UC-NRLF B H 2MS fi^fi I OUTLINES OF NATURAL PHILOSOPHY, BEING HEADS OF LECTURES DELIVERED IN THE UNIVERSITY OF EDINBURGH, BY JOHN PLAYFAIR, PROFESSOR OP NATURAL PHILOSOPHY IN THE UNIVERSITY OP EDINBURGH, FELLOW OF THE ROYAL SOCIETY OF LONDON, SECRETARY OF THE ROYAL SOCIETY OF EDINBURGH, AND HONORARY MEMBER OF THE ROYAL MEDICAL SOCIETY OF EDINBURGH. VOL. I. THIRD EDITION. EDINBURGH: PRINTED FOR ARCHIBALD CONSTABLE AND COMPANY, EDINBURGH, Af'D LONGMAN, HURST, REES, ORME & BROWN, AND CADELL & DAVIES, LONDON. 1819. .YfflOSCKIJHFI JA Printed by P. ft, /5V? ADVERTISEMENT. . JLHESE Outlines are offered to the Public, from a persuasion that it will be found of use to have the elementary truths of Natu- ral Philosophy brought into a small com- pass, and, though not accompanied with de- monstrations, yet arranged in the order of their dependence on one another. I have endeavoured to express them concisely, and with all the clearness and accuracy in my power, referring, at the same time, to the works in which they are more fully treated of, as often, at least, as these works were not so common as to be already in the hands of every body. Demonstrations have been occasionally added, when it appeared that an improvement might be made on those which are usually given. The 237 VI ADVERTISEMENT. The Second Volume treats entirely of Astronomy ; and the arrangement of the propositions in their natural or logical or- der, has been particularly attended to. The great variety of the subject, or rather, of the aspects under which the subject appears, ren- ders this a matter of some difficulty ; and, accordingly, there are very few, even of the best treatises of Astronomy, in which the arrangement is not obviously defective. The Systeme du Monde of LA PLACE, and the Traite Elementaire of BIOT, are almost the only exceptions to this rule. To both of these I am under great obligations ; to the first in particular, — the most valuable and philosophical compend, I believe, which ex- ists at present of any science. In Physical Astronomy I have been more full than any where else ; and on the subject of Central Forces, have given all the inves- tigations. - The ADVERTISEMENT. Vll The Third Volume is intended to contain the Principles of Optics, together with those of Electricity and Magnetism. This will complete a work which lays claim to no other merit, but the very humble one of expressing elementary truths in a distinct manner, and placing them in a right order. EDINBURGH COLLEGE, October 1814. NOTE. Sept. 27. 1819. This Edition is printed from a copy prepared for the Press by the Author. EDITOR. CONTENTS CONTENTS OF VOLUME FIRST. INTRODUCTION. SECT. 1. 2. Properties of Matter, 7 DYNAMICS. SECT. 1. Of the Measures of Motion, 19 — — 2. First Law of Motion, - #6 — 3. Second Law of Motion, - . 33 i 4. Motion equably accelerated or retarded, 43 — 5. Motion of Projectiles, - 49 — 6. Motion, accelerated or retarded by a va- riable Force, - 52 MECHANICS. SECT. 1. Of the Centre of Gravity, 58 Motion of the Centre of Gravity, 65 — — £. The Mechanical Powers. The Lever, 68 The Balance, 74 The Wheel and Axle, 77 The Pulley, 85 The x CONTENTS, The Wedge, *.... Page 86 The Screw, - 89 The Funicular Machine, 92 SECT. 2. Friction, . 93 ...... 4. Mechanical Agents, 107 — 5. Motion of Machines, 116 — — 6. Descent of heavy Bodies on plain and curved Surfaces, - 126 Centre of Oscillation, , 130 Heavy Bodies descending on a Cycloidal Surface, 134 — 7. Rotation of Bodies. Rotation about a Fixed Axis, - 136 Rotation on a Moveable Axis, - 143 APPENDIX (to MECHANICS). Construction of Arches. 147 Strength of Timber, - W > 157 HYDROSTATICS. SECT. 1. Pressure of Fluids, >5teM 168 — 2. Solid Bodies floating on Fluids, 176 - 3. Phenomena of Capillary Tubes, 184 HYDRAULICS. SECT. 1. Fluids, issuing through Apertures in the Bottom or Sides of Vessels, 194 - 2. Conduit Pipes, and open Canals, 199 " 3. Percussion and Resistance of Fluids, 206 — — 4. Undulation of Fluids, or the Formation of Waves, 211 5. Hydraulic CONTENTS. xi SECT. 5. Hydraulic Engines. Engines moved by the impulse of Water, - Page 2H Machines moved by the weight of Water, - 217 Machines moved by the re-action of Water, 219 AEROSTATICS. SECT. 1. Heat, - - 223 2. Equilibrium of Elastic Fluids, PNEUMATICS. SECT. 1. Air as accelerating or retarding Mo- tion. Machines for raising Water, 256 Steam-engine, 262 Motion produced by Gun-powder, 267 Bodies impelled by Currents in the Atmosphere, or by the Winds, 273 Resistance of the Air to Projectiles, 276 2. Air, as the vehicle of Sound. Vibration of Sonorous Bodies, 281 Propagation of Vibrations through the Air, - 287 " 3. Air as the vehicle of Heat and Moi- sture, 291 Wind, - - 305 Rain, - - OUTLINES OUTLINES NATURAL PHILOSOPHY- INTRODUCTION, SECT. I. 1. JL HILOSOPHY is the knowledge of the general laws observed by the phenomena of nature, whe- ther in the intellectual or material world. 2, In the material world, the action which takes place among bodies, either produces a permanent change in the internal constitution of those bodies, or it does not. In the former case, the phenomena belong to CHEMISTRY, in the latter to NATURAL PHILOSOPHY. The action of the same cause may, by a change of cir- cumstances, pass from being an object of the one of these sciences to become an object of the other. The action of heat, when it expands or contracts the di- mensions of bodies, belongs to Natural Philosophy ; VOL. I. (2) A when 2 OUTLINES OF NATURAL PHILOSOPHY. when the same power burns and consumes bodies, its action belongs to Chemistry. When it converts wa- ter into steam it may belong to either science. In all this we exclude the consideration of the organic structure of bodies. 3. WTien from a comparison of a number of facts known from experiment or observation to be true, the existence of a more general fact is inferred, the inference is said to be made by INDUCTION. It is from induction that all certain and accurate know- ledge of the laws of nature is derived. 4. When general principles or axioms have been established by induction, we can often, by the ap- plication of mathematical reasoning, deduce con- clusions from them, as clear and certain as the prin- ciples themselves. This great advantage which Natural Philosophy seems to possess almost exclusively, arises from the cir- cumstance, that the action which it treats of, ex- tends to large masses of matter, and to considerable distances^ such as can be measured by lines and numbers. 5. The branch of knowledge which collects and classifies facts, is called Natural History. Natural History is here taken in its most general sense, such as it is understood by BACON ; in common lan- guage its objects are confined to what are called the three kingdoms, the Mineral, Vegetable, and Ani- mal. 6. We INTRODUCTION. 6. We are said to explain a phenomenon, when we shew it to be necessarily included in some phe- nomenon or fact already known, or supposed to be known ; and we consider one phenomenon as the cause of another, when we conceive the existence of the latter to depend on some force or power resi- ding in the former. 7. A fact assumed in order to explain a certain set of appearances, and having no other evidence of its reality, but the explanation which it gives of those appearances, is called a HYPOTHESIS. 8. If the number of appearances explained is very great, or if the explanation is founded on facts known to be true from evidence independent of those appearances, the explanation is called a THEORY. A theory is often nothing else but a contrivance for comprehending a certain number of facts under one expression. In the explanation of natural appearances, and in all inductive reasonings, Facts, though equally certain, may not be of the same value for the discovery of truth. BACON has classified facts, and explained their peculiar advantages as instruments of investiga- tion, in the 2d book of the NOVUM ORGAXUM. 9. When one system of events or appearances is similar to another, and when we infer that the (3) A 3 onuses 4f OUTLINES OF NATURAL PHILOSOPHY. causes in these two systems are also similar, we are said to reason from ANALOGY. Such reasonings will be more or less conclusive, ac- cording as the similarity is more or less considerable. Explanations; therefore, or theories founded on ana- logy, may have all degrees of evidence, from the least to the greatest. Many theories are founded entirely on analogy, 10, When a theory has been discovered by in- duction, it may be made use of by reasoning in a reverse order, for discovering new facts, and pre- dicting the result of new combinations. This will be best illustrated by the examples that will occur in treating of Physical Astronomy. 11. The evidence of a theory, increases with the number of facts which it explains, and the preci- sion with which it explains them. It diminishes with the number of facts which it does not ex- plain, and with the number of different supposi- tions that will afford explanations equally pre- cise. In the analytical OF inductive method of BACON, the possible theories are all excluded but one or at most a small number, before the explanation is attempt- ed. Novum Organum, Lib. n. cap. 16. A theory may not deserve to be rejected, though it does not explain all the phenomena, if it explain a great number, and be not absolutely inconsistent with INTRODUCTION. 5 with any one. A single fact, inconsistent with a theory, may be sufficient to overturn it 12. In tracing the necessary connection of na- tural phenomena, this axiom is often of use : No- thing exists in any state that is not determined by some REASON to be in that state rather than in any other. Hence, two things of which the conditions are deter- mined by reasons that are precisely the same, are in all respects similar to one another. Hence, al- so, if there are two conditions, and no reason to determine a subject to be in one of them rather than another, we are to conclude that it is in nei- ther. This axiom has been called the principle of the SUFFICIENT REASON. It was used by ARCHI- MEDES in his demonstration of some propositions in mechanics; but it was first stated as a general principle of philosophic reasoning by LEIBNITZ. It may be used to great advantage for demonstrating the more simple propositions of geometry as well as of mechanics. 13. Experiment is not only necessary in the in- vestigation of truth, but it is useful for proving truths that have already been investigated. In this latter application of it, it serves three pur- poses: •i. It verifies the conclusion of our reasonings, and dis- covers whether any element has been left out, or any (> OUTLINES OF NATURAL PHILOSOPHY. any error committed in the course of the investiga- tion. u. It exemplifies the application of general principles to the explanation of particular facts. in. It impresses the truth both of the principles and the conclusion most strongly on the mind. Though an experiment can only prove a proposition in one particular case, yet from a combination of two or more experiments, such evidence of the ge- neral proposition may arise, as to fall little short of demonstration. 14. The study of Natural Philosophy is accom- panied with great advantages. i. It extends man's power over nature, by explain- ing the principles of the various arts which he practises. II. It improves and elevates the mind, by unfolding to it the magnificence, the order, and the beauty manifested in the construction of the material world. in. It offers the most striking proofs of the benefi- cence, the wisdom, and the power of the CREA- TOR. SECT. II. INTRODUCTION. SECT. II. PROPERTIES OF MATTER. 15. XJODY is a substance, extended in three dimen- sions, impenetrable, and moveable. Gravity or weight is often included in the description of body ; it is left out here, because, though it ap- pear to be universal, we can conceive a substance without gravity, which, nevertheless, should possess extension, impenetrability, and mobility. 16. By the Impenetrability of body is meant that no two particles of matter can at the same time oc- cupy the same identical portion of space. a. If bodies could compenetrate, all matter might be united into any space, however small, or it might be all annihilated. — MUSCHENBROEK, IntrocL ad Phil. Nat vol. i. § 81. b. The bodies that yield to pressure, and those that do not, serve equally to prove the impenetrability of matter. — BOSCOVICH, Theoria, Phil. Nat. § 41. 17. Body, being extended, is divisible without limit, or, as it is usually called, ad infmitum, sup- posing that the instruments of division are mere mathematical lines or points. a. For 8 OUTLINES OF NATURAL PHILOSOPHY. a. For it is easily proved, on this supposition, that the extension of lines ad infinitum, and their divisibi- lity ad infinitum, are necessarily connected with one another. 18. The actual division of body into parts, as the parts must be of a finite magnitude, must necessa-. rily be limited ; it is, nevertheless, capable of being carried to such extent, that the parts shall be of in- credible minuteness. a. We have examples in the gilding of silver-wire, in the propagation of odours, in the colours produced by chemical solutions, &c. — REAUMUR, Mem. Acad. des Sc. 1713, p. 204. BOYLE de Mird SubtiUtate Effluv'wrum, MUSCIIENBROEK, ubi supra, § 72. 19. All bodies are inclosed by one or more boun- daries, and therefore possess figure ; they have also the capacity of receiving an indefinite variety of figures. a. Bodies differ in their capacity for receiving and maintaining different figures. b. Some receive new figures with difficulty, but main- tain them easily. Such are the bodies usually call- ed solid. Others receive any figure easily, but cannot main- tain it without the assistance of other bodies. Such are the bodies usually called Jluid. c. Many bodies have figures, which, in their natural state, are peculiar to them. This is the case not only INTRODUCTION. 9 only with plants and animals, but with certain mi- neral and chemical productions called Crystals, usu- ally bounded by plain surfaces. 20. Motion, of which all bodies are susceptible, is the continued change of place. 21. It is a general law in the material world, that no body loses motion in any direction, without communicating an equal quantity to other bodies in that same direction ; and conversely, that no body acquires motion in any direction, without diminish- ing the motion of other bodies by an equal quanti- ty in that same direction. What relates to the motion of bodies, will be consi- dered more fully hereafter ; it is mentioned here merely for the sake of order, as being part of the definition of Body. 22. The force with which the parts of bodies re- sist any endeavour to separate them is called cohe- sion. Q. We may conceive a body to be made up of an as- semblage of small indivisible particles or corpuscles adhering to one another, with forces that are greater as the distance of the particles is less. These cor- puscles are called the elements of body. b. Hardness, softness, tenacity, fluidity, ductility, are modifications of cohesion. c. Smooth 10 OUTLINES OF NATURAL PHILOSOPHY. c. Smooth surfaces may be made to cohere, by bring- ing them very close to one another ; but, In general, cohesion cannot be produced between two bodies but by the intervention of fluidity. d. Cohesion is not a general property of body ; it does not belong to elastic fluids, which are kept together by pressure or by gravity. c. Adhesion is sometimes distinguished from cohesion ; the former being the force that connects the par- ticles of liquids ; the latter that which connects the particles of solids. They are no doubt modifications of the same force. 23. All bodies at the earth's surface, if left to themselves, descend in straight lines towards that surface. This tendency or disposition to fall is called weight, a. The weight of a given body may be employed tec measure the weights of all other bodies. 24. The directions in which bodies fall in diffe- rent places of the earth, tend nearly to the centre of the earth. a. At points not very far from one another on the surface, the directions in which bodies gravitate are nearly parallel, the distance of the earth's centre be- ing great in respect of the distance of the bodies from one another. If the distance is a nautical mile, that is, 6075 feet pearly, the angle made by the directions of gravity is one INTRODUCTION. 11 one minute; if 60 of those miles, it is a degree, &c. £. A plane at any place perpendicular to the line in which bodies gravitate, is called a horizontal plane ; and any plane passing through that line is called a vertical plane. 25. If we conceive the weight of bodies to be produced by a force applied to each of their ele- mentary particles, urging it downwards, this force is palled gravity. a. Gravity must be distinguished from Weight : the weight of a body is the product of the gravity of a single particle by the number of particles. Thus, if W be the weight of a body, g the gravity of a single particle, N the number of particles, 26. In all bodies the force of gravity is the same. «. This is proved by the fact, that all bodies, what- ever be their weight, fall at the same rate to the ground, when the resistance of the air is remo- ved. 27. The weights of two bodies are to one another as the quantities of matter in those bodies, a. Let 12 OUTLINES OP NATURAL PHILOSOPHY*. a. Let W and W be the weights of two bodies, M and M' their quantities of matter, _W_ M W rr M'* The quantity of matter in a body is called its mass. 28. Small empty spaces are disseminated through all bodies in greater or less abundance, and are called pores. This property of bodies is also called powsity. a. In solids, the pores can be seen sometimes by the naked eye, and almost always by the microscope. b. In fluids, the dissemination of vacuity through the mass is proved by experiments, though it is not perceived by the eye. — MUSCHENBROEK, § 91. N°3. c. The ratio of the quantity of matter to the quanti- ty of empty space contained within the superficies of any body is wholly unknown. It is probable, that even in the densest bodies, the quantity of solid matter is very small, compared with the quantity of empty space. NEWTON'S Op- tics. Book ii. Part iii. Prop. 8. d. From the porosity of bodies, it follows, that the particles of matter are not on all sides in contact with one another. They may perhaps only touch one another in a few points. 29. If INTRODUCTION. 13 29. If in two bodies of which the bulks are B and B', and the weights W and W, the weights W B' be proportional to the bulks, or ™- m ^-, the space occupied by the pores of these bodies bears the same proportion to that which is filled by solid matter. Bodies of this kind are said to have the same den- sity. 30. The weight of a body divided by its bulk, W or ip is called the specific gravity of the body. a. The weight of a body, in strictness, cannot be di- vided by its bulk, the two quantities being dissimi- lar, and incapable of comparison. But if we ex- press the bulk of all bodies numerically, by refer- ring them to the same unit, a cubic inch for ex- ample; and the weight of all bodies, by referring them to a similar unit, the weight of a cubic inch of water, any number in the first series may be di- vided by the corresponding number in the second : the quotient will be the specific gravity of the body to which the numbers belong. Hence the specific gravity multiplied by the bulk, gives the weight; or if S is the specific gravity r B x S = W. 31. A 14 OUTLINES OF NATURAL PHILOSOPHY. 31. A fluid is a body so constituted, that its parts are all ready to yield to the action of the least pressure. 32. Elasticity is a power by which a body, when its figure is altered by the action of any force, re- sumes that figure as soon as the force has ceased to act, This power is found both in solid and fluid bodies. 33. Light is something which has the power of rendering objects visible to the eye. is? Whether light is a substance, or merely a species of movement impressed on a certain substance, will be considered hereafter. In respect of light, bodies are either luminous, opaque, or transparent : Luminous, when they shine, or give out light without having received it from other bo- dies ; opaque, when they obstruct light ; and trans- parent, when they transmit it. 34. All bodies are subject, by expansion and contraction, to change their magnitude or volume, according to certain laws, and within certain li- mits. When expanding, they produce in us (un- der certain conditions) the sensation of heat ; and when contracting, the sensation of cold. They are therefore said to become hot or to acquire heat in INTRODUCTION. 15 in the former case, and to become cold or to lose heat in the latter. 35. Certain minerals have a power of attracting iron, and of communicating to pieces of that me- tal the power of attracting and repelling one ano- ther. The power thus communicated is called magnetism. Bodies having magnetism have also a tendency, when left free, to point toward a cer- tain quarter of the heavens. This is called pola- rity. Magnetism is peculiar to iron, and to two other me- tals, nickel and cobalt. Even in the bodies where it resides naturally, it is not always of the same in- tensity. 3(5. Some bodies acquire by friction the power of attracting and repelling certain other bodies. This power is called electricity. 37. The chemical action of metallic substances on one another, produces a like tendency in bodies which communicate with those substances accord- ing to certain conditions. This power is called galvanism. a. The four properties or powers last mentioned agree in this peculiarity, that they are communi- cated from one body to another by mere apposi- tion without any visible admixture, or transfusion of substance. In this they differ from the other pro- perties 16 OUTLINES OF NATURAL PHILOSOPHY. perties of body. Whether a passage of some invi- sible substance from the one body to the other it> not to be inferred in such instances, will hereafter be considered. b. As the action of bodies on one another generally involves motion, the properties or laws of motion constitute one of the main objects of natural phi- losophy. The doctrine which treats of those pro- perties is, however, modified by the constitution of the bodies to which motion is communicated. 38. When bodies are free to obey the impulses communicated to them, the science which treats of their motion is called DYNAMICS. a. Dynamics is the most elementary branch of the doctrine of motion, and the most general in its principles. The term signifies literally the doctrine of power ; power or force being known to us only as the cause of motion, and being measured by the motion it produces. b. In Dynamics, we abstract entirely from the figure of the body moved, and treat of it as if its matter were all concentrated in a single point. 39- When bodies, whether by external circum- stances, or by their connection with one another, are not left at liberty to obey the impulses given, the principles of dynamics must receive a certain modification before they can be applied to them. The INTRODUCTION. 17 The science of dynamics, thus modified, is called MECHANICS. «. According to this distinction, the motion of a body falling freely to the ground belongs to Dynamics ; the motion of the same body descending on an in- clined plane, belongs to Mechanics, &c. t. The doctrine of Machines belongs to mechanics; for in every machine, the connection of the parts prevents them from immediately obeying the im- pulses which they receive. 40. When the bodies to which motion is commu- nicated are fluid, another modification of the prin- ciples of dynamics takes place, which constitutes the science of HYDRODYNAMICS. In each of the three sciences just named, there are cases where the forces balance one another, and pro- duce, not motion, but rest. These cases in dyna- mics and mechanics constitute a particular branch, to which the name of Statics has been given. The similar cases in fluids form the branch of hydrodyna- mics, called Hydrostatics. Hydrodynamics is also subdivided, according as the fluids are incompressible, like water, or elastic, like 41. ASTRONOMY is .the science which describes the phenomena of the heavenly bodies ; and PHY- SICAL ASTRONOMY is the part of that science which VOL, L B applies 18 OUTLINES OF NATURAL PHILOSOPHY. applies the principles of Dynamics to explain those phenomena. 42. OPTICS is the science which treats of the pro- perties of Light, and of the laws of Vision. !;?•>>•' ?vi[-j*0~rf ' [rt DYNAMICS. • DYNAMICS. 19 DYNAMICS. SECT. I. OF THE MEASURES OF MOTION, 43. \V HEN a body changes its place continually, it is said to MOVE, or to be in MOTION. 44. A moving body cannot pass from one point of space to another without passing through every intermediate point ; that is, whatever be the path it describes, it must intersect all the planes that can pass any how in the interval between the two points. 45. The ideas of Space and Time are both neces- sarily involved in the idea of motion. 46. Time is conceived as a quantity consisting B2 of £0 OUTLINES OF NATURAL PHILOSOPHY. of parts which can be compared with one ano- ther, 47. Two events which are determined hy cir- cumstances precisely the same, are conceived to happen in equal portions of time. 0. A stone will fall to the ground from a given height in the same space of time to-day that it did yester- day, or that it will do to-morrow, or at any future period. It is not necessary that the stone should be of the same weight, for, as has been already pro- ved, the weight of the falling body does not affect the time of its descent (£6). It is on this proposi- tion, generalized and applied to the vibrations of a pendulum or a balance, that the going of a clock or watch is taken for a measure of time. Thus it is on the principle of the sufficient reason that time is divided into equal portions. b. The imperfections of mechanism require, however, that our chronometers should be frequently checked by a comparison with the motion of the heavenly bodies ; but as the circumstances that may influ- ence the duration of the events are not easily dis- covered in the case of those bodies, it has requi- red the highest improvements in science to derive an exact measure of time from astronomical obser- vation. 48. Time considered as a quantity increasing or flowing uniformly, without regard to the measure- ment DYNAMICS. gl ment or comparison of its parts, is called Absolute Time : when subjected to measurement or reckon- ing of any kind, it is called Relative Time. The least portion of time that we can measure is about one-fourth or one-fifth of a second ; yet, were we to trace the progress of nature minutely, a hun- dred or even a thousandth part of a second would be found to be distinguished by great changes. 49- The velocity of a moving body, or the rate of its motion, is said to be uniform, when the lines it passes over (or, as it is usually expressed, the spaces it describes) in equal times, are all equal to one another. a. In measuring the velocities of bodies, it is conve- nient to take a certain portion of time for the unit, in terms of which all other portions of time are to be expressed. The unit here assumed is one se- cond. 50. The space which a body moving with a uni- form velocity passes over in any time, is had by multiplying the time into the velocity, that is, by multiplying the number of seconds the motion has continued into the space moved over in one se- cond. a. Hence if S be the space passed over in the time T, with the uniform velocity V, (all of them being ex- pressed numerically), S = V x T. 51. In 22 OUTLINES OF NATURAL PHILOSOPHY. 51. In uniform motions, therefore, the velocity is S equal to the space divided by the time, or V ==?- 52. In uniform motions also, the time is equal S to the space divided by the velocity, or T = -. In all these theorems T is expressed in seconds, and V in some known measure of length, the same ia which S is also expressed. 53. The velocity of a body is sometimes estima- ted in a direction different from that in which it ac- tually moves. Thus, (fig. 1 .) if while a body B moves along the line AC, a perpendicular BD be continually drawn to the line AE, which makes with AC a given angle, the velocity with which D moves along AE is called the velocity of B in the direction AE. 54. The velocity of a body moving in a given direction, is to its velocity estimated in any other direction, as radius to the cosine of the angle which the two directions make with one another ; or if V be the velocity of the body, j8 the angle which the path of the body makes with another line, V cos & is the velocity reduced to the direc- tion of that line, If DYNAMICS. £ If S be the space described by the body in its own path, the space described by the perpendicular in the other line will be S cos /3. Hence also if the velocity of the body in its own path is uniform, it will be uniform when reduced to any other direction. 55. -The change which any variable quantity undergoes in an infinitely small portion of time, is called the Momentary Increment of that quan- tity. Thus if S is the space described by a moving body in the time t, and if V be its velocity, supposed vari- able, at the end of the time t, if we suppose t to be increased by an indefinitely small instant, the change of S and of V in that instant are called their Mo- mentary increments. The increments are denoted by the same letters that express the variable quantities, with a point over them. Thus S, V, T, are the momentary incre- ments of S, V and T. Though the variable quantities may decrease as well as increase, the momentary change is called an In- crement in both cases, but is accounted negative when the quantity diminishes. 56. Though the velocity of a body be variable, it may at any point be taken as uniform for an in- definitely small portion of time, and the increment of the space will be, just as in the case of uniform motion £4 OUTLINES OF NATURAL PHILOSOPHY. motion (47.), equal to the velocity multiplied into the indefinitely small portion of time, during which the motion is supposed uniform. Thus, if S he the increment of the space, t the correspond- ing increment of the time, and V the velocity* S zi V t ; hence These formulae apply to motion of every kind, whether accelerated or retarded, and express generally the re- lation between the velocity, the increment of the space, and the increment of the time. 57. The quantity of motion in a moving body is estimated by the product of the mass, or quantity of matter, multiplied by the velocity. The quantities of motion in two bodies are the same, when their velocities are inversely as their mas- ses. If M and M' be the masses of two bodies, V and V their velocities ; if M : M' : : V : V, the quantities of motion are equal, for M x V = M' x V. 58. When DYNAMICS. 25 58. When one body changes its place relatively to another, it is said to have a relative motion with respect to that other body. Two bodies if they move with equal velocities toward the same side, in the same or in parallel lines, will have no motion relatively to another. If two bodies move with equal velocities in the sanfe straight line, but in opposite directions, their rela- tive motion is the sum of their real motions ; if in the same direction, it is the difference of their real motions. The same is to be said of their relative velocities. The use of the signs + and — brings the two parts of this and all similar propositions under one enunciation. 59. The path of one moving body relatively ta another is determined, by supposing the latter to stand still in any point of its path, and by inqui- ring in what line the other must move, and with what velocity, so that it may approach to the for- mer (supposed at rest) or recede from it, at the same rate, and in the same direction, that it actually does- when they are both in motion. SECT. OUTLINES OF NATURAL PHILOSOPHY. SECT. II. FIllST LAW OF MOTION. 60. A BODY must continue for ever in a state of rest, or in a state of uniform and rectilineal mo- tion, if it be not disturbed by the action of an ex- ternal cause. a. If the body is at rest, it must remain at rest ; for if there is no action of another body, there can be no- thing to determine it to move in one direction more than in another. 1). If the body is in motion, it will continue to move in the same direction ; for there is nothing to de- termine its deflections to be in one line more than in another line. c. Lastly, it cannot change its velocity ; for if its velo- city change, that change must be according to some function of the time ; so that if C be the velocity which the body has at any instant, and t the time counted from that instant, V the velocity at the end of the time £, the relation between V, C, and t must be expressed thus, V — C -{- A t™ + B tn +, &c. Now, there is no condition involved, in the nature of the case, by which the coefficients A, B, &c. can be determined to be of any one magnitude rather than of any other ; for by the supposition, there is no other body which acts on the given body, and which, by any relation to it, can afford an equation for de- 3 termining DYNAMICS. 27 termining A or B, &c. ; each of these coefficients is therefore either indeterminate, or it is equal to nothing ; but no one of them can be indeterminate, or can admit of innumerable values, otherwise V itself would be indeterminate, and might have more values than one at the same instant, which is im- possible. A and B, &c. must therefore be each equal to 0, so that V = C ; in other words, the velocity remains uniform. D\ALEMBERT has given a different demonstration of this, Dy?iamique9 chap. !«. See also another by EULER, Mechanica, torn. 1. § 63. This proposition is the first law of motion, and, toge- ther with the proposition § 21. constitutes what is termed the INERTIA or INACTIVITY of matter. d. The inertia has been considered by some philoso- phers as merely the expression of a law of human thought, by which we are detennined, when we see a change, to infer the action of a cause. This seems to be inaccurate. That a change never hap- pens in the motion of any body, without an equal and opposite change in the motion of some other body, is as much a fact, independent of the mind that perceives it, as impenetrability, cohesion, or any other phenomenon of the material world. e. If a body move in a curve, the continued action of an external force must be inferred: if that action were to cease at any point, the body would continue its motion in a straight line touching its curvilineal path in that point. 61. The cause of motion is denominated Force, and a force is always measured by the effect, that is, 28 OUTLINES OF NATURAL PHILOSOPHY. is, by the motion which it produces in a given time. a. Two forces are said to be represented by two lines y when the motions which they singly produce are in the directions of those lines, and proportional to them. 62. Let a body at A (fig. 2.) be acted on by two forces at the same instant, one of which acting alone, would cause it to move uniformly over AB, in a given time ; and the other acting alone, would cause it to move over AC, at right angles to AB, in the same time. The velocity of the body in the one of these directions, will not be changed by the force impelling it in the other. For if the motion of A in the direction AB, is acce- lerated by the force AC, it will be retarded by the action of the opposite force AC'; but it must also be accelerated by AC', for AC and AC' are alike situated in respect of AB. Now, this is absurd ; therefore, &c. The velocity of a body in the direction of a line diffe- rent from its own path, is measured as in § 53. 63. If the lines which each of two forces, acting, singly, would have caused a body to describe in the same time, be at right angles to one another, the line which it will describe in that time, when both the forces act on it at once, is the diagonal of the rectangle under the two first-mentioned lines. 2 64. If DYNAMICS. 29 64. If the lines which each of two forces acting singly would have caused a hody to describe in the same time, make any angle whatsoever with one another, the line which the body will describe in that time, when both the forces act on it at the same instant, is the diagonal of the parallelogram under the two first-mentioned lines. a. This is the celebrated theorem known by the name of the COMPOSITION OF FORCES. The most remark. able demonstrations of it are by DAN. BERNOULLI, Comment. Petrop. torn. 1.; D'ALEMBERT, Opuscules^ torn. 1. cinquieme memoire ; LA PLACE, Mecanique celeste, torn. 1. § 1. POISSON, Mecanique, liv. i. § 18. & 14. All these, though ingenious, are too elaborate and difficult to be accounted elementa- ry, and are, besides, very remote from the train of reasoning by which the truth was originally disco- vered 65. If two forces be represented by the lines a and 6, which contain an angle = z9 the force compound- ed from them will be y a2 + 2 a b cos z + b2 : for this is the diagonal which, in a parallelogram ha- ving the sides a and b, and the contained angle z, subtends the supplement of z. Also the sine of the angle which this diagonal makes with a, is = b sin z. 66. If SO OUTLINES Of NATURAL PHILOSOPHY. 66. If three forces which impel a body at the same time, be proportional and parallel to the three sides of a triangle, so that the angle contained by any two of them is the supplement of the angle contained by the corresponding sides of the triangle, these forces will be in equilibrium. a. The converse of this is also true, viz. that if the three forces are in equilibrium, they are propor- tional to the sides of a triangle, &c. b. When three forces are in equilibrium, any two of them is greater than the third. p. If three forces are in equilibrium, they are all in the same plane. 67. If there be an equilibrium between three forces, AB, AC, AE, (fig. 3.) and if from any point F, in the direction of one of them, perpendiculars, FG, FH, be drawn to the other two, these perpen- diculars will be inversely as the forces AC, AB, on the directions of which they fall. For if FC, FB are drawn, the triangles FAC, FAB are equal, as they are on the same base AF, and have equal altitudes, the perpendiculars from B and € on the diagonal AE. Therefore FG x AC - FH x AB, or FG : FH : : AB ; AC. 68. If three forces are in equilibrium, and if perpendiculars be drawn to their directions, inter- secting one another in three points, the sides of the triangle so formed will be proportional to the forces. DYNAMICS. 31 forces, to the directions of which they are perpendi- cular. 69. If any number of forces act on a body, pro- portional to the sides of a rectilineal figure, and also parallel to them, in such a manner that the angle which the direction of each force makes with that of the contiguous force, be the supplement of the angle which the corresponding sides of the figure make with one another, these forces will be in equi-, librium. This proposition is true, whether the sides of the recti- lineal figure be all in one plane or not. If they inclose a space, that space may be extended either in two or in three dimensions. 70. If there be an equilibrium among any num- ber of forces, which are in different planes, but ap- plied to the same point ; if, through that point, three straight lines or axes, be drawn at right angles to one another, (one of them in a plane perpendi- cular to that of the other two^, and if every force be resolved into parts in the directions of the three axes, then shall the sums of the opposite forces, in the direction of each axis, be equal to one ano- ther. This theorem furnishes three independent equations. a. If F, F', F", F'", &c. be the forces; a, a', a", a'", &c. the angles which they make with one of the three 32 OUTLINES OF NATURAL PHILOSOPHY. three axes ; p9 /3', p"9 fl'"9 &c. the angles which they make with another of the axes, and y, y', y", y'", &c. the angle which they make with the third, F cos a + F' cos a' + F" cos a" + F'" cos a!" +, &c. = 0 F cos ft 4- F'cos /3' + F" cos ^s" + F"' cos /3'" +, &c. = 0 P cos y + F' cos y' + E" cos 7" + F/x/ cos y*" +, &c. = 0 When all the forces are in one plane, there are only two axes, and the equations are therefore reduced to the two first. Also as .£ is in this case the comple- ment of a, cos /3 = sin a, and so of the rest ; there- fore the equations are F cos a + F' cos of + F'' cos a" +, &c. =• 0 .and F sin a + F' sin a' + F" sin a' +, &c. — 0 SECT. DYNAMICS* SECT. III. SECOND LAW OF MOTION. 71. JL HE action and reaction of bodies on another are equal ; in other words, the changes in the quantities of motion of the bodies which act, and of the bodies which are acted on, are equal, and in opposite directions. This comprehends two cases, according as the bodies act merely in consequence of the motions impres- sed on them, as in impulse or collision ; or in con- sequence of some unknown agency, which commu- nicates motion from the one to the other, though they be distant, as in the cases of gravity, magne- tism, &c. Both kinds of action are subject to the same law : In the first, the quantity of motion gained by the one body, is just equal to that lost by the other in the same direction : In the second, the quantity of motion gained by the one in any direction, is just equal to that gained by the other in the opposite direction. This last proposition is only known from experience : the first may be pro- ved from the nature of Body, and depends on the theorem in the next article. VOL. I. c As* 34 OUTLINES OF NATURAL PHILOSOPHY. As we have already derived from the first law of mo- tion, the proposition usually reckoned the Third, viz. that the motion communicated, is always in the direction of the force "impressed, § 62. ; the laws of motion, according to this arrangement, are reduced to two, 72. If two bodies meet with velocities which are inversely as their masses, and are directly opposed to one another, they will remain at rest. If the bodies are equal, and have equal velocities, the proposition is evident. If A be double of B, but have only half the velocity of B ; then A may be divided into two equal parts to B, each of which will have half the velocity of B. One of these striking against B, would destroy half its velocity ; and the other striking against it, would destroy the other hah0. The same is true whatever multiple the one body be of the other, and therefore also whatever ratio the one bear to the other. — Vid. D^ALEMBEET, Dyna- mique, § 46. p. 51. Hence bodies that have equal quantities of motion,, have equal forces, or equal powers to produce mo- tion. 73. The quantity of motion in a system of bodies estimated in any given direction, is not 2 changed DYNAMICS. 35 changed by the action of those bodies on one ano- ther. This follows from § 71. 74. If A and B be the masses of two bodies that move in the same straight line toward the same parts with the velocities a and b ; and if A and B are hard or soft bodies, so that after colli- sion they go on together, with the velocity v, If b is negative, so that the bodies meet one another, the velocity after collision, or v = — £ " } ; and A. -f- A5 if A a = B 6, the velocity after collision is nothing, as before demonstrated. When the bodies are equal, these two formulas be- a -j- b , a — b come — ' — and -- . 2 2 If B is at rest, or b — 0, the formula is v = -- ~ . .A. -p jj If B is an immoveable obstacle, it may be considered as a body that is infinite in respect of A, so that the denominator A -j- B is infinite, and therefore c 2 75. If 36 OUTLINES OF NATURAL PHILOSOPHY. 75. If A overtake B, the velocity lost by A is B(a— b) .. , AB(a— b) —r— r-~ ; and the motion lost hy it is — -— — ^— ; xV -J- JL> A + J3 the velocity gained by B is . ,~>-^, and the ±> motion gained by it is — -~ — =~— , the same that .A. -j- Jj was lost by A. When B meets A, b becomes negative, and so the velocities gained or lost, are A X> 9 A. ~p -t* A (a + 76. If the two bodies before collision move in lines making a given angle with one another, their path after collision may be found, on the principle, that there is no change made in the total quantity of the motion of bodies by their action on one ano- ther, in whatever direction their motion be estima- ted. If AC (fig. 4.) represent the velocity and direction of the motion of A, BC the velocity and direction of the motion of B, before collision ; CG their joint path, and their velocity after collision: Let BD and GF be perpendicular to AC; then, because the quantity of motion in the direction AC re- mains DYNAMICS. 37 mains the same, and also in the direction BD, we have the equations, A . AC 4- B . DC = (A + B) FC ; and Hence CG may be found. 77. When the bodies between which the collision takes place are elastic, or such, that after suffering an alteration in their figure by collision, they re- sume that figure with a force which causes them to separate from one another, the result is considerably different from any of the preceding, A Jpody is ac- counted perfectly elastic, when, in restoring its fi- gure, as much motion is produced, as was destroyed in altering it. 78. If A and B are two perfectly elastic bodies, moving with the velocities a and b in the same di- rection, so that A overtakes B, the velocity of A Aa — Ba-f 2B& after the stroke is - . . ' - ; and that J\. -j- ±> P-D 2 A a— A+B A + The investigation turns on this ; that the body which overtakes the other must lose twice as much velo- city as if it were hard; and the other gain twice as much. If 38 OUTLINES OF NATURAL PHILOSOPHY. If the bodies move in opposite directions, one of the velocities, as b, must be made negative. If the bodies are equal, the velocity of A after the stroke is — 6, and that of B is — a. The bodies therefore exchange velocities. When the velocity of either body, after collision, comes out negative, we are to understand its direc- tion to be contrary to that which is accounted all .firmative. 79- The difference of the velocities of two per- fectly elastic bodies, is the same before and after collision ; but the body which had the greatest velo- city before collision, has the least after it. If A and B be the masses of the bodies, a and b their velocities before collision, a and b' their velocities after it; a — b = b' — a. Hence, also, a + ct — b -|- b'. Hence, too, the relative motion of the bodies is not changed by their collision. SO. In perfectly elastic bodies, the sum of the products made by multiplying each of the bodies into the square of its velocity, is the same after col- lision that it was before it. For DYNAMICS. 39 i For as the quantities of motion before and after colli- sion are the same, (73.) A a -|- B b _= A of + B l/9 or A(a — a') = B (^ — 5); and since (79) <* 4- a7 = 6' -t- b, therefore A (a2— Vs) = B (&* — £*), or Afl* + B6»- Aa" + B b*. See another demonstration, in MACI-ACRIX'S Account of NEWT ox's Discoveries, Book n. chap. iv. § 12. 81. If between two unequal elastic bodies A and C, a third B be interposed ; and if the least A, be made to strike with any given velocity on B, the motion communicated to C will be the greatest po«ble, when B k a mean proportional between A andC. It is easily shewn, from § 78. that the velocity com- municated to C is = 4ABa 4Afl (A + B) (B+C) This fraction is a maximum, when the denominator 1C A -f B -h \ - -f C is a minimum, that is, since B A and C are given, when B5 — AC, or when B is a mean proportional between A and C. Tbe 40 OUTLINES OF NATURAL PHILOSOPHY. The velocity of-C is A + ^ + c *v 4f A a , , . p rt lono 4 AC a If the number of the bodies in geometrical progres- sion be increased without limit, the quantity of motion communicated to the last, from a given quantity of motion in the first, however small, may also be increased without limit. Notwithstanding this, as all the bodies move backward after collision but the last, if they form an increasing series, the sum of all the motions in the direction of the first mover continues = A a. Also the sum of the pro- ducts of each body, into the square of its velocity, after collision, remains as it was before, equal to 82. If an elastic body strike against an im- moveable plane, it will be reflected from it, so that. its direction before and after collision will make equal angles with the plane, but toward opposite sides. If the elastic body strike on the plane CD (fig. 5.) in the direction AB, it will be reflected in the di- rection BE, so that the angles ABC, EBD shall be equal. 83. If DYNAMICS. 41 83. If two elastic spherical bodies A and B, (fig. 6.) moving in directions, and with velocities re- presented by the lines AA', BB', come into contact when at A' and B', and if AB' be the line which joins their centres at that moment : Then if the ve- locity of A be resolved into two, AC, A'C, the one perpendicular to AB', and the other parallel to it ; and if the velocity of B be resolved in like manner into BD, B'D ; after collision A will retain the velo- city represented by AC, and B that represented by BD, unchanged ; but the velocities A'C, B'D, will be changed by the collision, into others which may be found by article 78, and if these be a' and b' ; then the path of A after the stroke will be the dia- gonal of a rectangle under A'E (~AC) and a', and the path of B, the diagonal of a rectangle under B'F (zrBD) and V. Thus the paths of the bodies after collision are deter- mined. 84. If the bodies are imperfectly elastic, that is, if, in restoring their figures, they do not generate velocities equal to those which were destroyed in altering them, their motions after collision are de- termined, by supposing that the motion lost or gain- ed is not double, but has a given ratio to tjiat which would be gained or lost if the bodies were perfectly hard, as at $ 75, When 42t OUTLINES OF NATURAL PHILOSOPHY. When the bodies are imperfectly elastic, the sum of the products of the bodies into the squares of their velocities does not continue the same after collision that it was before it. The quantity of motion, however, estimated in a given direction, remains the same in every case. There is another law which extends to the collision of all bodies, whether elastic, unelastic, or imperfect- ly elastic. It is, that if each body be multiplied into the square of the change that has taken place in its velocity by the collision, the sum of these products is a minimum ; that is, using the prece- ding notation A (a — a')2 -f B (b — b')9 is the least possible, or is less in the case which actually takes place in nature, than if the difference between a and a', b and &', were other than they are. This law was discovered by MAUPERTUIS. It is not de- monstrated from principle, but is collected from the preceding propositions by induction. SECT. DYNAMICS. 43 SECT. IV. OF MOTION EQUABLY ACCELERATED OR RE- TARDED. 85. W HEN the motion of a body varies either in direction or velocity, as this cannot arise from any thing in the hody itself, it must he ascrihed to the action of an external cause or force. The simplest case is, when the direction remains the same, and when the velocity only varies. 86. When the velocity of a hody changes, the cause of that change is called an accelerating or retarding force, and is measured by the change of the velocity produced in an indefinitely small por- tion of time, or by the increment of the velocity divided by the increment of the time. If F be the accelerating force, v the momentary in- crement of the velocity, i that of the time, Hence also v = F t, and /=_.'. r It 44 OUTLINES OF NATURAL PHILOSOPHY. It has been disputed, whether this expression of the force be a necessary truth, or one known only from experience. D'ALEMBERT, Elemens de Phil. Mdan- geS) torn. 4. p. 197. It seems, however, to be in fact a definition, rather than a theorem. We have no distinct idea attached to the word FORCE, which we can compare with that conveyed by the formula -, in order to see whether there is a t necessary agreement between them or not. But as the quantity ^ is of great importance, and fre^- t quent recurrence in mechanical investigations, it is convenient to have a term to denote it. Though any term might be employed for this purpose; yet as the thing called FORCE, is conceived to be always greater, the greater the change of velocity which it produces in a given time, or to increase and diminish just as — does, we may, without diverting the word Force from its usual sig- nification, employ it to denote the quantity _, or the t momentary increment of the velocity divided by the correspondent increment of the time. FORCE, in dynamics, has, in reality, no other signification than this; the one expression may be everywhere substituted for the other, and thus an entire treatise of DYNAMICS, 45 of dynamics might be written, in which the term Force would not once occur. Retarding Forces are here included under the head of Accelerating Forces, as the same force may accele- rate or retard, according as the velocity of the mo- ving body is in one direction or in the opposite. 87. If a body be continually urged by the same accelerating force, in the same direction, its mo- tion will be uniformly accelerated, or its velocity will increase proportionally to the time. The momentary increment of the velocity will be al- ways the same, from the definition of accelerating force ; and as, by the first law of motion, none of the velocity once acquired can ever be lost, it there- fore increases continually at the same rate. This is applicable particularly to the case of bodies fall- ing by their gravity to the earth. The descent of these bodies, their velocity being found to increase uniformly, affords a direct and experimental proof of the first law of motion. The propositions that follow are all immediately appli- ed to falling bodies ; but they are equally applicable to all rectilineal motions produced by the constant ac- tion of the same force. 88. If the velocity which a heavy body falling from rest, acquires in the first second of its motion 3 be 46 OUTLINES OF NATURAL PHILOSOPHY. be called g, if after any time t, its velocity be vr v~gt\ and if s be the space through which it has descended, % s =s= ft2, or v = a. Since v=:gt, and therefore t^m 2s b. Hence if t = 1", and if s' be the space fallen through in one second, g = 2 s' ; or the velocity is equal to twice the space fallen through in one second. The space through which a body falls in the first se- cond of time is sixteen feet and one inch nearly, and therefore g — 32 J nearly, 89. If the velocity acquired by a body falling from rest, were at the end of any time to become uniform, the body, in an equal time, would move over twice the distance which it has actually fallen through. 90. If a body be projected directly upwards with any velocity c, its motion will be uniformly retarded, or it will lose in every second a velocity equal to g, the velocity generated by gravity in a second. After the time t, the velocity of the body is c — tg; and the time at which the body reaches its greatest height where ^ = 0, is -. 91. The DYNAMICS. 47 91. The greatest height to which a heavy body can ascend, when projected directly upwards with c2 the velocity c, is ^— , the same height from which it must have fallen to acquire the velocity c ; and in all points, both of its ascent and descent, the re- lation between the time and the height h is ex- pressed by the equation h=ct — ^g t*. For the space through which the body would have as- cended, with the velocity c in the time t, is ct; from which the retardation produced by the action of gravity in the same time, that is \ g /2, (§ 87. b) must be taken away. c _ . «. This equation gives t — - -f - ^/c2 — %gh. The cJ O reason why two different values of t correspond to the same value of h is, that the body is at the same height at two different times, one in its ascent, and another in its descent. 1). The sum of the two values of t is — , or twice the g time of the total ascent. The difference between them is - L/C* — 92. If two bodies A and B, of which A is the heavier, be suspended over a pulley move- able 48 OUTLINES OF NATURAL PHILOSOPHY. able about a fixed centre, A will descend, and will ^ jj be accelerated by a force proportional to -T-^T-TJ , or to the difference of the weights divided by their sum. If the natural force of gravity be expressed by g, the velocity which it generates in a second ; then the acceleration of the bodies will be A . -p x g- We may therefore diminish the intensity of gravita- tion at pleasure, or produce an accelerating force less than the force of gravity in any assigned ra- tio. It is on this principle that Mr ATWOOD'S machine for reducing the theory of accelerated motion to the test of experience, is contrived. The acceleration of falling bodies was discovered by GALILEO, Dlalogo in. de Motu locali, Ojpere, torn. 3. p. 98. edit. Padova, 1744. SECT. DYNAMICS* 49 SECT. V. MQTION QF PliQJECTILES. rj.fi •-.:.",] bl. 93, AF a heavy body be projected in the direction of a straight line, not perpendicular to the horizon, it >vijl describe a parabpla situate in the vertical plane passing tjirough that straight line, and ha- ving its axis perpendicular to the horizon. a. For it advances in the line of projection at a uni- form rate, and descends below it with a velocity uni- formly accelerated. b. The straight line in the direction of which the body is projected, touches the parabola in the point of pro- jection. c. Gravity is here supposed to have its intensity and its direction the same at all points, within the range of the same projectile. We also abstract from the resistance of the air. 4. This proposition "was discovered by GALILEO, and demonstrated, J)ialogo iv. de fyfotu Projectorum. 94. If from the poiijt of projection there be ta- ken upwards, in a vertical line, a part equal to — , (c being the velocity with which the body is VOL. i. i> projected.) 50 OUTLINES OF NATURAL PHILOSOPHY. projected,) the directrix of the parabola described by the projectile is a horizontal line drawn through that point, in the plane of the projection ; and the velocity of the projectile in every point of its path, will be the same that it would have acquired by falling freely from the height of the directrix. a. All the parabolas described by bodies projected from the same point, and with the same velocities, though with different elevations, have the same di- rectrix ; and have their foci in the circumference of the same circle. 95. If a body be projected with the velocity c9 and in a direction that makes an angle E with the horizon, the distance at which it will strike the ho- rizontal plane, (or what is called its horizontal range), will be equal to twice the perpendicular height to which it would be carried by its initial velocity, multiplied into the sine of double the angle E ; or = — sin 2 E. o a. Hence the greatest horizontal range is, when the direction of the projectile makes an angle of 45® with the horizon : Also with respect to every other range, there are two angles which will give the same range, the one as much less than 45° as the other is greater. 2 96. The DYNAMICS. 51 96. The greatest altitude above the point of pro- jection, to which a projectile can ascend, is equal to the height corresponding to its initial velocity, multiplied into the square of the sine of the eleva- tion. that is, to -!L. sin E2. Also the time of its flight from the moment of projection till it strike the horizontal plane passing through the point of 21 c projection, is — x sin E. o The greatest height to which the projectile ascends is therefore as the square of the velocity, multiplied in- to the square of the sine of the elevation ; and the time of the flight is simply as the velocity, multiplied into the sine of the elevation. 97. A projectile thrown with a given velocity, from a point in a plane, will go to the greatest distance on that plane, when its direction bisects the angle which the plane makes with the verti- cal. 98. The points of greatest distance at which a projectile, thrown with a given initial velocity, will strike the different planes in the last proposition, are all in a parabola given in position. OUTLINES OF NATURAL PHILOSOPHY. This parabola has its axis perpendicular to the hori- zon, and touches all the parabolas described by the projectile, when thrown in the same vertical plane, and with the same velocity, but at different eleva- tions. SECT. VI. 0? MOTION ACCELERATED OR RETARDED BY A VARIABLE FORCE. 99. JL HE changes of motion, whether in velocity or direction, are always made gradually, and ne- ver, as it is expressed, per saltuin. Though they may not, like those treated of in the last section, be exactly proportional to the time, they are always proportional to a function of the force and the time jointly, which function vanishes when either the time or the force is equal to nothing. This follows from the fundamental equation of Dyna- mics, F = - or v = F r . a. The above is called the LAW OF CONTINUITY, which, in what respects free motions, is never violated. This DYNAMICS. 53 This conclusion agrees perfectly with all the in- stances in which accelerated or retarded motion can be traced by the senses, and therefore we might conclude, even from analogy, that it holds in those cases where the progress of acceleration escapes ob- servation. When a ball, from being at rest, issues almost in an instant from the mouth of a cannon, with the velocity of 1800 feet in a second, it is not to be imagined, that it has acquired the whole of that velocity, or any part of it, suddenly, and with- out the lapse of time. Could we divide time into small enough portions, and did we perfectly under- stand the Law of the Force produced by the in- flammation of gunpowder, we should be able to de- termine the point of time, and the place in the in- terior of the gun, at which the ball had any as- signable velocity from 0 to 1800 per second ; when, for instance, it had a velocity of one foot per second ; of two, of 100, of 1200, &c. The same holds of the direction of motion ; a body from moving in one direction, does not come to move in another, without describing a portion of a curve, and taking in its motion every possible direction from the one to the other. b. The notion of the continuity of motion seems first to have occurred to GALILEO. LEIBNITZ introdu- ced it as a leading principle in his philosophy, and he proved the necessity of it by a metaphysical, but conclusive argument. If a body receive an in- crease of its motion without the lapse of time, then the same body, at the same indivisible instant, is in two different states, which is impossible. If, for in- stance, 54 OUTLINES OF NATURAL PHILOSOPHY. •stance, it is at the beginning of the motion that the saltus is made, the body is at the same instant both at rest and in motion. 100. Let AB (fig. 7.) be a straight line, along which a body is accelerated by forces directed from A to B ; let the perpendicular AD be equal to the velocity which a body, accelerated by the force at A, would acquire in one second ; and let DEF be a curve so related to AB, that at any point what- ever C, the ordinate CE may be to AD as the ac- celerating force at C to the accelerating force at A; then, if the body begin to descend from rest at A, the square of its velocity at any point C, is double of the curvilineal area ACED. For, by the construction, CE is proportional to the force at C, or is =-v, v being the velocity which the mo- t ving body has acquired at C, and t the time of the descent from A to C. Now C c is the momentary increment of AC the space, and is therefore = v i ; therefore, CE x C c= vv, and 2 CE x C c= 2 v v. But CE x Cc is the momentary increment of the area ACED, and 2 v v is the momentary increment of tr ; therefore the square of the velocity of the moving body, and twice the area of the curve ACED, in- crease at the same rate, and they also begin to exist &t the same time ; therefore they are always equal. j's Prin. Lib. i. Prop. 39. a. If DYNAMICS. 55 a. If the body begin to move from A with a certain velocity, then the difference between the square of its velocity at C and at A is equal to twice the area ACED. b. In like manner, if the body be projected upward with a given velocity, the difference between the square of the initial velocity, and the square of its velocity at any point C, will be double of the area BCEF ; and if the motion of the body be destroy- ed when it has ascended to a certain point A, the square of the initial velocity will be double of the area BADF. 101. If a body be accelerated or retarded by any number of forces acting in succession, and each continuing its action while tbe body moves over a certain distance ; the sum that is made up by mul- tiplying each of these forces into the distance over which it acts, and adding those products together, will be proportional to the difference of the squares of the velocities of the body at the beginning and end of the action of those forces. 102. Hence if the body is accelerated from rest, the sum collected, as in the last proposition, will be as the square of the velocity acquired ; and if it be retarded or resisted till it come to rest, the sum will be as the square of the initial velocity. Thus if F, F', F", F'", &c. are the forces that act in succession, s, 6*', s", s'", the distances or spaces over which 56 OUTLINES OF NATUllAL PHILOSOPHY, which they act ; v the velocity at the beginning of the action, tf at the end of it, F s + F' sf -f- F'' s» + F" s"' == + V* + t/«. And if v"' = 0, then F s -f FV + I1" *" -f tf '" *"> === » 2. 103. If the force which accelerates or retards in the last proposition j be uniform, the distance to which the body will go before it acquire or lose a given velocity* will be as the square of that velo- city. «. Oil this is founded that estimation of the force of moving bodies, which is known by the name of the vis VIVA, and which makes that force proportional to the square of the velocity. The truth is, that the effect of a body in motion may be measured either by the distance it goes to, or by the time that elapses before a resistance of uniform intensity re- duce it to rest. If the effect be measured in the first of these ways, it will be found to be as the square of the velocity ; if in the second, as the ve- locity simply. Both these measures may be consi- dered as correct, and are not inconsistent when light- ly understood. The former makes the effect F S, the latter F t, retaining the significations already given to these letters. The same term, however, ought not to be indiscriminately applied to two things so essentially different. It is mdst conformable to the use made of the word Force, in the fundamen- tal investigations of Dynamics, to t&ke the quantity DYNAMICS. ,57 5? t for the measure of it, and therefore to suppose it proportional to the velocity simply. This is the only way in which the language of the science can be rendered perfectly consistent and free from am- biguity. But as the distance to which a body goes before a given resistance destroys its motion, is often an object of inquiry, it may be proper to consider that distance as a measure of a certain power re- siding in the moving body. It has been proposed to distinguish this power by the term Impetus. b. This part of Dynamics would lead to the conside- ration of Central Forces ; but as this is a subject of some difficulty, and is most interesting when con- nected with astronomy, we shall defer treating of it till we enter on the explanation of the Planetary Motions. MECHANICS. 58 OUTLINES OF NATURAL PHILOSOPHY. MECHANICS. SECT. I. OF THE CENTRE OF GRAVITY. 104. IF a body fixed to a plane be preserved in equilibria by a force having a given direction, and attached also to a given point in the plane, it will be kept in equilibria by the same force, when at- tached to any other point whatever in the line of its first direction. Let the force be attached by a thread to a given point in the plane ; while the tension of the thread and its direction continue, the equilibrium must conti- nue. Suppose, then, that any other point in the thread is fixed down to the plane, it is evident, that the tension of the remaining part will not be changed by that circumstance ; and therefore the equilibrium must remain. C ARNOT has given a different proof of this proposition. Prmcipes de T Eguilibre, &c. § 77. 105. If MECHANICS. 59 105. If two heavy bodies be fixed to a plane moveable about a given point, the bodies will be in equilibria if that point divide the distance between them in the inverse ratio of their weights or quanti- ties of matter. A demonstration of this proposition may be derived from the composition of motion, as is done by NEW- TON, Principia Mathematica, Cor. 11. to the Laws of Motion ; and by VARIGNON, Nouvelle Mecanique; see also HAMILTON, Philosophical Essays^ p. 145. A demonstration on a different principle was given by ARCHIMEDES; it has been improved by MACLAURIN in his Account of Newton's Discoveries, and by VINCE in the Philosophical Transactions. See also WOOD'S Mechanics, Art. 73. 106. The effect of any heavy body, to produce in any system of bodies, an angular motion about a given centre, is proportional to the product of the mass of the body into the perpendicular drawn from the centre to a vertical line passing through the body. a. The axis of motion of any body is a straight line, which remains fixed while the body is free to revolve round it. 107. The product of any heavy body into its dis- tance, from a vertical plane passing through the axis of motion, is called the momentum of the bo- dy relatively to that axis; and in general, if, through 60 OUTLINES OF NATURAL PHILOSOPHY. through the axis of motion, a plane be made to pass parallel to the direction of any force, the pro- duct of that force into its distance from the same plane, is called the momentum of the force. 108. Any system of bodies being given, a point may be found, such, that if any plane pass through it, the sum of the momenta of the bodies on one side of the plane, relatively to the point found, shall be equal to the sum of the momenta of the bodies on the other side of the plane, relatively to the same point. a. The point thus formed is called the centre of gravity of the system. b. Every system of bodies, therefore, and every indivi- dual body, (as a body may be regarded as a system of particles or infinitely small bodies), has a centre of gravity ; and that centre remains the same while the bodies retain their position relatively to one an- other, however the whole system may change its place relatively to the horizon. 109. In any body, or system of bodies, if the centre of gravity be sustained, the whole will re- main at rest, In other words, the body or system of bodies has no tendency to angular motion round its centre of gra- vity. 110. The distance of the centre of gravity of any system of bodie.8 from a given plane, is equal to the MECHANICS. 61 the sum of the products of all the masses into their distances from the plane, divided by the sum of the masses. a. As a single body may be regarded as a system of particles, this proposition applies to such bodies, and is therefore universal. If the bodies be A, B, C, D, &c. and their perpendi- cular distance from a given plane a, b, c, d, &c. x the distance of the centre of gravity from the same plane, x = — . + B + C + D. 111. The effect of any number of bodies to pro- duce motion about a given point, is the same as if those bodies were united in their common centre of gravity. 112. When a heavy body is at liberty to move about a fixed point, it cannot come to rest till its centre of gravity is either the highest or the lowest possible. A body cannot rest on any base, unless a perpendicu- lar to the horizon from its centre of gravity fall with- in the base. The centre of gravity of plain figures, and of solids, may be found by the foregoing propositions. A few of the simplest and most elementary of these prob- lems may serve as examples. 113. 62 OUTLINES OF NATURAL PHILOSOPHY. 113. To find the centre of gravity of any number of bodies placed in the same straight line at given distances from one another. The centre of gravity is found by Art. 110. If the bodies are A, B, C, D, and their distances from a given point in the straight line in which they are supposed to be placed, a, 6, c, d, the dis- tance of their centre of gravity from the same point, A a + B b + C c + D d is A+B+C+D If the bodies are on different sides of the point to which their distances are referred, the distances on the one side being accounted affirmative, those on the op- posite must be accounted negative. 114. Any number of bodies being given in posi- tion, to find their centre of gravity. a. The bodies must be referred to three planes given in position, cutting one another at right angles, one of them horizontal, and, of course, the other two vertical. Let the bodies be A, B, C, and D, their distances from the given horizontal plane «, &, c, d; their distances from one of the vertical planes a', 6', c', d' ; and from the other a", 6," c ', d" ; then if we take Aa + B6 + Cc + Dd a? = - A + B-f C+D - ' centre of gravity of MECHANICS. 63 of the system is in a horizontal plane at the dis- tance x from the given horizontal plane. Take a,., = , and the centre of gravity is in a plane parallel to the first of the two vertical planes, and distant from it by the line x'. Lastly, take in the intersection of these planes, a point distant from the second vertical plane by a quanti- ty of' = Ad"+Bft" + Cc" + A+B+C+D This point will be the centre of gravity of the given bodies, as is evident from § 110. &. The problem in this article may also be resolved by means of this geometrical proposition: If from the centres of gravity of any number of bodies gi- ven in position, lines be drawn to their common centre of gravity, the sum of the products, formed by multiplying each of these lines into the weight of the body from which it is drawn, is a minimum, or is less than if the lines were drawn to any other point. This may easily be deduced from the 3d Cor. to the 5th Prop, of the 2d Book of the Loci Plani, APOL- LONIUS. See SIMSON'S edition, p. 180. 115. To find the centre of gravity of a triangular plane, all the points of which are supposed to gravi- tate equally. From OUTLINES OF NATURAL PHILOSOPHY. From two of the angles of the triangle, draw lines bi- secting the opposite sides. Their intersection is the centre of gravity. Each of these lines is divided by the point of intersec- tion, in the ratio of 2 to 1. 116. To find the centre of gravity of a given py- Draw a straight line from the vertex to the centre of gravity of the base, and divide it in the ratio of three to one, the greatest segment being next the vertex. The point so found is the centre of gravi- ty of the Pyramip!. This construction applies to a pyramid of any num^ ber of sides, anq. therefore also to a Cone. 117. To find the centre of gravity of any plane mechanically. Suspend it by a given point in or near its perimeter^ and when it is at rest, draw across it a vertical line passing through that point. Suspend it in like man- ner by another point, and draw a vertical line as be- fbrs. Tl>e intersection of these lines is the centre of gravity of the plane. If the body is of three dimensions, the same process may be followed ; but three suspensions will be ne- cessary. This construction is referred to by PAPPUS ALEXAX- DRINUS, Collect. Math. Lib. vm. Prop. i. Qf MECHANICS. 65 Of the Motion of the Centre of Gravity. In the preceding propositions, the centres of gravity of bodies or systems, in which the parts preserve the same relative position, have only been consider- ed. It is evident, however, that, though the parts do not preserve the same relative position, the centre of gravity may be found for any given state of the system. The properties common to all the centres so found, are of great importance in me- chanics. 117. If any number of bodies move uniformly in straight lines, their common centre of gravity will either move uniformly in a straight line, or remain at rest. NEWTONI, Principia, Lib. 1. Lemma 23. 118. The mutual action of bodies does not change the state of their centre of gravity, either as re- maining at rest, or as moving uniformly in a straight line. 119. The quantity of motion in any system of bo- dies estimated in a given direction, remains con- stantly the same, whatever may be the mutual ac- tion of these bodies on one another. VOL. T. E a. Both 66 OUTLINES OF NATURAL PHILOSOPHY. a. Both this and the preceding proposition are conse- quences of the equality that takes place between the action and re-action of bodies. The quantity of motion in any system is the same as if all the parts of it were united in the centre of gravity of the whole. b. Connected with this last, is a property of the centre of gravity, which has been found of use in the pure mathematics. If a plane figure be gene- rated by the revolution of a given line, or a solid by the revolution of a given plane figure ; the area in the first case, or the solidity in the second, is equal to the product of the generating quantity into the length of the line, described by its centre of gravity. The application of this theorem to the quadrature of curves, or the cubature of solids, constitutes what has been called the Centrobaric Method. The in- vention of the method is ascribed to GULDINUS, but the theorem above was known to PAPPUS, and no doubt the application of it also. 120. If there be a system of bodies acting any how on one another, and if at any point of time we compute the motions which these bodies would have in the succeeding instant, were they all free from their mutual action ; and if we also compute the motions, which, in consequence of their mu- tual action, they really have in that instant, the motions which must be compounded with the first of MECHANICS. 67 of these, in order to produce the second, are such as, if they acted on the system alone, would pro- duce no motion at all, or would be in equilibria with one another. a. The principle here laid down was discovered by D^ALEMBERT, Dynamique, partie 2de, chap. lr. It is a result of the equality of action and re-action, or rather, it is that very equality expressed with perfect generality and precision. It was supposed before, that the motions combined with the first of those above described, in order to produce the second, must, when added together, be equal to no- thing ; whereas it is not the motions simply, but their momenta, which, when added together, must be equal to nothing. I. This principle is of great importance for deducing the laws of constrained from those of free motion, and affords the only general method of determin- ing the manner wherein motion distributes itself through all the parts of a system to which it is any how communicated. It may very properly be called the Principle of 'the Distribution of Motion. A simple instance of its application is afforded by the following problem. 121. Let two bodies connected by a straight rod, without weight, be moveable about an axis passing through their centre of gravity ; and let a motion be communicated to one of them, which, were it de- tached and single, would give it a velocity c ; re- quired, the angular velocity with which the bodies will begin to revolve ? E Z SECT. 68 OUTLINES OF NATURAL PHILOSOPHY. SECT. II. OF THE MECHANICAL POWERS. 5. J\. MECHANICAL power, is an instrument by which the effect of a given force is increased, while the force itself remains the same. The simple mechanical powers into which more com- plex machines are resolved, are these : 1. The Le- ver; 2. The Wheel and Axle; 3. The Pulley; 4. The Wedge ; 5. The Screw : 6. The Funicular. Machine. Of the Lever. 123. DEF. — A lever is an inflexible rod, move- able about a centre or fulcrum, and having forces applied to two or more points in it. a. The simplest state of the lever is, when there are only two forces ; and of this there are two cases : in the first, the fulcrum is between the points where the forces are applied, and the forces are directed the same way; in the second, the points to uhich the forces are applied, are on the same side 'of the fulcrum, and the forces are directed opposite ways. k la MECHANICS. 69 I. In treating of the lever, it is usual to distinguish the forces by the names of the Power and the Weight, or the Power and the Resistance, — terms that have a reference to the intention with which the machine is used, not to any real difference in the action of the forces. We shall begin with abstracting entirely from the weight of the lever itself. 124. If two forces be applied to a lever having equal and opposite momenta, that is, tending to pro- duce motion in opposite directions, and being in- versely as the perpendiculars drawn to their direc- tion from the fulcrum, they will be in equilibrium with one another. When the forces are parallel, this proposition coincides with the property of the centre of gravity, § 105. When they are oblique, the case is reduced to the preceding by the resolution of forces. 125. If any number of forces be applied to a le- ver, there will be an equilibrium if the sums of the opposite momenta be equal to one another. This proposition also coincides with a property of the centre of gravity, (§ 108 and 109.) 126. When any number of forces are perpendi- cularly applied to a lever, the fulcrum is loaded with 70 OUTLINES OF NATURAL PHILOSOPHY. with the sum of the forces, if they are all in the same direction, and with their difference, if they are in opposite directions. 127. When there is an equilibrium in the lever, if each force be multiplied into the velocity which the point of its application would have, reduced to the direction of that force, if motion were to take place, the sum of all the products so formed is equal to nothing. a. Forces that act in opposite directions are here ac- counted positive and negative. This is always to be understood when the sums of forces or of momen- ta are said to be equal to nothing. b. This property of the lever is common to all the mechanical powers, and, indeed, to all machines whatsoever. It is known by the name of the prin- ciple of Virtual Velocities, and consists in this, that if the equilibrium of a machine be disturbed, by a quantity indefinitely small, and if the velocity of each force be multiplied into its quantity, the sum of these products, reckoning the forces which are in opposite directions, positive and negative with respect to one another, will be equal to nothing. This principle was suggested by BERNOTJILLI, and is of great use in mechanical investigations. It admits of a general, though indirect, demonstra- tion. The velocities compared, being not actual, but such as would take place if a certain event, the subver- 2 sion MECHANICS. 71 sion of the equilibrium, were to happen, are pro- perly denominated virtual. c. The different cases of the lever, whether it be straight or crooked, and the forces perpendicular or oblique, are all comprehended in the preceding pro- positions. The case of oblique forces admits of a very concise expression, in terms of the angles. 1 28. If two oblique forces be applied to the ex- tremities of a straight lever, there will be an equi- librium between them, if they be inversely as the lengths of the arms by which they act, multiplied into the sines of the angles which their directions make with the lever. It is evident that this proposition is general, and com- prehends the case of perpendicular forces. 129. If a beam of any kind be kept in equilibrium by three forces applied to different points in it, these forces must either be parallel, or must converge to the same point. a. If at the point to which the three lines converge, we take parts in those lines proportional to the forces, they will form the sides and the diagonal of a pa- rallelogram ; and if at the point in which this dia- gonal intersects the beam, we suppose a fulcrum to i>e placed ; the beam will be converted into an ordi- narv 72 OUTLINES OF NATURAL PHILOSOPHY. nary lever, and the weight sustained by the fulcrum will be to either of the other two forces, as the dia- gonal of the parallelogram to the side which cor- responds to that force : also the two extreme forces will be to one another inversely as the sines of the angles which their directions make with the above diagonal. 130. If we would allow for the weight of the le- ver itself, we must suppose its weight to be united in its centre of gravity, and to act there as a third force, added to the power or the resistance, accord- ing to the side of the fulcrum on which it is pla- ced. 131. When a beam carrying a weight, is sup- ported in a horizontal position by two props, the weights which the props sustain are inversely pro- portional to their distances from the centre of gra- vity of the weight. 132. If a weight W, be sustained on a horizon- tal plane by three props, (not in a straight line), the pressure on each will be the same as if a single weight were laid on it, so that the sum of all the three weights were equal to W, and their common centre of gravity the same with the centre of gravi- ty of that body. a. If MECHANICS. 73 a. If the props be A, B, C, (fig. 8.), and if W be placed with its centre of gravity at D, and if ADG, BDF, CDE, be drawn, the pressure on on on A B C is is is A-jiui AG DF BF BE CE X X X W, W, W. When D coincides with the centre of gravity of the triangle ABC, the pressure on each of the props is the same. b. If W be supported by more than three props, the problem appears to be indeterminate, or to admit of innumerable solutions. Nevertheless, if the centre of gravity of W, as it rests on the plane, be the same with the centre of gravity of the figure made by joining the tops of the props by straight lines, the pressures on the props are all equal to one another. c. The lever is the simplest of the mechanical powers, and appears to be the first that was attempted to be explained. ARISTOTLE has treated of it in his Me- chanical Questions, and has even endeavoured to re- duce some other of the mechanical powers to the lever. The first accurate explanation, however, was given by ARCHIMEDES, Lib. i. de Equiponde- rantibus, Prop. 6. d. The lever is used in mechanics in various forms and combinations. The compound lever is, where one 3 lever 74 OUTLINES OF NATURAL PHILOSOPHY. lever is made to turn another ; and then an equili- brium takes place, when the weight is to the power as the product of all the arms, taken alternately, beginning with that to which the power is applied, to the product of all the other arms. e. The common apparatus for raising a draw-bridge, is a combination of a lever of the first kind, with a lever of the second ; the two levers are parallel, and remain in equilibria in all situations. f. In the construction of the animal body, this me- chanical power is much employed. The bone is the lever, the muscle is the power, or rather the me- dium through which the power acts, and the joint is the fulcrum. The insertion of the muscle, or the point to which the power is applied, is usually nearer the joint or fulcrum than the weight to be raised is. A different structure might have produ- ced more strength, but would have diminished the activity of the animal, and the velocity of its mo- tion. Of the Balance. 133. The balance is a lever with equal arms, used for comparing weights with one another, and, when well constructed, must have the following properties : 1. It should rest in a horizontal posi- tion, when loaded with equal weights. 2. It should have great sensibility, that is, the addition of a small weight in either scale should disturb the equilibrium, and make the beam incline sensibly from the horizontal position. 3. It should have great MECHANICS. 75 great stability, that is, when disturbed, it should quickly return to a state of rest. That the first requisite may be obtained, the beam must have equal arms ; and that this equality may be more easily ascertained, the beam should be straight, or the point on which the beam turns, and the points to which the weights are attached, should be in the same straight line. The centre of suspension must also be higher than the centre of gravity. Were these centres to coincide, the beam, when the weights were equal, would rest in any position, and the addi- tion of the smallest weight would overset the balance, and place the beam in a vertical situation, from which it would have no tendency to return. The sensibility, in this case, would be the greatest pos- sible ; but the other two requisites, of level and sta- bility, would be entirely lost. The case would be even worse, if the centre of gravity were higher than the centre of suspension, as the balance, when deran- ged, would make a revolution of no less than a semi- circle. When the centre of suspension is higher than the centre of gravity, if the weights be equal, the beam will be horizontal ; if they be unequal, it will take an oblique position, and will raise the centre of gravity of the whole, making the momentum on the side of the lighter weight, by that means, equal to the momentum on the side of the heavier, so that an equilibrium again takes place. The second requisite, is the sensibility of the balance, or the small ness of the weight by which a given angle of inclination is produced. If a be the length of the arm of the balance, and b the distance between 76 OUTLINES OF NATURAL PHILOSOPHY. between the centre of suspension and the centre of gravity, P the load in either scale, and W the weight of the beam, the sensibility of the balance is as , fi) p - WV; ^ *3 t^leref°re greater, the greater the length of the arm, the less the distance^ between the two centres, and the less the weight with which the balance is loaded. Lastly, The stability, or the force with which the state of equilibrium is recovered, is proportional to (2 P -f W) b, the denominator of the preceding fraction. The diminution of Z», therefore, while it increases the sensibility, lessens the stability of the balance, but the lengthening of a will increase the former of these quantities, without diminishing the latter. The above formulas are of great practical utility, because, by means of them, one balance may be made having exactly the same sensibility and stability with ano- ther : it is only required, that the ratio of the lengths of the arms should be the same with that which is compounded of the ratios of the distances of the centres of gravity and suspension, and of the weights of the beams. A balance made by RAMSDEN for the Royal Society, is capable of weighing ten pounds, and turns with the ten millionth part of the weight, or little more than the two hundredth of a grain. YOUNG'S Lec- tures ', vol. i. p. 125. Descriptions of balances are always defective, where they do not give the values of 0, 6, arid W, the quantities on which the merit of the balance depends, and by the knowledge of which similar MECHANICS. 77 similar instruments might be constructed. In some of the nicest balances, b is made variable by means of a small moveable weight. 134. When a balance is false, or its arms une- qual, if any thing be weighed, first in the one scale, and then in the opposite, the true weight is a mean proportional between the weights thus found, or is equal to the square root of their pro- duct. Consult LA HIIIE, Tralte de Mechdtnique, prop. 33. MUSCHKNBKOEK, § 283, &c. EuLER, Comment. Pe- trop. torn. x. p. 1. &c. MAGELLAN, Journal de torn. xvii. (1781), p. 43, Of the JFheel and Axle. 135. The wheel and axle, consists of a wheel having a cylindric axis passing through its centre, and moveable in two grooves. The power is ap- plied to the circumference of the wheel, and the weight to the circumference of the axle. An equilibrium in this instrument takes place, when the radius of the wheel, multiplied into the power, is equal to the radius of the axle multiplied into the weight ; or when the power and weight are inversely as the radii of the circles to which they are applied. ft 78 OUTLINES OF NATURAL PHILOSOPHY. It is supposed here, that the power and the weight both act at right angles to the radius. a. The wheel and axle is nothing else but a lever, so contrived as to have a continued motion about its fulcrum. The principle of the virtual velocities is here obviously applicable. If the power act not at the circumference of the wheel, but at the extre- mity of a handspike inserted in the wheel, the dis- tance of that extremity from the centre of the axis, is to be accounted the radius of the wheel. b. The Capstan, the Windlass, and other contrivances of a similar nature, are nothing else than the wheel and axle adapted to particular circumstances. c. The mechanical contrivance called a Crank, is a spe- cies of wheel and axle. If the force which acts up- on a crank urges it directly down and up, alternate- ly, the effect is to that which would be produced if the force acted at right angles to the arm of the crank all round, as twice the diameter of a circle to its circumference, or as seven to eleven nearly. d. The combinations of wheels, so useful in mecha- nics, are generally reducible to the wheel and axle, though the wheel which turns the other is not al- ways upon the same axis with it. The motion, in such cases, is communicated from the one wheel to the other, either by belts or straps passing over the circumferences of both, or by teeth cut in those cir- cumferences, and working in one another. 136. When MECHANICS. 79 136. When one wheel moves another in either of these ways, the velocities of their circumfer- ences are equal ; and therefore their angular velo- cities, or the number of revolutions which they make in the same time, are inversely as their ra- dii. In combinations of wheels communicating motion to one another, it is usual to call a smaller wheel acted on by a larger one, a pinion ; and its teeth the leaves of the pinion. Sometimes the smaller wheel is a cylinder, in which the top and bottom are formed by circular plates or boards, connected by staves inserted at equal distances along their cir- cumferences, serving as teeth ; this is called a lan- tern. It is usual also to employ wheels and pi- nions, as in figure 9, where A is a large wheel, driving a pinion b on the same axle with the great wheel B ; B acts on the pinion c, which is on the same axle with the large wheel C ; and C drives the pinion J, &c. 137. When motion is communicated through a number of wheels and pinions, the angular veloci- ty of the first wheel, is to the angular velocity of the last pinion, as the product made by multiply- ing together the number of the teeth in every pi- nion, to the product made by multiplying together the number of the teeth in every wheel ; and that same ratio, compounded with the ratio of the ra- dius of the last pinion, to the radius of the first wheel, 80 OUTLINES OF NATURAL PHILOSOPHY. wheel, will give the ratio of the power to the re- sistance when an equilibrium takes place. The truth of the first part of this proposition is evi- dent from § 136. ; and the truth of the latter part follows from the principle of the virtual velo- cities. 138. When wheels act by teeth working in one another, notwithstanding that the length and po- sition of the levers by which they act, are conti- nually changing, the force of the one upon the other will remain constant, if the line which is drawn perpendicular to the surfaces of both teeth at the point of contact, pass continually through the same point of the line which joins the centres of the wheels. It is here necessary to distinguish between what is called the total, and the primitive radius of a wheel. The total radius extends from the centre to the ex- tremities of the teeth ; and the primitive extends only to the point where the teeth would touch, if they were to act merely by contact. If the distance be- tween the centres of the wheels be divided by the point O, (fig. 10.), in the ratio of the number of teeth which the wheels are required to have, AO and BO will be the primitive radii of the wheels, and O is the point through which the perpendiculars in the above proposition must be supposed to pass. 139. There MECHANICS. . 81 139. There are many different curves, accord- ing to which the teeth of wheels might be formed, so as to answer the condition in the last proposi- tion ; but that which seems most convenient, and which has been most generally adopted, is the epi* cycloid. Vid. EULER, Nov. Com. Petrop. v. p. 299. and xi. p. 207. One of the curves there proposed, is the evolute of the circle. The same is mentioned,, Encyc. Brit. If the circumference of one circle be made to roll along the circumference of another, the curve described by any given point in the first of these circumferences, is an epicycloid. The circle which rolls is called the generating circle^ and that on which it rolls is called the base of the epicycloid. As the former may be supposed to roll either on the outside or the inside of the latter, there are two kinds of epicycloids, distin- guished by the names of Exterior and Interior. 140. Let it be required, having given the mag- nitude of a wheel and pinion, and the numbers of teeth in each, to determine the figure of those teeth. Let CB (fig. 11.) be the primitive radius of the wheel, and BD the base of the tooth ; bisect BD in E, draw CE, and produce it indefinitely ; and with a generating circle, of which the radius is half the primitive radius of the pinion, describe an arch of an epicycloid on the primitive circum- VOL. I. F ferencc 82 OUTLINES OF NATURAL PHILOSOPHY. ference of the wheel as a base, beginning at the point B, and intersecting CE in F. Then BF is the figure of one face of the tooth, and in the same manner may the other FL be determined. When one tooth is formed, it may serve as a model for the rest. The teeth of the pinion are to be determined in the same manner ; the radius of the generating circle being made equal to half the primitive ra- dius of the wheel. See CAMUS on the Teeth of Wheels, Article 548, p. 53. et seq. of the English translation. Toothed wheels are frequently employed for changing the direction of motion : the directions of their axes are then inclined to one another, and the parts of their rims where the teeth are placed, are cut into frustums of cones, each of which has its vertex in the intersection of the axes of the two wheels. Wheels of this kind are called bevelled, or crown wheels. 141. The figures of the teeth of bevelled wheels, are determined by an epicycloidal surface, generated by a straight line, on the surface of a cone, while the cone rolls on the conical part of the wheel, its vertex being in the point where the axes of the wheels intersect. The line on the surface of the revolving cone passes through its vertex. Consult MECHANICS. $'/> ' Consult CAMUS on the Teeth of Wheels, Article 557. &c. The curve generated by a point on the surface of a revolving cone, is there called a Spherical Epi- cycloidj as being on the surface of a sphere. Dr BREWSTER, in the Appendix to Fergusons Meclia- nics9 vol. ii. p. 232. treats of the construction of be- velled wheels. a. Toothed wheels are employed in mechanics, not only for increasing or diminishing velocity, so as to adapt a given power to the purpose of producing a certain effect, as in a common mill, where the slow motion of the water-wheel is made to produce the rapid motion of the millstone, but they are also employed for th£ purpose of producing angular motions, that shall obtain, with great precision, gi- ven ratios to one another. This happens in clock- work, and it is there of importance to determine with accuracy the number of wheels, and the num- ber of teeth ill each. Suppose it is required to make one wheel turn exactly 240 times, while ano- ther turns once : The smaller wheel, which is to turn the fastest, cannot hate fewer than 6 leaves or teeth : The other, therefore, ought to have 6 times 240, or 1440> which is more than can conveniently be given to one wheel. Suppose, then, that three wheels ABC, and three pinions b c d, as in figure 9, are to be employed : Then the angular velocity of A is to the angular velocity of by as b x £ xd to AxBxC, $ 2 these 84 OUTLINES OF NATURAL PHILOSOPHY. these letters denoting the number of teeth in thei* respective wheels, therefore AxBxC:6xcxd::240:l. Let the number of teeth in the pinions be 7, 5, and 11, then A x B x C : 7 X 5 x 11 : : 240 : 1 ; and AxBxC = 7x5xllx 240, or resolving 240 into its simple factors, AxBxC = 7x5x11x3x2x2x2x2x5. These factors combined into three different groups in any order, will give three products that may be substituted for A, B, and C. We may take, for instance, 5x1 1, 7x3x2, and 2x2x2x5, which will give 55, 42 and 40, for the numbers of the teeth that are to work in the pinions, where the numbers are 7, 5 and 11, The angular velocity of A will thus be to the angular velocity of d in the ratio required. 142. The number of teeth in a wheel, and in a pinion which work together, should be prime to one another, that their coincidences may not recur in the same order after every revolution. a. The numbers expressing the ratio of the angular velocity of the wheel to that of the pinion, may sometimes be fractional, and may consist of so ma- ny places, when the fractions are taken away, that, to make the motions exactly in their ratio, would require more wheels, and a greater number of teeth, than can conveniently be allowed. It is then of consequence MECHANICS. 85 consequence to find two numbers that will express the ratio, not exactly, but more nearly, than can be done by smaller numbers. The method of finding such numbers, is explained in the Elements of Alge- bra. See EULER, Elem. cTAlgebre, torn. IT. Addi- tions, § 19. ; also WOOD'S Algebra, § 371. Of the Pulley. 143. The pulley is a wheel moveable on an axis with a groove cut in its circumference, round which a cord passes. The axis of the pulley is either fixed or moveable. When fixed, the pulley gives no mechanical advantage, but serves merely to change the direction of the power applied to the cord which passes over it. When the axis of the pulley is moveable, one end of the cord is made fast to a fixed point ; the power is applied to the other, and the weight hangs by the block in which the axis of the pulley is fixed. By a single pulley of this construction, a force is enabled to balance another twice as great. Various combinations of pulleys give various degrees of advantage ; but in every case there is an equilibrium, when the spaces that would be passed over by the power and the weight in the same time, are in the in- verse ratio of their quantities. It is obvious, that the pulley is reducible to the lerer of the second kind, or, still more directly, to the >case, § 128. of a body supported by two props. The 8t> OUTLINES OV NATURAL PHILOSOPHY. The combination of pulleys that has the most simpli- city, is that in which a number of pulleys hang above one another in an oblique line, each doubling the power of that which is under it, so that as the num- ber of pulleys increases in arithmetical progression, the advantage increases in geometrical progression. From being so extremely portable, and so applicable to cordage, the pulley is of all the mechanical powers the most useful at sea, Of the Wedge. 144. The wedge is a triangular prism, made of wood or of metal. It is usual to make the prism isosceles, and the angle contained between the equal sides more acute than the others. When instead of being isosceles, the prism is rectangular, and has one of the planes containing the right angle, placed horizontally, the wedge coincides with the inclined plane, which is sometimes con- sidered as a mechanical power distinct from the rest. a, There has been a considerable difference in the rules given by mechanical writers for determining the power of the wedge. This has arisen, from not attending sufficiently to the direction of the resis- tance, and to another circumstance, whether both the resisting bodies are moveable, or one of them only. The best way of considering the subject, is to resolve all the forces that act upon the wedge into parts parallel to two axes, at right angles to one another. MECHANICS. 87 another, arid one of them parallel to the back of the wedge, or the side to which the power is applied. In the case of equilibrium, the opposite forces, in the direction of these axes, must be equal to one another. In this way the following theorems are easily investigated. 145. When three forces are applied perpendicu- larly to the three faces of a triangular prism or wedge, they will be in equilibria, if their directions intersect in the same point, and if they are to ohe another as the lengths of the sides to which they are applied. 146. If the resistance to the motion of an isos- celes wedge be perpendicular to the back of the wedge, there will be an equilibrium, when the pow- er applied to the back of the wedge is to the sum of the resistances, as the thickness of the back to twice the height of the wedge ; and therefore if the bodies are equal, the power will be to the resis- tance of either of them, as half the base of the wedge to the height of it. This proposition is most easily demonstrated on the principle of the virtual velocities, (§ 127. b.) If one of the resisting bodies only is moveable, the power will be to the resistance as the base of the wedge to its height. The 88 OUTLINES OF NATURAL PHILOSOPHY. 'The more acute the wedge, the greater its power, or the smaller the force required to overcome a given resistance. • 147. When the wedge becomes an inclined plane, with its base horizontal, if the direction of the power be parallel to the plane, there will be an equilibrium when the power is to the resistance, as the height of the plane to its length, or as the sine of its inclination to the radius. 148. Universally there will be an equilibrium on an inclined plane, when the power is to the re- .sistance, as the sine of the inclination of the plane to the cosine of the angle which the direction of the power makes with the plane. Bossuf, Art. 263. Comment. Petrop. torn. u. p. 282. A wedge naay have the form of a pyramid, as well as of a prism. In the case of a pyramidal wedge, the ^forces that act must be resolved according to three axes, by which three equations will be obtained. 149. In a pyramidal wedge, where the triangu- lar faces are inclined, at the same angle i, to the base, to produce an equilibrium, the power must be to the sum of the resistances, as the cosine of i to the radius. Piercing MECHANICS. 89 Piercing instruments are all reducible to wedges of this kind, Nails, Bayonets, Stakes, Piles, &c. The resistance in some of these increases with the sur- face to which it is applied. Though a wedge be of a conical form, it is comprehended under the pre- ceding theorem. The common wedge is generally employed for clea- ving, and otherwise overcoming the force of cohe- sion in bodies : it may sometimes be used with advantage for raising great weights to a small height. All cutting instruments may be referred to it, Knives, Chisels, Scissors, Files, the Teeth of Animals, &c. A Saw is a series of wedges, on which the motion impressed is oblique to the resistance. A Wimble is a wedge to which the circular motion is given. The power applied to the wedge, when great resist- ance is to be overcome, is usually percussion ; and almost the only instance in which the wedge is used for the purpose of equilibrium, is in the construc- tion of arches, which are built of truncated wedges. As the consideration of this equilibrium would lead to too long a digression in this place, it will be treat- ed of in an Appendix. Of the Screw. 150. The screw is a spiral groove or thread, winding round a cylinder, so as to cut all the lines drawn on its surface parallel to the axis, at the same 90 OUTLINES OF NATURAL PHILOSOPHY. same angle. The spiral may be either on the convex or concave surface of the cylinder, and the screw is called accordingly, either the exterior or the interior screw. The screw is properly referred to the species of wedge called the Inclined Plane, and its principle, like that of the wedge, is easily reduced to the composition of forces. 151. If the power be applied parallel to the base of the screw, and perpendicular to the radius of the cylinder ; and if the weight press perpendicu- larly on the axis, an equilibrium is produced when the power is to the resistance, as the dis- tance between two threads of the screw to the cir- cumference described by the point to which the power is applied. MUSCHENBROEK, § 482. BoSSUT, § 284. SXrRAVE- SANDE, § 281. a. The screw is of great use for compressing bodies. A kind of percussion is sometimes added to it, as in the apparatus for coining. The great attrition or friction which takes place in the screw, is useful by retaining it in the state to which it has been once brought, and continuing the effect after the power is removed. The screw is also applied, with much ad- vantage, to raise great weights to a small height, and support them in that situation. b. The use of the screw to raise water, in the man- ner invented by ARCHIMEDES, is a very remarkable application MECHANICS. 9J application of it. This screw is usually formed by a spiral tube winding round a cylinder, and it may be applied to raise any body that can pass within the tube as well as a fluid. If a screw thus form- ed be placed obliquely, so as to make, with the ver- tical, an angle equal to that which the spiral makes with the lines parallel to the axis, there will be in each turn of the spiral a part parallel to the hori- zon, where, if a body were placed, it would be at rest. Jf, then, the screw be turned, the body will ascend, because the part of the screw behind it be- comes more inclined than the part before it, so that the body is urged forward, and consequently as- cends. c. If the screw used in this manner, be turned with great rapidity, the body may acquire a centrifugal force so great as to overcome its gravity ; in which case it will descend. On the subject of ARCHIMEDES'^ Screw, see BERNOU- ILLI Hydrodynamics Sect. ix. § 26. Also EULER, Nov. Com. Petrop. v. p. 259, &c. HENNERT, sur la Vis cF ARCHIMEDE. d. The screw is also employed in the division of ma- thematical instruments, and in reading off from them. The contrivance known by the name of the Micro- meter Screw, is used for measuring angles with great exactness. In RAMSDEN'S Dividing Instrument, the screw is applied for the same purpose. Of OUTLINES OF NATURAL PHILOSOPHY. Of the Funicular Machine. 152. If a body fixed to two or more ropes, is sustained by powers which act by means of those ropes, this assemblage is called the Funicular or Rope Machine. See VARIGNON", vol. i. p. 93, &c. ; also BOSSUT, Media- nique, art. 120. 153. If a body is sustained by two forces acting at the ends of two ropes, these forces will be in the inverse ratio of the sines of the angles which their directions make with the direction of the weight ; or directly as the secants of the angles which their directions make with the horizon. Hence the sum of the powers is to the weight, as the sum of the sines of the angles which the powers make with the direction of the weight, to the sine of the angle which the powers make with one ano- ther. 154. If the body is sustained by more than two cords, the powers, by the resolution of forces, may always be reduced to two. 155. If any number of points be given either in the same or in different planes, and if from their common MECHANICS. 9& eommon centre of gravity, lines be drawn to every one of the given points; the forces which draw in a direction of these lines, and are proportional te them, will be in equilibria. HUGENII, Opera varia, torn. i. p. 288. a. It follows from the above propositions, that by means of a rope, a small power may be made to raise a very great weight to a small height ; and also, that if a rope is stretched horizontally between two points, its own weight will prevent it from be- coming perfectly straight, whatever force be employ- ed to stretch it. The curve which a flexible chain or cord forms, when suspended by its two extremi- ties, and either hanging freely, or having any de- gree of tension applied to it, is called the Catenaria. SECT. III. OF FRICTION. 156. JJESIDES the resistance which the mecha- nical powers, and the machines compounded of them, are intended to overcome, an impediment to their motion is always observed when the moving parts are in contact with one another. This im- pediment to motion is called Friction. The 94 OUTLINES or NATURAL PHILOSOPHY. The subject of friction is treated by MUSCHENBROEK, §510,^533. BOSSUT, § 248,— 283. PHONY, Arch. Hydrauliqu?., Sect. v. BREWSTER, in his Appendix to FERGUSON'S Mechanics, vol. u. p. 334. GREGQ- RY'S Mechanics, vol. u. § 24, &c. 7< When a lieavy body is at rest oil ft hori- zontal plane, it is not in equilibria, or ready to obey the least impulse, in a direction parallel to the plane. A force must be applied to bring it into that state, with respect to any given direc- tion ; and this force is the measure of the fric- tion. 158. Friction destroys, but never generates mo- tion, and in this is unlike gravity, or any of the forces hitherto considered, which, if they retard motion in one direction, always accelerate it in the opposite. The force of friction, therefore, vio- lates the law of continuity, and cannot be accu- rately expressed by any geometrical line, or any algebraic formula. a. The retardation which friction opposes to motion, is nearly uniform, or the same for all velocities. COULOMB, in a series of experiments made with great accuracy, and on a large scale, found that bodies sliding on a plaile, oil which they Were tBO- ved by a constant force of IrdClwn^ were uniformly accelerated, which could not have been, if the im* peding as well as the impelling force had not been constant. MECHANICS. 95 constant. Journal dt Phys. torn. xxvu. p. 204, &c. COULOMB'S experiments received the prize of the Academy of Sciences in 1781, and are publish- ed in the tenth volume of the Memoires presentes. An abstract of them is given in the Journal de Phys. above quoted, and also in PEONY'S Arch Hyd* Sect. v. § 1089, &c. Motions retarded by friction are therefore subject to the laws explained, Articles 90 and 91 ; only that the retarding can never be converted into an acce- lerating force, so that the formulas of those articles cannot be applied in their full extent. 159. The force of friction is the greater, the" greater the roughness or asperity of the surfaces moving on one another ; it is also the greater, the greater the power by which these surfaces are pressed together ; but it is very little affbcted by their extent. a. That the quantity of friction does not depend on the extent of the surfaces that rub on one another, was affirmed by AMONTONS, and proved by a varie- ty of experiments. Mem. Acad. de Sciences, 1699, p. 208. The fact, however, has been since ques- tioned, particularly by LAMBERT, Mem. de Berlin, 1772; but the experiments of COULOMB, though they have pointed out certain exceptions, as those of some other writers have done, have shewn, that it holds in general, and that in practical mechanics* the ratio of the friction to the pressure, may be re- garded 96 OUTLINES OF NATURAL PHILOSOPHY. garded as constant, whatever be the extent of the surfaces that touch. Journal de Pliys. vol. xxvn, p. 296, at the end. From some very accurate experiments, Mr VINCE concluded, that friction increases in a less ratio than that of the pressure. Trans. 1785, p. 172. BOSSUT has remarked the same, when the weights become very great, and gives as a proof, the small inclination of the planks on which a ship slides when it is launched. Mecanique, § 256. 160. Friction may be distinguished into two kinds, that of Sliding, and that of Rolling Bodies. The force of the latter is very small compared with that of the former. Though the friction of rolling is small in respect of that of sliding, great force can be given to it by the application of pressure. In the passing of metals between rollers or cylinders, strongly compressed by springs, the friction is that of rolling, yet it is sufficient to communicate the motion of one of the cylinders to the plate of metal between them, in op- position to the great force which resists the exten- sion of the plate. 161. The preceding facts appear to prove, that friction arises from the asperities at the surfaces of bodies, the eminences of the one sinking into the cavities of the other, so that a certain force, when it acts parallel to the surfaces, is necessary to dis- engage the parts, by forcing the one body to rise 2 over MECHANICS, 97 over the other, or by bending and abrading the emi- nences which oppose the motion. MUSCHENBROEK, § 512. EuLER, Mem. de Berlin, 1748, p. 122. COULOMB, Mem. Presentes, torn. x. p. 164, &c. «. The difference between the friction of rubbing and rolling bodies, seems to arise from this, that the asperities do not require to be bent down or abra- ded in the latter case ; the eminences being lifted out of the cavities. b. When the rising up of the one body over the other is completely prevented, the friction becomes ex- tremely intense, as appears in what may be account- ed an extreme case, the drawing of wire, tubes, &c. where the extension of the metal is the effect of the friction acting all around. c. In bodies sliding on one another, the surfaces may be held together by the attraction of cohesion, so that the friction is not always diminished in pro- portion as the polish of the surfaces is increased. Professor LESLIE is of opinion, that the phenomena of friction are not fully explained by these causes, and that they arise chiefly from a tremulous or vi- bratory motion in the parts of bodies. Inquiry into the Propagation of Heat, p. 299 to 302. 162. The distance to which a given body will be moved by percussion, in opposition to friction, is as the square of the velocity communicated to it. VOL. I. G Tins 98 OUTLINES OF NATURAL PHILOSOPHY. This follows from § 99. DYNAMICS ; friction being a force by which motion is uniformly retarded. a. Some paradoxical appearances are explained on this principle. When a carpenter would drive the handle of an axe into the head, he strikes against the handle after it is slightly inserted into the head, holding it loosely in his hand, without any firm support, and the head perhaps downward ; at each blow the handle is driven farther than if the blow had been given to the head itself. The reason is, that the handle is the lighter body, so that the velocity which a blow communicates to it is greater than that which the same blow would communicate to the head. The friction is of consequence more effectually overcome. Thus, too, a nail is driven by a blow of no great force, into a piece of wood where the mere friction is suffi- cient to retain it against a great force applied to draw it out. b. The same thing is exemplified in the method some- times practised, of raising a stone by means of a tackle fastened to it by an iron plug, driven into a circular hole cut in the stone. A few blows of a small hammer are sufficient to fix the plug, so that it will serve not only to suspend the stone, though of several hundred weight, but to draw it out of the earth, in which it lies perhaps half-buried. When the stone is raised, a few blows of the hammer given obliquely to the plug, are sufficient to disengage it. The stones on which this experiment has been made have always possessed great toughness and indura- tion. The whole is explained by the great power of percussion MECHANICS. 99 percussion to overcome friction, compared with mere pressure or weight. 163. When motion begins, the intensity of fric- tion diminishes ; it does not, however, change after- wards as the velocity changes, but continues, as al- ready said, to retard with a uniform force. EULER makes the friction to be reduced to one-half, when the body is actually put in motion. Sur le Frottement des Corps Solides, Mem. Acad. de Berlin, 1748, p. 122, &c. § 13. The reduction appears in some cases to be much greater than this. COU- LOMB found the friction of wood sliding on wood to become less \\hen the body began to move, than it had been the instant before, in the ratio nearly of 2 to 9. Its intensity afterwards did not change. PHONY, Arch. Hydraulique, 1173, N° 2. 164. Friction may be measured by finding the force necessary to bring a body resting on a hori- zontal plane into such a state that it is ready to move on the application of the least force (§ 153.) : it may also be measured by placing the body on a plane of variable inclination, and increasing that inclination till the body begin to slide. As the ra- dius to the tangent of that inclination, so the weight of the body to its friction on the horizontal plane. If the weight be W, and the inclination of the plane, when the body begins to slide, t, the friction — W x tan ?*. G 2 165. Time 100 OUTLINES OF NATURAL PHILOSOPHY. 165. Time is often required for friction to come to its bearing, or to attain its maximum, and in this respect, different substances differ much from one another. d. i. COULOMB found, that in wood sliding on wood, without grease, the friction at first increased, but in a minute or two came to a limit, which it did not afterwards exceed. Oak, for example, sliding on oak, though the pressure was varied from 74 Ib. to 2474 Ib., had a friction, after a minute, always nearly 44 in the hundred. On diminishing the surface as much as possible, the friction was reduced no lower than 41 J per cent. b. ii. He found also, that when metal slides on me- tal, the friction is proportional to the pressure; in iron on iron, 28 \ in the hundred ; in iron on brass 26, &c. ; and this was the same whether the bodies were just beginning to move from rest, or had ac- quired any velocity whatever; different from the case just mentioned, of wood rubbing on wood. €. in. When heterogeneous bodies were made to slide on one another, as wood on metal, the friction in- creased slowly with the time, and did not come to its maximum in less than four or five days. Iron against oak, after 10", gave the force of friction = 7J per cent. ; at the end of four days it amounted nearly to one-fifth. In heterogeneous substances, too, the friction increases sensibly with the velocity, and follows nearly an arithmetical, while the velocity follows a geometrical progression. d. iv. When MECHANICS. 101 d. iv. When the surfaces are smeared with unctuous substances, the friction is diminished; but time is necessary for the attainment of the maximum. In a set of experiments, oak rubbing on oak, the sur- faces greased with tallow, and the weights being 1650, and 3280 pounds ; in the first, at the end of six days, the friction seemed to become stationary, and was 37 J per cent. ; in the second, the maximum arrived at the end of five days, and the friction was 47. e. In another set, with brass on iron, and fresh tal- low between them, the friction was four days in coming to its maximum, and it was then between 10 and 11 per cent. At the first moment it was 9 per cent., so that here the increase was inconsiderable. These are some of the principal results of COU- LOMB'S experiments, the most accurate and extensive that have been made on this matter : they shew a great difference in the laws of friction, that obtain among different substances. Consult PRONY and COULOMB, ubi supra. 166. Friction is diminished by the unctuous substances mentioned above. Those that are thin- nest and least tenacious are the best ; plumbago also, or Black Lead, as it is called, reduced to powder, and rubbed on the surfaces of wood, metal, stone, &c. serves greatly to diminish fric- tion. 167. The OUTLINES OF NATU11AL PHILOSOPHY. 167. The effect of friction may be diminished, by drawing a body in a line inclined at a certain angle to the plane on which it rests. Thus, if the weight of a body be to its friction, on a horizon- tal plane, as ft to 1, it will be drawn with the greatest ease in the direction which makes with that plane an angle, having for its tangent - . ft DAN. BERNOUILLI, Nov. Com. Petrop. torn. 13. p. 244. 168. Friction is diminished, when it is converted, by means of wheels or rollers, from the friction of sliding into that of rolling bodies. When each end of the axis of a wheel is made to turn, not in a groove, but on the circumferences of two wheels, every one moveable on its own axis, these last are called Friction Wheels, as serving very much to lessen the effect of friction. 169. The momentum of friction is diminished by friction wheels, in the ratio of the radius of the axis of any one of the wheels (they are supposed equal) to the perpendicular height of the axis that rests upon them, above the line joining their centres. EULER treats of the advantage of friction wheels, Mem. Acad. Berlin, 1743, p. 144. The theorem he has investigated is not so simple as the preceding, but may be reduced to it. In the Phil. Trans, vol. LIII. p. 155. MECHANICS, 108 p. 155. Mr K. FITZGERALD has described the con- struction of friction wheels, by which he reduced the power of friction from 425 Ib. to two and three- fourths. Rollers put under a heavy body, diminish the friction in the greatest degree possible, if they are spheres or cy- linders, without any axis on which they are constrain- ed to move. The conversion of friction from sliding to rolling is then complete. The wheels of carriages are contrivances of the same kind : but in them the conversion above referred to is imperfect : for though the friction at the circum- ference of the wheel is that of rolling, at the axis, it is that of rubbing ; yet as, in this last, the dis- tance from the centre of motion is small, the mo- mentum of the friction is small in the same pro- portion. 170. In wheel carriages, it is more advantageous that the wheel should turn on a fixed axle, than that the axle should turn with the wheel, because in the latter case, when the direction of the motion changes, one of the wheels must go backward, and must oppose the motion of the other, so that the friction of rubbing is substituted for that of rolling. 171. The plane on which a carriage moves, and the line of draught, being both horizontal, the ad- vantage for surmounting an immoveable obstacle, of a given height, is as the square root of the radius of the wheel. Let 104 OUTLINES OF NATUKAL PHILOSOPHY. Let the whole weight to be moved be W, the radius of the wheel r ; ,/^the force which, drawing horizon- tally, will raise the carriage over an immoveable ob- stacle of the height h ; then or 17%. When a machine is so loaded, that it would be in equilibria if there were no friction, it will not be ready to move till a part be added to the power, having its momentum equal to the momentum of the friction. The friction must be considered as a given force, op- posed to the power, and having a momentum pro- portional to its distance from the centre or axis of motion. 173. Let a lever turn on an axle of which the radius is r; let a be the length of the arm to which if the power P were applied, it would sustain the weight W, supposing there were no friction ; and let p be the addition to be made, in order to over- come the friction, the ratio of the pressure to the friction being that of n to 1 ; then is, jp= JL (P-fW). na -a. The friction will cause the equilibrium to remain while the power varies from P — p to P +p. If MECHANICS. 105 If a, P and W are given, p can only be lessened by diminishing r, or increasing n, that is, by diminish- ing the radius of the axle, or polishing its sur- face. For the application of a like correction to the other mechanical powers, see PRONY, Arch. Hyd. Sect. v. BOSSUT, Mech. § 261, to 283. VAN SWINDEN, Po- sitiones Phys. § 312—325. 174. The stiffness of ropes, or their resist* ance to bending, has a great analogy to friction. In different ropes, the forces requisite to bend them, are in the direct ratio of their diameters and their tensions jointly, and in the inverse ratio of the radii of the cylinders round which they are bent. Van SWINDEN, Pos. Pkys. 339. COULOMB, whose ex- periments ought to have great weight, holds the stiffness to follow the ratio of the section of the rope, that is, of the square, not of the simple power of its diameter. The decision of the question requires new experiments. 175. The friction of a rope wound round a cylinder, increases in geometrical progression, while the number of turns increases in arithmetical pro- gression. If the turns be represented by the numbers, 0, 1, 2, 3, 4, &c. ; the resistance made by the rope may be represented by the numbers 1, 2, 4, 8, 16, &c. Hence 106 OUTLINES OF NATURAL PHILOSOPHY Hence the great use of this kind of resistance to stop motion quickly, and to make fast one end of a rope when a great force or strain is to be applied to the other. 176. Though friction destroys motion, and ge- nerates none, it is of essential use in mechanics. It is the cause of stability in the structure of ma- chines ; and is necessary to the exertion of the force of animals. A nail, a screw, or a bolt, could give no firmness to the parts of a machine, or of any other structure, without friction. Animals could not walk, or ex- ert their force any how, without the support which it affords. Nothing would have any stability but in the lowest possible situation, and an arch which could sustain the greatest load when properly di- stributed, might be thrown down by the weight of a single ounce, if not placed with mathematical ex- actness at the very point which it ought to oc- cupy. SECT. MECHANICS. 107 SECT. IV. i OF MECHANICAL AGENTS. 177. A- FUNDAMENTAL distinction among me- chanical agents, or the powers which put bodies in motion, consists in this, that in some the intensity of their action, or the acceleration they produce in a given time, is the same whether the hody acted on be at rest or in motion; in others, it is greatest when the body acted on is at rest, and becomes less as its velocity increases. Gravity is the only force which is certainly known to act with equal intensity on bodies in motion and at rest. Magnetism has probably the same proper- ty. Every other power acts more forcibly on a body at rest than on one that has already acquired motion in the direction in which it acts. This happens with respect to the elasticity of springs, the impulse of fluids, and the strength of animals, the only powers, except gravity, which are employed to put machines in motion. The knowledge of the laws, according to which the action of these powers diminishes, as the motion they communicate increases, is of great im- portance in mechanics. 178, The 108 OUTLINES OF NATURAL PHILOSOPHY. 178. The absolute elasticity of a spring is near- ly proportional to its distance from its state of qui- escence, or that at which its elasticity does not act ; but the power of the same spring to impel a body, must depend not only on its absolute force, but also on the rate at which it overtakes the body. a. An arrow, for example, is impelled by the string of a bow, not simply with a force proportional to the distance of the string from the position in which it was at rest, (to which the absolute elasticity of the bow is probably proportional) but by a force which is less, the nearer the velocity of the string and of the arrow approach to an equality. This subject, however, has not been sufficiently exa- mined. b. A spring is sometimes made to act with uniform force by lengthening the lever to which it is applied in the same proportion as its tension becomes less, or as its distance from its state of quiescence dimi- nishes. This is done in the case of a watch-spring. The velocity acquired by the body impelled, is hers so small that it may be neglected. 179. When a moving body, whether splid or fluid, acts upon another, the motion communi- cated is less, the less the difference of their velo- cities. 2 The MECHANICS. 109 The determination of the law observed by this action, in the case of fluids, belongs to Hydrodynamics, and will be explained under that head. There remains, therefore, to be considered here only the law of ani- mal force. 180. The strength of men, and of all animals, is most powerful when directed against a resist- ance that is at rest : when the resistance is over- come, and when the animal is in motion, its force is diminished ; lastly, with a certain velocity, the animal can do no work, and can only keep up the motion of its own body. A formula having the three properties just mention- ed, will afford an approximation to the law of ani- mal force. Let P be the weight which the animal exerting itself to the utmost, or at a dead pull, is just able to overcome; W any other weight with which it is actually loaded ; and v the velocity with which it moves when so loaded ; c the velocity at which the power of drawing or carrying a load en- tirely ceases ; then W — P (1 J is an equation that has all the three conditions mentioned above. Not only, however^ has the formula P (1 — - j these conditions, but the square of it has the same, or, indeed, any function of it which vanishes when 1 — - vanishes, that is, when v =r c. We are left? c then* 110 OUTLINES OF NATURAL PHILOSOPHY. then, at liberty to choose any of these functions, and would assume the formula above as the sim- plest, if another condition did not seem necessary to be included. It is certain, that in all cases, when v approaches to r, or when the speed becomes great, a small variation in the weight is accom- panied with a great variation in the velocity. The simplest formula that corresponds to this condition, is, when 1 is raised to the square. 181. Therefore, till experience has led to a more accurate result, we may suppose the strength of animals to follow the law expressed by the formu- la, W = P (1 — - This equation, supposing W and v variable, is an equation to a parabola, the construction of which will serve to represent this law more clearly to the imagination. A formula for expressing the law of animal action, was first proposed by EULER, in a Dissertation on the Force of Oars. Mem. Acad. de Berlin, 1747. That which he employed was W = P (1 Y> C s different from both those we have mentioned, but a function of the first, such as to become 0, when v — c. EULER, however, changed this to another, Mem. Acad. de Berlin, 1752, and Nov. Com. Pe- 1 trop. MECHANICS. Ill trap. vui. p. 244, the same that is given in the last article. He appears to have done so merely on ac- count of the analogy thus preserved between the action of animals and of fluids. M. SCHULZE, of the Academy of Berlin, has since proved by direct experiments, that this formula is nearer the truth than the other. Mem. de TAcad. de Berlin,, 1783, p. 333, &c. 182. The effect of animal force, then, or the quantity of work done in a given time, will be t>Y proportional to W v, or t6 P v (1 — -) , and will c 4P be a maximum when v =: « , and W iz ~~ , that is, when the animal moves with the one-third of the speed with which it is able only to move itself, and is loaded with four-ninths of the greatest load it is able to put in motion. The quantities P and c can only be determined by ex- perience, and as they must differ for different indi- viduals, an average estimation of them is all that can be obtained. Even that average is but imper- fectly known ; EULER supposes, that for the work of men, P may be taken — 60 Ib. and c = 6 feet per second, or a little more than four miles an hour. A man, according to this estimate, when working to the greatest advantage, should carry a load of 27 Ib., and OUTLINES OF NATURAL PHILOSOPHY. and walk at the rate of two feet in a second, or a mile and one-third an hour. A horse, according to DESAGULIERS, drawing a weight out of a well, over a pulley, can raise 200 Ib. for eight hours together, at the rate of two miles and a half an hour. Supposing the horse, in this case, to work to the greatest advantage, P = 300x9 =450, and c = 2.5 x 3 = 7.5 miles 4 per hour. M. SCHULZE made experiments purposely to deter- mine the values of P and c, in the above formula. when applied to the working of men. He found by trials made on the strength of twenty men, that, ta- king a mean, P is to be estimated at 100 or 101 pounds avoirdupois, and c at 5.4 feet per second. For the labour of a man, we may therefore suppose W = 101 (1 -- ~Y. Thus, if 5.4/ W = 30, v = 5.4 (1 —T = 2.43 ; and in fact the common rate of a man's working may be near- ly estimated at this amount. If, however, the for- mula is correct, he does not work to the greatest ad- 4 vantage ; for then W would be - of P, or nearly 9 45 Ib., and v would be - of c, or 1.8 feet per se- , the height h = 0. With a load equal to twice a man's weight, he could not ascend. 184. The strength of a man being supposed to follow the law now laid down, its greatest effect in raising a weight, will be when the weight of the man is to that of his load as 1 to — 1 + V 3, or nearly as 4 to 3. Because h = - lh= w now / /&, or the weight multiplied into the height to which it is raised, is the measure of the effect, or of the work done, which, therefore, will be a maximum when the last formula is so, that is when / = «;(! -f V 3). H w (1 — - 5— T/. . 1 2 2o>x It in the equation /* = — -. —, we suppose 1.16 OUTLINES OF NATURAL PHILOSOPHY. h and I to be variable, the other quantities being constant, the locus of the equation is a hyperbola, which may be easily constructed. The theorems just given only differ from COULOMB'S, by being somewhat simpler, and free from all reference to any particular measure of length or of weight. On the subject of Animal Force, however, many more experiments are wanting : to determine, for instance, the friction of wheel carriages ; the difference be- tween the exertions required to walk on a horizontal plane, and on one of a given declivity ; the quantity of work done in a given time by the same animal, carrying different loads ; the difference between the effective exertion of a man's strength when he moves along with his load, and when he stands, as in turning a wheel, or sits, as in rowing a boat, Sec. &c. SECT. V. MOTION OF MACHINES. 185. JM.ACHINES either work with a variable or a uniform velocity. If the moving power is of the kind that, when the motion begins, diminishes in the intensity of its action, the machine, after a little time, will acquire a uniform velocity. If, MECHANICS. 117 If, however, the moving power be one that acts always with the' same force, and if the resistance is also uniform, the machine will be continually accelerated. But as a motion continually accelera- ted is incompatible with many of the purposes of machinery, it is either contrived, that, by an increa- sed resistance, the motion shall become uniform, or that the moving power shall at intervals cease alto- gether, or become a retarding force, — so that the velocity of the machine, though continually chan- ging, may be confined within certain limits. M. D'ALEMBERT'S principle (§ 120.) is very useful for determining the action of machines working with a variable velocity. One of the simplest examples of this is contained in the investigation of the follow- ing theorem. 186. Let A and B be two given weights, appli- ed to the ends of the arms of a lever, of which the lengths are a and b, and let A x a be greater than B x b, so that A may preponderate ; the space over which B will be raised in one second, (g being the velocity which a heavy body acquires in the first ,1 fi i . (Aa — ~Bb)b , second of its descent), is ^a24-B62 Xf g. Call x the velocity of B, then -L. will be the velocity of A. If the bodies were free to fall by their own gravity, the velocity of each would be g ; therefore the 118 OUTLINES OF NATURAL PHILOSOPHY. ax the velocity lost by A is g — ; and the veloci- ty gained by B is g -f x ; now the momenta of these must be equal by § 120, that is, A a (g ~\ b / = B b (g -f x) ; and hence ..) i fy."i3*^EO • ' '.'^f* *$ij '^iUC/Oi'l '*?! ' III 194 The moving power and the resistance be- ing both given, other things remaining as above, if a machine be so constructed that the velocity of the point to which the power is applied, be to the velocity of the point to which the resistance is ap- plied as 9 R to 4 P, the machine will work to the greatest possible advantage. The velocities of the points just mentioned are capa- ble of being adjusted to any given ratio, on princi- ples that have already been explained. Under the name of the Load, we suppose the fric- tion of the parts of the machine to be comprehend- ed, which must therefore be determined by experi- ment. 195. Care should be taken to give to every ma- chine the greatest possible regularity in its mo- tion. The contrivance known by the name of a Fly, is one of the best adapted for this purpose. When the impelling power is subject to alternate intention and remission, the inertia of a heavy wheel, by preserving MECHANICS. 125 preserving the velocity it has acquired, tends to les- sen the irregularity resulting from these causes. The effect of a fly, to remove irregularity in the mo- tion of a machine, its weight and diameter being gi- ven, is proportional to its velocity. 196. In wheel carriages, the equability of the motion is even of greater importance than in hy- draulic engines ; the sudden checks to motion ha- ving a more pernicious effect on an animal body than on an inert mass. The improvement of roads, and the diminution of fric- tion, are, in such cases, of the greatest use ; but that which corresponds most to the advantage derived from a fly, is the effect from supporting the weight by springs, or from giving elasticity to the parts by which the cattle are yoked to the carriage. The sudden action of the resistance, and the jerks which are the consequences of it, are thus prevented. The greater the velocity the carriages move with, the more advantage is derived from this application of elastic force. SECT, 126 OUTLINES OF NATURAL PHILOSOPHY. SECT. VI. OF THE DESCENT OF HEAVY BODIES ON PLANE AND CURVE SURFACES. 197. JL HE force which accelerates the motion of a heavy body on an inclined plane, is to the force of gravity, as the sine of the inclination of the plane to the radius, or as the height of the plane to its length. If/*= force accelerating the body on an inclined plane, of which the inclination is i> and if g= force of gra- vity, f=g sin i. Hence the motion of a body on an inclined plane, is a motion uniformly accelerated. SXrRAVESANDE, § 382. ; MuSCHENBROEK, § 608. ; GA- LILEO, Dial. III. Opere, torn. in. p. 106, &c. 198. If two bodies begin to descend from rest, and from the same point, the one on an inclined plane, and the other falling freely to the ground, their velocities at all equal heights above the sur- face will be equal. Hence the velocity acquired by a body in falling from rest through a given height, is the same, whe- £ ther MECHANICS. 127 ther it fall freely, or descend on a plane any ho'w inclined. 199. The space through which a body will de- scend OB an inclined plane, is to the space through which it would fall freely in the same time, as the sine of the inclination of the plane to the ra- dius. The diameter of a circle perpendicular to the hori- zon, and any chord terminating at either extremi- ty of that diameter, are fallen through in the same time. The velocities which bodies acquire by descending along chords of the same circle, are as the lengths of those chords. 200. If a body descend over a series of inclined planes, at each of the angular points, where it passes from one plane to another, it loses a part of the velocity it had acquired, proportional to the versed sine of the inclination of the planes. If v be the velocity it has acquired when it comes to any angle ?, v x vers. p is the velocity lost 201. Any angle being given, it may be divided into angles, so small, that the sum of the versed sines of these angles shall be less than any given magnitude. Hence 128 OUTLINES QF NATURAL PHILOSOPHY. Hence the number of planes may be so increased, and their inclination to one another so diminished, that though the change of direction between the first and the last be ever so great, the loss of velocity in the descending body shall be less than any given quantity. Therefore, if the body descend in a curve, it will suf* fer no loss of velocity. D^ALEMBERT, Traite de Dynamique, § 41. This is true of the motion of a pendulum, that is, of *>;•! a body suspended from a fixed point, and swinging freely forwards and backwards about that point. . When a pendulum vibrates in a circle, the force which accelerates it as it approaches the lowest point, is nearly proportional to its distance from that point. a. Hence the small vibrations of the same pendulunl are nearly isochronous, or of the same duration. 203. If a pendulum vibrate in a circle, and if g be the velocity acquired by a heavy body de- scending freely, in one second, the square .root of g is to the square root of the length of the pendu- lum; as the number which expresses the ratio of the circumference of a circle to its diameter, is to the time of. the least vibration of the pendulum, expressed in seconds. Let MECHANICS. Let I be the length of a pendulum, w the number that denotes the circumference of a circle, of which the diameter is 1, and t = the time of one vibration of the pendulum in seconds, t = As the weight of the body does not enter into the ex^ pression of the time, the vibrations of a pendulum are the same whatever be its weight. Hence the times of the vibrations of pendulums are as the square roots of their lengths. If tf = 1, I' = length of the seconds pendulum, and 1 — TT i / - ; therefore g^iP^l', or V = -^ • £? "^ • If it be required, having /', to find g, g— 7^ x I'. By help of this last formula, g is found more exactly than can be done by direct experiment. In London, Lat. 51° 31' 8", by Captain KATER'S experiments, the length of the seconds pendulum =39.1386 inches. Hence g = 32.193 feet. SIMPSON'S Fluxions, § 460. ; SAUNDEKSON'S Fluxions, p. 207. ; and CAVALLO, vol. i. p. 190. The values both of g and /' are somewhat different in different latitudes, as will be explained in Phy- sical Astronomy. In this proposition, the arch over which the pendu- lum vibrates, is supposed to be very small ; if the arch is considerable, let its versed sine (to the ra- dius 1) be v, and t the time of an entire vibration, VOL. I. I * = 130 OUTLINES OF NATURAL PHILOSOPHY, '42' 2.2.4.4 4 ' 2.2.4.4.6.6" qp SIMPSON, ibid.; FRANCOEUR, § 196. 3*ne edit. ..j '.,M ; Centre of Oscillation. 204. When a pendulum consists of two or more bodies, or of one body, from the figure and extent of which we are not permitted to abstract, it is called a Compound Pendulum. The axis of this pendulum is a Hire which passes through the point of suspension, and is vertical when the pendu- lum is at rest ; and its centre of oscillation is a point in this axis, or in the axis produced, in which, if a weight were placed, it would form a simple pen- dulum, vibrating with the same angular velocity as the compound. It vibrates, therefore, in the same time ; or is isochronous with the compound pendulum itself. 205. The distance of the centre of oscillation of a compound pendulum, from its centre of suspen- sion, is equal to the sum of the products of each body into the square of its distance from the centre of MECHANICS. 131 of suspension, divided by the sum of the products of each body into the simple power of its distance from that centre. If the bodies be A, B, C, D, &c. and their distances from the centre of suspension a, b, c, d, &c. and x the distance of the centre of oscillation from the same point, then Aa2 + B62 + C6'2+p^2 Aa+Bi+Cc+Dd This value of x is deduced from D^ALEMBERT'S prin- ciple, § 120. x is also the length of the simple pendulum, isochro- nous with the compound, or vibrating in the same time. If any of the bodies are above the point of suspen- sion, their distances from that point are accounted negative. 206. If a cylinder, of which the altitude is a, and the radius r, be suspended from its vertex, the distance of the centre of oscillation from the ver- . 2 a r* tex is . 3 2a If the radius r =z 0, so that the cylinder coincides with a straight rod of inconsiderable thickness, 52 a the distance of the centre of oscillation is -^- • o 207. If 132 OUTLINES OF NATURAL PHILOSOPHY. 207. If a cone of which the altitude is a, and the radius of the baser, be suspended from the vertex, the distance of the centre of oscillation f ^ 4 a r2 trom the vertex is -— 4- — - • 5 5a WOLFII Elementa Math. torn. n. § 459, 460. 208. If a sphere, of which the radius is r, be suspended by a fine thread, so that the distance of its centre from the point of suspension is a, the distance of the centre of oscillation from the same 2r2 point is a 4- -— . 5a HUGENII Horol. OsciL Prop. 22. 209. If in the axis of a pendulum, different points of suspension be taken, the distances of the centres of oscillation from the centre of gravity, will be reciprocally as the distances of the points of suspension from the same. This proposition holds of solids as well as of plane figures. 210. The centre of oscillation of a compound pendulum, may be found by comparing its vi- brations with those of a simple pendulum of a known length, the lengths of two pendulums being MECHANICS. 13S being inversely as the squares of the number of vibrations which they perform in the same time. Thus, if lr be the length of a simple pendulum vibra- ting seconds ; x the distance of the centre of oscilla- tion of a compound pendulum from its centre of sus- pension, and n the number of vibrations which it 3600 x V makes in a minute, x = 5 . n2 Because I' is nearly 39.1386 inches, 140898.9 . , .. a = 2 inches, nearly. In making this ex., periment, n may be measured by a watch. See BUTTON'S Math. Diet, article, Centre of Oscilla- tion. 211. The centres of suspension and oscillation are convertible, that is, if the centre of oscillation in any body or system of bodies be made the point of suspension, the former point of suspension will then become the centre of oscillation. The first author who treated of the pendulum and the centre of oscillation, was HUYGENS, in his Horolo- gium- Ostittatorium, Opera Varia, torn. i. p. 51. &c. See also S'GRAVESANDE, § 404, and 424, MUSCHENBROEK, § 640. BOSSUT, Mecanique, § 401, 405. Heavy 134 OUTLINES OF NATURAL PHILOSOPHY. r-.: ;:--' Heavy Bodies descending on a Cycloidal Surface* - 212. If a circle DHE (fig. 12.) roll along a line AB, until the point V in its circumference, which touched the line AB in A, come to touch it again in B, the curve AVB, generated by the point V, is called a Cycloid. The line AB is called the Base ; DV, which bisects AB at right angles, the Axis of the Cycloid ; and the circle DHE, the Generating Circle. If AC and BC be two semicycloids, or semicycloidal cheeks, each equal to the half of AVB, touching AB in A and B, and touching one another in C ; -and if there be fixed at C a pendulum P, hanging by a thread PTC, equal in length to the semi- cycloid, P in its vibrations will describe the cycloid AVB. 213. A pendulum being thus made to vibrate in a cycloid, is at every moment of its descent acce- lerated by a force proportional to the arch of the cycloid, between it and the lowest point. .Hence the vibrations of this pendulum, whether great or small, are all of the same duration* 214. If MECHANICS. 1S5 214. If / be the length of a pendulum vibrating in a cycloid, the time of a vibration, whether the vibration be great or small, will be #y -• o 215. The line in which a heavy body descends in the least time from one given point to another, not in the same horizontal plane, nor in the same vertical line with it, is an arch of a cycloid, having for its base a horizontal line drawn through the uppermost of the given points. Hence the cycloid is called the line of swiftest de- scent. The problem of finding the line of swiftest descent was first proposed by JOHN BERNOUILLI, Acta EruAito- rum, 1697. Though the solution of it requires the assistance of the higher geometry, elementary demon- strations have been given by several authors, parti- cularly by MACLAURIN, Fluxions, vol. n. § 574, &c. ; and by FRISIUS, Cosmographia, Introd. § 22f and 23. SECT. 136 OUTLINES OF NATURAL PHILOSOPHY. SECT. VII. ROTATION OF BODIES. Rotation about a Fixed Axis. 216. LET A, B, C, D, &c. (fig. 13.) be a sys- tem of bodies in one plane, so connected as to be immoveable in respect of one another, but moveable about an axis at right angles to the plane, and passing through a given point S ; let «, b, c9 d, be their distances respectively from S ; and let a force act on one of the bodies A, such that if A were unconnected with the system, it would receive the velocity u9 in a direction at right angles to the ra- dius a ; let v be the velocity which it actually ac- quires when it makes a part of the system ; That is, if each body be multiplied into the square of its distance from the axis of rotation, the sum of all these products, is to the product of A, into the square of its distance from the same axis, as the velocity which A would have had if it had beer? MECHANICS. . 137 been free to move by itself, to tbe velocity which it has as a part of the system. This is proved, by considering that A (w— - v) is the i A i , J$b v C cv D d v , motion lost bv A, and > ? — — , the a a a motions gained by B, C, and D ; and that these last must, by D'ALEMBERT'S principle (§120.), be in equilibria with the former ; whence the equation Aa(u — v) = (Bb* + Cc* + T>fP) -, from which a the value of v is obtained. Prom the demonstration, it is evident, that the mo- mentum of the system relatively to S, is found by multiplying each body into the square of its dis- tance from S, and the sum into the velocity at A. The angular velocity is found by dividing both side* by a ; this gives v __ A au a ~~ A «2 + B b2 + C c2 + D d* ' The quantity — is the value of the angle described round S in one second, expressed in parts of the ra- dius. If it is required in degrees and minutes, the above value must be multiplied by 57°.29578, the number of degrees ip an arch equal to the radius. These propositions hold not only of a system of bodies in one plane, but of a system in different planes, if we suppose perpendiculars drawn from them to the axis of rotation. S17. If OUTLINES OF NATURAL PHILOSOPHY. 217. If , a computed in this way is equal to 0, and thence the value of x, or SP. 218. Any system of bodies being given, a point may be found in which if all the bodies were collected, a force applied at any distance from the MECHANICS. 139 the axis, would communicate to the bodies thus collected the same angular velocity that it would have communicated to the system in its first con- dition. The point thus found is called the Centre of Gyra- tion. Its distance from the axis of rotation is A+B+C+D ATWOOD OTi Rectilinear and Rotatory Motion, p. As every body may be considered as a system of phy- sical points, it is evident that the preceding propo- sitions are of general application. We shall give examples in the cases of the cylinder and the sphere. 219. Let ADBE (fig. 15.) be a cylinder move- able about an axis passing through C its centre ; let the radius of the cylinder be given = r, and also its weight «= W ; and let another given weight F hang by a cord BF, wound round the cylinder. Required, the force accelerating the point B, and the velocity which B will acquire in the first se- cond of its iwotion. The 140 OUTLINES OF NATURAL PHILOSOPHY. The sum made by multiplying each particle of the cylinder, into the square of its distance from the axis, supposing the thickness of the cylinder, in the direction of that axis, to be m, is found by fluxions, or the quadrature of curves, to be \ K m r4, and therefore (by § 92.) the velocity acquired in the first second by the circumference of the cylinder is * m r9 = the solidity of the cylinder), = Q Tj' _, - — - — '-§~ , g being the velocity acquired by a heavy body in the first second of its descent. As £ F + W to 2 F, so the force of gravity to the force accelerating the cylinder at its surface. 220. The quantity A a2 + B 62, &c. in § 216, which is the sum of all the products made by mul- tiplying each particle into the square of its distance from the axis of motion, is called the momentum of the inertia of the revolving body. It is compu- ted for any given body, either by the method of fluxions, or the quadrature of curves. •'Saite-r ->ff 221. Let the circle ADB (fig. 15.) represent a sphere moveable about an axis that passes through the centre C, and is perpendicular to the plane of the circle ADB ; and let it be put in mo- tion MECHANICS. tion by the weight F hanging from B. The ra- dius of the sphere r, its weight W, and that of the hody F heing given, it is required to find the ve- locity which the point B acquires in one second, or the force by which it is accelerated. The momentum of inertia computed for the sphere, o is — v r5, and therefore the accelerating force at ^ Bis Fy3 •-&_=• F-£ .. As 2.^ Fr2+A*-r* F-f-^^r3 is the solidity of the sphere, if W be substituted F £• for it, the above expression becomes - '-£ - . F+1*W 4f Therefore F -{- ^ W is to F, as the force of gra- vity to the accelerating force at B, the equator of the sphere. . If the motion, instead of being produced by a body hanging from the sphere, and partaking of its velocity, were communicated by the impulse of a body F, moving with a given velocity c, in the direction of the tangent at B, and striking against the radius produced, the velocity communicated ni 15F-C would be -j-^-- The solution of the two last problems might be deri- ved from the supposition, that the whole mass was 3 collected 1421 OUTLINES OF NATURAL PHILOSOPHY. collected in the centre of gyration. The question wxmld then be reduced to the simplest case of § 211. where the system consists only of two bo- dies. The preceding solution is, however, more direct. 2.23. If in the two preceding cases, W and F be both supposed to be sustained by the axis when the whole is at rest ; the part of the weight of P of which the axis is relieved when the motion be- gins, is to the whole weight of F, as the Force by which it is accelerated to the force of gravity. Thus, in the instance of the sphere, the pressure on the axis is diminished by the quantity If W=3001b., F = 10 Ib. ; - ?~ - = \,so that F+ iW the diminution of pressure is 2 Ib., and therefore the whole pressure, when the motion takes place, is 308 Ib. If it were required to find directly the part of F pressing on the axis, it will come out W = - 4 - X F, which, in the last example, 15 would amount to 8 Ib. ATWOOD on Rectilinear and Rotatory Motion, § 6. p. 183., &c. FRJSII Mechanica, Opera, torn. u. p. 133., MECHANICS. 143 p. 133., Sec. VINCE DTI the Principles of Progres- sive and Rotatory Motion. Phil. Trans. 1780, p. 546., &c. Rotation on a Movedble Axis. . When the impulse communicated to a bo- dy is in a line passing through its centre of gravi- ty, all the points of the body move forward with the same velocity, and in lines parallel to the di- rection of the impulse communicated. But when the direction of the impulse communicated does not pass through the centre of gravity, the body acquires a rotation on an axis, and also a progres- sive motion, by which its centre of gravity is car- ried forward in the same straight line, and with the same velocity, as if the direction of the impulse had passed through the centre of gravity. The progressive and rotatory motion are independent of one another, each being the same as if the other had no existence. This is a consequence of the general law, That the quantity of motion estimated in a given direction is not affected by the action of the bodies on one another. The revolution of a body on its axb arises from an action of this kind, and can neither increase nor diminish the progressive motion of the whole mass. 2 The 144 OUTLINES OF NAT0EAL PHILOSOPHY. The progressive motion, if the moving force and the •\bodymoved are given, is determined by the prin- ciples of Dynamics; and the rotatory motion is computed, as if it were about a fixed axis, by the propositions just laid down. When one impulse only is communicated to the bo- dy, the axis on which it begins to revolve is a line drawn through its centre of gravity, and perpendi- cular to the plane that passes through that centre, arid the direction of the impulse. See Fmsius de Rotatlone clrcum Axem Motum, tlieor. v. -*-Opera, torn. it. p. 157. aaol) I^fi.j;iiiji;ii(Vioo sajuq/fli s»ij k* #oi&dtii> oii£ 225. When a body devolves on an axis, and a force is impressed, tending to make it revolve on another, it will revolve on neither, but on a line in the same plane With them, dividing the angle which they contain, so that the sines of the parts are in the inverse ratio of the angular velocities with which the body would have revolved about the said axes, separately. This proposition was discovered by FRISI, and is de- monstrated in his works, ubi supra, p. 134. See also the Cosmogra/phfra of the same author, vol. n. p. 34. ' If a force be impressed on anybody, by which it is made tfr levolve on "'ah axis, the quan- tity of its momentum, estimated by collecting in- to one sum all the products of the particles into their MECHANICS. 145 their velocities, and their distances from the axis of motion, is equal to the momentum of the force impressed, estimated in the same manner* Hence, though, in consequence of the rotation of a body on its axis, a change may take place in its figure, or in its internal structure, the total quantity of its momentum will continue the same, 227« A body may begin to revolve on any line as an axis, which passes through its centre of gravity, but it will not continue to revolve permanently about that axis, unless the opposite centrifugal forces exactly balance one another. A homogeneous sphere may revolve permanently on any diameter, because the opposite parts of the so- lid, being in every direction equal and similar, the opposite centrifugal forces must be equal ; so that no force tends to change the position of the axis. A homogeneous cylinder may revolve permanently about the line which is its geometric axis. It may also revolve permanently about any line that bisects that axis at right angles ; but it can revolve perma- nently about no other line, as the centrifugal forces cannot be equal. The same is true with respect to all solids of revolution. 228. In every body, however irregular, there are three axes of permanent rotation, at right VOL. I. K angles 146 OUTLINES OF NATURAL PHILOSOPHY. angles to one another, on any one of which, when the body revolves, the opposite centrifugal forces exactly balance one another. These are called the Principal Axes of Rotation. This singular theorem was first proposed by SEGNER in 1755, and first demonstrated by ALBERT EULER, in a Mtmoire crowned at Paris in 1760. LEON. EULER has also given a demonstration of it in his treatise de Motu Corporum rigidorum, prob. 27. See also FRISIUS, de Rotatione Corporum quorum- cunque, theorema vn. ubi supra, p. 194*. These three axes have also this remarkable property, that the momentum of inertia with respect to any of them, is either a maximum or a minimum, that is, is either greater or less than if the body revol- ved about any other axis. LEON. EULER, ubi su- pra. The momentum of inertia has already been defined, to be the sum of the products made by mul- tiplying every particle into the square of its distance from the axis of rotation, APP- MECHANICS. 147 APPENDIX. CONSTRUCTION OF ARCHES. . IF ABCD, DCEF, EFGH, &c. (Fig. 16.) be a series of truncated wedges, resting on two immoveable supports, and having the planes of the faces that press against one another perpendicular to a vertical plane, represented here by the plane of the paper, they will be in equilibria, if their weights be proportional to the differences of the tangents of the angles which these planes make with a vertical plane given in position. Thus if the weights be as the differences of the tan- gents which AB, DC, EF, &c. make with the ver- tical MN, the whole will remain in equilibria. This follows from the properties of the wedge already enumerated. PHONY, Architecture Hydraul. torn. I. §356. Truncated wedges disposed in this manner, form what are called Arches in architecture ; and the most ad- vantageous construction of them requires, that the parts should be so adjusted as to be in equilibria, or to balance one another by their weight only. An arch, of which the parts balance one another in this manner, is called an Arch of Equilibration. K 2 Some 148 OUTLINES OF NATURAL PHILOSOPHY. Some other definitions are necessary for understand- ing the construction of these arches. The truncated wedges of which the arch is composed, and which are usually of stone, are called the Voussoirs ; each course of these wedges forming one voussoir. The central voussoir is called the Key-stone. The sur- faces which separate the voussoirs from one another, are called the Joints. The interior curve of the arch is called the Intrados ; the exterior, or that which limits all the voussoirs, when they are in equilibrium, is called the Extrados. The Abutments, are the masses of masonry, at each end, that support the arch. The beginning of the arch is called the Spring of the arch ; the middle, the Crown ; the parts be- tween the spring and the crown, the Haunches. The part of the abutment from which the arch springs, is termed the Impost. The theorem above is sufficient for the construction of arches, inasmuch as, when the curve of the intra- dos is given, it determines the weight of every indi- vidual voussoir. The application of it, in particu- lar cases, admits, however, of being greatly simpli- fied. This happens remarkably when the arch is a circle, and when the joints, (which are usually made perpendicular to the curve of the intrados), intersect in the same point. 230. In a circular arch, the weights of the vous- soirs should be as the differences of the tangents of the arches, reckoned from the crown. This MECHANICS. 149 This is nothing more than the general proposition above, applied to a particular case. In Fig. 17. if B be the crown of a circular arch, the weight of the voussoir LKNM should be to that of the voussoir MNPO, as the tangent of BN minus the tangent of BK, to the tangent of BP minus the tangent of BN ; or if, from a given point D, DH be drawn perpendicular to AC, and if the joints KL, &c. be produced, to cut it in E, F, and G, the weights of the voussoirs must be as the segments DE, EF, FG, in order to pro- duce an equilibrium. As the stones themselves cannot always be made in the proportion thus required, the wedges, of which they make parts, are supposed to be extended up- ward, by courses of masonry. The whole mass in- cluded between the planes of the joints produced, as far as that masonry extends, is understood to make up the weight of the voussoir. It is the busi- ness of the theory to calculate this weight ; and to construct the curve which bounds the voussoirs, when so produced. 231. In an arch, of which the in trades is a circle given in position, the depth of the key-stone being given, it is required to describe the curve of the extrados. Let AGE (fig. 18.) be one-half of the arch, C the centre, AB the height or depth of the key-stone. From the centre E, with a distance equal to CB, intersect CA in D ; through D, draw DH per- pendicular to AC; and through G, any point in the 150 OUTLINES OF NATURAL PHILOSOPHY. the intrados, draw GC, intersecting DH'in'F. Let FL be perpendicular to CE ; join LB, and produce CG, till CK be equal to LB ; K is a point in the extrados; and in the same .way may innu- merable other points be found. The line CK is always greater than CF, but ap- proaches to it as the arch AG increases, so that DH is an asymptote to the extrados. The equation to the extrados is easily deduced from this construction. Draw KO perpendicular to CE. Let CO = x, OK =#, CA = a, CD = b, the equa- tion to the curve is jj.-^ + ^y-y4 ind The extrados, in the case of a circular arch, is, there- fore, a curve of the fourth order, very much resem- bling the Conchoid of NICOMEDES. It has an a- symptote DH, and also a point of contrary flexure, so that it coincides very nearly with the curve jn which a road is usually carried over a bridge. Instead of £r, it may be convenient to have its value in terms of a, and the depth of key-stone d ; viz. b* = Zad + d?. All this holds, whatever portion of a semicircle the .arch be supposed to consist of. In MECHANICS. 151 232. In an elliptic arch, or one of which the in- trados is a semi-ellipsis, if 2 a be the span of the arch, or the longer axis of the ellipsis, and b the height of the arch, or the semi-conjugate axis of the ellipsis ; then, if from any point in the curve, a perpendicular w be let fall on the longer axis, the weight of the key-stone is to the weight of the voussoir which has its middle at the point from which the perpendicular is let fall, as as to Hence, if W is the weight of the key-stone, and V that of the voussoir, V = _ ft5 x W. This is the same with the theorem given by BOSSUT, (RechercliessurTEquilibre des Voutes, § 14.), though the expression is somewhat changed, with a view to facilitate the numerical calculation. 233. If the weights of the voussoirs are all equal, the arch of equilibration is a catenarian curve, the same that a chain or cord of uniform thickness would assume if hanging freely, the horizontal distance of the points of suspension, being equal to the span of the arch, and the depth of the low- est point of the chain being equal to the greatest height of the arch. a. It 152 OUTLINES OF NATURAL PHILOSOPHY. a. It can be easily shewn, that if the figure of the chain were reversed, the joints being such, that the force which was a pull in the first situation becomes a thrust in the second, the chain would support itself, and remain in equilibria. b. The equation to the catenaria, if x be the abscis- sa, taken from the vertex along the axis or vertical line, and y the corresponding ordinate, is := A + hyp.log. c. The constant quantity A is the horizontal tension expressed by the length of a chain of the same sec- tion and the same specific gravity with that of which the catenaria itself is formed. If S be half the length of the catenaria, S= d Chain Bridges have been proposed, in which the road is to be carried along the catenarian curve sus- pended between two Piers. As in such bridges the curvature must always be small, and as the arch must consist of a small part of the curve near the vertex, an approximation more than sufficiently exact for practice may be obtained in very simple terms. e. This approximation gives These two formulas are sufficient for the construction of a chain bridge, when y half the span, and x the depression of the middle below the points of suspen- sion, are given. For MECHANICS. 153 For the properties of the catenaria, see J. BERNOUIL- LI Lectiones Cole. Integ. Lect. 36. Operum, torn in. p. 491. DAVID GREGORY, Phil Trans. 1697. COTES, Harmonia Mensurarum, p. 108., and Notes, p. 115. The catenaria is remarkable for this mechanical pro- perty, That a chain hanging in that curve, has its centre of gravity lower than if it were disposed in any other line, its length continuing the same, and also the points from which it is suspended. An arch, therefore, constructed in this form, has its centre of gravity the highest possible. 234. The pressure of an arch on the piers or abutments which support it, may be estimated by considering the parts of the arch which rest im- mediately on the abutments to a certain height, as parts of the abutments themselves ; and the re- mainder of the arch as a wedge, tending to se- parate the abutments from one another. Thus (fig. 19.) the parts ALMS, BONT, which would remain in their places, though there were no pressure from above, may be regarded as parts of the piers ; and LMDNO, the remainder of the arch, as a wedge tending to overthrow the piers by its pressure on the planes ML and ON. On these suppositions, the thickness of the piers may be de- termined, so that their weight shall enable them to resist this pressure. If from a point H, in the verti- cal EC produced, perpendiculars UK, HX, be drawn to the planes ML and ON, and if HU be taken to represent the weight of the arch MLOND, the parallelogram IV being described, HI and HV will 154 OUTLINES OF NATURAL PHILOSOPHY. will represent the two forces which compound the force UH, and each of them will represent the force with which the arch presses on its supports ML and ON. Let RK = HI, and let RK be re- solved into the two forces QK, KP, the one hori- zontal, and the other vertical ; KQ tends to overset the pier, supposed moveable about the point F, and KP adds to its stability. If FC be bisected in W, we may suppose the weight of the pier at W, to act by the lever F W, and the force KP by the lever FC, both tending to keep the pier in its place, while the force KQ acts by the lever KC or F Y to overthrow it. Farther, though the pier is properly only FGAG, yet it is usual to have it loaded with masonry to the height K, and even higher, so that its weight may be taken as proportional to the rectangle FK. This will give an equation between the forces, when their analytical expressions have been obtained. Angle UHI = ft, and VHI = 2/3, the area MLOND = a?, EC = r ; make HU = — , then IH = r r& — = KR. 2 r cos ft Hence QK = — . _ — °L anc[ t/J /« r»/"vo Q \^s o-i r\ /9 * """"" . (71 ^ ' Let CK = h, and FC = x, grcosl* then the rectangle FK = h x, and the line that re- presents the weight of the pier will be —. Therefore, MECHANICS. 155 Therefore, hx x -xi + This determination of the thickness of the pier, proceeds on a hypothesis usually employed for de- termining the resistance of walls, &c. ; but which, nevertheless, is not quite conformable to the fact. The hypothesis is, that the pier AF, if the weight of the arch were too great to be sustained, would fall, by turning round the point F as a fulcrum. Now this is not what would happen ; the part of the abutment behind SM would be thrust out in the horizontal direction, till the arch had room to fall ; it is therefore against the masonry immediately be- hind the part AM, and chiefly in a horizontal di- rection, that the force is exerted. The resistance made by a wall, is, in this view of it, analogous to the friction of a body resting on a rugged surface, and has, no doubt, a given ratio to the weight of the wall, and may perhaps be regard- ed as equal to it. Thus the mass of building im- mediately behind SM, should be of force sufficient to resist the thrust KQ ; or, the section of that part of the pier should be greater than We give this, however, only as a limit, to which it may be right to pay attention in the construction of such works. The 156 OUTLINES OF NATURAL PHILOSOPHY. The true principles on which the resistance of walls is to be estimated, are not sufficiently understood ; and, in the mean time, a skilful and cautious engi- neer, can do nothing better than compute the re- sistance on different hypotheses, and take care that the actual strength be greater than any thing that theory points out as necessary. 235. As the wedges made by extending the voussoirs, according to the preceding theorems, would be of an inconvenient length near the spring of the arch, it will be best to substitute for them such a resistance as does not press till it is pressed upon. Thus, if on the pier immediately behind the arch, there be raised a body of masonry formed of ho- rizontal courses of large stones, the voussoirs being continued no farther than to come in contact with the ends of those courses, the purpose of equili- brium will be answered, supposing the work of sufficient strength, because any of the voussoirs pres- sing against the horizontal courses, will have its own defect of weight immediately supplied by their re- action. Each voussoir will then receive, with mathe- matical exactness, the precise degree of pressure re- quired for the equilibrium, and can neither have more nor less than that quantity. The upper part of the arch is here supposed to be loaded in the manner that strict theory requires ; to which there does not appear to be any practical objection. If the pier be an intermediate one between two arches, the opposite pressure on the horizontal abutting courses will render their effect more certain, On MECHANICS. 157 On the subject of Arches, see HUTTON'S Principles of Bridges. BOSSUT, Recherches sur PEquilibre des Voutes, Mem. de TAcad. des Sciences, 1774 and 1766; also Mecanique of the same author, edit. 1802, p. 383., &c. PRONY, Architecture Hydrau- lique, torn. i. § 358., &c. AT WOOD'S Treatise on the Construction and Properties of Arches, published in 1801. This last work is particularly commend- able, for the great accuracy of the investigation, the minuteness with which the subject is consider- ed, and the various experiments brought in support of the theory. See also article Bridge, in the Edinburgh Encyclopaedia. STRENGTH OF TIMBER. 236. If a beam of wood, of which the section is any given figure, having its area = a2, have one end firmly fixed in a wall ; if s be the strength of each fibre estimated laterally ; d the distance of the centre of gravity of the section, from the edge where the fracture terminates, and round which the beam, when it is broken, turns as on a hinge ; let / be the length of the beam, measured from the wall from which it projects, to the end at which a weight W is hung, just sufficient to break it; Wxl=.sa?xd9 and Wzz — p. The weights, therefore, required to break different beams, are as the strength of the individual fibres, multiplied 158 OUTLINES OF NATURAL PHILOSOPHY. multiplied into the area of the sections, and into the distance of the centre of gravity of the sections from the points round which the beams turn in breaking, the whole being divided by the lengths of the beams. We are indebted to GALILEO for this theorem. He was the first who attempted to reduce the strength of the materials used in the mechanical arts, to the measures of geometry and arithmetic. Dialogo Se- condo, Opere, torn. in. p. 63., &c. See also ibid. p. 213., a Commentary on what GALILEO has writ- ten on this subject, begun by VIVIANT, and com- pleted by GUIDO GRANDE. It is obvious, that the hypothesis of the beam break- ing short over, and turning as on a hinge, is not the precise fact ; and therefore the conclusion dedu- ced from this hypothesis cannot be expected to be more than an approximation. It does not appear, however, that more refined views, or more compli- cated speculations, have led to results agreeing better with experiment than the simple hypothe- sis of GALILEO. . 237- In a beam, of which the section is a rec- tangle, having the breadth b, and the depth c, n Tj f£ W n -• — 7- . The strength, therefore, all other 2 / things remaining the same, is as the breadth mul- tiplied into the square of the depth. This proposition has been brought to the test of ex- periment by BUFFON, in a great number of trials, made with great accuracy, and on a large scale. The MECHANICS. 159 The beams he subjected to trial were of oak, from seven to twenty-eight feet in length, and from four inches square to eight inches. It appears from them, that the quantity called s in the above formulas, or the strength of the ligneous fibre, is nearly proportional to the specific gravity, or to the weight of a given bulk, for example, a cu- bic foot of the timber. The weights required to break the beams were found to be very nearly as the quantities b c2, the length remaining the same. The real strengths, however, fell a little short of the computed, and the more, the longer the beam. In beams of different lengths, the variation of strength did not follow the inverse ratio of the lengths near so exactly as it did that of the other quantities included in the formula. Histoire Natiir- relle par M. DE BUFFON : Supplement, torn. in. p. 255., &c. l&no edit. 238. In a beam, of which the section is a tri- angle, having the base &, and the depth or perpen- dicular p, when the base is uppermost, Wzz ^-£-; J / but when the vertex of the triangle is uppennost, or the edge of the prism, W = •*; . o / The strength of a prismatic beam is therefore twice as great when a face is uppermost, as when the op- posite edge is uppermost. 2 In 160 OUTLINES OF NATURAL PHILOSOPHY. In like manner, in a cylindric beam, of which the ra- ,. TX7 dius is r, W =s — — • I In a cylindric tube, the radius of the external surface being r, and of the internal r', w= vsrW The strength of the tube is therefore to the strength of the same quantity of matter, formed into a solid cylinder of the same length, as r to A tube may, therefore, be much stronger than the same quantity of matter in a solid form. This is known to be agreeable to experience. But it is also said, that a tube of metal has been found to support a greater transverse strain than a solid cylinder, of the same diameter ; or that a solid cylinder, when bored in the direction of the axis, and a considerable part taken away, was stronger than before. This must undoubtedly arise from a change taking place in the position of the fulcrum or hinge round which the fracture is made. In the case of a cylin- der, and, indeed, of all solids, the fulcrum is not the mere outward wedge, but a point in the interior ; on the one side of which the fibres are elongated, and on the other crushed together. The point, then, which serves as the fulcrum, will be found within the solid, at a greater or less distance, as the parts 3 resist MECHANICS. 161 resist lengthening more than crushing. The con- sequence of this is, that when the centre of gravity and the fulcrum are brought nearer to one ano- ther, the strength of the beam or bar is diminish- ed. When the heart of a solid mass is cut out, as is supposed of the cylinder, the fulcrum, or the axis of the fracture, is perhaps kept nearer to the surface than when the whole is a solid mass. This, at least, seems to be the most probable ac- count that can, at present, be given of a phenomenon not a little paradoxical, and not yet sufficiently ex- amined. In similar beams, or solids, of the s£me substance, the strength increases as the square of the lengths ; but the weight increases as the cube. There is of course a limit, which, if a beam of a given shape, and of given materials, were to reach, it could only bear its own weight, and would be incapable of farther increase. EMERSON'S Mechanics, scholium at the end of section vin. p. 111. 240. When a beam, instead of projecting from a wall, is supported at both ends, it must break in the middle, and the termination of the fracture will be on the upper side ; and hence, when a rec- tangular beam is supported at both ends, it is able to carry twice as great a load as when it is sup- ported only at one end. VOL. I. L When 162 OUTLINES OF NATURAL PHILOSOPHY. When the section of the beam is such, that the centre of gravity does not coincide with the centre of mag- nitude, as the termination of the fracture, by being brought from the under to the upper side, must change its distance from the centre of gravity of the section, a change of strength will take place al- so on that account. 241. When a beam is supported at both ends, the weight which it is able to bear at any point, is inversely as the rectangle under the segments into which it is divided by that point. A beam, therefore, supported at both ends, and of the same section throughout, is weakest in the middle. A beam supported at both ends, and of a given breadth, will be every where of the same strength, if its lon- gitudinal and vertical section be an ellipse, having the beam itself for its greater axis. 242. From a given cylinder, to cut out the strongest rectangular beam possible. Let the circle ACB (fig. 20.) be the base of the cylin- der ; at the extremity A, of the diameter AB, draw the tangent AH equal to the chord of 90°, draw HK to the centre K, and let it cut the circumference in C. If CE be drawn parallel to AB, and CD at right angles to it, the rectangle DE, under these lines, is the section of the strongest rectangular beam that can be cut out of the given cylinder. For it is easily shewn, that CD2 X CE, to which the strength of the beam is proportional (§ 237-), is greater in the rectangle thus determined, than in any other that can be inscribed in the circle. 2 As MECHANICS. 163 As from the construction CL2 = % LK2, if the radius be called r, LK = L. , and CL = r so CE = — , and CD - 2 r J- . CD is, there- V& * 3 fore, to CE, nearly as 7 to 5. In practice, as the tree from which the rectangular beam is to be cut, can hardly ever be truly cylin- dric, the first thing to be done is,, to inscribe, by trial, the greatest circle possible, in the section of the tree. The rectangle may then be found by the construction above. 243. If it is required to find the strongest rec- tangular beam under a given perimeter or circum- ference, divide the given circumference into six equal parts, and take two of these for the depth of the beam, and one for its breadth. These propositions are understood to be applicable, not only to beams of wood, but to bars of iron, or any other metal. 244. If a beam, having one end firmly fixed in a wall, project from it with a certain inclination to the horizontal plane, the weight which it will support at its extremity, is greater than that which it would support if horizontal, in the ratio of the square of the radius to the square of the co- sine of the inclination. L 2 That OUTLINES OF NATURAL PHILOSOPHY. That is, if W be the weight which a beam can support W in a horizontal position, -— TJ is the weight it can COS % support when inclined to the horizon at the angle i. The proof of this proposition proceeds on two prin- ciples, that the momentum of the weight is dimi- nished in the oblique beam, by its perpendicular distance from the fulcrum being diminished, and that the resistance of the beam is increased by the centre of gravity of the section being carried far- ther from the fulcrum. Each of these being in the inverse ratio of the cosine of the inclination, their joint effect is in the inverse ratio of its square. It is implied in this reasoning, that the resistance of each fibre is the same to the oblique and to the di- rect fracture. Though this seems probable, it is not altogether certain, and it would be useful to have experiments directed to the clearing up of this dif- ficulty. If the individual fibres resist fracture also in the inverse ratio of the cosine of the obliqui- ty, the strength of the beam will be in the inverse ratio of the cube of that cosine. This is observed by GUIDO GRANDI, in the treatise already referred to. Opere di GALILEO, torn. in. p. 271. 245. The theory in the preceding proposition being admitted, the force of a beam to resist the action of any strain, will be inversely as the square of MECHANICS. 165 of the sine of the angle which the direction of that strain makes with the direction of the longitudi- nal fibres of the wood. According to this, the strength of the beam to resist a force applied to it endwise is infinite. This is no doubt false ; but the force in the case of a longitu- dinal pull or thrust is so great, that no practical error will arise from it. The cause of the error in this extreme case is, that the beam does not, nor cannot, obey the law of continuity, which would re- quire, that when, by varying its inclination, it be- comes quite upright, its longitudinal section should be infinitely extended. From this theorem, it is evident, that great advantage is obtained by making the strain on all timber as oblique or as nearly longitudinal as possible. This is the great principle in the construction of roofs and centres of bridges. The principle of the Arch should also be combined with it, so that pressure downwards may be so applied as to produce a pull or thrust upward. See the article by Professor Ko- fi ISON 07i the Strength of Timber, in the Encyclo- padia Britannica. In the preceding theorems, no account is made of the weight of the beams, a circumstance that is often necessary to be considered. This may always be done by supposing the weight of the beam to be an addition to its load, collected in its centre of gravity. On this the following theorem is found- ed. 246. If 166 OUTLINES OF NATURAL PHILOSOPHY. 246. If a j-ectangular beam, fixed ia a -wal, project horizontally to the length I ; if the -depth of its section be a, and the weight of a cubic foot of the timber be p\ then if l=:HJ—9 the beam will be just able to support its own weight. To apply this theorem, the value of s must be found from experiment. In the case of a rectangular beam, we have sab. | = W x -, supposing the /w 2 ;beam supported at both ends, § 240. ; and therefore "W x I — 2-7—- I*1 BUFFON'S experiments, a and b a & s = were equal ; and in one where a was six inches, and I equal seven feet., W was 19250 Ib. Therefore s = 1925? x 7 = 19250 x 56 = 1078000. In this same experiment, the weight of the beam was 128 Ib., and therefore a cubic foot weighed 73.2 = », so that s = kj — = 85.8 feet nearly. ' p The following Table contains the mean results of BUFFON'S experiments. They were made on oak beams, cut from trees newly felled, and full of sap. The weight of the timber varied from 73 to 75 French pounds to the cubic foot ; its strength varied in the same proportion. TABLE. MECHANICS. 167 TABLE. Length of the Beams. Breadth or Thickness. 4 inches. 5 inches. 6 inches. 7 inches. Feet. 7 Ib. 5312 Ib. 11525 Ib. 18950 Ib. 8 4550 9787J 15525 26050 9 4025 33081 18150 22350 10 3612 7125 11250 19475 12 2987 6075 9100 16175 14 5300 7475 13225 16 4350 6362J 11000 18 3700 5562J 9245 3225 4950 8375 BUFFON, Hist. Naturelle, Supplement, torn. in. p. 260. 12mo edit. Paris 1779. For a comparison of the strength of different kinds of wood, &c. See EMER- SON'S Mechanics, section viu. scJwlium at the end. MUSCHENBROEK, § 1127, on the Strength of Tim- ber, and § 1129, &c. on the Strength of Metals. See also COULOMB. Application des Regies de Ma- ximis et Minimis a quelques Problemes relatifs a Architecture, Memoires Presentes, vol. vn. (1773), p. 343., &c. HYDRO- 168 OUTLINES OF NATUEAL PHILOSOPHY. HYDROSTATICS. SECT. I. PRESSURE OF FLUIDS. . A FLUID is a body so constituted, that its parts are ready to yield to the action of any pres- pure, however small, in any direction. NEWTON, Principia, lib. n. sectio 5. in initio. AR- CHIMEDES de Insidentibus Humido. Hydrodynamics is the science which applies the prin- ciples of Dynamics to determine the conditions of motion and rest in fluid bodies, and is divided into four parts, according as fluids are incompressible or elastic, and according as their equilibrium or their motion is considered. "\Vhen fluids are incompressible, like water, the sci- ence which explains the laws of their equilibrium is called HYDROSTATICS 169 called Hydrostatics, and that which explains the laws of their motion is called Hydraulics. When they are compressible and elastic, like air, the sci- ence which treats of the laws of their equilibrium is called Aerostatics, and that which treats of their motion is Pneumatics. The fluids here treated of, are supposed to gravitate in lines, which are either parallel, or directed to one centre, or to several centres. 248. The surface of every fluid when at rest, is horizontal, or perpendicular to the direction of gravity. If the directions of gravity are all parallel, the surface of the fluid is a plane : If they converge to a point, the surface of the fluid is a portion of a spherical surface, having that point for a centre. BOSSUT, Hydrod. torn. i. § 19. CAVALLO, vol. n. chap. ii. prop. &. Levelling is the art of drawing a line at the surface of the earth, to cut the directions of gravity every where at right angles. This line is nearly an arch of a circle, and if L be the length of any such arch in English miles, and D the depression, in feet, of one extremity of it, be- low a tangent drawn to it at the other extremi- tv D- 3 If a communication, by means of a tube or pipe, ei- ther straight or crooked, be made between the wa- ter 170 OUTLINES OF NATURAL PHILOSOPHY. ter in one vessel, and that in another, the surfaces of both will come to be at the same level before the water is at rest ; or if there is not water sufficient to bring both to a level, the whole will be accumulated in, the lowest. If water is permitted to flow freely from one vessel to another, it will never be at rest til it is the lowest possible. 249. The fluid contained in any vessel, being at rest, and subjected to the sole action of gravity, any particle of it is pressed in all directions, (ver- tically, horizontally, and obliquely,) by the same force, viz. the weight of the column of water per- pendicularly incumbent on it. BOSSUT, Hydrodynamique, vol. i. § 523. 250. If the fluid contained in any vessel be at rest, and subjected to the action of gravity only, the pressure on an indefinitely small area, at any point of the bottom or sides, is perpendicular to the plane of that area, and equal to the weight of a vertical column of the fluid, standing on it as a base, and reaching to the surface. Hence the pressure on any part in the bottom or sides of a vessel, depends entirely on the depth of the fluid at that point, and not at all on the extent of the fluid in a horizontal direction. 2151. The HYDROSTATICS. 171 251. The same being supposed, as in the last proposition, the whole pressure sustained by any portion whatever of the bottom or sides of the ves- sel, is equal to the weight of a column of the fluid having for its base the surface pressed on (extend- ed into a plane if necessary), and for its altitude the mean depth of the incumbent fluid. COTE'S Hydrostatical Lectures^ Lect. in. BOSSUT, Hylrodyn. vol. i § 26. The mean depth of any portion of the surface of the bottom or sides of the vessel, is the same with the distance of the centre of gravity of that portion be- low the surface of the fluid. The strength of the walls necessary to resist the pres- sure of a fluid of any given depth, is determined by this proposition. If the figure of a vessel containing a fluid be such as is represented, fig. 21. the pressure on every point of the bottom is the same as if it were entirely fill- ed with the fluid to the height FEG. Hence the pressure of a fluid on the bottom of a vessel may be very great, while the weight of the fluid is very small ; because the water in the narrow tube HE will press on the bottom CD with a force greater than its own weight in the proportion that the area of the bottom is greater than the horizontal section of the tube. This proposition is called the Hydrostatic Paradox. The 172 OUTLINES OF NATURAL PHILOSOPHY. The pressure in the narrow tube may be produced, not merely by the addition of a little water, but by the application of any kind of force, such as the working of a piston, &c. If the bottom or lid be made moveable, the pressure on either may be brought to bear on one point of an external body, and may then produce an enormous compression, as is very successfully done in the engine known by the name of BE AMAH'S Press. This property of fluids may therefore be said to fur* nish a SEVENTH MECHANICAL POWER. . A body immersed into a fluid is pressed upwards by a force equal to the weight of the fluid it displaces, and the direction of that force passes through the centre of gravity of the part immersed. The proposition holds, whether a body sink in a fluid or float on its surface. When a body floats, the weight of the water displa- ced is equal to the weight of the body. 253. The difference between the absolute weight of a body, and its weight when entirely immersed in a fluid, is the same with the weight of a quan- tity of the fluid equal in bulk to the body, If W be the weight of a body in vactio, (which we may suppose to be nearly the same with its weight in air), and if W be its weight in water, W — W 3 is HYDROSTATICS- 173 is the weight of a quantity of water equal in bulk to the body. It follows from § 30., that the weight of any body di- vided by the weight of an equal bulk of water, measures the specific gravity of the body. The specific gravity of a body weighed as above, is W w — w ' Hence the specific gravities of bodies are found by weighing them, first in air, and then in water. The instrument with which this experiment is made is tailed the Hydrostatic Balance. 254. Let it be required, by the hydrostatic ba- lance to determine the specific gravity of a body 60 light as not to sink in water. Weigh the body first in air, and then by a slender thread attach to it a heavier body, that the two to- gether may sink ; the heavier body being previous- ly weighed, first in air, and then in water. Let the compound body thus formed be weighed in water; then from the weight lost by it, subtract the weight lost by the heavier body, when weigh- ed singly ; the remainder is the weight lost by the lighter body, by which if its weight be divided, its specific gravity will be found. CAVALLO, vol. n. p. 61. The solution there is a little different, but leads to the same result. See the example, ibid. 255. If 174 OUTLINES OF NATURAL PHILOSOPHY. 255. If the same body be weighed in two diffe- rent fluids, the weights lost will be to one another as the specific gravities of the fluids. By help of this proposition, the specific gravities of different fluids may be compared together. The specific gravities of different fluids may also be found, by weighing equal bulks of them. If a body float on different fluids, the bulks of the part immersed will be inversely as the specific gra- vities of the fluids. It is on this principle that the hydrometer, aerometer, &c. are constructed. See Description of a Hydrometer, CAVALLO, vol. n. p. 66. ' 256. If the specific gravity of air be called m, that of water being 1, if W be the weight of any body in air, and W its weight in water, then W + m (W — W) is its weight in vacuo very nearly. In the mean state of the atmosphere, m — .00122 nearly. 257. If s be the specific gravity of a body, as- certained by weighing it in air and water, and m be the specific gravity of the air at the time when the experiment was made ; the correct spe- cific gravity, or that which would have been found if the body had been weighed in a vacuum instead of air, is s + m (1 — s). When HYDROSTATICS. 175 When the body is heavier than water, this correction is subtractive ; when lighter, it is additive. The manner of finding the specific gravity of air will be afterwards explained. 258. If B be the bulk of any body in cubic inches English, W its weight in grains, and S its specific gravity, that of water being 1 ; W T> --• . (252.576) S " This theorem is founded on an experiment of Sir G. SHUCKBOROUGH'S (Phil. Trans. 1798), by which the weight of a cubic inch of distilled water, of the temperature 66°, was found to be 252.422 grains, which, reduced to the temperature 60°, is 252.576. CAVALLO, vol. n. p. 98. The logarithm of 252.576 is 2.4023921. If W is expressed in Ib. Troy, it must be multiplied by 5760 ; if in Ib. avoirdupois, by 7002. If B is required in cubic feet, its value thus found must be divided by 1728. SECT. 176 OUTLINES OF NATURAL PHILOSOPHY, SECT. II. SOLID BODIES FLOATING ON FLUIDS, 259. A SOLID floating on a fluid specifically heavier than itself, will be in equilibria when it has sunk so far, that the weight of the fluid displaced is equal to the weight of the whole solid, and when the centres of gravity of the whole solid, and of the part immersed, are in the same vertical line. ARCHIMEDES, de Humido fnsidentibus, Kb. i. BOSSUT Hydrodyn. torn. i. chap. 12. Hence every solid of revolution, and in general eVery solid having an axis to which the opposite sides are similarly related, if it be specifically lighter than a fluid, and if it be placed in the fluid with its axis vertical, may sink to a position in which it will remain in equilibria. There are always two opposite positions of equili- brium in such bodies ; but there is only one of them in which the body can float permanently. 260. A prism, of which the triangle ABC (fig. 22L), is a transverse section, being supposed homogeneous, HYDROSTATICS. 177 homogeneous, and its specific gravity being less than that of a fluid in which it is immersed, but having a given ratio to it, required the different positions in which it will be in equilibrium. Let the prism float with the angle ABC downward, and let DE be the common section of the plane of the triangle with the surface of the fluid, or what is called the water line. The area of the triangle BDE is given, being to that of the triangle ABC, as the specific gravity of the prism to the specific gravity of the fluid ; therefore the line DE touches a given hyperbola, described with the asymptotes AB, BC, and is bisected in the point H where it touches that curve. Bisect AC in F, draw BF, BH ; take BG = f BF, and BK = f BH? G is the centre of gravity of the triangle ABC, and K that of the triangle DBE, and therefore, because of the equilibrium, GK is a vertical line, as also FH, which is parallel to it. Therefore FH is perpendi- cular to DE, and consequently to the hyperbola which DE touches in H. The position of FH, since the point F is given, may therefore be found ; and as many perpendiculars as can be drawn from that point to the hyperbola, so many positions are there in which the prism may float with the angle ABC downward. The different positions of FH are determined, by the roots of a biquadratic equation. From any point H' in the hyperbola, draw H'L perpendicular to BC, and let FE be also perpendicular to BC ; let BE = a, EF = fl, BL=^r, LH' = z/, and the gi- VOL. I. M ven 178 OUTLINES OF NATURAL PHILOSOPHY. ven area of the triangle BDE = 2c2; then FH2=(a — *)2+(6 — yY and (* — my) £ = c\ n where m is the cotangent, and n the sine of the angle B. Therefore taking the Fluxion of FH* substituting for x its value from the preceding equation, and supposing the whole ±s 0, we have (a — x) (x — 2 my) + b y — y* = 0. By the com- bination of this equation with the former, we may obtain either an algebraic solution, by a biquadra- tic equation, or a geometrical construction by the intersection of two hyperbolas. See the solution of other problems of this kind, AR- CHIMEDES de Humido Insidentibus, lib. n. BOSSUT, Hydrod. torn. I. chap. xii. § 166., &c» 261. When the equilibrium of a floating body is disturbed, or when the centres of gravity of the whole and of the part immersed, are not in the same vertical line, if a vertical plane be made to pass through both those centres^ the body will re- volve on an axis passing through its centre of gra- vity, and perpendicular to that plane. This follows from what has been proved concerning the rotation of bodies, § BOSSUT, Hydrod. torn. n. § 179. ATWOOD, Phil. Trans. 1796. HYDROSTATICS. 179 . A body may be so constituted, that in every position the centres of gravity of the whole, and of the part immersed, are in the same vertical line, and in such a body the equilibrium cannot be disturbed by any force which only tends to make it revolve on an axis. A homogeneous sphere is a body of this kind, and so also is a cylinder floating with its axis horizontal. These have no tendency to maintain one position more than another, and their equilibrium is called the equilibrium of indifference. 263. Some floating bodies, when their equili- brium is disturbed, return to their position after a few oscillations backwards and forwards. Others, when their equilibrium is ever so little disturbed, do not resume their former position, but revolve on their centres of gravity, till they come into another position, when they are again in equili- bria. The equilibrium in the first case is said to be stable, in the latter case to be unstable. In this last case, the bodies are said to overset. 264. When a floating body is made to revolve from the position of equilibrium, if the line of support (that is, the vertical through the centre of gravity of the immersed part) move so as to be on the same side of the line of pressure (or the M 2 vertical 180 OUTLINES OF NATURAL PHILOSOPHY. vertical through the centre of gravity of the whole) with the depressed part, the equilibrium is stable, and the body will resume its former posi- tion. 265. The same things being supposed, if the line of support is on the same side of the line of pressure with the elevated part, the equilibrium is unstable, and the body will overset. BOSSUT, Hydrod. torn. i. 266. If a body float on the surface of a fluid, the force tending to make it revolve about its centre of gravity is equal to the weight of the body, acting by a lever, the length of which is the horizontal distance between the line of pressure and the line of support. ATWOOD, Phil Trans. 1796, p. 61. When this distance is nothing, or when the two cen- tres are in the same vertical line, the force tending to make the body revolve, is equal to 0» as already stated, § 262. When the body is any how inclined from the state of equilibrium, and when the line of support is on the same side of the centre of gravity with the depressed part, the lever by which the weight acts is account- ed affirmative, and the force tends to establish the equilibrium. When H YDROST ATTCS. When the line of support is on the same side of the line of pressure with the elevated part, the force is accounted negative, and tends to overset the body. 267. If in ^, floating body, of which the trans- verse section is the same from one end to the other, W be the weight, a the length of the water line, 6>2 the area of the section of the immersed part, d the distance between the centre of gravity of the whole and the centre of gravity of the immersed part, and i an indefinitely small inclination from the position of equilibrium ; the momentum of the force tending to restore the equilibrium is _ _ d) W sin I If TTT-^ > ^» tne f°rce ten(ls to restore the body to 1 ^ f* its state of equilibrium, If jg— j = d, there is no force tending either to re- store or destroy the equilibrium. a3 l%c* < ^ ^e ^°rce ^ecomes negative, and tends to overset the body. When W remains the same, the stability is propor- tional to (~-^ — d ) sin i. When OUTLINES OF NATURAL PHILOSOPHY. When the centre of gravity of the body is lower than the centre of gravity of the immersed part, d is ne- 3 gative, and the quantity -^ — d is affirmative, X*v C' -««*;• whatever be the magnitude of r—z • If in the axis of the solid, or in the line passing through the two centres, there be taken a point dis- tant from the centre of the immersed part, by a3 i J«\.».' • = To~2' tms P°mt *s caUed tne metacentre, be- rno ft tTviuuiLkmi lUma vlsii ftfreJbiri n« * &n« ^ ma cause the centre of gravity must be lower than it, 81 £ in order that the body may float with stability. By the centre of gravity of the immersed part, is al- ways understood its centre of gravity, supposing it homogeneous. It is in fact the centre of gravity of the water displaced, and were perhaps better di- stinguished by the name of the centre of buoyancy. 268. When a floating body revolves about a gi- ven axis, the positions of equilibrium through which it passes, are alternately those of stability and instability. For between a state in which a body has a tendency to remain, and another in which it has also a ten- dency to remain, as these tendencies are opposite to one another, there must be an intermediate posi- tion, HYDKOSTATICS. 183 tion, in which the tendency to remain is equal to nothing. ATWOOD, Phil Trans. 1796, p. 63. 269. If a rectangular parallelepiped float in a fluid, with its altitude a perpendicular to the sur- face ; if its breadth be 6, and its specific gravity n, that of the fluid being 1, its stability will be as b*n(I — n)a\ When it has no stability, *L — n (1 — n) a1 = 0, and If n = - > as is nearly the case with fir, a = • & • = &!/ 1 = TT neaxly- The truth of T- V 3 6 J this conclusion may be shewn by experiment. ATWOOD, ubisuprayp.61. YOUNG'S Analysis of tfo Principles of Natural Philosophy, p. 212. § 15. 270. As the force impelling the floating body in its vibrations, is proportional to the sine of the in- clination 184 OUTLINES OF NATURAL PHILOSOPHY. •clination of its axis, or the angle which it makes with the vertical, and as this is the same law that regulates the vibrations of a pendulum in a small arch of a circle, a pendulum may be found, the vibrations of which are isochronous with the small vibrations of a floating body. ATWOOD, Phil. Trans, ubi supra, p. 123. BOUGUEK, Manoeuvre des Vaisseaux, lib. i. § iii. chap. 7. On the subject of this section, see also EULER/S Theo- ry of the Construction and Properties of Vessels, translated by H. WATSON, chap. 4, 5, & 6., &c. GREGORY'S Mechanics, vol. i. SECT. III. PHENOMENA OF CAPILLARY TUBES. 271. JLF a capillary tube of glass, that is, a tube of which the bore is less than one-tenth of an inch, be immersed in water, the water will rise within the tube to a greater height than it stands at on the outside ; and this height is nearly in the inverse ratio of the diameter of the tube. a. The amount of this ascent is very differently stated by different authors. The latest and most accurate experiments HYDROSTATICS. 185 experiments are those of Dr BREWSTER, made with a tube in which the diameter of the bore was .0561 of an inch ; the ascent was .587, and the product of the two, — .0327 of a square inch, a quantity that is constant, or the same for all capillary tubes. Dif- ferent fluids ascend to different heights ; water as- cends the highest of all. See Edinburgh Encyclo- paedia, art. Capillary Attraction. b. It follows also from the above, that the weights of the columns of water sustained in capillary tubes, are as the diameters of those tubes. c. Though the rise of water above its natural level is most manifest in small tubes, it appears, in a de- gree, in all vessels whatsoever, by a ring of water formed round the sides, with a concavity upwards. d. The height to which water rises in a capillary tube, is not affected by the thickness of the glass. - 272. Capillary suspension takes place, though the tube be not immersed in water, providing a drop of water adhere to the lower end. 273. The surface of the water in a capillary tube is concave upwards ; and if by taking the tube out of the water, and inclining it, the fluid be made to move along it, the concavity appears at both ends of the column, and is the same in figure and size whether the tube be held perpendicularly, horizontally, or obliquely. 2 * This 186 OUTLINES OF NATURAL PHILOSOPHY. This is not strictly true, except when the diameter of the tube is very small : when the bore is nearly one- tenth, the concavity becomes elongated on the lower '•'rf.4 ' 274. When by inclining the tube, the column of water is made to move, it appears to suffer re- sistance as it approaches either end, and does not completely reach the end, till the tube is either al- together, or very nearly inverted. When the tube is placed in the water, however far it be thrust down, the water within never reaches the top, till the tube is completely immersed. 275. If a capillary tube composed of two cylin- ders of different bores, be immersed in water, first with the widest part downwards, and afterwards with the narrowest, the water will rise in both cases to the same height. If the smaller end is such as to require the whole to be filled by suction, the water stands at the same height as if the whole tube were of the bore of the upper part. This experiment, however, does not succeed in vacuo ; and therefore the water in the wide part of the tube, must be considered as sustained by the pressure of the air. 276. If two plates of glass be kept parallel, and near to one another, and if their ends be immer- sed in water, the water will ascend between them to HYDROSTATICS. 187 to half the height it would rise to in a tube ha- ving its diameter equal to the distance of the plates. NEWTON'S Optics, Book in. query 31. HAUY, Traite Elemental™ de Physique, torn. i. § 333. edit. 1806. a. When the plates make an angle with one another, if they be immersed with the line of their intersec- tion vertical, the water will ascend between them, and form a hyperbola. b. LA PLACE caused experiments to be made with one tube placed within another, so that their axes coin- cided, and found that the water ascended in the space between them only half as high as it would have done in a single tube in which the diameter of the bore was equal to the distance of the two tubes from one another. HAUY, ibid. 277. If a glass tube of a small bore be immer- sed in mercury, the mercury does not rise within the tube to the height at which it stands on the outside : its surface within is convex, as it is also without, all round the tube. The convex surface of the mercury within the tube, intersects the surface of the tube at an angle of 40°. See Dr YOUNG'S Lectures an Natural Philo- sophy, vol. n. p. 649-, on the Cohesion of Fluids. 278. If 188 OUTLINES OF NATURAL PHILOSOPHY. 278. If into a conical capillary tube, held in a horizontal position, a drop of water be introduced, it will run toward the narrow end ; but if a drop of mercury be introduced, it will run toward the wide end. . LA PLACE, Mecanique Celeste, Lib. x. Supplement, p. 6. 279. Whether a fluid rise or fall between two vertical and parallel planes of glass, immersed in the fluid at their lower extremities, the planes tend to approach one another. a. It is from this tendency, that two small vessels of glass, of a parallelepiped form, floating on water or mercury, unite, whenever they approach near to one another. LA PLACE, ibid. §11. b. The phenomena here enumerated, clearly prove the existence of an attractive force between water and glass, but require to be carefully compared before they inform us of the manner in which that force is exerted. The fact that points most directly to the place where the force resides, is that of the concavity of the surface, mentioned at § 273. 280. From this fact it may be inferred, that a narrow ring or zone of glass immediately above the surface of the water, attracts the water with a force which, when the tube is of a small diame- ter, is able to suspend a thin film of the fluid in opposition HYDROSTATICS. 189 opposition to its gravity, the surface of which is concave upwards, in consequence of the attraction of the glass heing combined with the weight of the water, and the cohesion of its particles. Hence is derived the little meniscus of water, MIOKN (%. 23.), which terminates the column. This meniscus is, therefore, a body of water stretched across the tube, and sustained there by the attrac- tion of the glass, while it exerts its own attraction on the particles of the column immediately under- neath, by which means the gravity of those parti- cles is diminished, and the water rises in the tube above its level on the outside, to supply their defi- ciency of weight. a. This is the theory of LA PLACE. He has deter- mined the superficies of the meniscus to be nearly spherical, and its attraction to be equal to that of a spherule of water of the same diameter with itself, supposing the attraction to be insensible at all sen- sible distances. The ascent of the water, in the in- verse ratio of the diameter of the capillary tube, is thus accounted for ; the attraction of the meniscus being directly as its diameter, or as the diameter of the tube, that is, as the weight of the water sus- tained. b. It is understood in all this, that the attraction of the particles reaches beyond the nearest, but not to any sensible distance, c. In 190 OUTLINES OF NATURAL PHILOSOPHY. c In the case of mercury, glass either repels the fluid, or attracts it less than its particles attract one another. Hence the convexity of the surface that terminates the column, and the depression of that surface below the natural level. d. So also, in the movement of the drop to the small end of a conical capillary tube ; if the attraction of a particle of the tube be resolved into two, one in the direction of the axis of the tube, and the other at right angles to it, the former will be directed to- wards the vertex of the cone. If the force is re- pulsive, the part in the direction of the axis will be toward the base. On the same principle may be explained the tendency of two plates to unite, whether water rises or mercury sinks between them. e. The author of this theory considers the concave surface of the fluidj and the attraction which the meniscus exerts, as the principal cause of the phe- nomena of capillary tubes. Ibid. p. 5. We con- fessj that we cannot but think it more accurate to say, that the principal and primary cause of the sus- pension of the water, is the attraction of the glass, which sustains the meniscus, and enables it to act on the water below, without being drawn out of its place. It is not the concavity of its surface which makes the water in the tube press less on the bottom than if its surface were plain, but it is the attraction of the glass, which produces, in a manner equally direct, both the concavity and the diminution of the pres- sure. 281. The HYDROSTATICS. 191 281. The reason why the water between the two narrow plates of glass rises only to half the height it does in a capillary tube, of a diameter equal to the distance of the plates, is, that the quantity of the attracting surface does not go all round, but is nearly halved. a. The same holds of the tubes, inserted the one within the other. Any small column of the water between them, is supported by the attraction of the glass on- ly on two sides. 282. The adhesion of bodies to the surfaces of fluids, is an effect of the same forces which produce the phenomena of capillary tubes. a. This was first remarked by Dr YOUNG, Phil. Trans. 1805, and Lectures on Natural Philosophy, vol. n. p. 652. The figure of the drops of a fluid support- ed by a solid, or adhering to it, is connected with the same phenomena. SEGNER treated of this sub- ject, Comment. Soc. Reg. Gotting. torn. i. (1751) p. 301. He considers the surface of the drops as one formed like the catenarian curve, or like the surface of a sail exposed to the wind ; and he as- sumes the tension over all this surface to be the same. LA PLACE remarks, that this supposition is erroneous* b. The theory of LA PLACE, so far as the attraction of the tube is concerned, has a great affinity to that 3 of 192 OUTLINES OF NATURAL PHILOSOPHY. of Dr JURIN, Phil Trans. N° 363. art. 2. In what respects the form of the upper surface, it is founded on the same principles with CLAIRAUT'S ex- planation, Theorie de la Figure de la Terre, tiree des Principes de T Hydrostatique, § 59- See also SEG- NER, Comment. Soc. Reg. Gotting. torn. i. (1751,) . p. 301. CLAIRAUT has proved, that if the law by which the matter of the tube attracts the fluid, be the same with that by which the parts of the fluid attract one another, the fluid will rise above the level whenever the intensity of the first of these attractions exceeds half the intensity of the second. If it is exactly the half, the surface of the column within the tube will be a plane, on a level with the surface without. Theorie, &c. § 60. In some cases, the attraction of the bottom has an evi- dent effect ; when the lower end of the capillary tube is stopped by the finger, and the tube taken out of the water, if the finger is withdrawn, a drop formSj and the water stands higher in the tube than when it was immersed in the water. LA PLACE, ibid, § 15. This seems to arise from the action of the bottom and outside of the tube on the drop, by which the column of water in the tube is lifted up, as it were, and raised to a higher level. The HYDROSTATICS. 193 The circumstance that has so long rendered the theory of capillary tubes a matter of doubt, is the difficulty of finding a perfect experimentum crucis^ or a fact which can be explained by one theory only. It can- not be said, that even the theory of CLAIRAUT and LA PLACE possesses the demonstrative evidence which such an experiment would afford, ~i' VOL. I. N HYDRAULICS. 194 OUTLINES OF NATURAL PHILOSOPHY. HYDRAULICS, SECT. I. FLUIDS ISSUING THROUGH APERTURES IN THE BOTTOM OR SIDES OF VESSELS. 283. AF two bodies be uniformly accelerated over distances that are inversely as the accelerating for- ces, they will acquire equal velocities. This proposition is given as a lemma to that which follows, 284. The velocity with which a fluid issues from an infinitely small orifice in the bottom or side of a vessel that is preserved full, is equal to that which a heavy body would acquire by falling from the le- vel of the surface to the level of the orifice. BOSSTJT, Hydrod. torn. i. § 218. Therefore, HYDRAULICS. 195 Therefore, let d be the depth of the fluid ; g the velocity acquired by a falling body in one se- cond ; v the velocity with which the water issues ; v = If the velocity with which the fluid issues from the orifice, were turned directly upward, it would carry it to the level of the surface of the fluid. 285. The quantity of water that issues in one se- cond, through a given orifice, is equal to a column of water having the area of the orifice for its base, and the velocity with which the fluid issues for its altitude. a. Hence, if a2 be the area of the orifice, d the depth of the water above the orifice, and q the quantity of fluid running out in one second, q = a1 b. Of the quantities g, a and c/, any two being gi- ven, the third may be found : /v2 Q j Q- ii — — .=r--t , d rr — ± - . JZgd 2£«4 The quantity of water that issues through an orifice, is as the area of the orifice multiplied into the square-root of the depth. c. When the experiment is made, it is found that «2 must not be taken equal to the exact area of the orifice. The vein of water that issues through the small orifice, suffers a contraction, by which its N 2 section 196 OUTLINES OF NATURAL PHILOSOPHY. section is diminished in the ratio nearly of 5 to 7, Therefore we must take - of the above result for f* the true value of q ; so that q = - a? */2 g d. As* 5 . 1 - is nearly -= — , we may suppose q - a? d. As *Jg d is the velocity acquired by falling through - , it has been supposed by some writers, that the water issues only with the velocity that would be acquired by falling through half the depth of the fluid- CAVALLO falls into this mistake, vol. n. chap. 7. p. 16& 286. Water issuing through an oblique pipe in the side of a vessel which is kept full, describes a parabola in a vertical plane, passing through the axis of the pipe ; and the directrix of the parabola is the line of common section of the said vertical plane with the surface of the water produced. That the curve is a parabola, follows from the theory of Projectiles, § 93. ; and that the directrix is the same, whatever be the angle of elevation, follows from this, that the water issues with the same veloci- ty whatever direction the axis of the pipe has, viz. that which corresponds to the height of the surface of the fluid above the orifice, § 94. a. 3 287. Tbe HYDRAULICS. 197 287. The time that a cylindric vessel, (the area of its base being i2), requires to empty itself by a hole in the bottom, of which the area is a2, the depth at the beginning of this discharge being j? js J — ; and the time that the surface takes to sink, from the depth d to any other depth d', is DAN. BERXOUILLJ, Hydrodynamica, sect. iv. § 13. ; BOSSUT, Hydrodyn. torn. i. § 232. The construction of the clepsydra, or water-clock, de- pends on this proposition. If the whole depth through which the surface of the water sinks in 12 hours be divided into 144 parts, it will sink through 23 of these in the first hour, 21 in the second, 19 in the third, and so on, according to the series of the odd numbers. 288. The vein of water, as it issues out, is con- tracted ; and from that, and other causes, the actual discharge of water is not so great as it is computed to be in the foregoing theorems. This difference, however, affects only the absolute quantities, and not their proportions ; the quantities really dischar- ged 198 OUTLINES OF NATURAL PHILOSOPHY. ged being as the square-roots of the depths multi- plied into the areas of the orifices. a. According to some very accurate experiments of BOSSUT, the actual discharge through a hole made in the side or bottom of the vessel, is to the theo- retical as 1 to .6£, or nearly as 8 to 5. The theo- retical discharge, when computed, must, therefore, be diminished in this ratio to have the true one. ?s b. If the water issues, not through an aperture in the side or bottom of the vessel, but through a pipe from 1 to 2 inches in length, inserted in the aper- ture, the contraction of the vein is prevented, and the actual discharge becomes to the theoretical, as 8 to 10, or as 4 to 5. In this way, therefore, the discharge is increased nearly in the ratio of 4 to 3. BOSSUT, § 523. See also PHONY, Architect. Hyd. § 840. ' £. The theoretical discharge, the discharge through an additional tube, and that through a simple per- foration in the side, are as the numbers 16, 13, and 10 nearly. 289. Water thrown up in a perpendicular jet, ought to ascend to the height of the reservoir ; but on account of the resistance of the air, the fric- tion of the pipe, &c. it always falls short of that height ; and it is found by experiment, that the dif- ferences between the heights of the jets and of the reservoirs, are as the squares of the^heights of the jets themselves. If HYDRAULICS. 199 If H and H' be the heights of two reservoirs, h and h* the heights of the actual jets, H — h : H' — h' : : 7*2 : K* . This observation was made by MARIOTTE. BOSSUT, n. § 615. The water ascends highest when the jet is not quite perpendicular ; when it is perpendicular, the ascent is obstructed by the water falling back on the as- cending column. The height to which the water rises in the jet, is called the height of the effective head. The parabola which an oblique jet describes, has its di- rectrix not exactly as determined in § 286 ; but at the height of the effective head. SECT. II. OF CONDUIT PIPES AND OPEN CANALS. 290. W HEN the water from a reservoir is con- veyed in long horizontal pipes, of the same aper- ture, the discharges made in equal times are near- ly in the inverse ratio of the square roots of the lengths. It is supposed that the lengths of the pipes to which this rule is applied are not very unequal. It is an approximation not deduced from principle, but de- rived 200 OUTLINES OF NATURAL PHILOSOPHY. rived immediately from experiment. BOSSUT, torn. n. § 647, 648. At § 673. he has given a table of the actual discharges of water-pipes, as far as the length of 2340 toises, or 14950 feet English. 291. Water running in open canals, or in rivers, is accelerated in consequence of its depth, and of the declivity on which it runs, till the resistance, increa- sing with the velocity, become equal to the accele- ration, when the motion of the stream becomes uni- form. It is evident, that the amount of the resisting forces can hardly be determined by principles already known, and therefore nothing remains, but to as- certain by experiment, the velocity corresponding to different declivities, and different depths of water, and to* try, by multiplying and extending these ex- periments, to find out the law which is common to them all. The Chevalier Du BUAT has been successful in this re- search, and has given a formula for computing the velocity of running water, whether in close pipes, open canals, or rivers, which, though it may be call- ed empirical, is extremely useful in practice. Prin- cipes ffHydr antique, par M. LE CHEVALIER Du BUAT, 2 vol. 8vo. edit. %. Paris 1786. Professor ROEISON has given an abridged account of this book, in his excellent articles on Rivers and Water-works, in the Encyclopedia Britannica. Let V be the velocity of the stream measured Ipy the inches it moves over in a second; R a constant quantity, HYDRAULICS. 201 quantity, viz. the quotient obtained by dividing the area of the transverse section of the stream, express- ed in square inches, by the boundary or perimeter of that section, minus the superficial breadth of the stream, expressed in linear inches. The mean velocity is that with which, if all the parti- cles were to move, the discharge would be the same with the actual discharge. The line R is called by Du BUAT the radius9 and by Dr ROBISON the hydraulic mean depth. As its affinity to the radius of a circle seems greater than to the depth of a river, we shall call it, with the former, the radius of the section. Lastly, Let S be the denominator of a fraction which expresses the slope, the numerator being unity, that is, let it be the quotient obtained by dividing the length of the stream, supposing it extended in a straight line, by the difference of level of its two extremities ; or, which is nearly the same, let it be the co-tangent of the inclination or slope. 292. The above denominations being understood, and the section, as well as the velocity being sup- posed uniform, 307 R* — ~ A RJ 1 1 "; S_ og (s + 1.6) or 203 OUTLINES Or NATURAL PHILOSOPHY. TO- See Du BUAT, ubi supra, § 51., &c. At § 42. p. 65. the formula is reduced to English measure, as now given. When II and S are very great, V = ^ / 307 3\ R* I — : 5 — 1 nearly. ^•-JV*S ^ The logarithms understood here are the hyperbo- lic. The slope remaining the same, the velocities are as -. 10' The velocities of two great rivers that have the same declivity, are as the square roots of the radii of their sections. If R is so small, that R^ — ~ = 0, or R = JL, J.U J. UU the velocity will be nothing ; which is agreeable to experience ; for in a cylindric tube R = \ the ra- dius of the cylinder; the radius, therefore, equal /5 of an inch ; so that the tube is nearly capillary, and will not admit the water to flow through it. The HYDRAULICS. 203 The velocity may also become nothing, by the declivity becoming so small, that *£** "'' - is less than — -- S 200000 if _ is less than 3^7^, or than ^th of an inch to an English mile, the water will have a sensible motion. 293. In a river, the greatest velocity is at the sur- face, and in the middle of the stream, from which it diminishes toward the bottom and the sides, where it is least. It has been found by experiment, that if from the square root of the velocity in the middle of the stream, expressed in inches per second, unity be subtracted, the square of the remainder is the ve- locity at the bottom. Principes (THydr antique, par Du BUAT, § 67. Hence, if the former velocity be =: v9 the velocity at the bottom = v — 2 Jv + 1 . 294. It has been found by experiment, that the mean velocity, (or that with which, were the whole stream to move, the discharge would be the same with the real discharge,) is equal to half the sum of the 204 OUTLINES OF NATURAL PHILOSOPHY. the greatest and least velocities, as computed in the last proposition. The mean velocity is, therefore, — v — V v -f _ . The formulas of Du BUAT have been reduced to great simplicity by PHONY, so that they are free from logarithms, and require nothing more than the ex- traction of a square root. Reclierches Physico-Ma- tUematiques sur la Theorie des Eaux Courantes. Pa- ris 1804. Let V, as before, be the mean velocity ; A the length of an open canal, or of a close pipe ; z the difference of level of its two extremities ; R the radius, defined as in Du B,UAT^S propositions, in the case of a river or open cut ; D the diameter, in the case of a pipe ; h the height of the water in the reservoir above the up- per orifice of the pipe, and h' the height above the lower orifice, at which the water stands in the cis-, tern into which it is emptied, through that orifice. Let -, or the sine of the inclination — I ; and 295. The denominations above being supposed, we have for pipes, V ^ — * 1541131 + \/- 023751 + 32806 . 6 x r DK, 4 and HYDRAULICS. 205 and for rivers or open canals, V = — . 1541131 + \J • 023751 -f 32806 . 6 X HI . These formulas give the velocity in English feet ; those of PROXY give it in metres ; see the work just quot- ed, § 209, 210.; also Note prefixed to Table ii. p. 110. It must be observed, that when the water from the lower end of the pipe is discharged into the air, A'-O. From the comparison that has been made of these for- mulas with actual experiments, they appear to be very accurate. The numbers or constant quantities may perhaps require some correction, but there is no doubt that the form of the expression is exact. They must therefore supersede the use of Du BUAT*S more complicated theorems. 296. When the sections of a river vary, the quantity of water remaining the same, the mean velocities are inversely as the areas of the sec- tions. a. This must happen, in order to preserve the same quantity of discharge. 1). When the water in a river receives a permanent addition, the velocity is immediately increased. The increase of the velocity augments the action on the sides and bottom, in consequence of which the width is augmented, and sometimes also, but more rarely, the depth. The velocity is thus diminished, till the tenacity 206 OUTLINES QF NATURAL PHILOSOPHY. tenacity of the soil, or the hardness of the rock, af- ford a sufficient resistance to the force of the water. The bed of the river then changes only by insensible degrees, and, in the ordinary language of Hydrau- lics, is said to be permanent, though in strictness this epithet is not applicable to the course of any ri- ver. SECT. III. PERCUSSION AND RESISTANCE OF FLUIDS. 297. IF the sections of two streams be tbe same, the forces with which they strike on planes directly opposed to them, are as the squares of their velo- cities. For the force of a stream must be as the force of each particle, and as the number of particles that strike in a given time. Now, the force of each particle is as the velocity of the fluid, and the number of par- ticles that strike in a given time, the section being given, is also as that velocity. Therefore the whole force of the stream must be as the square of its ve- locity. Hence if v be the velocity of any stream, c? the area of the section, f the* force, and m a constant co- 2 efficient HYDRAULICS. 207 efficient, to be determined by experiment, 'jf= mcPv*', or if h be the height belonging to the velocity v9 so that %g li = a2, f-=. %md? gh. The quantity a? gli denotes the weight of the column, of which the base is a2, and the height h. The constant quantity m is found to be different, ac- cording to the manner in which the fluid is recei- ved by the plane on which it strikes. When the plane is indefinite in extent, m = 1, so that f=Qa?gh, or the force is double the weight of the column, of which the base is a2, and the alti- tude h. This agrees with experiment, and has also been demonstrated by DAN. BERNOUILLI, Comment. Petrop. torn. vin. p. 99-5 &c. See also Encyclopaedia Britanmca, 2d edit. Article Resistance, p. 109. When the plane is no greater in extent than the sec- tion of the stream, m — J, andf= a2 gh, only half the former column. Since, in two different streams, f:f' : : a2 a2 : &'2 ?/2, if a2 v9 or the quantity of water expended in a se- cond, be the same in both,^/"' : : v : v', or the forces are as the velocities simply. 298. If the plane struck by the stream be itself in motion, the impulse is as the square of the differ- ence of their velocities. If v be the velocity of the stream, and c the velocity with which the plane retires from it, then the force of percussion = m a2 (v — c)2. If £08 OUTLINES OF NATURAL PHILOSOPHY. If c is opposed to w, then the force is m a2 (v -j- c)*. and if v — 0, the force = m a2 c2. The plane, therefore moving against a fluid at rest, with the velocity c, suffers the same impulse as if the fluid were to move with the velocity c, and the plane to remain at rest. Hence the resistance of a fluid to a body in motion, is the same with the percussion of a fluid moving with the same velocity against the body at rest. Though this conclusion seems to be supported by very sound reasoning, yet, in fact, the resistance is less than the percussion, the velocity being in both cases the same, in the proportion of 5 to 6, DON JUAN, in his Examen Maritime, makes the difference much greater. It arises, no doubt, from the action of the fluid on the hinder part of the moving body, by which the resistance is in some degree counter- acted. It appears, however, to be only the abso- lute quantities, and not the ratios of the resistan- ces, that are thus affected ; the resistances being as the squares of the velocities. In the reasoning on which this proposition is found- ed, we have supposed the only resistance to arise from the inertia of the particles. There is, how- ever, another arising from the cohesion of these particles, which must be proportional to the quan- tity of that cohesion overcome in a given time, that is, to the velocity simply. This resistance is only perceived in slow motions, and is small in respect of the other, except in fluids of much viscidity. COULOMB HYDRAULICS. 209 COULOMB has made experiments on this kind of re- sistance, Mem. de TInstitut National, torn. in. p. 247., Sec. He found the part of the resistance proportional to the velocity to be 17 times greater in oil than in water. It must be remarked, that DON Gr. JUAN makes the resistance which ships meet with proportional nearly, cocteris paribus, to their velocities simply. Examen Maritime, torn. IT. § 194, 195. 299- If a stream strike obliquely on a plane, its force is less than if it struck directly on the same plane, in the ratio of the square of the sine of the obliquity to the square of the radius. If the area struck be «2, v the velocity of the fluid, and i the inclination or the angle which the plane makes with the direction of the stream, the force This conclusion does not agree with experience, ex- cept when the angle i is greater than 60 degrees. BOSSUT, Hyd. vol. i. § 375. It appears, from ex- periment, that the resistance to oblique planes con- sists of two parts, one of which is proportional to the square of the sine of 2, and the other to a certain power (3.J) of the angle i itself. BOSSUT found, that if a prow, in the form of a wedge, be drawn through a fluid, and if the 'complement of the angle which either face makes with the direction of the VOL. I. O stream 210 OUTLINES OP NATURAL PHILOSOPHY, stream measured in degrees be called #, and if the resistance to the base of the wedge be 10000, the resistance to the wedge itself will be 10000 x cos a* + 3.153 x BOSSUT Hydrod. torn. n. § 1019, & 1022. This formula, however, is to be considered only as an approximation to the truth. It is entirely empirical, and only agrees with experiment, when x is nearly a right angle, or when the angle i is very small. 500. Supposing the resistance to oblique sur- faces to vary, as the square of the sine of the incli- nation, the resistance to a sphere is half the resist- ance that would be made to the circumscribed cy- linder moving in the direction of its axis. NEWTONI Prin. Math. lib. n. prop. 34 This proposition cannot be expected to hold accurate- ly, as it is derived from a principle that is acknow- ledged to be erroneous. The propositions concern, ing the solid of least resistance, are faulty for the same reason. In general, though the theory of the perpendicular percussion and resistance of the dense fluids, agrees sufficiently with experiment for all practical purposes, that of their oblique percussion and resistance, as it stands at present, is not to be relied on. The experiments relative to percussion are HYDRAULICS. are more agreeable to theory than those that con- cern resistance. VINCE has found by experiment, that the force of the percussion of a fluid on an oblique plane, varies nearly as the sine of the incli- nation of the plane ; but their resistance to a plane moving in a fluid, is not subject to any law that ad- mits of a simple expression. Encyc. Britan. 3d edit. Article Hydrodynamics, p. 763. SECT. IV. UNDULATION OF FLUIDS, OR THE FORMATION OF WAVES. ' 301. WHEN the surface of water is unequally pressed on, in parts contiguous to one another, the columns most pressed on are shortened, and sink beneath the natural level of the surface, while those that are least pressed on are lengthened, and rise above that level. As soon as the former columns have sunk to a certain depth, and the latter have risen to a certain height, their motions are reversed, and continue so, till the columns that were at first most depres- sed have become most elevated, and those that O 2 were OUTLINES OF NATURAL PHILOSOPHY. were most elevated, have become most depress- ed. The alternate elevations and depressions thus produced are called waves. The water in the formation of waves has a vibratory or reciprocating motion, both in a horizontal and a vertical direction, by which it passes from the columns that are shortened to those that are length- ened, and returns again in the opposite direction. Progressive motion is not necessary to undula- tion. 302. The vibrations of water in the form of waves, may be compared to the reciprocations of the same fluid in a syphon or bent tube ; and it was from this that NEWTON deduced the velocity of waves, and the time required to an undula- tion. The time of an undulation, is the time from the wave being highest at any point, to its being highest at that point again. The velocity of the wave is the rate at which the points of greatest elevation or de- pression seem to change their places. 303. If a be the altitude of a wave, b half the breadth, *• the circumference of the circle, of which the diameter is 1, the time of an undula- tion HYDRAULICS. tion is ~ (a -f- 6)2, and the space which the wave o 86 appears to pass over in a second, is Examen Maritime, vol. i. § 816. In these theorems, 8 is put for V 2g-, to which it is nearly equal. If a he neglected, the velocity of the wave becomes - — > which is the velocity as determined by 7F NEWTON, Principia, lib. u. prop. 46. See also Bos- SUT, Hydrod. torn. i. § 312. 304. While the depth of the water is sufficient to allow the oscillation to proceed undisturbed, the waves have no progressive motion, and are kept, each in its place, by the action of the waves that surround it. But if by a rock rising near to the surface, or by the shelving of the shore, the oscillation is prevented, or much retarded, the waves in the deep water are not balanced by those in the shallower, and therefore acquire a progres- sive motion toward the latter, and form break- ers. Hence it is, that waves always break against the shore, whatever be the direction of the wind. Breakers 214 OUTLINES OF NATURAL PHILOSOPHY. Breakers formed over a great extent of shore, are di- stinguished by the name of Surf. The surf is greatest in those parts of the earth where the wind blows always nearly in the same direction. SECT. V. HYDRAULIC ENGINES. Engines moved by the IMPULSE of Water. 305. W HEN a wheel is put in motion by a stream of water striking against a float-board, the action of the water diminishes, as the velocity of the wheel increases, till at last the momentum of the water, or of the accelerating force, is just equal to the momentum of the resistance, or of the retarding force. The motion of the wheel then be- comes uniform. Mills with undershot wheels, or such as receive the impulse of the water at right angles to their radii, are machines of this kind. . 306. A machine driven by the impulse of a stream, produces the greatest effect, or does the most HYDRAULICS. most work in a given time, when the wheel moves with one-third of the velocity of the wa- ter. This theorem was first given by PARENT, Mem. de TAcad. des Sciences, 1704. See also MACLAURIN^S Fluxions, § 907. EULER, Nov. Com. Pet. torn. vnr. p. 230. BOSSUT, Hydrod. torn. i. § 390. If c he the velocity of the stream, v that of the float- hoard, a2 the section of the stream, or the area of the part struck, the impulse of the stream, by § 293. = ma?(c — v)2, and the effect or quantity of work done, in a given time, will be m c? (c -— v)2 v, which is a maximum when v = - . 3 The quantity ma? (c — z>)2, or the moving force he- comes then m a2 X - c2, which, making m = 1, as in y the first case of § 297., and writing %gh for c~, Q O becomes - a?gh, or -- of the weight of the co- y y lumn of water that falls on the wheel. If m = 1 , JC as in the second case of § 297., the moving force is 4 only - of the said column. According to the way y in which the water strikes the wheel, the moving force may therefore vary in the ratio of 2 to 1 . The 216 OUTLINES OF NATURAL PHILOSOPHY. mi a? A i • 8 y o *i that is, eight twenty-sevenths of the quantity of mo- tion expended on the machine. In different falls of water, that act on wheels moving with a velocity in a given ratio to that of the water, if the expenditure be the same, the forces will be as the velocities simply, (§ 297.), and the effects of the machines driven by them as the squares of the velocities. If the sections of two streams are the same, their forces will be as the squares of their velocities, or as the heights due to the velocity of the water, (§ 297.), and the effects of the machines driven by them, will be as the cubes of their velocities, or as the heights due to those velocities multiplied into their square roots. These conclusions, all except the first, are remarka- bly verified by the experiments of Mr SMEATON. He found, that an undershotwheel, when working to the greatest advantage, had a velocity which va- ried from one-third to one-half the velocity of the stream ; and was, in great machines, nearer to the latter of these limits than the former. Also, that the load, when the effect was greatest, approached very near to the weight of the whole column of wa- ter striking on the wheel. It is found above to he eight-ninths. When HYDRAULICS. 217 When the undershot-wheel worked to the greatest ad- vantage, its effect was about one-third of the motion o expended ; we have found it to be — . 27 Mr SMEATON also found, that the expence of water being the same, the effect is as the effective head, or the square of the velocity. And, lastly, that when the section of the water is the same, the ef- fect is as the cube of the velocity, both which re- sults are deduced above. By a mistake, for which it is difficult to account, SMEATON supposed these conclusions to be inconsistent with the theory of the percussion of fluids. Machines moved by the Weight of Water. 307. When a wheel receives the water into buc- kets, at or near the highest point, it is put in mo- tion by the weight of the water with which it is loaded on one side, and it is then called an overshot wheel. 308. An overshot wheel arrives at a state of uni- form motion, when the momentum of the water in the buckets is equal to the momentum of the resist- ance. The construction of the machine should be such, that when the motion of the wheel becomes uniform, the 218 OUTLINES OF NATURAL PHILOSOPHY. the buckets may contain all the water of the stream, and may carry it down as low as possible before they allow it to escape altogether. 309. If the section of that part of the wheel to which the buckets are fixed be called c2, and the perpendicular depth of the point where the water leaves the wheel, below that where it enters, be called p, the moving force is a column of water equal to c2 p ; and as this force acts by a lever = r9 its momentum = 310. If A fee the quantity of water issuing in a second, and h the height corresponding to the ve- locity of the circumference of the wheel, the ef- fect of the machine, (supposing the wheel to re- ceive the water, at or very near its highest point), is proportional to A (2 r — h). ALB. EULER, Enodatw Qutsstionis quomodo vis Aqua ad Moilas circumagendas cum maximo lucro impendi possit, 4to, Gottingen, 1754. BOSSUT, Hydrod. § 415., &c. It is evident, that the effect will be the greater the less h is, or the less the velocity of the wheel ; and as that velocity may be diminished indefinitely, 2 r A. is a limit to which the effect may approach nearer than any given quantity. The power of the overshot-wheel is greater, cateris paribus, than that of the undershot, nearly in the ratio of 27 to 8. The HYDRAULICS. 219 The maxim, however, that overshot wheels are the more powerful the slower they move, is found in practice to be subject to some limitations. Mr SMEATON'S experiments led him to conclude, that overshot-wheels do most work when their circum- ferences move at the rate of 3 feet in a second, and that when they move considerably slower than this, they become unsteady and irregular in their mo- tion. This determination is also to be understood with some latitude. He mentions a wheel 24 feet in diameter, that seemed to produce nearly its full effect, though the circumference moved at the rate of 6 feet in a second ; and another of the diameter of 33 feet, of which the circumference had only a velocity of 2 feet in a second, without any conside- rable loss of power. The first wheel turned round in 12".6, the latter in 51".9, SMEATON, Experimental Inquiry concerning the Natu- ral Powers of Water and Wind to turn Mills, &c. Phil Trans, vol. LI. (1759,) p. 100, &c. See par- ticularly p. 133, 134. Dr BREWSTER'S Notes on FERGUSON'S Mechanics. Machines moved by the Re-action of Water. 311. Some machines are moved, not directly by the impulse of a stream of water, but indirectly, by the relief from pressure which the motion of the stream occasions. 3 The 220 OUTLINES OF NATURAL PHILOSOPHY. The machine known by the name of BARKER'S Mill is of this kind. Let ABGH, (fig. 24.), be a hollow cylinder, moveable about a vertical axis CD, and having the horizontal boxes EB, EG communica- ting with it, so that the whole may be filled with water from the top. The boxes have each an open- ing in the sides, opposite to one another, through which the water issues. When these apertures are shut, and the whole filled with water, it remains in equilibria; but if the apertures are opened, the columns having these aper- tures for their bases, and the depth of the water for their altitudes, will cease to press on the sides ; and therefore the whole pressure on the sides where there are no perforations, will be greater than on the sides where the perforations are, by the sum of these two columns. A moving force, therefore, equal to the pressure of these two columns, will impel the machine in a di- rection opposite to that in which the water issues, and in that direction the machine will begin to move. 312. The moving force in this instance becomes greater, after the machine has begun to move ; for the water in the horizontal boxes acquires a cen- trifugal force, by which its pressure against the sides is increased. When the machine works to the greatest advantage, the centre of the perfora- ty _____ tions should move with the velocity — v hg, where HYDRAULICS. 221 where r is the radius of the horizontal arm, mea- sured from the axis of motion to the centre of the perforation, and r> the radius of the perpendicular tube, g being put for the force of gravity, or 32.J feet. As % * r is the circumference described by the centre Q /»•' of each perforation, — — is the time of a revolu- tion in seconds. The quantity — V hg is also the velocity of the efflu- r' ent water ; therefore, when the machine is working to the greatest advantage, the velocity with which water issues is equal to that with which it is car- ried horizontally in an opposite direction; so that, on coming out, it falls perpendicularly down. The effect of this machine is equal to A h, or the mo- mentum of the water expended ; so that it could, if there were no force lost by friction, raise up the whole of the water to the height from which it fell, and in an equal time. It is therefore, in theory, the most advantageous application of water that is known ; but, on account of the friction, which must be great, as the whole body of the water has to turn on the axis CD, (fig. 24.), it will probably be found in practice, that the overshot-wheel is, in ma- ny cases, preferable. In the construction of this machine for practical purposes, the water is made to 222 OUTLINES OF NATURAL PHILOSOPHY. to flow in pipes along a conical surface, diverging downwards, and is discharged through several aper- tures. On the construction of this machine, see EULER, Mem. de TAcad. de Berlin, torn. vi. (1750,) p. 311. ; and, again, torn. vn. p. 271. ALB. EULER, Eno- datio, &c. § 64. BOSSUT, Hydrod. torn. i. § 424. BREWSTER'S Notes on FERGUSON'S Lectures, vol. u. p. 205. AEROSTATICS. AEROSTATICS. 223 AEROSTATICS. SECT. I. OF HEAT. AN treating of Aeriform Fluids, we begin with the consideration of Heat, which, though it affect all material substances, acts more remarkably on elas- tic fluids than on other bodies. We shall treat of heat as affecting the Bulk, the Fluidity, and the Elasticity of bodies. 313. HEAT or CALORIC may be regarded as a substance which penetrates and expands all bo- dies, and produces in us the sensation of heat and cold. Though we conceive heat to be a substance, it is never found in an independent state, or existing otherwise than as a property of body. 314. As 224 OUTLINES OF NATURAL PHILOSOPHY. 314. As heat is known to us principally by its power of expanding bodies, we may take the ex- pansion of some known substance as a measure of the variations of heat. The substance selected for measuring heat is Mercu- ry ; and the Thermometer is an instrument indica- ting the variations of bulk to which we conceive the variations of heat to be proportional. It has been found, that the variation of the bulk of mercury, from being immersed in freezing, and in boiling water, bears always the same proportion to the whole bulk ; and, therefore, that variation di- vided into a certain number of parts, may be taken as a scale of the degrees of heat. In FAHRENHEIT'S thermometer, the interval above na- med is divided into 180°, and numbered at the lower end 32°, and at the upper end 212°. In the Centigrade or Centesimal Thermometer, the above interval is divided into 100°, and the degrees are numbered from the freezing point. The word temperature is used to denote the heat indi- cated by any degree of the thermometer, as exist- ing in the air or in any other body. 315. All bodies, whether solid or fluid, are ex- panded by heat ; but not all in the same propor- tion. 2 Expansion AEROSTATICS, Expansion of different Bodies for 180° of FAHRENHEIT, viz. from 32° to IN BULK IN LENGTH. Gold, .0042 1 ' 238 .0014 = 1 714 Platina, .0026 1 = 385 .00085 =± 1 1155 Silver, .006 1 " 160 .003 = 1 480 1 1 Copper, .0051 = 196 .0017 = 588 1 1 Brass, .0056 = 178 .00153 = 533 Zinc, .0093 1 r FOB .0031 = 1 322 Mercury, .02 i : 50 .0066 = 1 150 Water, .0466 i .0155 __ 1 ' 21.5 64 1 1 Salt waters, .05 J_ 20 .0166 = 60 Alcohol, - .01 1 r 100 .0033 = 1 300 All Gases, or per- manently elas- tic fluids. 1.376 1 - 2^6 .1253 =± 1 7.98 See DALTON'S New System of Chemical Philosophy, vol. i. p. 44. VOL, L S16. At OUTLINES OF NATURAL PHILOSOPHY. 316. At the heat of — 40° mercury freezes, and at 600° it boils, and is in both cases useless as a measure of heat. For measuring great cold, a thermometer of alcohol or spirit of wine is employ- ed ; for measuring great heat, the expansion of a solid is used, and the instrument is called a Pyro- meter. The best pyrometer hitherto in use, is that invented by Mr WEDGWOOD, consisting of a small cylinder of Cornish clay, which contracts continually by heat. The scale of this instrument begins at a red heat, of such intensity as to be visible in day-light, which, on FAHRENHEIT'S scale, could it be continued so far, would be marked 1077°. From this it ascends, (by degrees that are each of them equal to about 130° of FAHRENHEIT,) as far as 240°, or 32277° of FAHRENHEIT, which is its utmost Kmit; 28° mark- ing the melting heat of silver, 32° of gold, 125° the ^greatest heat of a glass-house furnace, and 150° the temperature at which pig-iron melts. 317. The fact that all bodies expand by heat, admits also of another exception. Water is most dense at a temperature between 37° and 39°, and itsv specific gravity diminishes both by the addition and subtraction of heat. When water is converted into ice a new arrangement takes place, and the specific gravity is considerably diminished. 318. Heat AEROSTATICS. 227 318. Heat diffuses itself on all sides, and passes continually from bodies in which the temperature is greater to those in which it is less ; and if a bo- dy be placed in a medium of a temperature differ- ent from its own, the momentary variations of its temperature will be as the differences between the temperature of the body and of the medium ; so that if the times, reckoned from any instant, be taken in arithmetical progression, the variations of temperature in the body, and also the differ- ences between its heat and the heat of the medium, will decrease in geometrical progression. This law of the heating and cooling of bodies, was first taken notice of by NEWTON, Scala Graduum Caloris, Phil. Trans. (1701,) N° 270. It was af- terwards proved by the experiments of Professor RICHMAN of St Petersburg. Novi Comment. Pe- trop. torn. i. (1747, 1748,) p. 174., &c. It follows from it, that if D be the difference between the temperature of the medium, and of the body . placed in it, and if — be the heat lost in one second, n then, at the end of V, %", 3", &c. the 'heats are D (i- D'' &c-; and the decrements of heat in each second -, D 1\ D , IV IT ( n/' 7T (1 ~~ n) > &C' ; and' therefore> if < P2 be 228 OUTLINES OF NATURAL PHILOSOPHY. be the time in seconds, reckoned from the moment when the difference of the temperature was D; and if A be the remaining difference of temperature at the end of the time t, we have D (1 — i Y — A, ^ «' and f- log n — log (n — 1) If this law is rigorously observed, no body can ever perfectly acquire the temperature of the medium in which it is placed, and this may be the fact, speak- ing with mathematical exactness, though the dif- ference of the two temperatures may soon become so small as to escape observation. In respect of us, the bodies are then of the same temperature. 319- The diffusion of heat through a fluid, is promoted by a hydros tatical principle, the heat rarifying the fluid, and so producing a motion and mixture of parts, by which the heat is communi- cated more rapidly through the whole, than it could be through a mass of which the parts were immovable in respect of one another. The .communication of heat in this way, is so rapidf that it renders the ordinary slow progress of acqui- ring and losing heat by contact almost impercep- tible, and has given rise to the error, that heat has no tendency to pass through fluids, except in con- sequence of the mixture of their parts. 320. Heat AEROSTATICS. 229 320. Heat escapes from bodies which are heated above a certain temperature, by radiation, or by passing in straight lines through the air with great rapidity. The heat thus emitted from bodies, is reflected from the surfaces of metallic specula, like the rays of light. A body heated, though not so as to shine, and placed before a concave speculum of metal, communicates heat instantaneously to a thermometer in the cor- responding focus. A cold body does the same ; and it is remarkable, that an effect so difficult to be explained, is, nevertheless, perfectly consistent with the law of continuity. The experiment of the re- flection of cold was first made by M. PICTET of Geneva. Essai sur le Feu, par M. A. PICTET, §69. Radiant heat from a luminous body, is refracted as well as reflected ; the rays from a candle or a fire produce heat in the focus of a lens. It does not appear that the same happens in the case of obscure heat, except when it comes from the sun, which, according to the experiments of Dr HERSCHELL, is concentrated in the focus of a lens. Phil. Trans. 1803, p. 298, 304. Dr HERSCHELL, in a series of very interesting experi- ments, separated the caloric from the light of the sun's rays, and shewed that the focus of the former was not the same with that of the latter. Phil. Trans, ibid. 321. The 230 OUTLINES OF NATURAL PHILOSOPHY. 321. The heat propagated by radiation from different bodies varies with the nature of their external surfaces, the quantity that flows in a gi- ven time from a body with a polished surface, being much less than would flow from the same body with a rough surface. This was discovered by Professor LESLIE, and proved by a variety of curious experiments. Experimental Inquiry into the Nature and Propagation of Heat, p. 17., &c. When the fluid contained in a vessel, is intended to retain its heat long, the vessel should be of metal, and it surface smooth and bright. If the fluid is required to cool as fast as possible, the surface should be rough covered with paper, char- coal, &c. D ALTON'S New System of Chemical Phi- losophy, Part i. p. 116. 322. If two portions of the same fluid, of which the masses are M and M', and the temperatures t and /', be mixed together, the temperature of the mixture will be M + M' This formula is investigated, on the supposition that no part of the heat is lost by the mixture. On making the experiment, the results are not found to agree exactly with those deduced from the for- mula. The expansions of mercury, therefore, do not AEROSTATICS. 231 not appear to observe the same law with the varia- tions of temperature ; and it is found, that, for equal variations of temperature, the variations of the bulk of the mercury constitute a series, of which the dif- ferences are in arithmetical progression, or in which the second differences are constant. Hence the rela- tion between the temperature and the expansion of mercury, may be expressed by a quadratic equation, or by the abscissae and ordinates of a line of -the se- cond order. 323. If D be any number of degrees reckoned from the freezing point of water in FAHRENHEIT'S thermometer, the real temperature corresponding to D is 207 (— 1 + v 72 This formula is derived from D ALTON'S experiments. See his System of Chemical Philosophy, Part i. p. 14. If D = — 72, which is known to mark the degree according to FAHRENHEIT'S thermometer, at which mercury congeals; the temperature, as deduced from this rule, is — 207, which, therefore, is the real distance between the freezing of mercury and the freezing of water. Hence the degrees on the mer- curial thermometer, reckoned from the degree which marks the congelation of mercury, are as the square of the real temperatures, reckoned from the same point of congelation. The OUTLINES OF NATURAL PHILOSOPHY. The correction of the mercurial thermometer, arising from the law of expansion, has not yet been intro- duced ; but the reality of the law itself seems suffi- ciently established. 324. When fluids of different kinds, and of dif- ferent temperatures, are mixed, as above, the tem- perature of the mixture is very far from corre- sponding with the theorem in article 322, in so much as to make it certain, that equal bodies, for equal differences of temperature, do not con- tain equal quantities of heat. Beside the masses M and M', therefore, we must introduce two in- determinate co-efficients c and c', before the for- mula can be applied ; we have then c" (M + M') for the temperature of the mixture, In this formula, c and c are the numbers that denote the differences of heat contained in two equal por- tions of these fluids, for equal differences of tempe- rature. They may be said to denote the capacities of the bodies for heat ; and their values must be determined by experiment. When equal weights of water and ice are put into the same vessel, the water being -at the temperature 176°, and the ice at 32°, the ice is in- stantly AEROSTATICS. 233 stantly melted, and the temperature of the whole is found to be exactly 32°. In this experiment, 140 degrees of heat have en- tirely disappeared ; and the same happens what- ever be the temperature of the water. Dr BLACK was the first who made this experiment; and as he found that the heat which had thus ceased to affect the thermometer, was not lost, but became sensible again on the congelation of the water, he said that it had become latent, — a term well adapted to express the fact, without any allusion to theory. 826. From the combination of this experiment with the formula in the last article, the beginning of the scale of heat, or the point to which the mer- cury in the thermometer would sink, if it were to lose all its heat, may be determined ; and it will be found, that if c is the capacity of water for heat, c' of ice, and x the number of degrees by which the beginning of the scale of heat is below the 32c'+11 beginning of FAHRENHEIT s, CD =2 —t 0 ~* C/ For, according to this notation, the total heat in ice, of the temperature 3£°, is c' (32 4- r'2 d h'. To a complete theory of the steam engine, much more is necessary than the knowledge, that r2 b h is greater than r'2 d h' ; for the rate of working de- pends on that excess, and must be determined as in the problem, § 186. The weight of the pump rods must be included, and also the effect of fric- tion. It is here supposed, that a perfect vacuum is produ- ced in the cylinder by the jet of cold water. This, however, is not the fact, for by the alternations of the heat and cold to which the cylinder is exposed, it can neither acquire the heat necessary to the full elasticity of the steam, nor the cold necessary for its complete condensation. On this account, the effect of the machine falls much under the compu- tation. 357- When the cylinder is full of steam, if a valve be opened, by which the steam is allowed to escape PNEUMATICS. 265 escape into another vessel, where a jet of cold wa- ter is introduced, the condensation is much more complete, and the heat of the cylinder being pre- served, the steam possesses its full elasticity. This improvement was made by Mr WATT, and com- pletely changed the character of the steam-engine. In the old engines, the power was reduced nearly to half its real value, so that the moving force, instead of amounting to 14 Ib. on every square inch of the area of the piston, was reduced to little more than seven. In Mr WATT'S engines, the moving force is not less than 12 Ib. on the square inch. 358. A farther improvement has been made on this engine, by injecting the steam into the cylin- der, alternately above and below the piston, so that the whole motion is produced by the elastici- ty of steam, and has no dependence on the weight of the atmosphere. This improvement is also due to Mr WATT, and could not have been made without the previous contri- vance of condensing the steam in a separate vessel. It is particularly accommodated to the production of a rotatory motion by means of a steam en- gine. In the double-stroke engine, the piston-rods require to be forced down as well as to be drawn up, in the same vertical line. The method by which Mr WATT has accomplished this depends on a geometri- cal theorem. 359. If 266 OUTLINES OF NATURAL PHILOSOPHY. 359. If the ends of a straight line given in magnitude, describe circles given in position, and having their convexities turned opposite ways, a point may be found in that line, which will de- scribe a curve, having, where it intersects the line of the centres, an arch of contrary flexure, not dif- fering sensibly from a straight line. The point to be found divides the given line in the ra- tio of the radii of the circles described by its extre- mities. A rectilineal vertical motion is also produced by ano- ther construction. Two of the adjacent angles of a parallelogram, are made to describe concentric circles, so that the side between them passes through their centre, and one of the remaining angles another circle, having its convexity opposed to that of the two former ; then the third angle of the parallelo- gram describes a line that differs insensibly from a straight line. PEONY treats of this motion, Arch. Hydraulique, torn. n. § 1478, &c. ; but has not given a complete theory of it. He says, on the authority of ADAMS, (Geo- metrical and Graphical Essays), that the contri- vance was suggested to Mr WATT by one of the in- struments for describing curves, invented by SOAR- DI. See Nuovi Istrumenti per la descrizione di diverse Curve, &c. del Conte GIAMBATISTA SOAR- DI, Padova, 1752. I have looked into this work with considerable attention, but have found nothing that has an affinity to the motions just described. 360. In PNEUMATICS. 267 360. In steam-engines of similar construction, the effects are nearly as the quantity of fuel con- sumed. In this estimate time is necessarily involved, as, in or- der to derive the greatest advantage from a given quantity of fuel, the combustion must neither be too quick nor too slow. It is computed, that an engine of the best construc- tion, will raise 20,000 cubic feet of water to the height of twenty-four feet for every hundred weight of good pit-coal. An engine with a cylinder of thirty-one inches diameter, and making twelve strokes in a minute, will do the work of forty horses, and will burn 11,000 Ib. of the best Staf- fordshire coal in a day. See Encyclopedia Britan- nica, article Steam, p. 769. PHONY, Arch. Hyd. § 1499, vol. ii. Motion produced by Gunpowder. 361. When gunpowder is fired, a permanently elastic fluid is generated, which being very dense, and much heated, begins to expand with a force at least 1000 times greater than that of air under the ordinary pressure of the atmosphere. ROBINS' Tracts, New Principles of Gunnery, Prop. vi. HUTTON'S Mathematical Dictionary, art. Gunpow- der. 1 In 268 OUTLINES OF NATURAL PHILOSOPHY. In the acceleration of a ball by the fluid thus genera- ted, three circumstances must be attended to. 362. 1. The elasticity is inversely as the space which the fluid occupies, and, therefore, as it forces the ball out of the gun, it continually dimini- shes. 2. The elasticity would diminish in this ratio, even if the temperature remained the same ; but it must diminish in a much greater ratio, both from the dispersion of heat, and the absorption of it by the fluid itself, during its rarefaction. 3. The air propels the ball by following it, and acts with a force that is, cceteris paribus, propor- tional to the excess of its velocity above the velo- city of the ball. The greater the velocity that the ball has acquired, the less, therefore, is its momen- tary acceleration. The effect of the elastic fluid must, for these reasons, decrease much faster than the space it occupies in- creases ; and a formula expressing the law of acce- leration, as depending on all these causes, more espe- cially on the latter, might be expected to be very complex. Nevertheless, it appears from Dr HUT- TON'S experiments, that the velocity with which a ball actually issues from the mouth of a cannon, other things PNEUMATICS. 269 things being " the same, is nearly as the square root of the weight of the gunpowder, or more generally, if v be the initial velocity of the shot, P the weight of powder, and B the weight of the ball, that / P v=:my ~-; m being a constant co-efficient, to be determined by experiment. Philosophical Transactions, 1778. Also BUTTON'S Course of Mathematics, vol. m. p. 270. / T* Now that v = mL/ — , is exactly the conclusion that would be deduced, from supposing the acceleration of the ball to depend simply on the expansion of the elastic fluid, without taking into account the dimi- nution of the impulse, arising from the velocity which the ball has already acquired. This may be shewn from the Principles of Dynamics ', § 100. That diminution, therefore, is inconsiderable, which arises no doubt from this, that the velocity acquired by the ball is very small, compared with the im- mense velocity with which the elastic fluid in the gun expands itself. The value of v given above, is only exact on the sup- position, that the piece is long enough to allow the elastic fluid generated by the gunpowder to pro- duce its full effect. The augmentation of the charge may so much lessen the space over which the fluid is to act, that the velocity of the ball shall 270 OUTLINES OF NATURAL PHILOSOPHY'. shall be less from a great charge than from a smaller. 363. The Balistic Pendulum is an instrument invented by Mr ROBINS, for determining the ve- locity with which balls are projected by ordnance of different kinds. The ball is made to strike a heavy block of wood, suspended from a centre ; the velocity is thus reduced to a quantity that is moderate, and easily admitting of being measured. From thence the original velocity is computed by help of the principles explained, Mechanics, Sect. 7. on the Rotation of Bodies, § 216. For a particular description of this instrument, see ROBINS' New Principles of Gunnery, p. 84. Also Eu LEU'S Notes on ROBINS' Gunnery. The rule for computing the velocity of the ball from the vibra- tion of the pendulum is there investigated, as it is likewise, BUTTON'S Tracts, p. 115., &c. The ini- tial velocity of shot estimated in this way, varies from 1600 to 2000 feet in a second. The latter is nearly as great as it is ever found. The velocity with which the elastic fluid, disengaged from the gunpowder, expands itself, is probably more than double of this. Dr HUTTON estimates it at 5000 feet per second. Course of Mathematics, vol. in. p. 270, 364. The PNEUMATICS. 271 364. The depth to which a ball penetrates into wood, earth, &c. is nearly as its weight multiplied into the square of its velocity. p Hence, since fl2 = w2x~5 BlfyzypPi so that the effect of a shot is nearly as the quantity of gun- powder. From the Table given by Dr HUTTON, Course of Ma- thematics, vol. in. p. 272. the value of m may be deduced nearly = 2000. The following proposition, which has been already mentioned, is applicable to many researches in Me- chanics and Hydrodynamics, and particularly to such as are now treated of. 365. When bodies, whether solid or fluid, act on one another by impulse or percussion, in such a manner that their action is subject to the law of continuity, the sum which is made up, at any in- stant, by multiplying every mass into the square of its velocity, and adding all the products toge- ther, is a constant quantity, or one which remains always of the same magnitude. The sum thus made up is the vis viva of the bodies, and this proposition is therefore called the conserva- tion, of the v is viva. 366. If OUTLINES OF NATURAL PHILOSOPHY. 366. If the bodies are subjected at the same time to the action of an accelerating force, the sum made up by multiplying every body into the square of its velocity, and adding all the products together, is equal to the sum made up by multi- plying every body into the square of its initial ve- locity, and adding to the amount the sum made up by multiplying every one of them into the square of the velocity which it would have acqui- red by the action of the accelerating force alone, if it had moved freely in the same line which it has actually described. a. See Dynamique par M. D'ALEMBERT ; where the whole of the 4th chapter is employed in demon- strating the different cases of these propositions. CLAIRAULT has also given a demonstration, Mem. Acad. des Sciences, 1742. DAN. BERNOUILLI had before founded an entire system of Hydrodynamics on the preservation of the vis viva, merely assuming it as true. Hydrodynamica sive de Viribus et Moti- bus Fluidorum, Argent. 1738. b. D'ALEMBERT has shewn, that the preservation of the vis viva depends on this principle, That when any number of forces are in equilibrium, the velo- cities of the points to which the forces are applied, estimated in the direction of the forces themselves, are in the inverse ratio of those forces. Dyna- mique, p. 267. edit. 2d. The conservation of the 2 vis PNEUMATICS. 273 vis viva is therefore an inference from the principle of the virtual velocities. Bodies impelled by Currents in the Atmosphere, or by Winds. 367. The impulse of a stream of air striking with the velocity V9 on a plane of the area a2, in- clined at an angle i to the direction of the stream, sin z2. 77i is a quantity to be determined from experiment, and is constant, while the density of the fluid con- tinues the same. This proposition coincides with that which is enun- ciated of non-elastic fluids, § 299. 368. The sails of a windmill are so disposed as to turn in a vertical plane round a horizontal axis, they themselves heing all inclined to that plane toward the same side ; when, therefore, the plane in which the sails turn, is placed at right angles to the direction of the wind, the force on each sail is resolved into two, one at right angles to its sur- face, and the other parallel to it. The latter has no effect, hut a part of the former impels the sail, on the side toward which it makes an acute angle VOL. I. S with OUTLINES OF NATURAL PHILOSOPHY. with the direction of the wind. Thus a circular or rotatory motion is given to the sails. The sails are so constructed, as to have different in- clinations to the plane of their motion at different distances from the axis, greatest nearer the centre, and least at their extremities. This is called the weathering of the sails, and is done in order to make the momentum of the wind nearly the same at all different distances from the centre of motion. 369. Supposing the sail of a windmill to be a plane, and the inclination of that plane to the axis or the direction of the wind to be i9 the effect of the wind to turn the sail, in a plane at right angles to its axis, will be the greatest when cos i x sin i* 1 is a maximum, or cos i ~ — - . V & This gives £ = 54° 44', and therefore the inclination of the sail to the plane of its motion, or what is called the angle of weather, = 35° 16'. This is true only when the sail is at rest, or just beginning to move. When the sail is in motion, and of course near the extremities of the sail, where it moves faster, the angle of weather must be less. MACLAURIN has given a formula for this case, Fluxions, vol. u. § 913, 914. MACLAURIN'S theorem makes the weather to vary from 26° 34', at the point of the sail nearest the centre, PNEUMATICS. centre, to 9° at its extremity. SMEATON made some corrections on this rule, from experiment. See Experimental Inquiry concerning Mills, p. 45, and 370, From SMEATON'S experiments it appears, that a windmill works to the greatest advantage, when it is so constructed that the velocity of the sails, is to their velocity when they go round with- out any load, as 6.5 to 10 nearly; and also that, the load, when the mill works in this manner, is to the load that would just keep it from moving, as 8.5 to 10 nearly. Experimental Inquiry, p> 49 and 50. 371. With different velocities of wind, the load that gives the maximum effect, varies nearly as the square of the velocity of the wind, and the effect itself nearly as the cube. Ibid. p. 52. The effect is always measured by the product of the velocity of the load into its weight. The velocity of the load varies in the simple and direct ratio of the velocity of the wind. Resistance 876 OUTLINES Of NATURAL PHILOSOPHY. Resistance of the Air to Projectiles. 37%. Though we should be led, as in hydraulics, to conclude that the resistance which air makes to moving bodies* is as the square of their velocities, experiment appears to prove, especially when the velocity is great, that the resistance is partly pro- portional to the square, and partly to the simple power of the velocity. The resistance to the same ball, its velocity being vy is m v*+nv ; where m and n are given co-efficients, BUTTON'S Course of Mathematics, vol. in. p. 278. In this formula m = .00002576 = -- , and n = — .00388 = — — -, the diameter of the ball being 1.965 inches. Hence for any ball of which the diameter is d, m = . 00000666, and w=r — .001, that is, m= Sooooo >andw = -Io6o; and PNEUMATICS. 277 and the resistance in avoirdupois pounds, or ~ 1000 (,300 " For this theorem, so useful in gunnery, and so well accommodated to practice, we are indebted to Dr HUTTON of Woolwich, ubi supra. A ball is often resisted by a force that is many times its own weight. An iron ball 3 Ib. weight has its diameter — 2.78 inches, and when it is thrown with a velocity of 1800 feet, it is resisted by a force equal to 176 Ib., more than 58 times its own weight. 373. Supposing the air to resist according to the law just assigned ; the height to which a ball of the weight w9 and the diameter d, will ascend, when projected perpendicularly upwards with any velocity c, will be, 2 n w m md? met* If this expression be reduced, by substituting for m its value -5236 d3, and using the common loga- 27S OUTLINES OF NATURAL PHILOSOPHY. It may be shewn by this, that a ball 1.05 lb., dis- charged with a velocity of 2000 feet, will ascend only to the height of 2920 feet, or little more than half a mile, whereas in vacuo it would have ascend- ed to the height of eleven miles and three-fourths. HUTTON, ibid. p. 284. 374. If a body descending in the atmosphere, has acquired such a velocity, that the resistance is equal to its weight, the accelerating and retarding forces being equal, its motion will become uni- form. Hence, since .5286 d5 is the weight of an iron ball of the diameter d, if we make 523.6 d — ZJ-L — v, we have a quadratic equation, oOO from the solution of which v may be found. The velocity thus found is called the terminal velocity of the falling body, or of the projectile. For an iron ball of 1 lb., the terminal velocity is 244 feet ; for one of 42 lb., it is 456. Vid. TABLE, BUTTON'S Course, vol. in. p. 291. In strictness this velocity is not acquired till after an infinite time, and a descent infinitely long: the time PNEUMATICS. 279 time of acquiring a velocity differing from it, by any quantity, however small, is finite, and can ea- sily be assigned. 375. The great problem in gunnery, viz. having given the weight, the magnitude, the direction, and the velocity of a projectile, to determine its path through the air, supposing the law of resis- tance to be known, is very difficult, and the solu- tions of it hitherto given, have not led to results easily applicable to practice. An approximation has been given by NEWTON, and complete solutions by EULER and LE GENDRE. Prin. Math. lib. n. Prop. 10. EULER'S Remarks on ROBINS, BROWN'S translation, p. 322., &c. LE GENDRE'S solution is in FRANCOEUR'S Mecanigue, p. 196., &c. 3d edit. Dr HUTTON of Woolwich is in possession of many va- luable materials relative to this problem, furnished by his own experiments, and there is reason to ex- pect a more useful solution than has yet been gi- ven, from one who unites profound mathematical knowledge with great practical skill. D'ANTONI has proposed a method of determining by experiment an indefinite number of points in the path of the same projectile. If a series of stations, one above another, be taken on the declivity of a hill, at the bottom of which there is 280 OUTLINES OF NATURAL PHILOSOPHY. is stretched out a plain of considerable extent ; and if the same piece of artillery be carried successively to these stations, and fired at the same angle, and in the same circumstances in all respects, and the points marked where the shot in all these cases strikes the plain, it is evident, that all these are points in the curve described by the same shot. The same might be done still more easily, by remo- ving the gun from one distance to another in the plain, and firing against the side of the hill. If the points from which the shot were fired, and those which they struck, were all marked on an accurate profile of the ground, they would enable us to de- termine as many points in the path of the pro- jectile as there were experiments made. It is unnecessary to observe, that at each station a great number of shot might be fired at different elevations, so that many curves might be determi- ned from the same set of experiments ; and that at each particular elevation many shots ought to be fired, and a mean taken among the points where the balls struck. 376. The ranges of the same ball, with the same elevation, but different charges, are nearly as the square roots of the initial velocities ; and the times of flight are nearly as the ranges. HUTTON^S PNEUMATICS. 281 BUTTON'S Mathematical Dictionary, article Gunnery, p. 567. Both these propositions are empirical, or deduced solely from experiment. Besides the works already quoted, see the article Re- Distance, by Professor ROB i SON, in the Encyclopce- dia Britannica. SECT. II. AIR AS THE VEHICLE OF SOUND. Vibration of Sonorous Bodies. 377. -I HE bodies which we consider as sonor- ous or as causes of the sensation of sound, at the time when they produce that sensation, are observ- ed to be in a state of tremor or vibration. This is exemplified in a bell, and in a drinking glass, when sound is produced by rubbing its edge with a wet finger, in a musical string, &c. The latter affords the simplest case of this fact, and that in which the laws of the vibration are most easily inves- tigated. 378. If a musical string stretched between two fixed points, be struck any where between those points, 282 OUTLINES OF NATURAL PHILOSOPHY. points, so as to be forced out of the straight line by a small quantity, it will vibrate backwards and forwards on each side of that line, and the curves into which it will successively pass in the course of these vibrations, will have their curvature at •every point proportional to the distance from the straight line joining the fixed points ; the accele- rating force at each point will also be proportional to that distance, and the great and small vibrations will be performed in the same time. It is usual to reckon the vibrations of a string differ- ently from those of a pendulum ; the passage from the highest point on one side, to the highest on the other, is reckoned a vibration of a pendulum. The passage from the farthest distance on one side to the farthest on the other, and back again to its first position, is accounted a vibration of a musical string* It is properly a double vibration. r >'?ft 'f^rft (TfM\ rf MtTTf"* 379. The figures which a musical string assumes in its vibrations, constitute a series of elastic curves, or elongated cycloids. The construction of these curves is easy. See the Notes in LE SEUR, and JACQUIER'S Commentary on the Principia, lib. u. prop* XLII. the Note 301. Al- so SMITHES Harmonics. nwrnt & 380. If PNEUMATICS. 283 380. If L be the length of a musical string of uniform thickness, B its weight, S the weight by which it is stretched, or tne measure of its tension, t the time of a double vibration, t — t) as usual is measured in seconds, and g is the velo- city acquired in 1" by a falling body, expressed in the same measure with L. The most convenient unit in this case is an inch ; so that g = 386 . The value of t may be made more convenient, by supposing w to denote the weight of an inch of the string ; so that P = w L, then t = ±3-^_ V#s If N be the number of vibrations in a given time, for instance in one second, N — _^~£L — . 2 L V w If the lengths and weights of two chords are the same, their times of vibration will be inversely as the square roots of the forces by which they are stretched ; and the number of vibrations which they perform in the same time, directly as those square roots. So also, the tension and the weight remaining the same, the celerity of the vibration is inversely as the square-root of the length; or, the tension and weight 284 OUTLINES OF NATURAL PHILOSOPHY. weight per inch remaining the same, the celerity of vibration is inversely as the length. The problem of the musical chord was first resolved, and the preceding theorems investigated, by BROOK TAYLOR, in his Meihodus incrementorum, prop. 21. 23., Sec. His solution, though very ingenious and extending to most of the cases that occur in nature, was not general, nor complete, when mathemati- cally considered. The first complete solution was given by D'ALEMBERT, in the Berlin Memoirs for 1747, and it led him at the -same time to discoveries which form a great era in the history of the Diffe- rential and Integral Calculus. Mem. de Berlin, 1747, p. 214., &c. 381. When the sounds of different musical strings are compared, a certain difference between them is perceived by the ear, which is called dif- ference of tone; and this difference is also expres- sed by saying, that the one sound is graver, and the other more acute ; the variations of tone are found to have a constant or fixed relation to the comparative celerities of vibration. As the string performs more vibrations in a given time, the sound it yields becomes more acute ; and as it vibrates more slowly, the sound is graver. This is easily brought to the test of experiment. The strings that vibrate faster, either from the greater tension, or their smaller length and weight, invariably PNEUMATICS. 285 invariably produce sounds that are more acute, &c. 382. If eight strings be such, that the number of vibrations which they perform in a given time are as the numbers 24, 27, 30, 32, 36, 40, 45, 48, the sounds of the first seven will be perceived as in- creasing in acuteness one above another, from the first to the last, and will yield the notes from the combinations of which all musical effects are pro- duced. The tone is not affected by the extent of the vibra- tions, but merely by their time. The loudness of the sound is supposed to depend on the greater ex- tent of the vibrations. Noise and discordant sounds arise from a want of isochronism of vibration. When the vibrations of a sonorous body are isochronous, the sound is always musical. The last of the strings will sound what is called the octave above the first, and the same series may be repeated again between the number 48 and its double 96, and each note will be an octave above its corresponding note in the first interval ; the num- bers of vibrations will be 54, 60, 64, 72, 80, 90, 96; and it is evident that this series may be continued either up or down without limit. The musical scale thus formed, is called the Diatonic Scale. The 286 OUTLINES OF NATURAL PHILOSOPHY. The pleasure derived from the successive or simulta- neous perception of the sounds of this series, ap- pears to be an ultimate fact which can be no farther analysed, but must be referred to the original con- stitution of the mind. The selection of the com- bination of these notes, capable of affording high degrees of pleasure, is the object of the musical art. '•Mifroii . .;•••;•. m 383. All other sonorous bodies, at the time they emit sound, vibrate in like manner, but according to laws less simple. In wind instruments, the sounding and vibrating body is the air itself. The number of vibrations performed in a given time, by any sonorous body, may be determined by com- paring its sound with the note which is sounded by a musical string of a given length, weight and tension. The ear is sufficient to decide what string in a harp- sichord is in unison with the given sound. The num- ber of vibrations performed in a given time by the former, is equal to the number of those performed in the same by the latter. 384. All musical sounds are computed to be con- tained within ten octaves ; so that the number of vibrations in a given time that yields the gravest note, is to that which yields the most acute, as 1 to 210, or as 1 to 1024. 3 Propagation PNEUMATICS. 287 . Propagation of Vibrations through the Air. 385. The vibrations of sonorous bodies are com- municated to the air, and by the impression thus made on the ear, excite the sensation of sound. That air is necessary to the production of sound, is evident from including a bell in a receiver, exhaust- ing the air, and making the clapper strike on the bell : the sound is hardly audible. 386. It is not every kind of vibratory motion produced in the air that is the cause of sound ; a musical string may vibrate, but if it is touched by a bit of cloth, or any soft body, no sound is heard. The vibrations in the air that produce sound must be communicated by some elastic substance. Sound is produced by the explosion of gunpowder, that is, by the sudden extrication of a fluid mass, dense, and highly elastic. It is produced also by the sudden rushing in of air to supply a vacuum. The crack of a whip appears to be an example of this last. 387. Though the vibrations of the air which produce the sensation of sound, are no doubt al- ternate 288 OUTLINES OF NATURAL PHILOSOPHY. ternate condensations and rarefactions of that fluid, in consequence of which the particles go and re- turn, or oscillate backwards and forwards for some time, even though there is no renewal of the im- pulse of the sonorous hody, yet every such oscilla- tion acts only once, or by a single impulse on the ear. AVere it otherwise, sound would be always something inarticulate and ill defined. 388. The velocity with which vibrations are propagated through the air, is the same that a heavy body would acquire by falling through half the height of the homogeneous atmosphere, or that which the atmosphere would be reduced to, if it were every where of the same density, and the same temperature with the air at the surface of the earth. The height of this homogeneous atmosphere has been formerly computed at 4343 fathoms, when the tem- perature is that of freezing. If this height be call- ed H, then v9 the velocity of the aerial vibrations, = V^^H. Hence V — 1057, which is too small, as it is found by experiment to be 1142 feet per second. The nature of the vibrations of an elastic fluid, was first considered by NEWTON, and their velocity de- fined, as in the preceding proposition, by compa- ring them with the vibrations of a pendulum, 2 Princip. PNEUMATICS. 289 Princip. Math. lib. ii. prop. 47. & 49- The sound- ness of this reasoning, however, was questioned by EULER ; and it was afterwards shewn by CRAMER, that the same argument might be used to prove conclusions very different from that at which NEW- TON had actually arrived. Commentary on the Prin- cipia, lib. ii. vol. ii. p. 364. LA GRANGE after- wards gave an accurate solution of the problem, by which, though he pointed out the error of NEW- TON'S reasoning, he confirmed the truth of his conclusion. 389. The velocity of the pulses propagated in an elastic fluid, are as the square root of the elasti- city divided by the density of the fluid. Prin. Math. ibid. prop. 48. The velocity of sound, computed as in the last ar- ticle, comes out less than by actual experiment. LA PLACE has suggested a very probable explana- tion of this, viz. that the condensation of the undu- Ice, which must take place in these vibrations, pro- duces a degree of sensible heat, by which the elas- ticity is increased, or, to speak more correctly, the density diminished, while the elasticity remains the same. The heat required for that effect has been shewn by BIOT to be within the probable li- mits which analogy, in the absence of direct expe- riment, would lead us to assign. HAUY, Lemons de Phys. § 478. VOL. I. T 390. There 290 OUTLINES OF NATURAL PHILOSOPHY. 390. There is perhaps, in the economy of na- ture, no contrivance more wonderful, than that by which things apparently so little susceptible of pre- cision as the impulses communicated to an elastic fluid, become the means of conveying to the mind such a multitude of distinct impressions as it re- ceives through the ear ; the finest modulations of harmony, and the nicest distinctions of articulate language. 391. An Echo is a repetition of sounds produced by the reflection of the aerial pulses that convey sound to the ear. A wall, a rock, a grove of trees, may be so placed, as to cause an echo. That an echo may return one syllable as soon as it is pronounced, the reflecting surface should be 80 or 90 feet distant ; for a dissyl- labic echo, 170 feet, &c. The sound, whether di- rect or reflected, appears to proceed nearly at the rate of 1142 feet in a second. 392. The Speaking Trumpet is an instrument intended to transmit the sound of the voice in some particular direction, to a greater distance than it would otherwise reach. The best form of a speaking-trumpet, is found to be that of a hollow cone, with a mouth-piece at the narrow PNEUMATICS. 291 narrow end, to receive the lips, and confine the voice of the speaker. The beat of a watch may be heard to twice the dis- tance through a speaking-trumpet, that it can be heard at without one. This experiment is said to succeed equally whether the trumpet is cylindrical or conical. See the Edinburgh Encyclopaedia, ar- ticle Acoustics, Part n. Sect. ii. The instrument seems to produce its effect, by preventing the im- dulae generated in the air, from diffusing themselves all round, by which means they are longer subjected to an impulse in the same direction. SECT. III. AIR, AS THE VEHICLE OF HtEAT AND MOISTURE. 393. JL HE Earth and the Atmosphere, taken generally, receive at all times nearly the same quan- tity of heat and light from the sun. As the whole of one side of the earth is constantly turned to the sun, the only difference in the quan- tity of heat and light which the earth receives in a given time, must arise from the changes which take place in its distance from the sun at different sea- T 2 sons 292 OUTLINES OF NATURAL PHILOSOPHY. sons of the year. In consequence of these changes, the quantities of light and heat received by the earth are not proportional to the times, but to the angles described by the earth round the sun in those times. These variations, however, are but inconsiderable, and as they are annual, they do not produce any ine- quality in the whole heat or light of one year compa- red with those of another. 394. Though the earth is thus receiving heat continually, and nearly at the same rate, its ave- rage temperature appears to remain invariable ; as much heat as comes from the solar rays, flying off constantly into the space, whether empty or occupied by subtle matter, which surrounds the earth. 395. While the general temperature of the earth remains invariable, the distribution of heat over its surface is extremely unequal, being different in different places, and in the same place subject to variations, both regular and irregular. 396. The causes which determine the distribu- tion of heat over the earth's surface, are either the direct influence of the solar rays, or the communi- cation of heat by the air from one part of the earth's surface to another. 397. The PNEUMATICS. 293 397. The first of these depends on the latitude of the place, or its distance from the equator, by which the intensity of the heat and light from the sun, and also the length of the day, are determined for different seasons of the year. The intensity of the sun's rays, when they strike on any plane, is as the quantity that falls on a given space, or as the sine of the sun's elevation above the plane. The nearer the sun is to the zenith of any place, at a given moment, the greater the intensity of heat produced by his rays. The heat for an entire day, depends also on the length of the day, and as the day is longer where the dis- tance from the zenith is greater, the inequality in the distribution of heat arising from the one of these causes, compensates that proceeding from the other, and brings their combined effects nearer to an equa- lity than might be imagined. FONT AN A has shewn, that the heat of the day of the summer solstice at Pavia, is greater than the heat of the same day at Petersburgh, in a ratio not greater than that of 63° to 62°, though the latitude of the former be 45° 11', and of the latter 59° 56'. The same author finds, that when the sun's declination exceeds 18°, or from about the 10th of May to the 30th of July, the heat in twenty-four hours proceed- ing from the sun's rays is greater at the north pole than at the equator. GREG. FONTANA, Disquisi- tiones Physico-Mathematicce, lma & 2da. The OUTLINES OF NATURAL PHILOSOPHY. The distribution of heat, therefore, if only the direct influence of the sun were to act, would be very diffe- rent from that which takes place in nature. The intensity of the solar heat is less in low eleva- tions than is supposed in these calculations, on ac- count of the rays coming through a large mass of air, and of air more loaded with vapour, so that a great quantity of them is intercepted. BOUGUER, Traitc cTOptique^ sur la Gradation de la Lumiere, Liv. in. Sect. iv. 398. The effects of the direct influence of the sun, are greatly modified by the transportation of the temperature of one region into another, in con- sequence of that disturbance in the equilibrium of the atmosphere which the action of those rays ne- cessarily produces. In order that there may be an equilibrium in a fluid like the air, every stratum of air that is level or ho- rizontal all round, ought to be of the same density. It ought, therefore, also to be everywhere of the same temperature, which, not being the case, the constant motion of the air is the necessary conse- quence of heat being unequally distributed. The columns of air that are lighter, are displaced hy those that are heavier, and hence a general tenden- cy in the air to move from the poles toward the equator. This general tendency, which is calcula- ted to moderate the extremes of temperature, is al- so greatly modified by local circumstances. 399. The PNEUMATICS. 295 399. The sea is preserved of a moderate tempe- rature by the statical principle which makes the heavier columns of a fluid displace the lighter. A more uniform temperature is thus given to the sea, which communicates itself to the air incumbent on it, and to that on every side. 400. Conversely, the effect of great and unbro ken continents, is favourable to the extremes of heat or of cold. The constitution of the surface may' tend to increase and sometimes to diminish this effect. High moun- tains especially, if covered with snow, may enforce the rigour of a cold climate, or temper the heats of a warm one. Forests tend to increase the cold, by preventing the sun's rays from striking on the ground. Evapora- tion produces cold ; and marshes and lakes are there- fore favourable to the severity of the weather. The congelation of water produces heat, and moderates the cold ; the melting of ice, on the other hand, in- creases the capacity for heat, and so produces cold. 401. Height above the level of the sea, causes a diminution of heat at the constant rate of 1° for 270 feet nearly, when not far from the surface of the earth. It 296 OUTLINES OF NATURAL PHILOSOPHY. It has already been remarked, that this decrease seems to be somewhat slower as we ascend, but not very considerably, as far as our observations have ex- tended. 402. The combination of these causes gives to every place a mean temperature, which remains always nearly the same, and which decreases from the equator to either pole, according to a law that has been determined by observation. 403. Let t be the mean temperature of any pa- rallel of which the latitude is L, M the mean tem- perature of the parallel of 45°, and M + E the mean temperature of the equator ; then is t = M + E cos 2 L. In this formula, M=r58°, and E=27°. When % L > 90, cos 2 L is negative. This theorem was first given by MAYER, Opera ine- dita, vol. i. p. 4., &c. Also KIBWAN, Estimate of the temperature of different Latitudes, p. 18. From this a geometrical construction, for finding the mean temperature, may be readily deduced. In the line AC (fig. 27.), divided into equal parts, num- bered from A, so as to represent the scale of a ther- mometer, let AC = 85, and AB — 58. From the centre PNEUMATICS. 297 centre B, with the distance BC or 27, describe the semicircle CGH ; take the arch CG equal to the double latitude of any parallel ; and from G drasv GO perpendicular to AC ; then is AO the mean tem- perature of that parallel, according to FAHRENHEIT'S scale. The mean temperatures thus found, agree very well with observation. Springs, in which the water does not considerably change its heat from one sea- son of the year to another, afford an expeditious and accurate way of ascertaining the mean tempe- rature. 404. If the place is at any height H above the TT level of the sea, t = M — — - + E cos 2! L. H is understood to be expressed in English feet. 405. On ascending into the atmosphere, at a cer- tain height in every latitude, a point is found where it always freezes, or where it freezes more than it thaws, so that the mean temperature is below 321°. The curve joining all these points, from the equa- tor to the pole, is called the line of perpetual con- gelation. The equation to it will be found, by TT making 32! = M -- - -h E cos 2 L. This 298 OUTLINES OF NATURAL PHILOSOPHY. This line at the equator is elevated 15577 feet above the level of the sea ; so that for determining n we n KPJ77 have 32 = 58 — — — + 37 or n _ 394 w Thus, H = 7642 + 7933 cos 2 L, and this seems nearly to agree with actual observation. Professor LESLIE has given a different equation, found- ed on the law of the diminution of heat mentioned at § 339. In the table calculated from it, heights come out rather below what observation requires. Elements of Geometry, 2d edit. p. 495. On great elevations, the variations of temperature from day to night, and from summer to winter, ap- pear to be less than at the surface. SAUSSURE, torn. iv. p. 406. The temperature of the latter end of April is observed, at least in the temperate zone, to be nearly the mean temperature of the year. From that time the heat increases, and is at its maxi- mum about the 21st of July ; it goes on decreasing from that time till it come to the mean in the end of October, and it passes from thence to the great- est cold about the 21st of January. All these vi- cissitudes may be nearly represented by the for- mula in which G is the mean temperature for the given place, F a constant co-efficient to be found by observation, X the mean longitude of the sun computed from the first of Aries, for any day of the year, the mean temperature of which is y. Thus, PNEUMATICS. 299 Thus, y = G 4- F sin (A — 30°). This supposes the mean heat to take place about a month after the equinox, and the extremes about a month after the solstices. In this latitude we may suppose F = 15° . 407. From this formula, compared with the two former, one theorem may be deduced, including the effects of the latitude, the elevation above the sur- face, and the season of the year, viz. n(A — 30°); 270 Where y is the mean temperature for any day of the year, in the latitude L, and at the elevation H. 408. The formula for the mean temperature, as laid down above, would probably hold over the whole globe, if it were every where covered by the ocean ; but it agrees strictly only with the At- lantic Ocean, and the western part of the Old Con- tinent. The editor of MAYER'S Posthumous Works observed, that his theorem applied accurately only to the 3 portion 300 OUTLINES OF NATURAL PHILOSOPHY. portion of the globe contained between the parallels of Stockholm and of the Cape of Good Hope, and between the meridians of Stockholm and Mexico. This does not seem quite exact ; no part of North America having its mean temperature the same with that of places of the same latitude in Europe. It would require at least 10° to be taken from M, to adapt the formula of the last article to the New Con- tinent. See KIR WAN'S Estimate of the temperature of different Latitudes^ p. 15. 409- As we go eastward from the shores of the Atlantic, the mean temperature of any parallel be- comes lower, at a rate that may perhaps, for the north part of the temperate zone, be estimated at a degree for 150 miles. At St Petersburgh, lat. 59° 56', about 750 miles' from what may be accounted the shores of the Atlantic, the temperature is 5° 5' below the standard. The medium temperature of January is no more than 10°. By computation from the formula above, it ought to be greater than 3£°. The winter lasts from October to April, and the cold is sometimes as great as the freezing point of mercury, or — 39° • From a mean of several years, the mean of the winter cold is — 25°. KIRWAN, ibid. p. 61. It was at Krasnojark, lat. 56° 30', long. 93° E. that mercury was first known to freeze by natural cold. If PNEUMATICS. 301 If we were to begin where any parallel intersects the shore of the Atlantic, and draw on the map a line along which the mean temperature should be con- stantly the same as at the first-mentioned point, it would incline greatly to the south. The point, for instance, in the meridian of Petersburgh, which has the same temperature with the standard belonging to the parallel of that city, is about 5° south of it, or in the latitude of 54° 30' nearly. At Irkutz, latitude 52° 15', longitude 105° east, the mean temperature from October to April has been known to be as low as — 6°. 8, a temperature which for severity and duration exceeds any thing that has yet been observed elsewhere. 410. This increase of the severity of the winter, and the consequent diminution of the mean tem- perature, on going eastward, holds in all the lati- tudes north of the parallel of 30° ; but the diminu- tion is slower as we approach that parallel ; to the south of 30° the mean heat increases on retiring from the ocean. This diminution takes place all the way to the shores of the Pacific, or very near them. The climate of Pekin is vastly more severe than that of the same parallel (39° 54') in Europe. 411. In all the part of the New Continent which is to the north of the Tropic of Cancer, the 302 OUTLINES OF NATURAL PHILOSOPHY. the mean temperature is much lower, and the se- verity of the winter much greater than in the same latitudes in Europe. At Prince of Wales' Fort, Hudson's Bay, lat. 59°, long. 92° west, the mean temperature is 20° under the standard ; at Nain in Labrador 16° ; at Cam- bridge in New England (lat. 42° 25',) 10 degrees. Mercury has been supposed to be frozen by the na- tural cold as far south as Quebec, lat. 47°. A very low mean temperature, and extreme cold in winter, are characteristic of the climate of North America. 412. In the higher latitudes of the Southern Hemisphere, the temperature is lower than in the same latitudes of the Northern Hemisphere. FORSTER describes a small island on the coast of South Georgia, latitude 54° south, which, in the middle of summer, was covered almost entirely with frozen snow, to the depth of several fa- thoms. The South Pole is surrounded to the distance of 18 or 19 degrees with a barrier of solid ice, through which even the skill and intrepidity of Captain COOK could not force a passage. It is known also, that detached masses of ice float down in that hemisphere as low as the latitude of 46°, PNEUMATICS. 303 46°. The cause of this extraordinary cold is by no means sufficiently understood. 413. The varieties of temperature to be met with on the surface of the earth, appear to be con- fined between the limits of + 100 and — 40°. No degree of natural cold much below — 40* has ever been observed, even when thermometers have been, employed containing a fluid not liable to congelation. Sir CHARLES BLAGDEN on the Congelation of Mer- cury, p. 61. The heat of 100° is rare, but a heat approaching to 90°, is found in the summer of most countries within the limits of the temperate zones. There is hardly any climate, even in the frigid zone, where a temperature between 60° and 50° is not oc- casionally experienced. The greatest heat is much less above the mean tem- perature than the greatest cold is below it. The mean temperature for the whole surface may be ta- ken at 58° ; the greatest summer heat is only 4£* above this; the greatest winter cold is 98° un- der it. 414. There is reason to think that the climates of Europe were more severe in ancient times than they are at present, and that the change which has taken place may with great probability be ascribed to 304 OUTLINES OF NATURAL PHILOSOPHY. to the better and more extensive cultivation of the ground. CESAR says, that the vine could not be cultivated in Gaul on account of the severity of the winter. The rein-deer was then an inhabitant of the Pyrenees. The Tiber was sometimes frozen over, and the ground about Rome covered with snow for several weeks together. See HUME cm the Populousness of Ancient Nations, Essays., vol. i. p. 451. edit. 1772. DAINES BARRINGTON, Phil. Trans, vol. Iviii. (1768} p. 58., &c. Cultivation may improve a climate, lmo, By the draining of marshes, and lessening the evaporation, which is so great a cause of cold ; 2do, By turning up the soil, and exposing it to the rays of the sun ; Stio, By thinning or cutting down forests, which, by their shade, prevented the penetration of the sun^s rays. The improvement which is continually taking place in the climate of North America, proves that the power of man extends to the phe- nomena, which, from the magnitude and variety of their causes, appear most beyond its reach. At Guiana in South America, the rainy season has been shortened by the clearing of the country, and the warmth has been greatly increased. It thun- ders continually in the woods; rarely in the culti- vated parts. BUFFON, Sup. torn. ix. p. 346. oc- tavo. 3 Wind. PNEUMATICS. 305 Wind. 415. The principal cause of those currents of air to which we give the name of Winds, is the dis- turbance of the equilibrium of the atmosphere by the unequal distribution of heat. In order that an equilibrium may take place in an elastic fluid, circumfused about a solid, to which it gravitates, every level stratum of the fluid, that is, every stratum which, when continued round, cuts the directions of gravity every where at right angles, should be of the same density, and therefore of the same temperature. As this is not the case, the equilibrium of the atmosphere is inconsistent with the actual distribution of heat on the eartrTs sur- face. The general tendency, in such circumstances, is for the heavier columns to displace the lighter, and for the air at the surface to move from the Poles to- ward the Equator. The only supply for the air thus constantly abstracted from the higher latitudes, must be produced by a counter-current in the up- per regions of the atmosphere, carrying back the air from the Equator toward the Poles. The quan- tity of air transported by these opposite currents, is so nearly equal, that the average weight of the air, as measured by the barometer, is the same in all places of the earth. VOL. I. U If 306 OUTLINES OP NATURAL PHILOSOPHY. If the surface of the earth were wholly covered with water, so that there were no part of it more dispo- sed than another to obstruct the motion of the air, or no part which had a greater capacity than ano- ther, of acquiring or communicating heat, the air would probably circulate continually from the Poles to the Equator, and back again, without any irregu- larity whatsoever. 416. In consequence of the rotation of the earth on its axis, another motion is combined with that of the currents just described. The air, which is constantly moving from points where the earth's motion on its axis is slower, to those where it is quicker, cannot have precisely the same motion eastward with the part of the surface over which it is passing, and therefore must, relatively to that surface, describe a curve, having its convexity turn- ed to the east. The two currents, therefore, from the opposite hemispheres, when they meet toward the middle of the earth, having each acquired an apparent motion westward ; and as their opposite motions from south and north must destroy one another, nothing will remain but this motion west- ward, by which they will go on together, and form a wind blowing directly from the east. This is the cause of the Trade Wind, which (with certain exceptions) blows continually between the Tropicsy PNEUMATICS. 307 Tropics, or rather between 30° on the one side of the Equator and 3C° on the other. The Trade Wind declines somewhat from due east toward the parallel to which the sun is vertical at different seasons of the year. As the sun ap- proaches the southern tropic, the Trade Wind is directed somewhat to the south ; and as he ap- proaches the northern, somewhat to the north. The cause usually assigned for the Trade Wind is the constant motion, toward the west, of the spot to which the sun is vertical, and where of course the rarefaction is greatest. This, it is supposed, draws along with it the air from the east. This, however, is by no means a satisfactory explanation, and it seems certain, that if the Trade Wind were produced in that way, it must blow with great force, instead of being a gentle breeze, moving at the rate of seven or eight miles an hour. The opinion that the Trade Wind is produced by the air, in its motion southward, falling back toward the west, is mentioned, but rejected by H ALLEY. It has since been espoused by FRANKLIN and LA PLACE, and is, on the whole, less objectionable than any other. The matter is here stated somewhat differently from what is done by those authors, particularly the ef- fect of the currents from the opposite hemispheres, U2 in 308 OUTLINES OF NATURAL PHILOSOPHY. in determining the motion to be wholly from the east. 417. The attractions of the sun and moon have sometimes been considered as among the general causes of the winds. They have no doubt a ten- dency to produce in the atmosphere an undulation backwards and forwards, like the tides which they cause in the ocean. It does not appear, however, that they could produce any continued progressive motion of the air, similar to that of the Trade Winds. Their effects also, are too minute to be perceived, amid the action of so many more power- ful causes. D'ALEMBERT in his Recherches sur les Causes Ge- nerales des Vents, has treated of the forces of the sun and moon to produce currents in the atmo- sphere. His essay is more remarkable for the re- source and ingenuity it displays in the management of the calculus, than for the physical conclusions to which it leads. 418. The superior current above described, re- stores the air carried from the higher latitudes to the lower, with such a degree of equality, that the average weight of the atmosphere, as measured by the barometer, is nearly the same in all climates. This restoration is, however, subject to great local and temporary irregularities, from the different degrees PNEUMATICS, 309 degrees of resistance which the air meets with in passing over the surface, and the different capaci- ties of that surface for receiving and communica^ ting heat. The motion of the inferior and superior currents, may be seen exemplified on opening a door, between two apartments of different temperatures. The flame of a candle near the ground, will shew the stream that sets from the colder room to the war- mer ; near the top it will indicate a s£ream in the op- posite direction. As the average quantities of the air carried by these opposite currents are equal, the surface that sepa- rates them is probably not far different from that at which the barometer would stand at 15 inches, or half its medium height at the surface. If we suppose the mean temperature is 3£°, this ele- vation will be found — 3010 fathoms, or 18,060 feet = 3.425 miles, not so high as the summits of the Cordilleras. The upper stream in each hemisphere being directed to one point, the pole of that hemisphere, there must arise a considerable condensation, and acceleration of the air, as the currents approach that point, and hence the causes of irregular winds must be increased on approaching the poles. The general direction of the upper current must be to the westward, for the same reason that that of the lower was toward the east. 419. The "310 OUTLINES OF NATURAL PHILOSOPHY. 419. The Trade Wind itself is subject to cer- tain irregularities. As the sun advances into the Northern Hemisphere, the Trade Wind becomes irregular ; and about the middle of April, in all the tract between Africa and the peninsula of In- dia, and much farther to the east, it changes from north-east to south-west, and continues to blow in that direction, till the sun returns to the Southern Hemisphere. ( The cause of this change is difficult to be assigned. It seems probable, that by the sun's entry into the Northern Hemisphere, he communicates great heat to the sandy deserts of Africa, which lie to the west or south-west of the seas just mentioned. The great heat acquired by the sand of those deserts, produces a rarefaction in the columns of air incumbent on them, and consequently a tendency, in the columns that are near them, and more moderately heated, to flow in and displace the heated air. The air of the Atlantic is most likely to do this, and in passing over Nigritia, &c., to acquire a velocity that carries it on eastward through the Indian Ocean. Perhaps the direction of the eastern coast of Africa has a share in producing this effect. The advance of the sun into the Northern Hemisphere, would naturally give the Trade Wind an inclination to the north, when, meeting obliquely with the coast of Africa, and the elevated tracts in the interior, it is inrned yet more to the north. Here, again, recei- 2 ving PNEUMATICS. 311 ving an impulse from the great body of air de- scending from the north, it goes round wholly to the west, and becomes, as it were, an eddy of the Trade Wind. These periodical winds are called Monsoons. The change, or the setting in of the monsoons, does not happen all at once. In some places the shift- ing of the wind is accompanied with calms ; in others with variable winds, heavy rains, thunder, and vio- lent storms. 420. The tract from the parallel of 30° to the Pole, in each hemisphere, is the region of variable winds ; and their unsteadiness and violence seem to increase, the nearer they approach the Polar Circles. The irregularity of winds proceeds from inequalities in the motion of the general currents above mentioned, and from a variety of local causes ; also from the chemical changes that are carried on in the air, such as the solution and precipitation of moisture, and the action of the electric fluid on the different gases that compose the atmosphere. 421. Sudden and strong gales of wind appear almost always to arise from a diminution of the weight of the air in the tract where the wind pre- vails, and are accompanied, or preceded, by a fall of the barometer. The OUTLINES OF NATURAL PHILOSOPHY. The sudden sinking of the barometer almost always indicates a gale of wind, though a gale that is some- times at a considerable distance. When the baro- meter begins to rise, it is a symptom that the gale has reached its height ; and though it may still con- tinue to blow for a long time, it is usually with de- creasing violence. 422. Notwithstanding these irregularities, there is in most countries a tendency to periodical winds, more or less remarkable, according to the steadiness of the climate. Even with us, where an insular situation, with a great Continent on one side, and a great Ocean on the other, unites all the causes of a variable climate, the East wind usually prevails in the spring, from the vernal equinox to the summer solstice, and beyond it ; during the rest of the year, the Westerly winds prevail, though not without frequent incursions of the east, by which our most unpleasant weather is always produced. The Etesian, or northerly wind, prevails very much in summer all over Europe. PLINY describes it as blowing regularly in Italy, for forty days after the summer solstice, lib. ii. cap. 47. It is part of the great current that carries the atmosphere of the higher latitudes down to the tropical regions. 423. The PNEUMATICS. 313 423. The velocity of the wind varies from one that is hardly sensible, to one of 100 miles in an hour. The force of the wind, is as the density of the air multiplied into the square of its velocity. SMEA- TON has given a table of its force, expressed by its pressure in avoirdupois pounds, on a plane of one foot square directly opposed to it. Phil. Trans. vol. li. p. 165., &c. Also CAVALLO, vol. ii. p. 287. A gentle light breeze is there estimated as moving at the rate of four or five miles an hour, and pressing with a force of about two ounces : A brisk but pleasant gale from ten to fifteen miles ; with a force from half a pound to a pound : A high wind goes at the rate of thirty or thirty-five miles, and pres- ses with the force of five or six pounds : A hurri- cane, that tears up trees, and blows down houses, has a velocity of one hundred miles an hour, and a force of forty-nine pounds on a square foot. The velocity of the wind may be estimated by the ve- locity of clouds, or of light bodies carried by it. Rain. 314 OUTLINJES OF NATURAL PHILOSOPHY. Rain. 424. The vapour that rises from water uniting itself to the air, ascends into the higher regions of the atmosphere, and is carried by the winds to great distances. It cannot be doubted, that the humidity raised in this manner is chemically dissolved in the air. The humidity does not lessen, but increases the trans- parency of the air, and cannot be withdrawn from it but by substances which attract it powerfully. 425. The power of air to dissolve humidity in- creases in a greater ratio than that in which its temperature increases. It appears from SAUSSURE'S experiments, that while the temperature increases in arithmetical progres- sion, the humidity which the air is able to contain increases in geometric progression. A cubic foot of air, of the temperature 66°, is able to hold in solution 11 or 12 grains of water : the air itself weighs 570 grains : so that air of the temperature 66°, dissolves about a 50th part of its own weight of water. Essai sur THygrometrie, xi. chap. x. p. 167., &c. 426. Hence, PNEUMATICS. 315 426. Hence, if two portions of air, of different temperatures, and both saturated with humidity, be mixed together, a precipitation of humidity must necessarily take place. If the abscissae AB, AB' (fig. 28.) be the tempera- tures of two equal portions of air, and the or- dinates BC, B'C' represent the quantities of humi- dity contained in them, the curve CFC', which re- presents the dissolving power, corresponding to dif- ferent temperatures, will be convex toward AB, and will, so far as experiment has shewn, be a lo- garithmic curve. Now if CC' be joined, and if BB' be bisected in D, the perpendicular DE will represent the humidity contained in a given weight of the mixture of the two portions of air, AD the temperature of the mixture, and DF the quantity of humidity which the air of the temperature AD is able to contain. The quantity precipitated from each portion of the air is therefore EF ; so that the whole quantity precipitated is 2 EF. In general, if T and t are the temperatures of two equal portions of air, H and h the humidity con- tained in them when saturated, the quantity of hu- midity precipitated from the mixture will be, H + h — % V H h. 427. If, therefore, large portions of the atmo- sphere, of different temperatures, and saturated, or nearly 316 OUTLINES OF NATURAL PHILOSOPHY. nearly saturated, with humidity, be driven against one another hy contrary winds, the consequence must be a precipitation of humidity, or the forma-, tion of clouds. This simple and ingenious theory of the formation of clouds, was a discovery of the late Dr HUT- TON. See Edinburgh Transactions, vol. i. p. 41. It is not necessary to the theory, that the curve which terminates the perpendiculars BC, B'C', &c. should be a logarithmic curve ; it is sufficient that it be a curve convex towards the axis AB. Of this Dr HUTTON had satisfied himself by the obser- vation of natural phenomena. If the line termi- nating the perpendiculars was straight, and much more were it concave, no precipitation could ever take place. Professor LESLIE has shewn, that the collision of op- posite currents of air of different temperatures, may furnish a supply sufficient for the production of the heaviest rain. Experiments on Heat and Moisture^ p. 130. 428. The clouds thus formed, are not disposed equally over the whole atmosphere, but occupy a peculiar region, elevated at an average between two and three miles above the earth. The mixture of different portions of air is likely to take place most frequently when the two opposite 3 currents PNEUMATICS. 317 currents already mentioned come in contact with one another. This is at the height of 18,000 feet and upwards, which agrees very well with the me- dium height of the clouds. 429. The clouds thus formed, have their parti- cles united into larger masses or drops by different causes, such as the mutual attraction of aqueous particles, the force of the wind, or the operation of electricity, and so fall down in rain on the surface of the earth. The intimate connection between rain and the exist- ence of different currents of air, is evident from ma- ny appearances. 1. When the Trade Winds blow uniformly, hardly any rain falls ; but when the monsoon changes, heavy falls of rain seldom fail to take place. 2. In the tropical climates, the rainy season is always on the sun's approach to the zenith, at which time also the winds are most variable. 3. There are some spots of continual rain, which seem to be where opposite streams of air constantly meet one another. 4. There are several tracts on the earth's surface, where it hardly ever rains. These are usually far inland, and are extensive plains, without any of those 318 OUTLINES OF NATURAL PHILOSOPHY". those inequalities of surface that promote the mix- ture of air. 5. In the midst of such deserts, where hills oc- cur, moisture is precipitated, sometimes in the form of rain, but most frequently of dew, so that springs of fresh water arise, and great fertility is produ- ced. 6. There is in our climate hardly any instance of rain without a change of wind, and very rarely a change of wind without rain in a greater or smaller quan- tity. 7. The lowness of the mercury in the barometer is a sign of rain. It is a certain indication of the sub- version of the equilibrium of the atmosphere, and makes it probable, that before the equilibrium is restored, winds from different quarters, and of dif- ferent temperatures, must come into collision with one another. 430. The Hygrometer is an instrument intend- ed to measure the quantity of humidity contained in the air at any time. The power of evaporation to produce cold has been very happily applied by Professor LESLIE, to the construction of an instru- ment of this kind. 431. The vapour raised up into the air, is of a quantity sufficient to afford all the rain that falls, or PNEUMATICS. 319 or to supply all the springs, and of consequence all the rivers derived from them, on the surface of the earth. Dr HALLEY shewed, that the evaporation from the sea alone is a sufficient supply for all the water that the rivers carry into it. His calculation was founded on a very complex view of the subject, and liable to several objections. BUFFON took a more simple view of the matter, by selecting one of those lakes that sends out no stream to the ocean, and shewing that the probable evaporation from the sur- face of the lake is equal to all the water carried in- to it. It may be also satisfactory to shew, that all the wa- ter annually emptied by a river into the sea, is less than the rain, which falls on the surface drained by it. Thus, according to Dr HALLEY'S computa- tion, the water which the Thames carries down through Kingston Bridge, to which the tide does not reach, is 25344000 cubic yards, or 684288000 cubic feet per day, which gives 249765120000 cu- bic feet for the quantity of water which the Thames discharges annually into the sea. Now, the surface drained by the Thames and its branches, appears to be about equal to a circle of 40 miles radius; that is, nearly to 5036 square miles, or 140395622400 square feet. But if we suppose that the depth of rain which falls annually, at an ave- rage, over this surface, is two feet, we have a quan- tity which exceeds that carried down by the Thames by 320 OUTLINES OF NATURAL PHILOSOPHY. by 31026124800, or nearly an eighth part of the whole, which is therefore left to be taken up by the evaporation. MARIOTTE has made a similar computation for the Seine, in his Traite de Mouvemeni des Eaux. 432. The disputes that prevailed so long con- cerning the origin of fountains and rivers, chiefly arose from the difficulty of conceiving, how a pre- carious and accidental supply could be rendered equal to a regular and great expenditure. The opinions of the ancients, concerning the origin of fountains, are to be found in SENECA, Nat. Quocst. lib. in. See also VARENIUS, sect. iv. chap. xvi. prop. 5. 433. The quantity of rain that falls in different places in the same year, and in the same place in different years, is extremely various ; and even in the temperate zone runs between the extremes of 18 and 100 inches. In places not very distant from one another, the dif- ference in the quantity of rain is often very great. The neighbourhood of the sea on one hand, and mountains on the other, is most favourable to the production of rain ; from the first is derived a hu- mid PNEUMATICS. mid atmosphere, and from the second the preva- lence of winds of different temperatures. When the rain exceeds 25 inches a-year, the climate is to be accounted moist. In the year 1809, the rain that fell at Dalkeith, near Edinburgh, was 28.5 ; at Glasgow 25.1 ; at Largs, at the mouth of the Clyde, 40.6; and at Gordon Castle, on the east coast of Scotland, 24.5: At London, in the same year, 20.7 ; in the preceding year, the fall was but 18.8. In some places of this island, such as Kendal and Keswick, the rain amounts to 64 and 68 inches in a year. 434. The disposition of the rocks in strata, con- tributes much to the collecting of the waters un- der the surface, and the Conveying them without waste, as if in close pipes, till they are united in fountains, lakes, rivers, &c. BERNARD PALLASSY seems to have been the first who made this remark ; he even brought it to the test of experiment. 435. The region of the air in which the preci- pitation of humidity takes place, is one where the temperature is frequently below freezing ; the fro- zen particles then uniting, as in the case of melt- ed vapour, form flakes of Snow, which reach the ground in that state, when the cold of freezing con- tinues down to the surface. VOL. I. X When OUTLINES OF NATURAL PHILOSOPHY. When the aqueous particles first form a drop, and are afterwards frozen in their descent, they be- come Hail, which is sometimes found crystallised with some degree of regularity. The whiteness and opacity of the hail, is probably owing to the congelation being performed where the air is very rare. Professor LESLIE has remarked, in the cu- rious experiments he has made on the production of ice by evaporation, in a receiver where the air was considerably rarefied, that the ice is more porous, and less transparent, than that which is formed under the ordinary pressure of the atmo- sphere. 436. Dew is a precipitation of humidity from the lower strata of the atmosphere, which does not disturb the transparency of the air, and to which the mixture of different streams of air does not seem necessary. When air containing humidity cools below a certain point, it must begin to deposite its humidity. It is in this way that dew is formed in warm wea- ther, when, on the sun's going down, the heat of the air at the surface is greatly diminished. The temperature at which the deposition of dew takes place, is therefore an indication of the quantity of humidity contained in the air. BRUCE has remark- ed, that during the dry season no dew falls in Egypt, but that it is usually observed in the Delta five or six days before the inundation. 437. Besides PNEUMATICS. 323 437- Besides these deposites from the atmosphere, which are all of one substance, there are others that cannot he traced to the same origin. These are the Stones which have often heen said, and have of late years been ascertained beyond all doubt, to fall down from the air ; on examination, they are found to be composed in a manner similar to one another, but in many respects unlike to any thing that is found either on the surface or in the bowels of the earth. For an account of Meteoric Stones, see HOWARD, Phil. Trans. 1802. IZARN, Lithologlc Atmosplie- rique, Paris, 8VO, 1803 : and CAVALLO, vol. iv. p. 372., &c. In the absence of all analogous appearances, it is per- haps unphilosophical to offer any explanation of the Meteoric Stones. We would only suggest as a mere possibility, that gaseous substances may be thrown up into the air, from the numerous volca- noes on the earth's surface, and may carry with them certain elements of metallic and stony bodies from the mineral regions ; these, while floating about in the atmosphere, may sometimes be col- lected in considerable quantities into one place, where, being subjected to electric or galvanic ac- tion, they are united into a solid mass. The fact, that many of these stones have fallen during thunder storms, seems to limit the place of their formation to the atmosphere of the earth. END OF VOLUME FIRST. DIRECTIONS TO THE BINDER, VOL. I. has Three Plates, to be placed at the end, in the order of the Figures. Plate I. contains from Fig. 1. to Fig. 10, II. Fig. 11. to Fig. 19. III. Fig. 20. to Fig. 28. 21. -$ B A flff 28. B O