EM, the corresponding chord of the 
 generating circle. 
 
 (284.) CoR. The whole semi-cycloidal arc EB is 
 equal to twice the axis EE. 
 
 Prop. LXXI. 
 (285.) To make a body oscillate in a given cycloid. 
 
 Let A VB be the given cycloid, placed with it's vertex 
 downwards, and it's axis DV perpendicular to the horizon. 
 Produce VD to C, making DC = VD ; complete the 
 rectangles DE, DF; upon AE describe a semi-circle 
 AGE, and with A as the generating point, and base EC, 
 describe a semi-cycloid ATC ; this will pass through the
 
 OF BODIES. 
 
 151 
 
 point C, because the semi-circumference AGE = DIIV = 
 AD = EC ; in the same manner, describe an equal semi- 
 cycloid between C and B. Then, if a body P be suspended 
 
 'Q 
 
 from C by a string whose length is CF or CTA (Art. 284.), 
 and made to vibrate between the cycloidal cheeks CA, CB, 
 it will always be found in the cycloid AVB. 
 
 Let the string be brought into the situation CTP, and 
 since it is constantly stretched by the gravity, and the 
 centrifugal force of P, it will be a tangent to the cycloid 
 at the point T where it leaves the curve. From T and 
 P draw TGW, PHR, parallel to AD; join AG, GE, 
 DH, HV; and through K draw ,vKy perpendicular to 
 TG or PH. Then, since the chord JG is parallel to TP 
 (Art. 281.), and TG is parallel to AK, the figure GK is 
 a parallelogram, and AG—TK, GT=^AK; and because 
 the length of the string is equal to CTA, and the part CT 
 is common to the string and the cycloidal arc, TP = AT = 
 2AG (Art. 283.) = 2 T^; or TK = KP; hence, the tri- 
 angles TKx, PKy, are similar and equal, and Kcc = Ky ; 
 also, Koo=^AW SiX\& Ky = DR., thereioYe AW=DR, and 
 AE — DV; hence, the arc AG = the arc DH; and the 
 angle GEA = the angle HVD; or, the angle GAK = the 
 angle KDH (Euc. 32. iii.) ; consequently, AG is parallel 
 to DH; and therefore, TP is parallel to DH, and the 
 figure KPHD is a parallelogram ; hence, KD = PH.
 
 152 OK THE OSCILLATIONS 
 
 Again, since the arc AG=GT (Art. 280.) = AK, the arc 
 DH = JK ; and the semi-circumference DHV = AD; 
 therefore, the arc VH=KD = HP; that is, P is in the 
 cycloid, whose axis is DF, and vertex V (Art. 280.). 
 
 (286.) Cor. l. Since DH is parallel to TP, and 
 VH to the tangent at P, the angle contained between TP 
 and tlie tangent, or between TP and the curve, is equal 
 to the angle DHV; that is, TP is always perpendicular 
 to the curve. 
 
 (287-) CoE. 2. If Pp be an evanescent arc, the per- 
 pendiculars to the curve at P and p, ultimately meet in T; 
 and Pp may be considered as a circular arc whose radius is 
 TP. 
 
 (288.) CoR. 3. An evanescent arc at the vertex of 
 the cycloid may be considered as a circular arc whose 
 radius is CV. 
 
 (289.) Def. If a body begin to descend in a curve, 
 from any point, and again ascend till it's velocity is de- 
 stroyed (Art. 272.), the time in which the motion is 
 performed is called the time of an oscillation. 
 
 Prop. LXXII. 
 
 (290.) If a body, vibrating in the cycloid AVB, begin 
 to descend from L, the velocity acquired at any point M 
 varies as ^VL" — VM"; or, as the right sine of a circular 
 arc whose radius is equal to VL, and versed sine to LM. 
 
 From the points L and M, draw LOR, MQS, at right 
 angles to DV, meeting the circle DOV in O and Q; join 
 OV, QV; with the radius VI = VL, describe the semi-circle 
 IZp, and take hn = LM; draw m.r, VZ, at right angles 
 to VI; and join Va;.
 
 OF BODIES. 153 
 
 The velocity acquired in the descent from L to My is 
 
 equal to the velocity acquired in falling from R to S 
 (Art. 270.) ; and therefore it varies a s ^^ (Art. 241.); 
 that is, oc JRV - SV oc ^ DV x RV - DV x SV 
 (because DV is invariable), oc ^ VO^ — VQ^ 
 ^^VO^ -. 4 "rQ^ oc: J VU - VM^ (Art. 283.), 
 J F/2- Vm^ oc ^F^^-Fwi" oc ^^wa?' oc wa?. 
 
 (291.) CoR. The velocity at M : the velocity at 
 V :: mo) : VZ :: mou : Va?. 
 
 oc 
 oc 
 
 Prop. LXXIII. 
 
 (292.) The time of an oscillation in the arc LVP, is 
 equal to the time in which a body would describe the semi- 
 circumference IZp, with the velocity/ acquired at V con- 
 tinued uniform.
 
 154 ON THE OSCILLATIONS 
 
 Let MN be a very small arc, and take mn = MN; 
 draw nt, wr, respectively parallel to mx and VI; and sup- 
 pose a body to describe the circumference IZp with the 
 velocity acquired at V continued uniform. Then, when 
 MN is diminished without limit, the velocity with which 
 it is described : the velocity with which a)t is described 
 :: mo! : VZ (Art. 291.) ; therefore, the time of describing 
 
 ^ , ., . MN xt wr set 
 
 MN : the time 01 describing oct v. : — — :: : — - 
 
 moG VZ mx Vx 
 
 (Art. 15.). Now, the triangles Vxm, xrt are ultimately 
 
 similar, and Vx : mx :: xt : xr; therefore = — - ; 
 
 mx Vx 
 
 consequently, the time of descent down MN, is equal to 
 
 the time of describing the corresponding circular arc xt 
 
 with the velocity VZ ; and the same may be proved of all 
 
 other corresponding arcs in the cycloid and the circle ; 
 
 therefore the whole time of an oscillation is equal to the 
 
 time of describing the semi-circumference IZp, with the 
 
 velocity acquired at V continued uniform. 
 
 Prop. LXXIV. 
 
 (293.) The time of an oscillation in a cycloid is to the 
 time of descent down ifs axis, as the circumference of a 
 circle to ifs diameter. 
 
 If a body fall down the chord V, the velocity acquired 
 at V is equal to the velocity in the cycloid at V (Art. 270.); 
 and with this velocity continued uniform, the body would 
 describe 2 OF, or VL, or VI, in the time of descent down 
 OV (Art. 237.); that is, in the time of descent down DV 
 (Art. 264.). It appears then, that the time of an oscillation 
 is equal to the time of describing IZp with the velocity ac- 
 quired in the cycloid at V (Art. 292.) ; and that the time 
 of descent down the axis DF is equal to the time of de- 
 scribing VI with the same velocity ; therefore, the time of 
 an oscillation : the time of descent down the axis :: the
 
 OF BODIES. 155 
 
 time of describing the circumference IZp, with the velocity 
 
 VZ : the time of describing VI with the same velocity 
 :: IZp : VI (Art. 13.) :: 2lZp : 2Vl :: the circumference 
 of a circle : it's diameter. 
 
 (294.) Cor. 1. The time of an oscillation in a given 
 cycloid, at a given place, is the same, whether the body 
 oscillate in a greater or a smaller arc. 
 
 For, the time of an oscillation bears an invariable ratio 
 to the time of descent down the axis, which, in a given 
 cycloid, at a given place, is given. 
 
 (295.) CoR. 2. The time of an oscillation in a small 
 circular arc whose radius is CF, is to the time of descent 
 down ^ CV, as the circumference of a circle to it's diameter. 
 
 For, the time of an oscillation in this circular arc is 
 equal to the time of an oscillation in an equal arc of the 
 cycloid AVB (iVrt. 288.).
 
 156 ON THE OSCILLATIONS 
 
 (296.) Coil. 3. The time of an oscillation in a cycloid, 
 or small circular arc, when the force of gravity is given, 
 varies as the square root of the length of the string. 
 
 For, the time of an oscillation varies as the time of 
 descent down half the length of the string ; that is, as the 
 square root of half the length of the string, or as the square 
 root of it"'s whole length. 
 
 Ex. 1. To compare the times in which two pendulums 
 vibrate, whose lengths are 4 and 9 inches. 
 
 Since Toe J~L, we have : 7" : ^ :: /4 : I/9 :: 2 : 3. 
 
 Ex. 2. If a pendulum, whose length is 39.2 inches, 
 vibrate in one second, in what time will a pendulum vibrate 
 whose length is L inches "i 
 
 .2 : JL:: I : T = \/Ji 
 
 39.2 
 the time required, in seconds. 
 
 Ex. 3. To compare the lengths of two pendulums, 
 whose times of oscillation are as 1 to 3. 
 
 Since Tocy/% T\ocL; therefore, 1 : 9 -i L : l. 
 
 (297.) CoR. 4. The number of oscillations, which a 
 pendulum makes in a given time, at a given place, varies 
 inversely as the square root of it's length. 
 
 Let n be the number of oscillations, t the time of one 
 oscillation; then, nt is the whole time, which, by the 
 
 supposition, is given; therefore, w«^: i^^g- Art, 206.), 
 and (oc^L, consequently, noc — ^.
 
 OF BODIES. 157 
 
 Ex. 1. If a pendulum, whose length is 39.2 inches, 
 vibrates seconds, or 60 times in a minute, how often will 
 a pendulum whose length is 10 inches vibrate in the same 
 time ? 
 
 Since n oc —,^ , we have 
 
 s/L 
 
 s/^O : ^39'^ '' 60 : 60 X ^3.92 = 118.8, nearly, 
 
 the number of oscillations required. 
 
 Ex. 2. If a pendulum, whose length is 39.2 inches, 
 vibrate seconds, to find the length of a pendulum which 
 will vibrate double seconds, or 30 times in a minute. 
 
 Since n oc -—=^ , we have, -L oc -r, ; ,^nd in this case, 
 
 (30)" : (60)- :: 39.^^ : L = 4 X 39.9. = 156.8 inches, the 
 length required. 
 
 Ex. 3. To find how much the pendulum of a clock, 
 which loses one second in a minute, ought to be shortened. 
 
 Since the pendulum vibrates b9 times, whilst a pen- 
 dulum of 39.2 inches vibrates 60 times, it'*s length may be 
 found as in the last example ; {^9Y '• (^^Y '•' ^9-2 : 40.5, 
 it''s kngth; and it ought to be 39.2 inches; therefore, 
 40.5 — 39.2, or 1.3 inches, is the quantity by which it ought 
 to be shortened, in order that it may vibrate seconds. 
 
 (298.) Cor, 5. If the force of gravity be not given, 
 the time of an oscillation varies as the square root of the 
 length of the pendulum directly, and as the square root 
 of the force of gravity inversely. 
 
 For, the time of an oscillation varies as the time of 
 descent down half the length of the string ; and in general, 
 
 the time of descent through any space oc \/ — (Art. 241 .) ; 
 
 r 
 
 in this case, aS" = J Z. ; therefore S oc L, and the time
 
 158 ON THE OSCILLATIONS 
 
 of descent ex. y ~ ; hence T, the time of an oscillation, 
 
 OC 
 
 sAi 
 
 (299.) Cor. 6. If the length of the pendulum be 
 
 ^1 - „ 1 
 
 given, T OC ; and F oc — . 
 
 The time in which a given pendulum vibrates, increases 
 as it is carried from a greater latitude on the earth's surface 
 to a less ; therefore, the force of gravity decreases as the 
 latitude decreases. 
 
 (300.) Coil. 7. The force of gravity at the equator 
 : the force of gravity at any proposed latitude :: the length 
 of a pendulum which vibrates seconds at the equator : the 
 length of a pendulum which vibrates seconds at the pro- 
 posed latitude. 
 
 For, Toe V - ; if therefore T be given, ^Tc<z^, 
 or F OC L. 
 
 (301.) CoR. 8. If the chord BV* be drawn, the time 
 of descent down the cycloidal arc B V : the time of descent 
 down the chord :: DB : BV. 
 
 For, the time of descent down the arc BV is equal to 
 half the time of an oscillation (Art. 272) ; therefore, the 
 time of descent down the arc BV; the time of descent 
 down DV :: half the circumference of a circle : it's dia- 
 meter :: DB : DV; also, the time of descent down DV 
 : the time of descent down the chord BV :: DV : BV; 
 therefore, ex cEquo, the time of descent down the arc B V 
 : the time of descent down the chord :: DB : BV. 
 
 The line BF is wanting in the figure.
 
 OF EODIES. 159 
 
 Prop. LXXV. 
 
 (302.) The space through which a body falls by the 
 force of gravity in the time of an oscillation in a cycloid, 
 or small circular arc, is to half the length of the pendulum, 
 as the square of the circumference of a circle to the square 
 of ifs diameter. 
 
 The spaces through which bodies fall by the action of 
 the same uniform force are as the squares of the times 
 (Art. 241.) ; and since the time of an oscillation : the time 
 of descent down half the length of the pendulum :: the 
 circumference of a circle : it^s diameter, the space fallen 
 through in the time of an oscillation : half the length of 
 the pendulum :: the square of the circumference of a circle 
 : the square of it's diameter. 
 
 Ex. To find how far a body will fall by the force of 
 gravity in one second, where the length of the pendulum, 
 which vibrates seconds, is 39.2 inches. 
 
 The circumference of a circle : it's diameter :: 3.14<159 
 
 : 1 ; consequently, the space fallen through in one second 
 
 30.2 
 : :: (3.14159)^ : 1' ; hence, the space fallen through 
 
 is 19.6 X (3.14159)" = 193 inches, or l6^ feet, nearly. 
 
 If the arc, in which a body oscillates, be diminished, 
 the effect of the air's resistance is diminished; and when 
 the arc is very small, this resistance does not sensibly affect 
 the time of an oscillation. By observing, therefore, the 
 length of a pendulum which vibrates seconds in very small 
 arcs, we determine the space through which a body would 
 fall in vacuo in one second, with sufficient accuracy for all 
 practical purposes. 
 
 Prop. LXXVI. 
 
 (303.) The time of descent to the lowest point in a 
 small circular arc is to the time of descent down ifs chord, 
 as the circumference of a circle to four times the diameter.
 
 160 
 
 ON THE OSCILLATIONS 
 
 Let AB be the arc, C it's center, BD the diameter 
 perpendicular to the horizon ; T the time of descent down 
 the arc AEB, t the time of descent down the chord, C the 
 circumference of a circle, D it's diameter. Then, 2 T is 
 the time of an oscillation of the pendulum CB ; therefore. 
 
 CB 
 
 2 T : the time down :: C : D (Art. 295.) ; 
 
 CB 
 
 and time down — : time down DB, or AB, :: 1 : 2 
 
 2 ' ' 
 
 {Art. 241.); therefore, 2 T : the time down AB (t) :: C : 2 A 
 and Tit'.-.C : 4.D. 
 
 Prop. LXXVII. 
 
 (304.) The force, which accelerates or retards a body's 
 motion in a cycloid, varies as the arc intercepted between 
 the body and the lowest point. 
 
 Let DV represent the whole force of gravity, (See Fig. 
 in p. 151.) from P draw PH parallel to AD meeting the 
 circle DHV in H -, join DH, HV. 
 
 Then, the whole force DV, which acts upon the body 
 at P, may be resolved into the two DH, HV; of which 
 DH is in the direction of the string, and therefore neither 
 accelerates nor retards the motion of P; and ^F is in the 
 direction of the tangent at P (Art. 281.), and therefore 
 wholly employed in accelerating or retarding the motion
 
 OF BODIES. 161 
 
 ill the curve ; consequently, 
 
 the force of gravity : the accelerating force :: DV : HV ; 
 
 and since the force of gravity, and DV, are invariable, 
 the accelerating force oc HVoc2HVoc PV (Art. 283.). 
 
 (305.) Cor. l. If a body move in any line, and be 
 acted upon by a force which varies as the distance from the 
 lowest point, the motion of this body will be similar to the 
 motion of a body oscillating in a cycloid. 
 
 For, if an arc, measured from the vertex of a cycloid, 
 be taken equal to the line, and the accelerating forces, in 
 the line and the cycloid, at these equal distances from the 
 lowest points, be equal, they will always be equal, because 
 they vary according to the same law ; and the bodies, being 
 impelled by equal forces, will be equally accelerated, and 
 describe equal spaces in any given time. 
 
 (306.) CoR. 2. The time of descent to the lowest 
 point in the line will always be the same, from whatever 
 place the body begins to fall. 
 
 (307.) CoR. 3. If the distance of the body from the 
 lowest point at the beginning of the descent, be made the 
 radius, the velocity acquired will be represented by the 
 right sine, and the time by the arc, whose versed sine is 
 the space fallen through. 
 
 Prop. LXXVIII. 
 
 (308.) If a body vibrate m a circular arc, the force 
 which accelerates or retards ifs motion varies as the sine 
 of ifs distance from the loicest point. 
 
 Let a body oscillate in a circular arc whose radius is 
 AC', from the center C, and A the place of the body, draw 
 CB, AE perpendicular to the horizon, and take AE to 
 represent the force of gravity ; draw AD perpendicular 
 
 L
 
 162 
 
 ON THE OSCILLATIONS 
 
 to CB, and EF perpendicular to AF, which is a tangent 
 
 to the circle at A. Then, the force AE is equivalent to 
 the two AF, FE; of which FE is perpendicular to the 
 tano-ent AF, or in the direction of the radius CA, and can 
 neither accelerate nor retard the motion of the body ; the 
 other, AF, is in the direction of the tangent, and is wholly 
 employed in accelerating or retarding the body's motion; 
 hence the force of gravity : the accelerating force :: AE : AF, 
 that is, from the similar triangles AEF, CAD, :: CA : AD; 
 and consequently, 
 
 . « gravity x AD 
 the accelerating force = — ; 
 
 in which expression, gravity and the radius AC are in- 
 variable ; therefore, the accelerating force varies as AD. 
 
 (309.) CoR. 1. If the accelerating force were pro- 
 portional to the arc, the oscillations, whether in greater 
 or smaller arcs, would be performed in equal times (Art. 
 306.); but, since the sine does not increase as fast as the 
 arc, the force in the greater arc is less than that which 
 would be sufficient to make the time of oscillation equal 
 to the time in the smaller arc ; therefore, the time of os- 
 cillation in the greater circular arc is greater than in the 
 less.
 
 OF BODIES. 163 
 
 (310.) Coil. 2. Call F the force in the direction JiJ; 
 then, since AC is invariable, the accelerating force in the 
 
 AB 
 
 cuYveocFxAD; and if FxADoc AB. ov Foa , the 
 
 AD' 
 
 accelerating force varies as AB, and the times of oscillation 
 
 in different arcs are equal (Art. 306.). 
 
 SCHOLIUM. 
 
 (311.) In this Section we have considered the vibrations 
 of a simple pendulum only, or of a single particle of matter, 
 suspended by a string, the gravity of which is neglected. 
 The Propositions are indeed applicable in practice, when 
 the diameter of the body is small with respect to the length 
 of the string by which it is suspended, and the v/eight of 
 the string inconsiderable when compared with the weight 
 of the body. That the conclusions are not strictly true in 
 this case, is evident from the consideration that two par- 
 ticles of matter, at different distances from the axis of 
 suspension, do not vibrate in the same time (Art. 296.); 
 and consequently, that when they are connected together, 
 they affect each other's motion ; thus, the time of vibration 
 of the two particles when united, is different from the time 
 in which either would vibrate alone. 
 
 The method of determining the time of vibration of a 
 compound pendulum the Reader will find in the Priiiciples 
 of Fluxions, Art. 65 ; to which place he is also referred for 
 the investio-ation of the rules for determining the centers of 
 Gyration and Percussion; questions properly belonging to 
 Mechanics, but inserted in that part of the Work, because 
 the rules cannot easily be applied to the determination of 
 those points, even in the most simple cases, without the 
 assistance of the fluxional calcidus. 
 
 (312.) To avoid the introduction of analytical de- 
 monstrations in subjects professedly geometrical, Sir I. 
 Newton and other Writers, have had recourse to inde- 
 
 L2
 
 164 SCHOLIUM. 
 
 finitely small or evanescent increments, which continually 
 approximate to the true increments of the quantities whose 
 finite values are required. This method may be applied 
 with success in all cases where the differejice between the 
 assumed and the true increments continually decreases, 
 and at length vanishes, with respect to the increments 
 themselves; or, which amounts to the same thing, when 
 the ratio which the sum of the differences bears to one 
 of the increments, does not exceed a finite ratio : for, by 
 observing; the limit to which the sum of the assumed in- 
 crements approaches, when their number is increased and 
 their magnitudes are dimished in infinitum, it is evident 
 that the sum of the real increments is obtained. In the 
 same manner, when there are two ranks of quantities, in 
 which the assumed increments continually approximate to 
 the real increments, as in the former instance, and the 
 limiting ratio of the sums of the assumed increments in 
 these cases, when their numbers are increased and their 
 magnitudes diminished without limit, is obtained, the exact 
 ratio of the quantities themselves is obtained. These pro- 
 positions are laid down by Sir I. Newton in the first 
 Section of the Principia, Lem. 3d and 4th, and the same 
 mode of reasoning has been applied in Art. 292, to compare 
 the time of an oscillation in the cycloid BVA, with the time 
 of describing the arc IZp with the velocity acquired at V 
 continued uniform. In tliis Art. it is supposed that the 
 time of describing MN, with the uniform velocity mx, 
 is the increment of the former time, and that the time 
 of describing xt, the side of a triangle similar to Vxm, 
 with the velocity VZ, is the increment of the latter ; these 
 assumed increments, it is manifest, differ from the true 
 increments of the times under consideration ; but when 
 they are diminished without limit, they differ from them 
 by quantities which are evanescent with respect to the 
 whole increments, and therefore by determining the limiting 
 ratio of the sums of the assumed increments, we obtain the 
 ratio of the actual times of describing the corresponding 
 arcs.
 
 SECTION IX. 
 
 ON THE MOTION OF PROJECTILES. 
 
 Prop. LXXIX. 
 
 (313.) A BODY projected in any direction^ 7iot per- 
 pendicular to the horizon, will describe a parabola, on 
 supposition that the force of gravity is uniform, and acts 
 in parallel lines, and that the motion is not affected by 
 the resistance of the air. 
 
 Let a body be projected from A in the direction JE, 
 from which point draw ABF perpendicular to the horizon ; 
 also, let AE be the space over which the velocity of pro- 
 jection would carry the body in any time, T, and AB the 
 space through which the force of gravity would cause it 
 to descend in the same time; complete the parallelogram 
 AC; then, in consequence of the two motions, the body 
 will be found in C at the end of that time. For, the 
 motion in the direction AE can neither accelerate nor 
 retard the approach of the body to the line BC (Art. 29.) ; 
 therefore, at the end of the time T, the body will be in 
 the line BC ; and, by the same mode of reasoning, it 
 appears that it will be in the line EC at the same time ; 
 consequently, it will be at C, the point of their intersection, 
 at the end of the time /'. Now, since AE is the space
 
 166 ON THE MOTION 
 
 which would be described in the time T, Avith the velocity 
 
 of projection continued uniform, JEoc T (Art. 13.); and 
 BC=AE; therefore BCozT, and BC^<ycT\ Also, since 
 AB is the space through which the body would fall by the 
 force of gravity in the time T, ABocT^, (Art. 241.); 
 hence, AB oc BC" ; and this is the property of a parabola, 
 in which AF is a diameter, and BC an ordinate to the 
 abscissa AB. 
 
 (314.) Cor. 1. The axis, and all the diameters of 
 the parabola described, being parallel to AF, are perpen- 
 dicular to the horizon. 
 
 (315.) Cor. 2. The direction of projection AE is 
 parallel to the ordinate 5C, and therefore it is a tangent 
 to the curve at A. 
 
 (316.) Cor. 3. The time in which a body describes 
 the arc ^C is equal to the time in which it would fall from 
 A to B hy the force of gravity, or describe AE with the 
 first velocity continued uniform. 
 
 (317) CoR. 4. If the Z EAf be made equal to the 
 Z EAb, the line Af will pass through the focus of the 
 parabola described.
 
 OF PROJECTILES. 
 
 167 
 
 (318.) Cor. 5. The body will always be found in 
 the plane ACB which passes through the direction of 
 projection and the perpendicular to the horizon. 
 
 (319.) Cor. 6. If the motion in the direction AE be 
 produced by the action of an uniform force, AEoc TocAB; 
 or ABo<zBC; in which case, the locus of the point C is a 
 right line. 
 
 Prop. LXXX. 
 
 (320.) The velocity of the projectile, at any point 
 in the parabola, is such as would be acquired in falling 
 through one fourth part of the parameter belonging to 
 that point. 
 
 Let ABhe the space through which a body must fall 
 by the force of gravity to acquire the velocity of the pro- 
 jectile at A : and AE the space described with that velocity 
 continued uniform, in the time of falling through AB ; then 
 2AB = AE (Art. 236.) ; and, completing the parallelogram 
 AC, 2AB = BC; hence, ^AB' = BC\ Also, since C is 
 a point in the parabola, and BC an ordinate to the abscissa 
 
 AB (Art. 313.), if P be the parameter belonging to the 
 point A, PxAB^ BC ^ iAB' 
 
 therefore J5 = l P.
 
 168 
 
 ON THE MOTION 
 
 (321.) Cor. 1. If the velocity at A be given, the 
 parameter at that point is the same, whatever be the 
 direction of the body's motion. 
 
 (322.) CoR. 2. If AB be the space through which 
 a body must fall to acquire the velocity at A, and a circle 
 be described from the center A with the radius AB, the 
 focus of the parabola described will lie in the circumference 
 of this circle, whatever be the direction of projection ; since 
 the distance of any point in the parabola from the focus, is 
 one fourth part of the parameter belonging to that point. 
 
 f 
 
 (323.) Cor. 3. The velocity in a parabola varies as 
 the square root of the parameter. 
 
 Since the velocity is such as would be acquired in 
 falling through ^ P, it varies as sj ^ P (Art. 241.), or 
 as ^. 
 
 (324.) Cor. 4. The velocity is the least in the vertex 
 of a parabola ; and at equal distances from the vertex the 
 velocities are equal. 
 
 For, the parameter belonging to the vertex is the least ; 
 and the parameters at equal distances from the vertex are 
 equal. 
 
 Prop. LXXXI. 
 
 (325.) To find the direction in which a body must he 
 projected from a given point, with a given velocity, to hit 
 a given mark. 
 
 Let A be the point from which the body is to be 
 projected, C the given mark; join AC, and from A draw 
 AB parallel, and AP perpendicular to the horizon; take 
 AP equal to four times the space through which a body 
 must fall to acquire the given velocity of projection, (de- 
 termined by Art. 248, Case 3.); then will AP be the
 
 OF PROJECTILES. 
 
 169 
 
 parameter belonging to the point A of the parabola de- 
 scribed (Art. 320.). Draw AK perpendicular to AC ; 
 
 B IM 
 
 bisect PA in G, and draw KGH perpendicular to AP, 
 meeting AK in K; join KP; then the triangles KGP, 
 KG A, being similar and equal, KP — KA. From K as 
 a center, with the radius KA, or KP, describe a circle 
 AHP, cutting KGH in H; through C draw C£/ parallel 
 to AP, and cutting the circle in E and /; join AE, AI ; 
 and if a body be projected, with the given velocity, in 
 the direction AE or Af, it will hit the mark C. 
 
 Let the body be projected in the direction AE ; join 
 PE, and complete the parallelogram AECX; then AX 
 is a diameter of the parabola described; and XC, which 
 is parallel to the tangent AE, is in the direction of an 
 ordinate to the abscissa AX; if then XC be the length 
 of the ordinate to this abscissa, C is a point in the parabola. 
 Now, since the angles AEC, EAP, are alternate angles, 
 and the angle EAC — ihe angle EPA, because AC is a
 
 17^ 0^' i'HE MOTIUX 
 
 tangent to the circle at A (Euc. l6. iii.), the triangles 
 EPA, EAC, are similar; and AP : AE :: AE : EC; 
 or substituting for AE and EC their equals ^C and AX, 
 AP : JTC :: XC : JX; that is, XC is a mean pro- 
 portional between the parameter and the abscissa; and 
 therefore it is the ordinate belonging to that abscissa; 
 hence, C is a point in the parabola which the body de- 
 scribes. 
 
 In the same manner it may be shewn that, if the body 
 be projected with the same velocity in the direction AI, it 
 will hit the mark C 
 
 (326.) CoR. 1. Join AH, HP; then the angle HAP 
 = the angle HP A = the angle HAC 
 
 (327.) CoR. 2. Because KH is drawn through the 
 center of the circle perpendicular to the chord EI, it 
 bisects it, and consequently it bisects the arc IHE 
 (Euc. SO. iii.) ; therefore, the angle I AH = the angle 
 HAE. That is, the two directions AE, AI, make equal 
 angles with AH, which bisects the angle PAC. 
 
 (328.) CoR. 3. Draw HLM touching the circle in H; 
 then, when the point C coincides with L, the two directions 
 AE, AI, coincide with AH. • 
 
 (329.) CoR. 4. If the point C be taken in the plane 
 AL, beyond L, the line CEI will not meet the circle. In 
 this case, the velocity of projection is not sufficient to carry 
 the body to the distance AC 
 
 (330.) CoR. 5. Bisect AC in r, and draw tvr parallel 
 to AP; then tvr is in the direction of a diameter, to which 
 AC is a double ordinate. Also, rt is the sub-tangent; and 
 if rt be bisected in v, this point is in the parabola.
 
 OK PROJECTILES. 
 
 171 
 
 (331.) Cor. 6". A tangent to the parabola at v is 
 parallel to the ordinate AC; therefore, v is the point in the 
 parabola which is at the greatest distance from AC*. 
 
 (332.) CoR. 7. The greatest height of the projectile 
 above the plane, measured in the direction of gravity, 
 is I EC. For, rv = \rt, and Tf = ^EC; therefore, 
 rv = ^EC. 
 
 Prop. LXXXII. 
 
 (333.) Having given the velocity and direction of 
 projection, to find where the body will strike the horizontal 
 plane which passes through the point of projection. 
 
 Let AC (Art. 325.) coincide with the horizontal line 
 AB; then AK coincides with AG; and PHA is a semi- 
 
 circle; also, HAC is an angle of 45°, and AE, AI, are 
 equally inclined to AH. 
 
 * The properties of the parabola here referred to, may be found 
 in any Treatise of Conic Sections.
 
 172 ox THK MOTIOX 
 
 From the last Proposition it appears, that if the velocity 
 of projection be such as would be acquired in falling through 
 ^ PA^ and AE^ or AI, be the direction of projection, the 
 range is AC. From E draw ED parallel to AC^ or per- 
 pendicular to AP , join EK ; then ED^ or it's equal AC, 
 is the sine of the Z EKA to the radius KA ; and the 
 /^ EKA = 2 A EPA = 2 A EAC ; therefore, AC is the sine 
 of 2/.EAC to the radius KA ; and the sine of a given 
 angle is proportional to the radius; consequently, 
 
 rad. : KA :: sin. 2 Z EAC : AC ; 
 
 ^^ sm.^AEACxKA 
 
 hence, AC = = . 
 
 rad. 
 
 If V be taken to represent the velocity of projection, P the 
 parameter AP, and 7n = \6^, then 
 
 ^^ sin. 2 I EAC xP sin. 2 ^ EAC x V~ 
 
 AC= = ^ (Art. 248.). 
 
 2 rad. 2 rad. x m 
 
 In the samd manner, if AI be the direction of projection, 
 _ sin. 2 z lAC xP _ sin. 2 z /JC x V~ 
 2 rad. 2 rad. x m 
 
 (334.) CoR. 1. Hence, ^Cocsin. 2 z ^JC x V-. 
 
 (335.) Cor. 2. If the velocity of projection be in- 
 variable, the horizontal range varies as sin. 2 Z EAC. 
 
 (336.) Cor. 3. The range is the greatest when sin. 
 
 2 Z EAC is the greatest ; that is, when the z EAC is 45". 
 
 ■r 1 . . ^ rad X P , „ 
 
 In this case, AC = ~ = iP. 
 
 2 rad. ^ 
 
 (337.) Cor. 4. If the angle EAC be 1.5°, or 75°, 
 sin. 2 Z 1:JC = 1 rad. and AC = I P. 
 
 (338.) Cor. 5. If the range and the parameter be 
 given, the angle of elevation may be found.
 
 For, AC = 
 
 OF projectti.es. 
 
 sin. 2 z EAC X P 
 
 173 
 
 2 rad. 
 
 therefore sin. 2 Z EAC = 
 
 2jCxrad. 
 
 (339.) Cor. 6. If JC and the angle EAC be known, 
 
 2 AC X rad. . ,^„ 2 rad. x m x ^C 
 
 P = ~ — - — ^z—..; and V~= — r 
 
 sin. 2^ EAC 
 
 sin. 2 z J5JC 
 
 Prop. LXXXIII. 
 
 (340.) The same things being given, to find the time 
 of flight. 
 
 The time in which the velocity of projection, F", would 
 
 be acquired, or the time of descent down 4- PA. is — " 
 
 ^ 2m 
 
 (Art. 248.) ; hence, the double of this time, or the time of 
 
 descent down PA, is — " (Art. 241.); which is also the 
 m 
 
 A C 
 
 time of descent down EA (Art. 264.). Let T bo the time
 
 174 ox THK MOTION 
 
 of descent down PA, or EA, t the time of descent down 
 EC, or the time of flight (Art. 3l6.) ; 
 
 then T : t :: EA : EC (Art. 262.); 
 
 that is, T : t :: rad. : sin. Z EAC, 
 
 sin. z EAC X T sin. z EAC x F 
 
 and # = 
 
 rad. rad. x m 
 
 (341.) CoR. 1. If the velocity of projection be in- 
 variable, the time of flight oc sin. z EAC 
 
 (342.) CoR. 2. Hence, the time of flight is the great- 
 est, when sin. z EAC is the greatest. In this case, the 
 
 ran x 'J ' 
 time becomes — '—z — , or T ; that is, the greatest time of 
 rad. 
 
 flight is equal to the time of descent down the parameter. 
 
 Prop. LXXXIV. 
 
 (343.) The same things being given, to find the 
 greatest height to which the projectile rises above the 
 horizontal pla7ie. 
 
 The greatest height is ^ EC, or ^ AD ; and AD is the 
 versed sine of the z AKE, or 2 z EAC, to the radius AK\ 
 consequently, since the versed sine of a given angle varies 
 as the radius, 
 
 rad. : AK (IP) :: the versed sine of 2 Z EAC : AD. 
 
 ver. sin. 2 z EAC x P 
 
 Hence, AD — 
 
 and \AD, the greatest height. 
 
 2 rad. 
 
 ver. sin. 2 z EAC x P 
 
 8 rad. 
 
 ver. sin. 2 z jE^C x V" 
 8 rad. x m 
 
 (344.) Cor. l. The greatest height ex: ver. sin. 
 2 z EAC X r^ ; and when V is given, the greatest height 
 oc ver. sin. 2 z EAC.
 
 OF PROJECTILES. 
 
 175 
 
 (345.) Cor. 2. The versed sine of an arc varies as 
 the square of the sine of half the arc ; therefore the greatest 
 height oc (sin. z EACf x V\ 
 
 (346.) CoR. 3. A body, projected with a given ve- 
 locity, will rise to the greatest height above the horizontal 
 plane, when the angle of elevation EAC is a right angle. 
 In this case, ver. sin. 2 z EAC = 2 rad. and the greatest 
 
 ^ . , . 2 rad. X P P 
 
 altitude IS ; = — . 
 
 8 rad. 4 
 
 Prop. LXXXV. 
 
 (347.) The velocity and direction of projection being 
 given, to find where the body will strike a given inclined 
 plane which passes through the point of projection. 
 
 It appears from Art. 325, that if a body be projected 
 from A, in the direction AE, with the velocity acquired in 
 
 ;* 
 
 ■7 
 ''. / 
 
 [> 
 
 H 
 
 X 
 
 G A 
 
 / \ 
 
 
 I 
 
 i 1 
 
 vl 
 
 falling down ^ PA, it will strike the plane AC in the point 
 C\ Let / be the angle of inclination CAB ; E the angle
 
 176 ON THE MOTION 
 
 of elevation EAC ; Z the angle EAP. Then, in the 
 triangle EAP, AE : AP :: sin. Z EPA : sin. z AEP; 
 and the z EPA = the z J5JC = E; also the z AEP 
 = the Z JECJ = the supplement of the Z ACS; hence, 
 
 AE : AP :: sm. E : cos. /; therefore, AL— — . 
 
 COS. / 
 
 Again, in the triangle EAC, 
 
 AC : AE :: sin. Z AEC (sin. Z) : sin. Z^C^ (cos. /); 
 
 therefore, AC= — ; 
 
 COS. / 
 
 sin. E X AP 
 and by substituting for AE it's value 
 
 AC = 
 
 cos. / 
 sin. E X sin. Z x AP sin. E x sin. Z x V~ 
 
 (cos. /)^ (cos. /)' X m 
 
 sin. ^ X sin. Z x V~ 
 
 (348.) CoR. Hence, ACoc 
 
 Prop. LXXXVI. 
 
 (cos. iy 
 
 (349.) TOe 5ttme ^Am^« being given, to find the time 
 of flight. 
 
 Let T be the time of descent down PA, t the time of 
 descent down EC or the time of flight ; then, 
 
 T^ : f :: PA : EC ; 
 
 and since, in the similar triangles PAE, AEC, 
 PA : AE :: AE : EC, 
 therefore PA' : AE' :: PJ : £:C :: T' : f' ; 
 and PA : AE :: T : t; 
 
 but P^ : ^£ :: sin. Z PEA : sin. Z ^P^ 
 :: sin. Z -ECJ : sin. z JE^C :: cos. / : sin. E;
 
 OF PROJECTILES. 177 
 
 therefore, T : t :: cos. / : sin. E, 
 
 - sin. E X T sin. E x V 
 
 and t = 
 
 (350.) Coit. Hence, ^ oc 
 
 COS. / cos. I X m 
 
 sin. Ex V 
 
 and, if V be invariable, toe 
 
 Prop. LXXXVlI. 
 
 cos. / 
 
 sin. E 
 
 cos. / 
 
 (351.) The same things being given, to find the 
 greatest height of the projectile above the plane AC, 
 measured in the direction of gravity. 
 
 The greatest height is \ EC (Art. 332.) ; and in the 
 
 triangle AEC, EC : AE :: sin. E : cos. /; therefore, 
 
 „^ sin. E X AE , , , . . „ ^^.^ 
 EC= — ; and, by substituting tor AE it s value, 
 
 COS. / 
 
 sin. ExAP.^ ^ ^ ^ ^ (sin. Ef x AP 
 
 (Art. 347.), we have CE =^ r r^o ; 
 
 COS. 1 (cos. ly 
 
 , , ^^ (sin. Ef X AP (sin. Ef x V ^ 
 
 and i J5:C = ^-- -—- = ) ' , the greatest 
 
 4 (cos. /)- (cos. /)• X 4m 
 
 height required. 
 
 (352.) Cor. The greatest height varies as 
 
 (cos. ly 
 
 SCHOLIUM. 
 
 (353.) The theory of the motion of projectiles, given 
 in this section, depends upon three suppositions, which are 
 all inaccurate ; 1st. that the force of gravity, in every point 
 of the curve described, is the same; 2d. that it acts in 
 parallel lines ; 3d. that the motion is performed in a non- 
 resisting medium. The two former of these, indeed, differ 
 insensibly from the truth. The force of gravity, without 
 
 M
 
 178 SCHOLIUM. 
 
 the Earth's surface, varies inversely as the square of the 
 distance from the center ; and the altitude to which we 
 can project a body from the surface is so small, that the 
 variation of the force, arising from the alteration of the 
 distance from the center of the Earth, may safely be 
 neglected. The direction of the force is every where 
 perpendicular to the horizon ; and if perpendiculars be 
 thus drawn, from any two points in the curve which we 
 can cause a body to describe, they may be considered as 
 parallel, since they only meet at, or nearly at, the center 
 of the Earth. Even the resistance of the air does not 
 materially affect the motions of heavy bodies, when they 
 are projected with small velocities. In other cases, how- 
 ever, this resistance is so great as to render the conclusions, 
 drawn from the theory, almost entirely inapplicable in 
 practice. From experiments made to determine the motions 
 of cannon-balls, it appears that when the initial velocity is 
 considerable, the air^s resistance is 20 or 30 times as great 
 as the weight of the ball ; and that the horizontal range is 
 often not ^ part of that which the preceding theory leads 
 us to expect. It appears also, that when the angle of 
 elevation is given, the horizontal range varies nearly as 
 the square root of the velocity of projection ; and the time 
 of flight as the range ; whereas, according to the theory, 
 the time varies as the velocity, and the range as the square 
 of the velocity of projection (Arts. 340, 334.) These ex- 
 periments, made with great care, and by men of eminent 
 abilities, shew how little the parabolic theory is to be 
 depended upon, in determining the motions of military 
 projectiles. See Robin's New Theory of Gumiei-y, and 
 Hutton's Mathematical Dictionary^ article Gunnery. 
 
 Besides diminishing the velocity of the projectile, the 
 air's resistance will also change it's direction, whenever 
 the body has a rotatory motion about an axis which does 
 not coincide with the direction in which it is moving. 
 For the velocity with which that side of the body, strikes 
 the air, on which the rotatory and progressive motions
 
 SCHOLIUM. 179 
 
 conspire, is greater than the velocity with which the other 
 side strikes it, where they are contrary to each other ; and 
 therefore the resistance of the air, which increases with the 
 velocity, will be greater in the former case than in the 
 latter, and cause the body to deviate from the line of it's 
 motion ; this deviation will also be from the plane of the 
 first motion, unless the axis of rotation be perpendicular 
 to that plane. 
 
 Upon this principle Sir I. Newton explains the ir- 
 regular motion of a tennis-ball * ; and the same cause has 
 been assigned by Mr. Robins for the deviation of a bullet 
 from the vertical plane f. Mr. Euleii, indeed, in his 
 remarks on the New Theory of Gunnery^ contends that 
 the resistance of the air can neither be increased nor di- 
 minished by the rotation of the ball ; because such a motion 
 can produce no effect but in the direction of a tangent to 
 the surface of the revolving body ; and the tangential 
 force, he affirms^ is almost entirely lost. In this instance, 
 the learned writer seems to have been misled by the common 
 theory of resistances, according to which the tangential 
 force produces no effect ; whereas, from experiments lately 
 made, with a view to ascertain the quantity and laws of 
 the air's resistance, it appears that every theory which 
 neglects the tangential force must be erroneous. 
 
 * Phil. Trans. Vol. VI. p. 3078. Maclaurin's Newton, p. 120. 
 t Tracts, Vol. I. pp. 151. 198. 214. 
 
 M 2
 
 APPENDIX. 
 
 ON THE EFFECTS PRODUCED BY 
 
 WEIGHTS ACTING UPON MACHINES IN MOTION, 
 
 AND 
 
 ON THE ROTATION OF BODIES. 
 
 The investigation of the effects produced by bodies 
 when the machines on which they act are in motion, has 
 not usually been introduced into elementary Treatises; 
 but as the theory depends upon the principles already laid 
 down, and may, by the help of the simplest analytical 
 operations, be easily deduced from them, it may not im- 
 properly be added, by way of Appendix, here. 
 
 Prop. LXXXVIII. 
 
 (354.) To find what weight x, 'placed at A upon 
 a machine in motion^ resists the rotation as much as y 
 placed at B. 
 
 Let a and b be the velocities of the weights; then <'va 
 and y h are their momenta ; and since these momenta pro- 
 duce equal effects on the machine, or, are sufficient to 
 balance each other, xa . yh :: h : a (Art. 14.9.) ; there- 
 fore xa^ = yh^i and ,t? = — r- . 
 
 a
 
 ACTION OF BODIES, &C. 
 
 181 
 
 Prop. LXXXIX. 
 
 (355.) If two weights acting upon a wheel and axle 
 put the machine in motion^ to determine the velocity 
 acquired by the descending body, and the tension of the 
 string by which it acts. 
 
 Let C be the center of motion ; CJ, CB the radii of 
 the wheel and axle ; p and q the two weights, of which p 
 descends ; CA = a, CB = b, then a and b are proportional 
 to the velocities of j) and q. And let i^ = the weight which 
 
 op 
 
 q would sustain at p ; and w = the weight which placed 
 at p would resist the communication of rotation as much 
 as q resists it ; v = the velocity generated in the time t ; 
 
 m=l6^ feet. 
 
 7 lO 
 
 Then a : b :: q : % = — , and .^ = Ar (Art. S54^.)\ 
 a a^ 
 
 hence, p = the force at p to move the machine, and 
 
 a 
 
 /) + -^ = the inertia to be moved, neglecting the inertia 
 
 P- 
 
 qb 
 
 of the machine ; consequently, ,„ = the accelerating
 
 182 ACTION OF BODIES ON 
 
 force, that of gravity being represented by unity (Art. 268.) ; 
 and since v = 9.mft (Prop. Lxi. Cor. 7.), we have, in this 
 qh 
 
 a pa' — gab 
 
 case, v= -^ X 2m t = -^ x 2mt. 
 
 qb" va-^qb" 
 
 A • • pa' — qab . , , . ^ 
 
 Again, since - — ^ — ^-^ is the acceleratins; force at 
 jia -\-qb' * 
 
 p, the moving force, which generates ^/s velocity, is 
 
 pa^ — qab par-qab 
 
 1 -r^ X p ; therefore p x p is that part 
 
 pa^ + qb- ^ ^ pa" + qb~ ^ ^ 
 
 of ^'s weight * which is sustained, or the weight which 
 
 ^ ^, , . 1 . pqb'+pqab (a + b).bpq 
 
 stretches the string ; that is, ^^ ., ., , or -^ ^ — f^ 
 
 pa'~\-qb' pa' + qb- 
 
 is the weight which stretches the string AP. 
 
 (356.) CoR. 1. The tension of the string AP is just 
 
 sufficient to sustain the tension of the string BQ, ; therefore 
 
 (a + b).bpq {a + b).apq , ^ . . , 
 
 b : aw — - : — — — = the tension of the 
 
 pa~-\-qb- pa^ + qb 
 
 string BQ. 
 
 (357.) Coil. 2. The pressure on the center of motion is 
 
 the sum of the tensions of the strings JPand BQ (Art. 101.), 
 
 (a + b).b (a + b).a (a + bf.pq 
 
 or, , . j. Xpq+ —T> — '—-^xpq= ^ TT-^J' 
 
 pa~ + qb- pa'-j-qb^ ^^ pa" + qb^ 
 
 (358.) Cor. 3. When a and b are equal, the pressure 
 
 on the center is 
 
 7> + ^ 
 
 (359.) Cor. 4. Since s, the space which p descends 
 
 from rest in t seconds = mft' (Prop. Lxi. Cor. 7.), 
 
 pa^-qab 
 s — - — ^ — —— X mr. 
 pa~ -f qb' 
 
 * In this operation, the moving force and the quantity of matter are, 
 respectively, represented by the weight.
 
 MACHINES IN MOTION. 183 
 
 (360.) Cor. 5. The same reasoning may be applied 
 when the bodies act upon any other machine. 
 
 (361.) Cor. 6. If the inertia of the machine is to be 
 
 taken into consideration, let r be the weight, determined by 
 
 experiment or calculation, which when placed at p, would 
 
 resist the communication of rotation as much as the whole 
 
 qb 
 machine resists it ; then p is the moving force at p, 
 
 gb~ 
 and r-\-p-\ ~ is the quantity of matter to be moved; 
 
 pa~ — qab . . 
 
 therefore, —^ — is the accelerating force at «, 
 
 ra'+pa^ + qb' ^ ^' 
 
 the accelerating force of gravity being represented by 
 
 unity. 
 
 Prop. XC. 
 
 (362.) If a string be wrapped round a hollow cylinder 
 G, and one endjixed at S, to find the tension of the string 
 when the cylinder is suffered to desceiid. 
 
 Let a — the weight of the cylinder, collected in the 
 circumference ; a? = the tension of the string. 
 
 Then, since the motion of the center of gravity of the 
 cylinder is the same at whatever point of the body the 
 
 Sr 
 
 A 
 
 G 
 
 force is apphed (Art. 182.), a — ^v hi the moving force by
 
 184 
 
 ACTION OF BODIES ON 
 
 which the the center of gravity of the cylinder descends, 
 
 and is the accelerating force. Again, cc is the 
 
 a 
 
 weight, or moving force, which applied at the circum- 
 
 ference of the cylinder, generates the rotation, and - is 
 
 the accelerating force; and since accelerating forces are 
 proportional to the velocities generated in the same time, 
 and, from the nature of the case, the center of gravity 
 of the cylinder descends as fast as the string is unfolded, 
 that is, the velocities of the center of gravity and rotation 
 
 — X X a 
 
 = - ; hence, a? = - ; or 
 
 a a 2 
 
 are always equal, we have 
 
 the tension of the string is half the weight of the cylinder, 
 (363.) CoR. The accelerating force is = - 
 
 the accelerating force of gravity being represented by 
 unity. 
 
 Prop. XCI. 
 
 To Jljid the tension when the string passes over a 
 fixed pulley and a weight is attached to it. 
 
 (364.) Let p be the weight of the body attached to 
 
 r^ 
 
 p6 
 
 <3 
 
 the string; x the tension; a the weight of the cylinder.
 
 MACHINES IN MOTION. 185 
 
 Then p—x is the moving force on p ; , the accelerating 
 
 P 
 
 force ; , the force which accelerates the cylinder ; 
 
 5 the accelerating force which produces the rotation ; 
 
 which quantities are proportional to the velocities generated 
 
 in the same time. Also, the spaces descended by j) and A 
 
 are, together, always equal to the length of the string dis- 
 
 1 1 o P~'^ a-OG w 2ap 
 
 engaged, thereiore, 1 = - ; hence ; x = . 
 
 ^ "^ ' pa a 2p-\-a 
 
 (365.) CoK. 1. The pressure on the center of the 
 
 2p + a' 
 
 p —X 
 
 pulley is 2 a?, or 
 
 (366.) CoR. 2. The accelerating force on p = 
 
 r 
 2p — a 
 
 9.p -\- a 
 
 (367-) Cor. 3. The accelerating force on the cylinder 
 
 a — w a 
 
 2p + a 
 
 (368.) CoR. 4. If p = (f, the body and the cylinder 
 are equally accelerated ; that is, they descend at the same 
 rate. 
 
 (369.) CoR. 5. If ^ = - , the accelerating force on 
 p vanishes, and p remains at rest. 
 
 (370.) Cor. 6. If the cylinder be solid and of uniform 
 
 density, it will appear, nearly in the same manner, that the 
 
 2ap 
 tension of the string is ; the force which accelerate.^ 
 
 '& 
 
 3p-^a
 
 186 ACTION OF BODIES ON 
 
 the cylinder, ^ ; and the force which accelerates «, 
 
 ^ 3p + a 
 
 3p — a 
 
 Sp + a' 
 
 Prop. XCII. 
 
 (371.) To find the tension of the string, when the 
 iveight of the pulley is taken into the account. 
 
 Let c be the weight which, placed at the circumference 
 of the pulley, would resist the communication of motion as 
 much as the pulley ; and let y = the tension of the string 
 
 p — y 
 
 SP, cc = the tension of SA. Then = the accelerating 
 
 P 
 
 force on «, = the accelerating force on A ; 
 
 a c 
 
 = the accelerating force on the circumference of the pulley ; 
 
 and - = the accelerating force which produces the rotation 
 a 
 
 of the cylinder. Then, as in the last Proposition, 
 
 p — y a — w X 
 p a a 
 
 also, because the circumference of the pulley always moves 
 as fast as p, 
 
 p-y ^ yjzZ- 
 
 p ~ c ' 
 
 2ap + ac 
 
 from which equations .%' = 
 and y = 
 
 2p -\- a -\- 2 c 
 (a-{-c).2p 
 
 2p-\-a -\-2c 
 (372.) Coil. 1. The force which accelerates the 
 
 cylinder is = ; and the force which 
 
 -^ a 2p + a-j-2c 
 
 1 P — y 2p — a 
 
 accelerates p^ ^ ~ 
 
 p 2p-\-a-^2c
 
 MACHINES IN MOTION. 187 
 
 (373.) CoK. 2. The accelerating force being known, 
 the space, time, and velocity, may be found in terms of 
 each other (Prop. Lxi. Cor. 7.). 
 
 Prop. XCIII. 
 
 (374.) //' any weights A, B, C, act upon a machine 
 and put it in motion, and x, y, z, he the spaces described 
 in the direction of gravity, a, b, c, the actual velocities of 
 the zceights, m = 1 6^ feet, 
 
 then 4m x (Ax + By + Cz) = Aa- + Bb- + Cc". 
 
 Let AB be the direction of A's motion, AC perpen- 
 dicular to the horizon; take AB = s, the space described 
 by J in a very small time; draw BC parallel to the 
 
 horizon, and CD perpendicular to AB ; let / be the force 
 which accelerates A''s motion ; F the acceleratinsr force in 
 the direction of gravity. 
 
 Then 2mfds= ada * (Vince's Flux. Art. 82.); 
 
 and F : f :: AC : AD :: AB : AC :: ds : d.v; 
 
 therefore, fds = Fdx, and 27nFd.v — '2mfds = ada, 
 
 ada 
 
 consequently F = 
 
 2m dx 
 
 hence, the effective moving force on A, in the direction ol' 
 
 Aada Aada . , p v, i 1 
 
 p-ravity, = -— , and A — - is that part 01 ^ s whole 
 
 ^ '' 2mdx 2mdx ^ 
 
 The differential notation is used'
 
 188 ACTION OF BODIES ON 
 
 weight, or moving force, which is sustained by the action 
 of the other bodies in the system ; that is, with wliich A 
 urges the machine in the direction of gravity. In the same 
 
 manner, B is the part of 5's moving force sustained ; 
 
 2mdy ^ ^ 
 
 Bhdh 
 
 and the weight at A which would balance this : B — 
 
 ° 2may 
 
 Bdii Bhdh . , 
 :: dy : dx (Art. 149-), therefore — -^ — is the 
 
 weight at A which would balance 5's pressure upon the 
 
 ^Bbdh Bdy . , ^ ^, . „ 
 
 machme ; and ; is that part of ^ s moving lorce 
 
 2mdx dx 
 
 Cede Cd% 
 
 which is sustained by B. In the same manner ; — 
 
 -^ 2mdx dw 
 
 is that part of A\ moving force which is sustained by C; 
 
 consequently 
 
 Aada Bhdh Bdy Cede Cdz 
 2mdx 2mdx dx 2m dx ax 
 
 and 2m X {Adx ■\- Bdy -\-Cd%) = Aada-t Bhdh ^ Cede, 
 
 and taking the integrals, which require no correction, 
 
 r. r. X ^«' ^^' ^^~ 
 
 2m X (Ax ■{■ By-\- C^) = j ! , 
 
 2 2^ 
 
 or 4m X (Ax -\- By-\- C%) = Aa~-^Bh'-\- Cc" *. 
 
 (375.) Cor. If any of the bodies move in a direction 
 opposite to that which is here supposed to be positive, the 
 space described must be reckoned negative. 
 
 * For this very concise demonstration, the Author is indebted ta 
 the suggestions of the Rev. D. M. Peacock.
 
 MACHINES IN MOTION, 
 
 189 
 
 Ex. 1. If the weights A and B be attached to the 
 lever AB, to find the velocity acquired by A during the 
 motion of the lever, round the pivot C, from an horizontal 
 to a vertical position. 
 
 Let CA = a, CB = 6, v = the velocity acquired by A ; 
 then a : b :: v : — = the velocity acquired by B ; therefore, 
 
 by the Proposition, 4fmAa — 4>mBb = Av"-\ — , and 
 
 „ Aa-Bb J, . ^ 
 
 v- = 4<ma~x — -^ — ^v^; and by extracting the square root 
 
 of this quantity, v is obtained. 
 
 Ex. 2. If the weight p raise q by the wheel and axle, 
 and descend through s feet, to find the velocity acquired 
 by^.. 
 
 OP 
 
 Let CA = a, CB = 6, and v = the velocity required.
 
 190 
 
 ACTION OF BODIES ON 
 
 Then a : h •: V '. — = q'% velocity, also 
 a 
 
 a : b :: s 
 
 = the space through which q is raised ; 
 
 , _ bos „ b-v~ 
 
 thereiore, A^mps — im x z=:pv -\- q x — — 
 
 a a- 
 
 , 9 pa' — qab 
 
 and V =4?H.9X 
 
 pa- -\-qb~ 
 
 -; whence i' is known. 
 
 Ex. 3. If two equal weights p, p, attached to a string 
 which passes over the pullies A and B^ raise the weight w 
 through the space WD, to find the velocity communicated 
 to w. 
 
 Suppose AB to be parallel to the horizon, and WDC 
 perpendicular to it. Take BC — CA = a ; WC = b ; 
 WB = c ; DC = x ; v = the velocity of w ; y = the velocity 
 
 of p ; DF the small increment of WD ; draw FG at right 
 angles to DB. 
 
 o 
 
 Then V : y .: DF : DG :: DB : DC, 
 
 andv- : y" :: DB' : DC' :: a^ + ^v' : .^•-, 
 
 therefore ?/' 
 
 .17' V 
 
 Also, H^Z), or h — x is the space 
 
 ar -f ,v- 
 
 which 7^ lias described in a direction perpendicular to the 
 horizon, and WB — DB, or c — ^ cr -^ .v', is tlie space
 
 MACHINES IN MOTION. 191 
 
 through which p has descended ; therefore, by the Pro- 
 position, 
 
 . A— 7, „^ 2pos'v" 
 
 4'm.2p,(c— ^ a- -\-x') — 4<7nw . (h — .v) = -= + wir, 
 
 a^ + <p 
 
 and v = ^m. (a" + c*-"^) x ~ — ^ r; ^^ ; 
 
 whence v is known. 
 
 THE END.
 
 UNIVERSITY OF CALIFORNIA LIBRARY 
 
 Los Angeles 
 This book is DUE on the last date stamped below. 
 
 MAY 2 4 1984 
 
 ^^^ 
 
 JVJ>^ 
 
 ■ k-l-il- 
 
 iS,V-' 
 
 \.\^ 
 
 ■asA 
 
 Fcrm L9-10Cm-9,'52(A3105)444 
 
 TR^ TTT^RARY 
 
 UNIVEHEj. ' ..' '. J (V\LiFORNIA
 
 7 Engineering 
 
 Library 
 
 n'?53