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 UNIVERSITY OF CALIFORNIA. 
 
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 M'l cha. Acts. 
 
 
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 THE 
 
 CAMBRIDGE COURSE 
 
 ELEMENTARY NATURAL PHILOSOPHY; 
 
 BEING THE DEMONSTRATIONS OF THE PROPOSITIONS IN 
 
 MECHANICS AND HYDROSTATICS 
 
 [N WHICH THOSE PERSONS WHO ARE NOT CANDIDATES FOR 
 HONOURS ARE EXAMINED FOR THE DEGREE OF B.A. 
 
 THE THIRD EDITION. 
 
 0^t?2^ 
 
 OF TITS 
 
 CAMBRIDGE : 
 
 US 17 BE SIT 7 
 
 PRINTED BY JOHN W. PARKER, UNIVERSITY PRINTER. 
 
 THOMAS STEVENSON, CAMBRIDGE; 
 A. H. BA1LY AND CO., LONDON. 
 
 M.DCCC.XLIV. 
 

 y/ u-s/S-/ 
 
ADVERTISEMENT. 
 
 The subjects in which those persons who are not 
 Candidates for Honours are examined for the degree 
 of B.A. are, 
 
 I. The Acts of the Apostles in the original Greek. 
 
 II. One Greek Classic ; appointed in the last week 
 of the Lent Term in every year for the year next 
 but one following. Thus the Classical subjects 
 for the Examinations of 1844- were apppointed in 
 the last week of the Lent Term of 1842. 
 
 III. One Latin Classic ; appointed in the same man- 
 ner as the last subject. 
 
 IV. Paley's Moral Philosophy. 
 
 V. Such Mathematical Subjects as are contained in 
 the following Schedule*. 
 
 * The Examiners are required to publish the names of the persons 
 whom their respective Colleges permit to offer themselves for Examination 
 at the Bachelors' Commencement, in a list, which is to be arranged in 
 alphabetical order, and separated into two Divisions. The distribution 
 of the different subjects, and the times of Examination, are fixed by the 
 following Table. 
 
 
 Div. 
 
 Hours of Examination. 
 9 to 12. 
 
 Div. 
 
 12* to 3^. 
 
 Wednesday .. 
 Thursday .... 
 
 Friday 
 
 Saturday 
 
 Monday 
 
 Tuesday 
 
 Wednesday .. 
 
 1 
 1 
 
 1 
 
 1 
 1 
 1 
 
 Acts of the Apostles 
 
 2 
 2 
 2 
 2 
 
 2 
 
 2 
 
 2 
 
 Greek Subject. 
 Acts of the Apostles. 
 Latin Subject. 
 Euclid. 
 
 Moral Philosophy. 
 C Mechanics and 
 ( Hydrostatics. 
 Arithmetic and Algebra. 
 
 
 
 c Mechanics and i 
 t Hydrostatics \ '" 
 Moral Philosophy 
 
 Arithmetic and Algebra.. 
 
Schedule of Mathematical Subjects of Examination, for the 
 Degree of B.A., of Persons not Candidates for Honours. 
 
 Arithmetic. 
 
 Addition, subtraction, multiplication, division, reduction, rule of three ; 
 the same rules in vulgar and decimal fractions ; practice, simple and 
 compound interest, discount, extraction of square and cube roots, 
 duodecimals. 
 
 Algebra. 
 
 1. Definitions and explanation of algebraical signs and terms.. 
 
 2. Addition, subtraction, multiplication, and division of simple algebraical 
 quantities and simple algebraical fractions. 
 
 3. Algebraical definitions of ratio and proportion. 
 
 4. If a : b :: c : d then ad = bc, and the converse : 
 
 If a : b : 
 
 also b : a : 
 and a : c : 
 and a + b : 
 
 : c : 
 :d : 
 : b: 
 
 b :: 
 
 d then f 
 c + c/ : d. 
 
 If a : b :: c 
 
 and c : d :: e 
 
 then a : b :: e 
 
 : <?, 
 :/, 
 
 
 If a : b :: e 
 
 and b : e :: d 
 
 id, 
 =7, 
 
 
 then a : e :: c : f. 
 
 7. Geometrical definition of proportion. (Euclid. Book v. Def. 5.) 
 
 8. If quantities be proportional according to the algebraical definitio 
 they are proportional according to the geometrical definition. 
 
 9. Definition of a quantity varying as another, directly or inversely, 
 as two others jointly. 
 
 Euclid. 
 
 Books i. ii. in. ; 
 
 Book vi. Props. 1. 2. 3. 4. 5. 6. 
 
[The remaining portion of the Schedule forms the Table of Contents 
 of this Book.] 
 
 CONTENTS. 
 
 MECHANICS. 
 
 Chapter I. 
 
 ARTS. 
 
 1_15. Definitions of Force, Weight, Quantity of Matter, Density, Mea- 
 sure of Force. 
 
 Chapter II. The Lever. 
 
 17. Definition of Lever. 
 
 18. Axioms. 
 
 19. Prop. 1. A horizontal prism or cylinder of uniform density will pro- 
 duce the same effect by its weight as if it were collected at its middle 
 point. 
 
 20. Prop. 2. If two weights acting perpendicularly on a straight lever 
 on opposite sides of the fulcrum balance each other, they are inversely 
 as their distances from the fulcrum ; and the pressure on the fulcrum is 
 equal to their sum. 
 
 21. Prop. 3. If two forces acting perpendicularly on a straight lever in 
 opposite directions and on the same side of the fulcrum balance each 
 other, they are inversely as their distances from the fulcrum ; and the 
 pressure on the fulcrum is equal to the difference of the forces. 
 
 23. Prop. 4. To explain the different kinds of levers. 
 
 24. Prop. 5. If two forces acting perpendicularly at the extremities of 
 the arms of any lever balance each other, they are inversely as the 
 arms. 
 
 25. Prop. 6. If two forces acting at any angles on the arms of any lever 
 balance each other, they are inversely as the perpendiculars drawn 
 from the fulcrum to the directions in which the forces act. 
 
 26. Prop. 7. If two weights balance each other on a straight lever when 
 it is horizontal, they will balance each other in every position of the 
 lever. 
 
 Chapter III. Composition and Resolution of Forces. 
 
 27. Definition of Component and Resultant Forces 
 
 1-3 
 
6 CONTENTS. 
 
 ARTS. 
 
 com- 
 
 28. Prop. 8. If the adjacent sides of a parallelogram represent the 
 
 ponent forces in direction and magnitude, the diagonal will represent 
 the resultant force in direction and magnitude. 
 
 Prop. 9. If three forces, represented in magnitude and direction by 
 the sides of a triangle, act on a point, they will keep it at rest. 
 
 29. 
 
 Chapter IV. Mechanical Powers. 
 31. Definition of Wheel and Axle. 
 
 31. Prop. 10. There is an equilibrium upon the wheel and axle when the 
 power is to the weight as the radius of the axle to the radius of the 
 wheel. 
 
 32. Definition of Pulley. 
 
 32. Prop. 11. In the single moveable pulley where the strings are parallel, 
 there is an equilibrium when the power is to the weight as 1 to 2. 
 
 33. Prop. 12. In a system in which the same string passes round any 
 number of pulleys and the parts of it between the pulleys are parallel, 
 there is an equilibrium when Power (P) : Weight (W) :: 1 : the 
 number of strings at the lower block. 
 
 34. Prop. 13. In a system in which each pulley hangs by a separate 
 string and the strings are parallel, there is an equilibrium when 
 P . W :: I : that power of 2 whose index is the number of move- 
 able pulleys. 
 
 35. 36. Prop. 14. The weight (W) being on an inclined plane and the 
 force (P) acting parallel to the plane, there is an equilibrium 
 when P : W :: the height of the plane : its length. 
 
 37. Definition of Velocity. 
 
 38. Prop. 15. Assuming that the arcs which subtend equal angles at the 
 centers of two circles are as the radii of the circles, to shew that if P 
 and W balance each other on the wheel and axle, and the whole be put 
 in motion, P : W :: W's velocity : P's velocity. 
 
 39. Prop. 16. To shew that if P and W balance each other in the 
 machines described in Propositions 11, 12, 13 and 14, and the whole be 
 put in motion, P : W :: Ws velocity in the direction of gravity : P's 
 velocity. 
 
 Chapter V. The Center of Gravity. 
 
 40. Definition of Center of Gravity. 
 
 41. Prop. 17. If a body balance itself on a line in all positions, the 
 center of gravity is in that line. 
 
CONTENTS. / 
 
 ARTS. 
 
 42. Prop. 18. To find the center of gravity of two heavy point? ; and 
 to shew that the pressure at the center of gravity is equal to the 
 sum of the weights in all positions. 
 
 43. Prop. 19. To find the center of gravity of any number of heavy 
 points ; and to shew that the pressure at the center of gravity is 
 equal to the sum of the weights in all positions. 
 
 45. Prop. 20. To find the center of gravity of a straight line. 
 
 46. Prop. 21. To find the center of gravity of a triangle. 
 
 47. Prop. 22. When a body is placed on a horizontal plane, it will 
 stand or fall, according as the vertical line, drawn from its center of 
 gravity, falls within or without its base. 
 
 48. Prop. 23. When a body is suspended from a point, it will rest with 
 its center of gravity in the vertical line passing through the point of 
 suspension. 
 
 HYDROSTATICS. 
 
 Chapter I. 
 1 — 3. Definitions of Fluid; of elastic and non-elastic Fluids. 
 
 Chapter II. Pressure of non-elastic Fluids. 
 
 4. Prop. 1. Fluids press equally in all directions. 
 
 5. Prop. 2. The pressure upon any particle of a fluid of uniform den- 
 sity is proportional to its depth below the surface of the fluid. 
 
 6. Prop. 3. The surface of every fluid at rest is horizontal. 
 
 7. Prop. 4. If a vessel, the bottom of which is horizontal and the 
 sides vertical, be filled with fluid, the pressure upon the bottom will 
 be equal to the weight of the fluid. 
 
 10. Prop. 5. To explain the hydrostatic paradox. 
 
 11. Prop. 6. If a body floats on a fluid, it displaces as much of the 
 fluid as is equal in weight to the weight of the body ; and it presses 
 downwards and is pressed upwards with a force equal to the weight 
 of the fluid displaced. 
 
 Chapter III. Specific Gravities. 
 
 13. Definition of Specific Gravity. 
 
 14. Prop. 7. If M be the magnitude of a body, S its specific gravity, 
 and W its weight, W=MS. 
 
 15. Prop. 8. When a body of uniform density floats on a fluid, the 
 part immersed : the whole body :: the specific gravity of the 
 body : the specific gravity of the fluid. 
 
o CONTENTS. 
 
 ARTS. 
 
 16. Prop. 9. "When a body is immersed in a fluid, the weight lost : whole 
 weight of the body :: the specific gravity of the fluid : the specific 
 gravity of the body. 
 
 18. Prop. 10. To describe the hydrostatic balance, and to shew how to 
 find the specific gravity of a body by means of it, 1st, when its 
 specific gravity is greater than that of the fluid in which it is weighed ; 
 2dly, when it is less. 
 
 19. Prop. 11. To describe the common hydrometer, and to shew how to 
 compare the specific gravities of two fluids by means of it. 
 
 Chapter IV. Elastic Fluids. 
 
 20. Prop. 12. Air has weight. 
 
 21. Prop. 13. The elastic force of air at a given temperature varies as 
 the density. 
 
 22. Prop. 14. The elastic force of air is increased by an increase of 
 temperature. 
 
 24. Prop. 15. To describe the construction of the common air-pump, 
 and its operation. 
 
 25. Prop. 16. To describe the construction of the condenser, and its 
 operation. 
 
 26. Prop. 17. To explain the construction of the common barometer, 
 and to shew that the mercury is sustained in it by the pressure of 
 the air on the surface of the mercury in the basin. 
 
 27. Prop. 18. The pressure of the atmosphere is accurately measured 
 by the weight of the column of mercury in the barometer. 
 
 28. Prop. 19. To describe the construction of the common pump, and 
 its operation. 
 
 29. Prop. 20. To describe the construction of the forcing -pump, and its 
 operation. 
 
 30. Prop. 21. To explain the action of the siphon. 
 
 31. Prop. 22. To shew how to graduate a common thermometer. 
 
 32. Prop. 23. Having given the number of degrees on Fahrenheit's 
 thermometer, to find the corresponding number on the Centigrade 
 thermometer. 
 
THE ELEMENTS 
 
 OF 
 
 MECHANICS AND HYDROSTATICS. 
 
 The explanatory articles and paragraphs which are enclosed in 
 brackets, form no part of the Course adopted by the University. Those 
 in smaller type are illustrative of the Definitions and Propositions they 
 immediately follow. The whole of the extra matter so introduced forms 
 but a very small, and a very easy, portion of the book ; and the student 
 who comes to this subject for the first time had better not omit it. 
 
 THE ELEMENTS OF MECHANICS. 
 CHAPTER I. 
 
 DEFINITIONS ; EXPLANATION OF STATICAL FORCES ; THE MAN- 
 NER IN WHICH THEY ARE MEASURED, AND REPRESENTED. 
 
 1. The science of Mechanics investigates the causes 
 that. prevent or produce motion in bodies, or that tend to 
 produce or prevent motion. ■I^l.tuLcj 
 
 It is divided into two parts. The one, which investi- 
 gates the con dition s fulfill ed when a body is in a state 
 of rest , is called Statics. The other, which treats of the 
 causes and the effect s of motion, is called Dynamics. 
 
 Thus it is the province of Statics to shew how the 
 roof of a building is supported by the cross beams and 
 the walls. If the roof gave way and fell, it would belong 
 to Dynamics to account for the circumstances attending 
 the fall; to explain why the motion took place in one 
 direction rather than in another, to determine the time 
 
1 MECHANICS. 
 
 elapsed in falling, and the swiftness of motion at any in- 
 stant. 
 
 The part of Mechanics treated on in the following 
 pages is The Elements of Statics. 
 
 2. Def. Whatever the cause may be that produces 
 or prevents motion, or that tends to produce or prevent 
 motion, it is called a Force, fli+4- Ta-y- yf. j\ • 
 
 [If a heavy body, as a stone, be laid on the open hand, experience 
 shews that to prevent the stone from falling the hand must make some 
 effort. Again, to set a ball rolling along the ground requires some 
 exertion. The effort, or exertion, is called in either case a force : and 
 although the effect produced be not great enough to prevent entirely the 
 fall of the stone, or to communicate motion to the ball, yet it is still 
 denominated a force. 
 
 From the definition of Statics given in Art. 1. it will readily be 
 understood, that in that branch of Mechanics the conditions are in- 
 vestigated which are fulfilled by those forces only which keep a body at 
 rest.] 
 
 3. Defs. (1) All bodies, if left to themselves, would 
 fall to the earth's surface. This power that resides in the 
 earth, of drawing all substances to its surface, is called 
 the force of gravity. 
 
 (2) The act of a body's pressing downwards to the 
 earth when laid on any substance is called the gravi- 
 tation of the body. 
 
 (3) The precise amount of gravitation residing in any 
 particular body is termed the weight of that body. 
 
 4. [The weights of different bodies may be thus compared. If 
 two bodies, when successively attached in the same manner to a spring, so 
 that they may act upon it by their weights in the same way, produce the 
 same effects, — (by bending the spring to the same extent), — the weights of 
 the bodies will be equal. Any other body that produces the same effect 
 on the spring by its weight as both the former bodies when applied at the 
 same time do by their weights, is double the weight of either of them. 
 And by means of such a contrivance as this spring, bodies might be 
 shewn to be three, four, or any number of times the weight of the first 
 body. 
 
 The weight of any body is thus measured. The weight of a certain 
 bulk of some particular substance is first fixed upon as a standard. Thus 
 the weight of a piece of lead of a certain size being called a pound, any 
 other body whose gravitation produces the same effect as four, or six, or 
 ten such pieces of lead, will be four, or six, or ten pounds in weight, as 
 the case may be.] 
 
MECHANICS. 1 1 
 
 5. Def. The substance, (material, or stuff,) of which 
 any body is^made, is called matter. {Pud-ft^ /<#..?*. 
 
 [As all bodies have "Weight, the property of having Weight is to be 
 considered as necessarily and inherently belonging to all Matter; and it 
 is with respect to this its property of having Weight that Matter is made 
 the subject of mathematical investigation. The existence of Matter, un- 
 der whatever form it may appear, is therefore tested by that form having 
 Weight ; and in the same ratio or degree that one body has more Weight 
 than another, it is concluded that it contains more Matter.] 
 
 G. Def. The Quantities of Matter which dif- 
 ferent bodies contain are proportional to the weights of the 
 bodies. Jh^^U- %^- /£"- /I . 
 
 [Thus, if a body A weigh one pound, and another body B weigh three 
 pounds, the quantity of matter contained in A is said to be to the quantity 
 contained in B as 1 to 3 ; or B is said to contain three times as much 
 matter as A does. 
 
 The exact quantity of matter contained in any body is measured by 
 comparing its weight with the weight of some particular body that has 
 been fixed upon for a standard. Thus if a cubic inch of ivaterbe previously 
 taken as the body to measure the quantities of matter contained in all other 
 bodies by, and the quantity of matter in this cubic inch of water be called 
 one, then the quantity of matter in any other body would properly be said 
 to he five, if the weight of that body were five times as great as the weight 
 of the cubic inch of water.] 
 
 7- Def. When the densities of different substances 
 are spoken of, the closeness is referred to with which the 
 matter composing them is packed, as it were, in the sub- 
 stances. 
 
 To compare this closeness, let equal bulks, or sizes, of 
 two different substances be taken; and suppose the sub- 
 stances to be Water and Lead. If the bulk of water which 
 is taken weigh one pound, it will be found that the piece of 
 lead of equal size with it will weigh 11£ pounds. There - 
 is evidently, therefore, 11^ times as much heavy matter in 
 a piece of lead as there is in an equal bulk of water ; and 
 this fact is expressed or described by saying, that " The 
 density of Lead is to the density of Water, as 1 lj 4 is to 1 ;" 
 or by saying, "The density of Lead is 11^ times that of 
 Water." 
 
 Cor. 1. If the density of water be called 1, i.e. if water 
 be taken to measure Density, then the density of lead will 
 be properly called 11 ±, or 11-4. JWV- Jl^i~ /£.-(• 
 
 [Con. 2. In the same manner as it has been explained how the 
 density of lead is estimated with respect to the density of water, the 
 densities of any other substances, whether solid or fluid, may be deter- 
 mined with respect to that of water. J 
 
12 MECHANICS. 
 
 8. To explain how the precise amount of a Sta- 
 tical Force is described ;\ that is, to shew how sta- 
 tical FORCES ARE MEASURED. (Pu^if j£ J, 
 
 The amount of a Statical Force is described by stating 
 the number of pounds it would support if the Force were 
 made to act directly opposite to the force of gravity. Thus 
 if the weight of a body were P pounds, and it were pre- 
 vented from moving towards the earth's surface by a hand 
 placed beneath it, the resistance offered by the hand to the 
 communication of motion, (that is, the force exercised by 
 the hand), would be P pounds; and if this same pressure 
 were produced by the hand in any other direction, it would 
 be described in the same manner, by saying that it "was 
 equal to P pounds," or that it "was" "P pounds." 
 
 9. [[Forces are, in Statics, also called Pressures. 
 In whatever direction a force tends to produce motion, its 
 magnitude, (as has already been stated} is measured by the 
 weight of the body which would exert the same effect to 
 produce motion downwards, as the force under consider- 
 ation exerts in the line in which it endeavours to produce 
 motion. (And that such a method of measuring Forces 
 is allowable appears from this consideration^ viz., that the 
 effect produced by the w T eight of a heavy body may be 
 made to take place in any direction whatever; horizontally, 
 as in the case of a string being attached to an object lying 
 on a table and kept stretched by a heavy body ( IV) that 
 hangs over the side of the table; 
 or vertically upwards, as in fig. 1, 
 by passing the string over a peg 
 A and attaching the end to a ring 
 B, so that BA may be vertical ; 
 or in any other direction (fig. 2), 
 by making the heavy body pull 
 the string in a line BA which ^jj 
 is inclined at any angle to that 
 (A IV) in which it acts itself. 
 
 1<». (1) The point at which a force acts upon a body 
 is called the 'point of application' of the force. 
 
 (2) The line in which a force produces, or tends to pro- 
 duce, motion, is called Hhe line of the force's action' A and 
 
MECHANICS. 
 
 13 
 
 ai 
 
 uj line that is parallel to the line of a force's action is 
 said to be "in the direction of the forces action", or "in the 
 direction of the force." fu^UH^^ fir. n 
 
 [When the direction of a force's action, or as it is generally called 
 " the direction of the force," is indicated by a line, either the very line 
 is given in which the force acts, or some line which is parallel to it. 
 " The line of a force's action" and " the direction of the force" must 
 by no means be confounded together. If the former be known, the latter 
 is necessarily known also; but if only the latter be given, the precise 
 tide in which the force produces, or tends to produce, motion, is a matter 
 of uncertainty, and all that can be said respecting it is, that the line of 
 action of the force is either that line, or some other line which is parallel 
 to it.] 
 
 (3) If two or more forces be applied to a body, or 
 at some point, and no motion is produced, they are said 
 to " counteract" or to " balance" one another, or to be " in 
 equilibrium'' 
 
 11. Forces may be properly represented by lines. 
 
 Since lines may be drawn of any length and in any 
 direction from a point, the lines in which forces act, and 
 the ratios which the forces bear to one another, may be 
 represented by drawing lines which coincide with the lines 
 in which the forces act, and whose lengths bear to one an- 
 other the same ratios that the forces themselves bear to one 
 another. %e^J- f^ /5~, Zs . 
 
 [Among other advantages which attend this method of indicating the 
 magnitudes and directions of forces,\ the addition and subtraction of such 
 forces as act at a point in the same straight line are easily effected. 
 Thus, if a certain force act at A in the line AH, and AB be taken to re- 
 present it, and another force, half the size of the former, act at A in the 
 same line, and also tend to move the body from A towards H, then by 
 taking BC equal to the half of AB the 
 line A C will represent the whole pres- 
 
 sure at A, both with respect to the mag- K ■ • • ^ H 
 
 n itude of that pressure and to the line 
 in which it acts. And in like manner, 
 
 if a force equal to half the original force AB act at A in the line AH, 
 but tends to move the body at A from A towards K, half the pressure 
 of the former force will be counteracted by this new force. Cutting off 
 from the line AB, therefore, a part BD equal to the half of AB, the 
 effective pressure still remaining will be properly represented by AD, 
 both with respect to its magnitude and to its line of action.'] 
 
 12. £Note. It will be gathered from the last Article, 
 that a force AB applied at A has not the same effect as 
 a force BA applied at that point; for a force AB would 
 
 2 
 
A 5 5 
 
 H 
 
 14 
 
 MECHANICS. 
 
 tend to move a body at A in the line KH towards H, 
 but a force BA would tend to move a body at A in the 
 line KH from A towards K. It is not, therefore, indif- 
 ferent whether the words "a force AB" or "a force BA" 
 be used ; since, though the two forces represented by AB 
 and BA be the same in magnitude, and also act in the same 
 straight line, yet they tend to produce motions directly 
 opposite to one another; the force AB tending to move 
 the body at A towards H, and the force BA tending to 
 move the body at A towards K.~} Pi^U- ?a^ /g- ? 
 
 13. [The effect produced at a point by any force 
 is the same at whatever point in its line of action 
 the force is applied. Jl^i j^ /5~ 3 Q (j 
 
 For if a heavy body P be suspended by a string CP, the £ 
 force necessary to prevent P falling to the earth is found to be 
 the same whether that force be applied at A, or 2?, or C ;— the 
 weight of the string being either neglected, or the weight of that 
 portion of it which is supported along with the heavy body P, A- 
 being counterbalanced. ] 
 
 14. [Def. If a string, fastened at one end, be pulled 
 by a force applied at the other end, the resistance to motion 
 made by the string at any point in it is called the tension 
 of the string at that point. 
 
 If the string be supposed to be without weight, it will 
 follow from Art. 13 that the tension at every point of it is 
 the same ; namely, the force by which the string is pulled.] 
 
 15. [To recapitulate the substance of this Chap- 
 ter ,- \ 
 
 (1) Matter is the substance of which ^all bodies are 
 composed. It is found, by universal experience, to have a 
 tendency to move towards the earth. Art. 5. 
 
 (2) The precise amount of the pressure with which 
 any particular body endeavours to move towards the earth , 
 is called the weight of that body. 
 
MECHANICS. 15 
 
 This weight of a body is measured by comparing the 
 tendency of the body to move towards the earth with that 
 of some other given body, (of a certain size and formed of 
 a certain material), which is taken for a standard, and to 
 which the name of a grain, an ounce, a pound, or a ton is 
 given, as the case may be. Art. 3. 
 
 A (3) The quantities of matter contained in different 
 
 bodies (whether the bodies be great or small, rare like gas 
 or dense like lead), is proportional, (not equal), to the 
 weights of the bodies. Art. 6. 
 
 1 
 
 n 
 
 (4) The densities of different substances are propor- 
 tional, (not equal), to the weights of equal bulks of the 
 substances. Art. 7. 
 
 (5) i. Whatever be the cause which moves, or tends 
 to move, matter existing under any form whatsoever, it is 
 called force, ft/l^". %. 
 
 *\ Forces which prevent motion taking place, — that is, 
 
 statical forces, — are measured by the number of pounds 
 they would support if they acted vertically upwards. Arts. 
 2, 8. 
 
 ? ii. The line of a force's action is the actual line 
 
 in which the force tends to produce, or to prevent, motion. 
 Art. 10. 
 
 n in. The direction of a force is indicated either 
 
 by the line of its action, or by any line which is parallel to 
 the line of its action. Art. 10. 
 
 t3 iv. The magnitudes of forces, and the lines (or 
 
 directions) in which they act, may be represented by means 
 of straight lines. Art. 1 1 . 
 
 * v. If a straight line AB represent a force acting on a 
 
 material point placed at A, and BA represent another force 
 acting on the same point, the two forces AB and BA are 
 equal, and tend to move the point in the same straight 
 line, but in opposite ways from A. Art. 12. 
 
 D 
 
 vi. A Statical Force produces the same effect at what- 
 ever point in its line of action it is applied. Art. 13 
 
J. flu, L'n-t- »i dile^Ucirn. /V urk<id it~ cLtb. 
 1 6 MECHANICS. 
 
 VII. To investigate the effects produced by a force 
 there must be given; $<* 
 
 1st, The magnitude of the force ;-(-which is known if 
 the number of pounds be known which the force would 
 support ;~) 
 
 2nd, The point at which the force is applied ; 
 
 3rd, The line in which it acts, or the direction ; (for 
 knowing the direction of the force and the point it is 
 applied at, the line in which the force acts may be de- 
 termined by drawing a line through the given point pa- 
 rallel to the given direction.] 
 
 CHAPTER II. 
 
 THE LEVER. 
 16. DEFS. (1) A PLANE, Or A PLANE SUPERFICIES, is 
 
 that superficies, or surface, in which if any two points be 
 taken, the straight line joining them lies wholly within the 
 superficies. Euclid i; Def. 7- 
 
 (2) A solid is that which hath length, breadth, and 
 thickness. Euclid xi; Def. 1. 
 
 (3) Parallel Planes are such as do not meet one 
 another, however far they may be produced. Euclid xi ; 
 Def. 8. 
 
 (4) A Prism is a solid bounded by plane rectilineal 
 
 figures; two of them, which J> d 
 
 are parallel to each other, being 
 
 similar and equal triangles q ( 
 
 squares or polygons, having 
 
 the similar angles in each op- J* b 
 
 posite, and all the remaining plane rectilineal figures which 
 
 bound the prism being rectangles. 
 
 [Thus, let ABCD and abed be two equal and similar quadrilateral 
 figures, placed with their planes parallel, and with their similar angles 
 opposite to each other; and let all the figures, such as ABba, which are 
 formed by joining the equal angles of ABCD and abed, be rectangles; 
 then the solid included by these rectangles and by the ends, or bases, 
 ABCD and abed, is called a prism. 
 
 The length of the prism is any one of the edges A a, Bb, &c. ; which 
 lines, being the sides of adjacent rectangles, are all equal to one another.] 
 
MECHANICS. 17 
 
 (5) A Cylinder is a solid de- D 
 scribed by a rectangle (ABCD) 
 revolving round one of its sides 
 (AB) which remains fixed. 
 
 [The side A B, which remains fixed, is the length of the cylinder, and 
 is called its axis. 
 
 The surfaces described by the two sides AD and BC of the rect- 
 angle which are adjacent to the fixed side, are circles in form, and are 
 parallel to each other; they are called the ends, or bases, of the cylinder.] 
 
 17. Def. A lever is a rigid and inflexible rod, 
 moveable in one plane round a point in the rod called 
 
 the FULCRUM. 
 
 In the following propositions the thickness of the rod 
 is neglected, and the lever is considered to be a geometrical 
 line ; it is also supposed to be without weight. 
 
 18. The investigation of the properties of the Lever 
 will be made to depend on the following Axioms, which 
 the mind readily admits as true. 
 
 Axiom I. If two equal weights, or forces, act perpen- 
 dicularly upon a horizontal straight lever, they will balance 
 round a fulcrum placed at the middle point between the 
 points at which they are applied; the pressure on this 
 fulcrum will be the sum of the weights, and it will act in 
 the same direction as the weights act. 
 
 Axiom II. If any two weights, or forces, acting per- 
 pendicularly on a horizontal straight lever, on opposite 
 sides of a fulcrum, balance each other, the pressure on 
 the fulcrum is equal to the sum of the weights. 
 
 Axiom III. The efforts made by two equal forces, act- 
 ing perpendicularly at the extremities of equal straight arms 
 of any lever, to turn the lever round, are equal. 
 
 2—3 
 
IS 
 
 MECHANICS. 
 
 er 
 
 19. Prop. i. A horizontal prism, or cylind 
 of uniform density, will produce the same effect by 
 its weight as if it were collected at its middle point. 
 
 Let P and P be two equal weights acting perpen- 
 dicularly at M and N on 
 the straight horizontal lever /[\ 
 
 AB which is moveable a ?I 
 
 1 °" 1 
 
 round any fulcrum in it; 
 
 They produce the same 
 
 effect as if they were ap- & 
 
 plied at C the middle point between M and N. 
 
 At C let a weight 2 P act perpendicularly downwards, 
 and also let a pressure equal to 2 P act upwards at C. 
 Since these two forces are equal and opposite they will 
 counteract each other; and whatever the tendency of the 
 lever was to turn round the fulcrum, it will remain unaltered. 
 
 Now since P at M and P at N would produce a pressure 
 downwards of 2 P on a fulcrum at C, (Axiom I.), the press- 
 ure upwards exerted by such a fulcrum would be 2 P. 
 The pressure therefore of 2P, which has been supposed 
 to act upwards at C, will serve instead of such a fulcrum, 
 and will therefore balance P at M and P at N. Wherefore, 
 the three forces, P at M, P at N, and 2 P acting upwards 
 at C, have no tendency whatever to turn the lever about ; 
 but its tendency to turn about, (which has been shewn to 
 be the same as when the lever was acted on by no other 
 forces than P at M and P at N), arises entirely from a 
 force 2P applied at C, and acting downwards. 
 
 Now supposing* that a horizontal prism (or cylinder) of 
 uniform density, would produce the same effect by its weight, 
 as if the whole matter composing it were collected into a 
 number of small equal weights placed at equal distances 
 along a horizontal straight line of the same length as the 
 prism (or cylinder), then, since it appears from what has 
 been proved above, that every pair of weights which are 
 equidistant from the center of the line, would produce the 
 
 * A uniform prism, or cylinder, may be divided, by means of equally 
 distant planes which are parallel to its ends, into equal small prisms or 
 cylinders lying on each side of the plane passing through its center. Now 
 if these be supposed to be condensed and placed at equal distances along 
 a straight line equal to the length of the original prism, or cylinder, the 
 supposition made in the text will not appear unreasonable. 
 
MECHANICS. 19 
 
 same effect as if it were collected at the center, the whole 
 prism therefore, or cylinder, (being made up of such p c airs 
 of equal weights), will produce the same effect by its 
 weight as if it were collected at its center, q. e. d. 
 
 Cor. Hence it follows that a horizontal prism, or 
 cylinder, of uniform density, will balance on its middle 
 point ; and the pressure on a fulcrum placed (there?! will 
 be the weight of the prism, or cylinder. at ^ *"*&*- V- 
 
 20. Prop. it. If two weights, acting perpen- 
 dicularly on a straight lever, on opposite sides of 
 the fulcrum, b alance each other, they are i nvers ely 
 as their distances from the fulcrum ; and the pr essur e 
 on the fulcrum is equal to their sum. 
 
 . Let there be an uniform cylinder whose length is AB, 
 and whose weight is P + Q. M y „ 
 
 If it be placed horizontally 1 t ~^ 1 15 
 
 it will balance on a fulcrum I ^ 
 
 C placed at its middle point 
 
 (Prop. i. Cor.) ; and the pressure on C will be the weight 
 
 of the cylinder, namely P + Q. 
 
 Let AB be divided in E so that AE : AB :: P : P+Q; 
 then, since the cylinder is supposed to be of uniform 
 density, Weight of the part AE of the cylinder : weight 
 of the whole :: AE : AB 
 
 :: P : P + Q, by construction; 
 
 but, Weight of the whole = P + Q ; .-. weight of the part 
 AE = P; and .-. weight of EB=Q. 
 
 ^ But if the parts AE and EB be collected at their middle 
 points M and TV, and act perpendicularly on the lever, they 
 will produce the same effects by their weights as beforeja^ 
 Prop. i. Therefore, the weights P and Q, acting perpen- ^' 
 dicularly at M and N, will balance about C, and the pressure ^ 
 on C will be P + Q. Now y at " 
 
 P _ AE AB-BE 2BC-2BN BC-BN CN 
 Q EB AB-AE 2 AC - 2 AM " AC -AM" CM' 
 *' ~&^m' or P : Q :: CN : CM. 
 
 Wherefore, " If two weights, &c." q. e. d. 
 
 AC -AM. ' ' 3C-IM ___£Ii 
 
 * ' ACT-AM. - Clfr 
 
20 MECHANICS. 
 
 21. Prop. hi. If two forces, acting perpen- 
 dicularly on a straight lever in opposite directions 
 and on the same side of the fulcrum, balance each 
 other, (l) they are inversely as their distances from 
 the fulcrum, and (2) the pressure on the fulcrum is 
 equal to their difference. 
 
 Let the forces P and Q, acting P^ iR 
 
 perpendicularly at M and N on 
 
 the straight lever MC in opposite **■ T AC 
 
 directions and on the same side of *** A9- 
 
 the fulcrum C, balance each other; % jz Vc 
 
 Then P : Q :: CN : CM, and, rf 1 
 
 (Q being the force which is the x 
 
 nearer to the fulcrum), the pressure on the fulcrum = Q - P. 
 
 Let the fulcrum at C be removed, and the place of its 
 resistance (R) supplied by a force equal to R, and acting 
 perpendicularly to the lever in the direction of P's action. 
 
 Since P and R are counterbalanced by Q, they must 
 produce a pressure at N equal and opposite to Q. Let Q 
 be removed, and its place supplied by a fulcrum on the 
 contrary side of the lever to that on which Q acted, sus- 
 taining the pressure (namely Q) produced by P and R. 
 
 The case is now that of two forces acting perpendicu* 
 larly on opposite sides of the fulcrum, and balancing each 
 other ; and therefore, by Prop, u, 
 
 P : R :: CN : NM ; 
 .-. P : P+R :: CN : CN + NM :: CN : CM. 
 
 But Q, the pressure on the fulcrum which has been sup- 
 posed to be placed at N, is (by Prop, u.) = P + R. 
 
 .: P : Q :: CN : CM. 
 Also since Q = P+R, .-. R=Q- P. 
 
 Wherefore, "If two forces, &c." Q. e. d. 
 
MECHANICS. 21 
 
 22. [From the last two Propositions it appears, that 
 if a straight lever, which is acted on perpendicularly by 
 two weights, or other forces, P and Q, respectively applied 
 at the distances CM and CN from the fulcrum C, be at 
 rest, then, whether P and Q act on the same or on different 
 sides of the fulcrum, the proportion P : Q :: CN : CM is 
 always true. 
 
 When the Lever is used to balance a given weight or 
 force Q, by the application of another force P, Q is usually 
 called "the Weight," and P " the Power." 
 
 If CM, the perpendicular distance from the fulcrum at 
 which the Power acts, be greater than CN, the distance at 
 which the Weight acts, the Power required to balance the 
 Weight is less than it. In this case "force" is said " to be 
 gained" by the application of the lever. But if CM be 
 less than CN, the Power required to balance the Weight 
 is greater than it, and "force" is then said "to be lost ."] 
 
 23. Prop. iv. To explain the different kinds of 
 levers. 
 
 Levers are divided into three classes. 
 
 (1) Where the Power and the Weight act on opposite 
 sides of the fulcrum. 
 
 (2) Where the Power and the Weight act on the same 
 side of the fulcrum, but the perpendicular distance from the 
 fulcrum at which the Power acts is greater than that at 
 which the Weight acts. 
 
 (3) Where the Power and the Weight act on the same 
 side of the fulcrum, but the perpendicular distance from 
 the fulcrum at which the Power acts is less than that at 
 which the Weight acts. 
 
 -&- 
 
 Of the first class the poker, when used to raise the 
 coals, is an instance; the bar of the grate on which the 
 poker rests being the Fulcrum, the force exerted by the 
 hand the Power, and the resistance of the coals the Weight. 
 In the common balance, the Power and the Weight are 
 equal forces perpendicularly applied at the ends of equal 
 
 \\ /T . 
 
 RSIT7i 
 
22 MECHANICS. 
 
 arms. In the steelyard, the Power and the Weight are 
 perpendicularly applied at the ends of unequal arras. 
 Pincers, scissars, and snuffers, are double levers of this 
 kind, the rivet being the fulcrum in each. 
 
 Since CM may be either greater or less than CN, the 
 Power in levers of this class may be either less or greater 
 than the Weight, and consequently " Force" may be either 
 "gained" or "lost" by using them. 
 
 Of the second class, a cutting blade, such as those used 
 by coopers, moveable round one end, which is fastened by a 
 staple to a block, by means of a handle fixed at the other, is 
 an example. An oar is also such a lever; the Fulcrum being 
 the extremity of the blade, (which remains fixed, or nearly 
 so, during the stroke), the muscular strength and weight of 
 the rower being the Power, and the Weight being the re- 
 sistance of the water to the motion of the boat, which is 
 counteracted and overcome at the rowlocks. A pair of nut- 
 crackers is a double lever of the second class. 
 
 , Here, since CM is greater than CN, the Power is always 
 less than the Weight, or " Force is gained" by using levers 
 of the second class. 
 
 An example of the third class is the board which the 
 turner (or knifegrinder) presses with his foot to put the 
 wheel of his lathe in motion ; the Fulcrum being the end 
 of the board which rests on the ground, the Power being the 
 pressure of the foot, and the Weight being the pressure pro- 
 duced at the crank put on the axletree of the wheel. Fire 
 tongs, and sugar tongs, are double levers of this kind; the 
 Weight in either case being the resistance of the substance 
 grasped. The limbs of animals are also such levers : thus, 
 if a weight be held in the hand and the arm be raised round 
 the elbow as a Fulcrum, the weight is supported by muscles 
 fastened at one extremity to the upper arm, and again at- 
 tached to the forearm, after passing through a kind of loop 
 at the elbow. 
 
 Here, since CM is less than CN, the Power is greater 
 than the Weight, or " Force is lost" by making use of levers 
 of the third class. 
 
MECHANICS. 23 
 
 24. Fkop. v. If two forces acting perpendicularly 
 at the extremities of the (straight) arms of any lever, 
 balance each other, they are inversely as the arms. 
 
 Let the forces P and Q, * [ 
 
 acting perpendicularly at the p^~ 
 extremities of the straight arms 
 CM and CN of any lever whose M > 
 fulcrum is C, balance each other; I" 
 They are inversely as the arms, p'y ^q' 
 
 Draw the horizontal line M'N' through C, and take 
 CM' = CM, and CN'= CN. Let a force P', equal to P, act 
 perpendicularly at M'; and a force Q', equal to Q, act per- 
 pendicularly at N', 
 
 Now since CM f =CM and P'=P, P' will produce 
 the same effect as P on the lever; Axiom in. Art. 18; 
 
 But, by supposition, P balances Q; 
 
 .-. P' would balance Q. 
 
 And since CN'=CN and Q'=Q, Q! will produce the 
 same effect on the lever as Q; Axiom in. 
 
 But Q would balance P'; 
 .-. Q' would balance P'; 
 wherefore, by Prop, n., P' : Q' :: CN' : CM', 
 and .-. P : Q :: CN : CM. 
 
 Wherefore, " If two forces, &c." q. e. d. 
 
 25. Prop. vi. If two forces acting at any angles 
 on the arms of any lever, balance each other, they 
 are inversely as the perpendiculars drawn from the 
 fulcrum on the directions, (i. e. the lines), in which 
 the forces act. 
 
 Let P and Q be two forces which acting at any angles 
 on the arms CA and CB of any lever ACB balance each 
 
24 
 
 MECHANICS. 
 
 other about the fulcrum C; They 
 are inversely as the perpendicu- 
 lars CM and CN drawn from the 
 fulcrum on the lines in which 
 they act. 
 
 Since a force produces the same effect at whatever point 
 in its line of action it is applied (Art. 15), the force P may 
 be supposed to be applied at M ; and in order that it may 
 be so applied, let a rod (CM A) without weight be fastened 
 to CA. In like manner Q may be supposed to be applied 
 at N perpendicularly to the part CN of the rod CNB 
 which is added to CB. 
 
 And, since P acting at M perpendicularly to CM 
 balances Q acting at N perpendicularly to CN; by Prop, v., 
 
 P : Q :: CN : CM; 
 
 and therefore when P and Q acting at A and B in the lines 
 A MP, BNQ balance, they are as CA" : CM. 
 
 Wherefore, " If two forces, &c." q. e. d. 
 
 If two weights balance one an- 
 
 26. Phop. vii. 
 other on a straight lever when horizontal, they will 
 balance each other in every position of the lever. 
 
 Let P and Q, be two 
 weights that balance each other 
 round the fulcrum C on the 
 lever ACB when it is hori- 
 zontal ; They will balance each 
 other on the lever when it 
 comes into any other position, 
 as A'CB'. 
 
 Produce QB' to cut AB in L 
 
 N. 
 
 IM 
 
 1 
 
 VC X 
 
 ,1 
 
 V 
 
 ^^ 
 
 
 > 
 
 35' 
 
 ©( 
 
 > 
 
 ■ 
 
 Q 
 
 Since weights act perpendicularly to the horizon, A'P 
 and NB'd are each perpendicular to the horizontal line 
 ACB; 
 
MECHANICS. 25 
 
 .*. angle CM A' = right angle = angle CNB', 
 
 and angle A'CM = opposite angle B'CN; Euc. I. 15. 
 
 .-. also angle CA'M = angle CB'N, and the triangles are 
 equiangular, and therefore similar. 
 
 Therefore, by Euc. vi. 4, 
 
 CN : CB' :: CM : CA' ; 
 
 .-. CN : CM :: CB' : CA' :: CB : CA. 
 
 But, v P and Q balance on ACB, .% CZ? : CJ :: P : Q; 
 
 .-. CiV : CM :: P : Q; 
 
 But C3/ and CN are the perpendiculars from C on 
 the lines in which P and Q act when they are hung at 
 A' and B' ; therefore, by Prop, vi., P and Q will balance 
 on J'C^. 
 
 Wherefore, " If two weights, &c." Q. e. d. 
 
 CHAPTER III. 
 
 THE COMPOSITION AND RESOLUTION OF FORCES. 
 
 27. Definition of Component and Resultant 
 Forces. 
 
 It is found, by experiment, that a body which is acted 
 on by two forces applied at the same instant to the same 
 point of it, instead of having a tendency to move in either 
 of the lines the forces act in, has a tendency to move in 
 a line lying between them. Whence it appears, that the 
 two original pressures by their combined action produce 
 a third pressure; which pressure is called, from the cir- 
 cumstance of its " resulting," out of the actions of the 
 o 
 
26 
 
 MECHANICS. 
 
 original pressures, their " Resultant" with respect to them ; 
 while they are called, with respect to it, its " Components." 
 
 The Resultant (R), which produces the same effect as 
 the compound action of the original forces P and Q applied 
 at the same point at the same instant, is said to be " com- 
 pounded" of P and Q. This Resultant (R) also, if con~ 
 ceived to be the sole original force, may be supposed to be 
 "resolved" into the two forces P and Q; since those two 
 forces, acting in the manner described, (namely, at the same 
 point, and at the same instant), produce exactly the same 
 effect on the body as the single force R does. 
 
 [An instance of the "Composition of Forces'* is a boat moored in 
 a stream by ropes attached to the boat and to either bank ; the Compo- 
 nent forces here being the tensions of the ropes, and their Resultant being 
 the force by which the pressure of the stream on the boat is counter- 
 acted.] 
 
 28. Prop. viii. If the adjacent sides of a paral- 
 lelogram represent the component forces phat act at a 
 point at the same instant], in direction p. e. in their lines 
 of action] and magnitude, the diagonal will represent 
 the resultant force (l) in direction pine of action], 
 and (2) in magnitude. 
 
 Let AB and AC re- 
 present, in direction and 
 
 magnitude, the two com- 
 ponent forces that act at 
 A. Complete the paral- 
 lelogram BC, and draw 
 the diagonal AD. Then 
 AD will represent the 
 resultant of AB and AC, 
 (1) in direction, and (2) 
 in magnitude. 
 
 From D draw DM and DN perpendicular to AB and 
 AC, produced if necessary. 
 
 (1) Then in the triangles DBM, DCN, 
 angle DM B = right angle = angle DNC, 
 
MECHANICS. 27 
 
 and angle DBM = angle BAC (since BD, AC are parallel 
 and MBA cuts them), = angle DCN ; 
 
 /. the third angle, BDM, of the one triangle = the third 
 angle, CDN, of the other ; and the triangles are equiangular 
 and similar. 
 
 .-. CD : DN :: BD : DM, 
 
 and .-., alternately, CD : BD :: DN : DM. 
 
 Now if there be a lever AD whose fulcrum is D, 
 which is acted on by the forces AB, AC applied at A ; 
 since 
 
 Force in the line AM : force in line AN :: AB : AC 
 
 :: CD : BD, since BC is a parallelogram, 
 
 :: DN : DM; 
 
 the two forces acting on the lever AD are inversely as the 
 perpendiculars from the fulcrum on their lines of action, 
 and therefore the lever will be kept at rest about D by 
 them (Prop. vi.). Wherefore the lever will also be kept 
 at rest by the Resultant of those forces ; because that single 
 force produces the same effect as they do when they act 
 at the same point and at the same instant. 
 
 This Resultant therefore must act in the line AD, for 
 it keeps the lever at rest, which it could not do were it 
 to act at A and make any angle with the lever AD on either 
 side of it. 
 
 (2) Again: Having shewn that the resultant of AB 
 and AC acts in the line of the diagonal, next to prove 
 that the diagonal represents it in magnitude as well as 
 in direction. 
 
 Produce DA, and suppose a force AE to be taken in 
 it equal and opposite to the Resultant of AB and AC. 
 The joint effect of AB and AC will now be counteracted 
 by AE ; and the point A, which is acted on by the three 
 forces AB, AC, and AE, will remain at rest. Whatever 
 be the effect, therefore, produced by the joint action of 
 AE and AC, it is counteracted by AB ; that is, AB must 
 be equal and opposite to the Resultant of AE and AC. 
 
28 
 
 MECHANICS. 
 
 Complete the parallelogram EC, and draw the diagonal 
 AF. By the first part of 
 the Proposition, ^.Fisthe 
 line in which the result- 
 ant of AE and A C acts ; 
 and since the force AB 
 is equal and opposite to 
 that resultant, AF must 
 be in the same straight 
 line with AB, and, there- 
 fore, it is parallel to 
 CD. 
 
 Wherefore, EC and FD are parallelograms ; and AE = 
 FC = AD. The Resultant, therefore, of AB and AC (which 
 is equal and opposite to AE) will be properly represented 
 in magnitude by AD, the diagonal of the parallelogram of 
 which AB and AC are the sides. 
 
 Wherefore, " If the adjacent sides, &c." q. e. d. 
 
 29. Prop. ix. If three forces represented in 
 magnitude and direction by the three sides of a tri- 
 angle [when taken in order], act on a point, they will 
 keep it at rest. 
 
 Let the sides AB, BC, .,, 
 
 CA, taken in order*, of the 
 triangle ABC, represent in 
 magnitude and direction (see 
 Art. 15), three forces that act 
 on the point A; They will 
 keep A at rest. 
 
 Complete the parallelogram BD. 
 
 Then AD is parallel and equal to BC ; and it will, 
 therefore, represent in magnitude and in line of action 
 (Art. 15) the force which acts at the point A in the direction 
 BC, and is represented in magnitude by BC. 
 
 * By the expression "taken in order" is meant, that if ABC be the 
 triangle and AB be one of the forces, BC (and not CB) is the next, and 
 CA (not AC) is the third ; so that the forces are described in the same 
 direction round the triangle, proceeding from A to B, from B to C, and 
 from C to A again. 
 
 X> 
 
mechanics: 2J> 
 
 Now the forces AB and AD acting at A will produce a 
 resultant AC. Prop. vin. 
 
 If, therefore, a force CA act at A, the force AC will 
 be counteracted, and the point A will remain at rest ; 
 Wherefore, if three forces, represented in magnitude and 
 direction by AB, AD, CA, — (or, which is the same thing, 
 if they be represented by the three sides AB, BC, CA, 
 taken in order, of the triangle ABC) — act on a point A, 
 they will keep it at rest. q. e. d. 
 
 [[Cor. It appears from this proof, that if two sides of 
 a triangle ABC, as AB and BC, taken in order, represent 
 in magnitude and direction two forces which act at the 
 same instant at a point A, the third side of the triangle AC, 
 not taken in the same order as AB and BC, represents their 
 Resultant, in magnitude and direction.] 
 
 CHAPTER IV. 
 
 MECHANICAL POWERS. 
 
 30. The Mechanical Powers are certain instruments 
 by which a lesser weight, called the Power, may be made 
 to balance a greater, called the Weight. The properties 
 will now be investigated of three of them, called the Wheel 
 and Axle, the Pulley, and the Inclined Plane. They will 
 be supposed to be rigid bodies and without weight*. 
 
 31. The Wheel and Axle consists of a cylinder 
 (or Axle) AB, terminating in another cylinder (or Wheel) 
 CD of greater base. The two cy- 
 linders have a common axis EF, 
 which is supported in a horizontal 
 position. The Power (P) is a heavy 
 body which hangs freely by a string 
 coiled round the Wheel; and the 
 Weight (W) is another body, hang- p ^ 
 ing by a string coiled round the Axle, and tending to turn 
 the machine round in the opposite direction. 
 
 * The other Mechanical Powers are the Lever, the Toothed Wheel, 
 the Wedge, and the Screw. 
 
 3-3 
 
80 MECHANICS. 
 
 Pitop. x. There is an equilibrium on the Wheel 
 and Axle when the Power is to the Weight as the 
 radius of the Axle is to the radius of the Wheel. 
 
 Suppose the strings by which the Power and the 
 Weight hang, to be in the planes of the circles which 
 are those ends of the two cylinders that lie in the same 
 plane*. Since the cylinders have a com- 
 mon axis, their circular ends have a com- 
 mon center; let be it, and join with 
 M and N the points in which the strings 
 leave (and therefore touch) the circum- 
 ferences of the circles. OM and ON are 
 therefore perpendicular to MP and NW, 
 the lines in which the Power and the 
 Weight act. 
 
 Now P and W act perpendicularly on the arms OM 
 and 02V, and balance each other ; and therefore, supposing 
 MON to be a lever moveable round as fulcrum, 
 
 P : TV :: ON : OM :: radius of axle : radius of wheel. 
 
 When, therefore, this ratio of the Power to the Weight 
 is fulfilled on the Wheel and Axle, there will be equili- 
 brium. Q. E. D. 
 
 £Cor. Since OM and ON are both perpendicular to 
 the vertical lines in which P and W act, and also pass 
 through the same point 0, MON is a horizontal straight 
 line.] 
 
 32. Def. The Pulley consists of a grooved wheel 
 moveable round an axis passing through the centre of the 
 wheel, and whose ends are fixed in a frame called the Block. 
 
 When the Pulley is used as a Mechanical Power, a 
 string, which is fastened at one end, is passed round part 
 of the circumference of the wheel, and the Power is applied 
 at the other end of the string. The Weight is attached 
 to the Block. 
 
 Prop, xi. In the single moveable pulley where 
 the strings are parallel, there is an equilibrium when 
 the Power is to the Weight as 1 to 2. 
 
 * These circles are HD and AG in the figure on p. 29. 
 
MECHANICS. 
 
 SI 
 
 Let PA, which touches the pulley at A, be 
 the line in which the Power (P) acts. Join A 
 with C the center of the pulley. Produce AC 
 to B. Draw BR touching the circle ; it is per- 
 pendicular to AB and parallel to AP; this, there- M 
 fore, will be the other part of the string. Let y 
 W, the weight, act at C by means of the block, 
 in the line CfV. iw 
 
 A 
 
 The whole being at rest, the string RBDAP must be 
 equally stretched throughout, or motion would ensue. The 
 force upwards, therefore, exerted at the fixed point R is P ; 
 for it is the same as that which keeps AP stretched. Art. 14. 
 
 Now considering ACB as a straight lever kept at rest 
 round C by forces P and P acting perpendicularly upwards 
 at A and B, the pressure upwards on C will be 2 P 
 (Prop, n); wherefore the weight (W) which acts down- 
 wards at C and is supported by this pressure, is = 2P; 
 
 or P : W :: 1 : 2. 
 Wherefore, "In the single moveable pulley, &c." q.e.d. 
 
 £Cor. The pressure produced at C by the forces P 
 and P acting at A and B is perpendicular to AB, or 
 parallel to AP and BR (Art. 18. Ax. i.); wherefore the 
 line of action of the Weight is perpendicular to AB, or 
 parallel to AP and BR. And therefore since in a case 
 of equilibrium the strings are parallel to the line in which 
 the weight acts, if the Weight be a heavy body hanging 
 freely from C, AP and BR are vertical.] 
 
 33. Pitop. xii. In a system in which the same 
 string passes round any number of pulleys, and the 
 parts of it between the pulleys are parallel, there is 
 an equilibrium when Po.ver (P) : Weight (TF) :: 1 : 
 number of strings at the lower block. 
 
 [In this Proposition, as in Prop. XI., the Power and the Weight are 
 supposed to act in a direction parallel to that of the strings.] 
 
32 
 
 MECHANICS; 
 
 Because there is equilibrium, the portions 
 of the string which lie on either side of any 
 pulley must be pulled by equal forces ; and 
 since the outside portion, to which the 
 power is applied, is acted on by a force P, 
 each portion is therefore pulled by a force 
 P. 
 
 Wherefore the part of the weight that is 
 sustained at A is (Prop, xi.) 2P; and the 
 same is true with respect to the pulley B ; 
 and so on for every other pulley at the 
 lower block. If, therefore, there be n pul- 
 leys at the lower block, W, the whole weight 
 supported, will be n x 2P; or, W=2n P ; 
 
 P_J_ 
 
 ""; w~2?i' 
 
 i.e. P : W :: 1 : 2?i :: 1 : number of 
 strings at the lower block. 
 
 Wherefore, "In a system, &c." q. e.d. 
 
 34. Prop. xiii. In a system where each pulley 
 hangs by a separate string and the strings are parallel, 
 there is an equilibrium when P : W :: 1 : 2 raised 
 to that power whose index is the number of move- 
 able pulleys. 
 
 [The weight W is supposed to act in a line parallel to the strings.] 
 In the figure, C being a moveable pulley J? 6 k 
 with parallel strings, there is equilibrium if ( 
 the pressure downwards at C = 2P. 
 
 .-. Tension of PC = 2 P; 
 
 .•. Weight supported at moveable pulley 
 B = 2 x (2 P) = 2 2 P = tension of AB ; 
 
 So Weight supported at moveable pulley 
 
 ^=2x(2 2 P) = 2 3 P. 
 
 Wherefore, if W, "the Weight," be-2 3 P, 
 there is equilibrium on the system of Ihree 
 moveable pulleys here represented. 
 
MECHANICS. 33 
 
 And in the same manner, if n were the number of move- 
 able pulleys, it would appear by the same mode of reasoning 
 that there would be equilibrium when W=2 n P; 
 
 P 1 
 
 • — -— • or P - JV ~ 1 - Q n 
 ' ' W~~ 2 n ' . 
 
 Wherefore, " In a system where each pulley hangs by 
 a separate string, &c." q.e.d. 
 
 35. Def. If a prism whose ends are similar and equal 
 
 right-angled triangles, (C and c b ^ 
 
 being the right angles), be placed 
 so that either of the sides of the 
 prism adjacent to the right an- 
 
 gles (as ACca) be horizontal, the A a 
 
 slant side (ABba) is called an Inclined Plane, 
 
 When the Inclined Plane is used as a Mechanical Power, 
 the Power and the Weight both act in a plane parallel to 
 either end of the prism, and the Inclined Plane is represented 
 by a triangle such as ABC, of which the side AB is called 
 the Length of the plane, AC its Base, and CB its Height. 
 And the plane, by being supposed to be perfectly rigid and 
 inflexible, will be capable of counteracting, and entirely 
 destroying, the effect of any force which acts upon it per- 
 pendicularly. 
 
 36. Pitor. xiv. The weight (W) being on an 
 Inclined Plane and the force (P) acting parallel to 
 the plane, there is an equilibrium when P : W :: 
 the height of the plane : its length. 
 
 Let AB be the length of the plane; AC its horizontal 
 base; and BC, perpendicular to AC, 
 its height. Let the Weight (PV) act 
 vertically at D, and the Power (P) act 
 in the line DB which is parallel to the 
 plane; Then if P : W :: BC : AB, 
 there is equilibrium. 
 
 From D draw BE at right angles to AB, meeting the 
 base in E ; and from E draw EF vertical, or at right angles 
 to AE. 
 
34 MECHANICS. 
 
 Then in the triangles EFD and ABC, 
 
 angle DFE wangle ABC, y .Fis, 1?C are parallel, and 
 
 AFB cuts them ; 
 
 and angle FDE = right angle = angle I? C-d, 
 
 therefore angle DEF = angle C^4J5, and the triangles are 
 equiangular and similar. 
 
 Now, by hypothesis, 
 
 P : W :: BC : AB 
 
 :: DF : FE, by equiangular triangles. 
 
 But the same effect that is produced on D by the two 
 forces which are represented in magnitude and direction 
 by DF and FE, will be produced by their resultant. But 
 their resultant DE (Art. 29- Cor.), being perpendicular 
 to the plane AB, will be entirely counteracted and destroy- 
 ed by that plane, and no motion can be communicated by 
 it to the body at D. The body at D, therefore, will remain 
 at rest on the inclined plane when acted on by the two 
 forces P and W which bear to each other the ratio of 
 BC to AB. 
 
 Wherefore, "The Weight, &c." q.e.d. 
 
 37^ Def. By the Velocity of a body in motion is 
 meant the degree of quickness, or speed, with which the 
 body is moving. 
 
 £This "degree of quickness" is described, or measured, 
 by saying how long the line is that the body moves through 
 in some given portion of time. Thus a clear notion would 
 be conveyed of the Velocity of a coach, if it were said to 
 be nine miles in an hour; the space moved over by the 
 coach being nine miles, and an hour being the portion of 
 time during which the motion takes place. If only the 
 space that is described were mentioned, and nothing were 
 said about the time taken up in describing that space, or 
 if the time alone were given, it is evident that no idea at 
 all could be formed of the degree of quickness, that is, of 
 the Velocity, of the coach's motion. 
 
 The Velocity of a body is measured, in mathematical 
 investigations, by the number of feet passed over by the 
 body in a second of time. 
 
MECHANICS. 35 
 
 Cor. Since the quicker a body moves the more space 
 it will pass over in a given time, it will follow from the 
 observations just made, that The Velocities of two bodies 
 that move during any (the same) time, are as the spaces 
 which the bodies respectively describe in that time.~\ 
 
 38. Prop. xv. Assuming that the arcs which 
 subtend equal angles at the centers of two circles 
 are as the radii of the circles, to shew that if P and 
 W balance each other on the "Wheel and Axle, and 
 the whole be put in motion, P : W :: IP's velocity 
 : P's velocity. 
 
 Suppose P and TV to act at the circumferences of the 
 Wheel and the Axle, in the same plane, 
 perpendicularly to the radii OM and ON 
 (as in Prop, x.); and let the whole be 
 put into motion round the axis so that y 
 mOn may become the horizontal dia- 
 meter. P will now act at m at right 
 angles to Om, and TV will act at n at 
 right angles to On; and the velocities pV 
 of P and TV will be as Mm to Nn, since 
 the former of these arcs is the length through which P 
 will have descended in the same time that TV has ascended 
 through Nn. 
 
 .\ Vel. of P : vel. of TV :: Mm : Nn, (Art. 37. Cor.), 
 
 :: OM : ON Q>y the assumption 
 
 made in the enunciation], 
 
 :: TV : P, by Prop, x; because, 
 
 by the hypothesis, P and TV originally balanced each other. 
 Wherefore, " Assuming, &c." q.e.d. 
 
 39. Prop. xvi. To shew that if P and W 
 balance each other on the Machines described in Pro- 
 positions xi, xn, xin, and xiv, and the whole be 
 put in motion, P : W :: Ws velocity in the direc- 
 
36 MECHANICS. 
 
 tion of gravity : P's velocity [in the direction in which 
 it acts.] 
 
 (1) Let C, the center of the moveable pulley (see Fi^. 
 Prop, xi.), be raised through any height, as an inch ; TV 
 will thereby be also raised through an inch, and each of 
 the strings RB and AP will have been shortened an inch ; 
 so that if P continue to keep the string tight, it will have 
 moved through two inches in the time that TV has been 
 raised one inch. 
 
 Now P : W :: 1 : 2, Prop, xi, 
 
 :: 1 inch : 2 inches, 
 
 :: TV's velocity : P's velocity; Art. 37* Cor. q.e.d. 
 
 (2) In the system where the same string passes round 
 all the pulleys, and the parts of it between the pulleys are 
 parallel, as in Prop, xn, (see Fig.), if the lower block be 
 raised an inch, each of the strings between the upper and 
 lower pulleys will be shortened an inch, and, there being 
 2n of such strings, P must have moved through 2n inches 
 in the time that TV moved through one inch, in order to 
 have kept the string tight. 
 
 And P : TV :: 1 : 2w; Prop, xu, 
 
 :: 1 inch : 2 n inches, 
 
 :: TV's velocity : P's velocity; Art. 37. Cor. q.e.d. 
 
 (3) In the system where each pulley hangs by a sepa- 
 rate string and the strings are parallel (Prop, xin), if TV 
 be raised through an inch, and P have also moved through 
 such a space that the strings are kept tight, A will have 
 been raised through one inch, and B through two inches 
 (by the first case proved in this Proposition). And B 
 having been raised through two inches, C (by the first 
 case) will have moved through 2x2, or 2 2 , inches ; and 
 the next moveable pulley (the fourth) would have been 
 raised through 2 x 2 2 , or 2 3 , inches. By the same kind 
 of reasoning, if n were the number of moveable pulleys, 
 the highest of them would have moved through 2 n-1 inches, 
 and the end therefore of the string by which P acts would 
 have moved through 2" inches ; but 
 
II G^ 
 
 P^ 
 
 /^ 
 
 \i 
 
 ^>^u 
 
 
 
 c 
 
 MECHANICS. 37 
 
 P : JF :: 1 : 2", Prop, xiii.; 
 :: 1 inch : 2" inches 
 :: TV's velocity : P's velocity ; Art. 37- Cor. q.e.d. 
 
 (4) Let the weight {TV) be kept at rest at D on the 
 inclined plane AB by the power (P) which acts parallel 
 to the plane by means of a 
 string DP. Suppose P, by 
 moving through the space P p, 
 raises TV through an equal 
 space DG. Through G draw . 
 GH horizontal, and through 
 D draw DH vertical. Then, by being moved through DG, 
 TV has been raised through a vertical height DH ; DH 
 being the vertical line which cuts the horizontal lines pass- 
 ing through the two positions of TV. Since, therefore, in 
 the time that P moves through a space equal to DG, the 
 vertical height that TV moves through is DH, DG is to 
 DH as the velocity of P in the direction of its action is 
 to the velocity of TV in the direction of gravity. 
 
 Now, in the triangles GDH and ABC, 
 
 since GH is parallel to AC, and AG meets them; 
 
 .*. angle DGH = alternate angle BAC ; 
 
 and since DH, being vertical, is parallel to BC, and DB 
 meets them ; 
 
 .'. angle GDH '= alternate angle ABC; 
 
 .-. also angle DHG = angle BCA, and the triangles are 
 equiangular, and .'. similar. 
 
 And P : TV :: BC : AB ; Prop, xiv.; 
 
 :: DH : DG, by similar triangles. 
 
 :: TV's vel. in direction of gravity : P's vel. 
 in the direction of its action. 
 
 Wherefore, " If P and TV balance, &c." Q. e. d. 
 
CHAPTER Y. 
 
 THE CENTER OF GRAVITY. 
 
 40. Def. The center of gravity of a body, or 
 of a system of any number of bodies connected together, 
 by inflexible bars which are without weight," 1 is that point 
 upon which the body, or the system, will balance when 
 placed in any position whatever; the point itself being 
 maintained at rest, and the body, or the system, being acted 
 on by no other pressure than that arising from the weight 
 of the matter composing it. 
 
 [The Center of Gravity of a body is, as it were, that 
 fulcrum, round which the body, when placed in any po- 
 sition^iias, of itself, no kind of tendency whatever to turn, 
 although the body be capable of being moved in any plane, 
 and in any direction, about that fulcrum.] 
 
 41. Prop. xvii. If a body balance itself upon 
 a line in all positions, the center of gravity of the 
 body is in that line*. 
 
 If possible, let G, the center of gravity of the body 
 AB, not lie in the line CD on C 
 which the body balances in all posi- 
 tions ; and let the body and line be 
 so placed, that G and CD may be 
 in the same horizontal plane. From 
 G draw GH perpendicular to CD, 
 and GF vertical. 
 
 Now since (by Def. Art. 40,) if G be supported the 
 body remains at rest when turned round the line into any 
 position, the vertical pressure, therefore, which arises from 
 
 * That is to say ; If there be a line round which a body can be made 
 to revolve, such that if when the line is put into any position, the body, 
 after being made to revolve round the line into any position, remains at 
 rest without having any tendency to move, the center of gravity of the 
 tody lies in that line. 
 
MECHANICS. 39 
 
 the weight of the matter composing the body, must always 
 pass through G, or else that pressure could not be rendered 
 inoperative (as it is supposed to be) by G being maintained 
 at rest. Wherefore, in the position of the body represented 
 in the figure, this pressure will act perpendicularly at an arm 
 GH, and have a tendency to turn the body round the point 
 H, which (since H is a point in a line CD fixed in position) 
 may be considered to be a fixed fulcrum. But, by hypo- 
 thesis, when CD is supported the body balances in all 
 positions upon it. Wherefore G does not lie out of the 
 line CD, i.e. it lies in it. q. e. d. 
 
 42. Prop, xviii. To find the center of gravity 
 of two heavy points*, and to shew that the pressure 
 at the center of gravity is equal to the sum of the 
 weights in all positions. 
 
 Let P and Q be the weights of two heavy points A 
 and B. Join AB by a straight inflexible bar without 
 weight, and take 
 
 AC : AB :: Q : P+Q; 
 and .-. AC : AB- 
 :: Q : (P + Q) 
 
 Let ACB be in any position. Through C draw MCN 
 horizontal; and through A and B draw the vertical lines 
 MAP and BNQ; these last are the lines in which the 
 weights P and Q act. 
 
 Then, the angles at M and N being right angles, and 
 the angle ACM being equal to the opposite angle BCN, 
 the angle CAM is equal to the angle CBN, and the tri- 
 angles A CM and BCN are equiangular, and .-. similar. 
 
 Now, supposing P to act on the horizontal lever MCN 
 at M at right angles to CM, (Art. 15), and Q to act at 
 ^ N at right angles to CN, since 
 
 * The "heavy points" spoken of in this Proposition and the next, 
 are exceedingly small material bodies, and not geometrical points, which 
 (being denned by Euclid to be "without parts") do not possess Length, 
 or Breadth, or ThicknesSj and consequently can be of no AVeight, since 
 they can contain no Matter. 
 
40 
 
 MECHANICS. 
 
 P : Q :: CB : CA 
 
 :: CN : CM, by similar triangles, 
 
 therefore P and Q will balance on a fulcrum placed at C, 
 and the pressure on C is P + Q, acting vertically. Props, u. 
 and vu. 
 
 And this, being true for any position, is true for all 
 positions of A and B. Wherefore, by the definition of the 
 Center of Gravity, C is the Center of Gravity of A and 
 B ; and the pressure at C, in all positions of A and B, 
 is P+Q, acting vertically downwards. 
 
 43. Prop. xix. To find the center of gravity 
 of any number of heavy points ; and to shew that 
 the pressure on the center of gravity is equal to the 
 sum of the weights in all positions. 
 
 Let A, B, C, three heavy points whose weights are 
 P, Q, R, be connected together and 
 placed in any position. 
 
 Join AB, and take D a point in A 
 AB such that 
 
 F+Q 
 
 AD : AB :: Q : P+Q; p y 
 
 .-. AD : AB-AD, or DB, :: Q: (P + Q)- Q, or P; 
 
 therefore D is the center of gravity of A and B, and 
 the pressure produced by P and Q in all positions of the 
 system, is a pressure P+Q acting vertically at D. Prop. 
 
 XVIII. 
 
 Join DC, and take in DC a point E such that 
 
 DE : DC :: R : P+Q + R; 
 
 r.DE : EC :: R : P+Q; 
 
 therefore E is the center of gravity of the weights P+Q 
 acting at D and R acting at C ; and if E be supported, 
 those weights are supported in any position of the system. 
 Since therefore the system will balance itself in all positions 
 on E, that point is its Center of Gravity; — and the Pressure 
 on E is P+Q + R. 
 
MECHANICS. 41 
 
 The construction here applied to a system of three 
 bodies is applicable to a system of any number of bodies. 
 
 Wherefore, " The center of gravity of any number of 
 heavy points may always be found, and the pressure on 
 the center of gravity is equal to the sum of the weights." 
 
 Q. E.D. 
 
 44. [By the definition given in Art. 40,, of " the Center 
 of Gravity" of a body, it will be understood that to have a 
 center of gravity a body must have Weight. Now in the 
 next two Propositions it is required to find the centers of 
 gravity of a line, and of a plane; the former of which is 
 defined by Euclid to have length merely, without either 
 breadth or thickness; and the latter, though possessing 
 length and breadth, is defined to be without thickness. A 
 geometrical line, or plane, therefore, can have no weight; 
 since there can be no weight where matter does not exist, 
 and when matter exists under any form whatever, it is of 
 three dimensions, or has length, and breadth, and thickness. 
 The line, therefore, and the plane, of which it is required to 
 find the centers of gravity, are not the line and plane of 
 geometry. 
 
 But the line of which the center of gravity is deter- 
 mined in the next Proposition, is supposed to be formed 
 of very small equal heavy bodies placed, at the same dis- 
 tance one from the other, along the whole length of the 
 line. And the plane triangle referred to in the next Propo- 
 sition but one, is supposed to be made up of such lines 
 arranged parallel to any one of the sides of the triangle, 
 and at equal distances one from the other.]] 
 
 45. Prop. xx. To find the center of gravity of 
 a straight line. 
 
 Let AB be a straight line composed of small equal 
 heavy bodies ranged at equal distances one from the other 
 from end to end of it. 
 
 Bisect AB in C, and let P and Q be two of the small 
 heavy bodies equally distant from C. 
 
 Then P and Q will balance in every '^-E c 9 fi 
 
 position on C. And since the same ^ 
 
 is true of all such pairs of heavy bodies equidistant 
 from C, the whole line will balance on C in every 
 
 4—3 
 
 
 k 
 
42 MECHANICS. 
 
 position, and therefore C is the center of gravity of the 
 line. 
 
 46. Pitoi\ xxi. To find the center of gravity of 
 a triangle. 
 
 Let ABC be a triangle formed of lines ranged, at 
 equal distances, parallel to any 
 one of the sides, the lines them- 
 selves being made up of small 
 equal heavy bodies placed at 
 equal distances. Let bfc, paral- 
 lel to BFC, be one of these lines, a^ r — "^c 
 
 Bisect AB in E and BC in F; join AF, CE by lines 
 intersecting in G; — G is the center of gravity of the tri- 
 angle. 
 
 For Af : fb :: AF : FB, (from the equiangular tri- 
 angles, ABF, Abf), 
 
 :: AF : FC, v BF = FC; 
 
 :: Af : fc, (from the equiangular triangles, 
 AFC, Afc). 
 
 And since the first and third terms of this proportion 
 are the same, fb is also equal to fc; and therefore the straight 
 line b c would balance in any position on/. Prop. xx. 
 
 In the same manner all the other lines parallel to BC 
 might be shewn to balance in any position on the points 
 in which they are cut by AF; therefore the whole triangle 
 would balance on AF in any position, and therefore the 
 center of gravity of the triangle is in AF, 
 
 Similarly it may be shewn that the center of gravity 
 of the triangle is in the line CE. 
 
 Wherefore G, the intersection of AF and CE, is the 
 center of gravity of the triangle ABC. 
 
 47. Prop. xxii. When a body is placed on a 
 horizontal base, it will stand or fall, according as 
 the vertical line drawn from its center of gravity 
 falls within or without the base. 
 
 Let GH be a vertical line drawn through G the center 
 
vzLP 
 
 V 
 
 MECHANICS. 43 
 
 of gravity of the body, and meeting the horizontal plane 
 
 on which the body is 
 
 placed in H ; and let > 
 
 ABH be any hori- 
 zontal line drawn 
 
 through H and ter- " ' A B"""|lT 
 
 minated at A and B, 
 points in the bound- 
 ary of the base of the body. 
 
 (1) Let H fall within the base; it is therefore between 
 A and B. 
 
 Then AB being considered as a horizontal lever acted 
 on at H (Art. 13) by the weight of the body/^no motion 
 can take place round A, because the tendency of the body's 
 weight would be to draw the lever downwards round A, 
 a motion which the resistance of the plane will prevent. 
 For the same reason no motion can take place round B ; 
 and therefore the body will fall over neither at A nor B. 
 
 In the same manner it may be shewn that the body 
 will not fall over at either extremity of any other hori- 
 zontal straight line which is drawn through H, and is 
 terminated by the boundary of the base. 
 
 Wherefore the body will remain at rest, when the ver- 
 tical drawn from its center of gravity falls within the base. 
 
 (2) If H fall without AB,(b.s in the second figure, no 
 motion of the lever ABH can take place round A as ful- 
 crum, because the effect of the weight acting at H would be 
 to draw AH downwards, a motion which the plane pre- 
 vents. But if B be considered as the fulcrum^ the effect 
 of the weight at H would be to make the line AB to 
 move upwards from the plane round B, and as there is 
 no force to prevent this motion taking place, the body, 
 though it were prevented from falling over at every other 
 point of the boundary of the base, would still fall over 
 at B. When, therefore, H falls outside the horizontal 
 base, the body cannot stand, but must fall over at some 
 point or other in the boundary of the base.* 
 
 Wherefore, " When a body, &c." q.e.d. 
 
 * The demonstration of the second case requires it to be understood 
 that no line whatever can be drawn through II, meeting the circumference 
 of the base in points which are on opposite sides of H. 
 
44 MECHANICS. 
 
 48. Prop, xxiii. When a body is supended from 
 a point [round which it can swing freely], it will rest 
 with its center of gravity in the vertical line passing 
 through the point of suspension. 
 
 Let AB be the body in any position, C the point from 
 which it is suspended, G its center of gra- 
 vity, NGK a vertical line drawn through 
 G, CN a perpendicular from C on NGK. 
 
 Then since the weight of the body acts 
 in the vertical line which passes through G 
 the center of gravity, it may be supposed 
 to be applied at A T ; and, CN being con- 
 sidered as a lever moveable round C as ful- 
 crum, since there is no force to counteract 
 W acting perpendicularly at N on CN, 
 motion must ensue. 
 
 But if NG pass through C, CN vanishes, the weight 
 is sustained by the immoveable fulcrum C, and the body 
 is at rest. 
 
 Wherefore, "When a body, &c." q.e.d. {*JL . 
 
 fuutti C (j> J , £c C<y7i^£ tL£Z<^t~Ljct , suLi^t if a fit. 
 
THE ELEMENTS 
 
 HYDROSTATICS. 
 
 The explanatory articles and paragraphs which are enclosed in 
 brackets form no part of the Course adopted by the University. Those 
 in smaller type are illustrative of the Definitions and Propositions they 
 immediately follow. The whole of the extra matter so introduced forms 
 but a very small, and a very easy, portion of the book ; and the Student 
 icho comes to this subject for the first time had better not omit it. 
 
 CHAPTER I. 
 
 1. The Science of Hydrostatics investigates the 
 conditions fulfilled by the pressures produced by fluids 
 wh en in a state of rest. 
 
 2. Def. A Fluid is a material body of which the 
 parts can be divided in any direction, and moved among 
 one another by any force, however small. 
 
 [[Mercury, Water, Air, Gas, Steam, are all instances 
 of fluids.] j 
 
 3. Fluids have been divided into Elastic, and Non- 
 clastic. 
 
 (1) Elastic Fluids are those of which the dimensions 
 are increased or diminished when the pressure upon them is 
 diminished or increased. 
 
 £Air is an elastic fluid, as will hereafter be proved. 
 So also is Gas, and Steam.] 
 
 (2) Non-Elastic Fluids are those of which the di- 
 mensions are not diminished by the addition of pressure, 
 or increased by the withdrawal of pressure. 
 
46 HYDROSTATICS. 
 
 QWater, Mercury, and probably all other liquids, are 
 compressible, but in a slight degree only. The resistance, 
 however, which they offer to compression is so great, that 
 the conclusions obtained on the supposition of their being 
 entirely incompressible are free from any sensible error, 
 except in cases where the pressure is exceedingly great.] 
 
 CHAPTER II. 
 
 PRESSURE OF NON-ELASTIC FLUIDS. 
 
 4. Prop. i. Fluids press equally in all direc- 
 tions. 
 
 Let a close vessel of any shape be filled with fluid, 
 and let A, B, C and D, similar and equal 
 portions (of any size and shape) of the sides 
 of the vessel, be removed, and their places 
 supplied by plugs fitting the orifices exactly 
 and acted upon by pressures just sufficient 
 to keep them at rest. Then if an additional force P be 
 applied to any one of the plugs, it is found, by experi- 
 ment, that an equal additional pressure must be applied 
 to every other of the plugs, to prevent the fluid from burst- 
 ing out. 
 
 This proves that a pressure communicated to the sur- 
 face of a fluid at rest is transmitted by the fluid, undi- 
 minished and unimpaired, to every other equal and similar 
 portion of the surface by w r hich the fluid is bounded. 
 
 Also, whatever be the directions in which any of these 
 plugs are inserted through the sides of the vessel, the 
 same quantity of pressure is required to keep each of them 
 at rest; which shews that, at the same point in a fluid, 
 the fluid presses with equal force in all directions. 
 
 Wherefore, "Fluids press equally in all directions." 
 
 Q. E. D. 
 
HYDROSTATICS. 47 
 
 5. Prop. ii. The pressure upon any particle 
 of a fluid of uniform density is proportional to its 
 depth below the surface of the fluid. 
 
 Let the vertical lines PM and QN be drawn through 
 P and Q two particles situated in the ODMN 
 
 interior of an uniformly dense fluid which \ 1/ I 
 
 is at rest, and let them meet the surface C\_*Rpl Q, 
 of the fluid in the points M and N. 
 
 Suppose the pressures on P and Q to be produced by 
 the weight of fluid particles, of the same magnitude as 
 P and Q, placed touching each other along the lines PM 
 and QN. Then, since the fluid is of uniform density, 
 the weights of such lines of particles will be proportional 
 to their lengths. 
 
 .•. Pressure on the particle P 
 
 : pressure on the particle Q 
 
 :: weight of the line PM of particles 
 
 : weight of the line QN of particles 
 
 :: PM : QN. 
 
 Again;,- the fluid being at rest, the pressure of any part 
 of it will not be affected if the particles (which are at rest) 
 of the fluid upon which the pressure acts, be supposed 
 to become connected with one another. Whence it will 
 follow, that whether there be an uninterrupted line of fluid 
 particles (as RO) reaching from a particle R to the surface 
 of the fluid, or part of the line be cut off by a portion ACD 
 of the fluid becoming solid, the pressure upon the particle 
 R will still be the same. 
 
 Wherefore, " The pressure, &c." q. e. d. 
 
 6. Pitor. in. When a fluid is at rest its sur- 
 face is horizontal. 
 
 [The manner in which the matter composing non-elastic fluids acts is 
 twofold ; first, by the particles pressing with their Aveights on those im- 
 mediately below them ; and, second, by the property these particles pos- 
 sess of communicating, in all directions and without diminution, any 
 pressure to which they are subjected.] 
 
Q 
 
 48 HYDROSTATICS. 
 
 Imagine two contiguous particles P and Q, situated in 
 the same horizontal plane in a fluid at rest. Since a & 
 the action of gravity on these particles is perpen- 
 dicular to the plane in which they lie, its action on 
 either of them can produce no effect in that plane. 
 It is in consequence, therefore, of the action of the 
 fluid which surrounds them, that the particles P and Q 
 have no horizontal motion: and this action must be the 
 same on each, or motion would ensue. 
 
 But since fluids press equally in all directions (Prop, i.), 
 and the horizontal pressures on P and Q cannot but be 
 equal, the vertical pressures on them, which are as the 
 distances Pa and Qb below the surface of the fluid 
 (Prop, n.), must also be equal; and therefore 
 
 Pa = Qb; 
 
 .'. a b is parallel to PQ, and is .*. horizontal. 
 
 And since, in like manner, the line joining any two ad- 
 jacent particles in the surface of the fluid may be shewn to 
 be horizontal, the surface itself is horizontal. 
 
 Wherefore, " When a fluid, &c." q. e. d. 
 
 7. Prop. iv. If a vessel, the bottom of which is 
 horizontal and the sides vertical, be filled with fluid, 
 the pressure on the bottom will be equal to the weight 
 of the fluid. 
 
 The sides of the vessel being vertical, and the whole of 
 the fluid being conceived to be made up of vertical straight 
 lines of fluid particles, each of these lines will press vertically 
 with its weight, and the sum of these vertical pressures will 
 be the weight of the fluid in the vessel. Now the base of 
 the vessel, being horizontal, will counterbalance all the ver- 
 tical pressures upon it, and destroy them entirely. The 
 pressure, therefore, sustained by the horizontal base of a 
 vessel whose sides are vertical, is equal to the weight of the 
 fluid in the vessel. Q. e. d. 
 
 8. [[From Propositions 11 and iv the following con- 
 clusion may be drawn. 
 
HYDROSTATICS. 49 
 
 The pressure exercised by a fluid on any horizontal 
 plane placed in it, is equal to the weight of a column 
 of the fluid whose base is the area of the plane, and 
 whose height is the depth of the plane below the hori- 
 zontal surface of the fluid. 
 
 Let AB and CD be two equal areas in the same ho- 
 rizontal plane. Since the pressure is the 
 same on every point in the same horizontal 
 plane, and (the areas AB and CD being 
 equal) the number of particles in the plane 
 CD is the same as that in the plane AB, therefore the 
 pressure on AB = pressure on CD. 
 
 Let AB be an area such that a vertical line of fluid 
 jiarticles reaches from every particle in it to the surface, 
 and suppose the rest of the fluid to become solid; EABF 
 then becomes a vessel with vertical sides and horizontal 
 base, and the pressure on AB = weight of the column EB 
 of fluid. Prop. iv. 
 
 Therefore, the pressure on any horizontal area CD = 
 weight of a column of the fluid whose base is CD, and 
 whose height is the depth of CD below the horizontal plane 
 of the surface of the fluid in the vessel, q. e. d/] 
 
 9. QFrom Art. 43 of the Mechanics it appears that the 
 pressure on the Center of Gravity of a system of bodies con- 
 sidered as heavy points, is the same as if the weights of the 
 bodies were collected at it. So long therefore as the Quan- 
 tity of Matter in the system remains the same, the amount 
 of Pressure produced by the weights of the several parts of 
 the system is the same whatever be the manner in which 
 they may be arranged. 
 
 Now though at first sight it might appear probable, that 
 the pressure on the bottom or the sides of a vessel filled with 
 fluid would depend, somehow or other, on the quantity of 
 the fluid by which that pressure is produced, such is by no 
 means the case ; for it is found, both from experience and by 
 theory, that the pressure produced by a fluid, at rest, on 
 the surface of the vessel containing it, is not dependent on 
 the quantity of the fluid; a fact apparently so much at 
 variance with the law governing the amount of pressure 
 produced by Solid bodies, that it has been called the 
 Hydrostatic Paradox.! 
 5 
 
50 HYDROSTATICS. 
 
 10. Prop. v. To explain the Hydrostatic Pa- 
 radox. 
 
 The Hydrostatic Paradox is this; that "any Pressure, 
 however small, may, by means of a fluid, be made to counter- 
 balance any other Pressure, however large ; and whether the 
 quantity of fluid employed be great or small, it is possible to 
 produce this result." 
 
 The top and bottom of a vessel AD are boards which 
 are connected together by leathern sides. 
 The vessel communicates with a vertical 
 tube EF of small uniform bore, by means 
 of a horizontal pipe CE. Let (A) be the 
 number of square inches in the area of the 
 board AB, and {a) the number in the area 
 of a horizontal section of the pipe EF. 
 
 W 
 
 Let AB be held in a horizontal posi- 
 tion, and water be poured into the tube E c d 
 until it just rises in AD to AB. The water will therefore 
 rise in the tube to G, a point in the same horizontal plane 
 as AB; Prop. in. 
 
 If now there be a heavy weight W laid upon AB, it is 
 found that it can be supported at rest by a small quantity 
 FG of water poured into the tube. To shew the reason of 
 this. 
 
 Since the sides of the tube FG are vertical, the pressure 
 on the horizontal section of the tube at G is the weight of 
 the fluid column FG. Suppose w to be this weight, 
 
 Then, since the pressures at every point in the same 
 horizontal plane of a fluid at rest are the same, 
 
 .\ Pressure on the area (A) of AB : pressure on the area 
 («) at G :: A : a, 
 
 i. e., W : ru :: A : a ; 
 
 If therefore W and A be given, w may be made as small 
 as may be wished by diminishing a. 
 
HYDROSTATICS. 51 
 
 Whether the pressure on the area (a) at G be produced 
 by the weight of the fluid column GF, or by means of a 
 plug acted on by some force, the pressures on the areas (a) 
 at G and (A) at AB will still bear to one another the ratio 
 of a to A ; a ratio which is wholly independent of the 
 quantity of fluid contained in the vessel. 
 
 11. Prop. vi. If a body float on a fluid, it 
 displaces as much of the fluid as is equal in weight to 
 the weight of the body ; and it presses downwards, 
 and is pressed upwards, with a force equal to the 
 weight of the fluid displaced. 
 
 Let ABCD be a body that displaces the portion 
 BEDCB of the fluid on which it floats. A 
 
 The weight of the floating body pro- y~^~^* 
 
 duces a pressure which acts vertically ^ ^J - - .— ~ -)12-.--_ 
 
 downwards. Therefore, the effective jjlj^ ^c > fc^J C 
 pressure of the fluid, by which the body 
 
 is kept at rest, must act vertically upwards, and be equal 
 to the weight it balances. 
 
 Now let the floating body be removed, and the space 
 BEDCB be filled with fluid the same as that on which 
 the body floated. The equilibrium of the fluid will not 
 be disturbed, neither will the pressure of that part of it 
 which was formerly in contact with the surface of the 
 floating body be altered, if the parts of the fluid now in 
 BEDCB be supposed to become permanently connected 
 with one another, and so to form a solid. 
 
 Let this take place. The pressure downwards of 
 BEDCB is its weight; and since this pressure is counter- 
 acted by the same sustaining power as that which balanced 
 the weight of the floating body, it follows that the weight 
 of the floating body is equal to the weight of the fluid it 
 displaces. 
 
 Wherefore, " If a body float on a fluid, &c." Q. e. d. 
 
CHAPTER III. 
 
 SPECIFIC GRAVITIES. 
 
 12. The Bulk, or Volume, or Content, or Mag- 
 nitude of a solid body is measured by the number of 
 times the body is greater than some particular body pre- 
 viously fixed upon to compare the sizes of other bodies by, 
 which is called " the unit of solid measurement." 
 
 A cube of which an edge is an inch in length is called 
 "a cubic inch." In the following pages a cubic inch will 
 be taken for the unit of measurement ; so that when it is 
 said that M is the bulk, volume, content, or magnitude, of a 
 body, it is meant that the body is M cubic inches in size. 
 
 13. Def. The Specific Gravity of any substance 
 is the weight of an unit of its magnitude. 
 
 If, fas stated in the last Article,^ the magnitude of a 
 body be measured by the number of cubic inches it con- 
 tains, and the weight of a cubic inch of the substance 
 the body is composed of be given in grains, then it will 
 follow, that, S being taken to represent the Specific Gravity 
 of any substance, S is the number of grains that one cubic 
 inch of that substance weighs.* 
 
 * The Tables commonly called " Tables of Specific Gra- 
 vities" give the ratios which the weights of bulks of various substances 
 bear to equal bulks of Water. 
 
 It having been found, by experiment, that the weight of a piece of 
 Iron : the weight of a bulk of Water of the same size :: 7'8 : 1, and 
 that the weight of a piece of Silver : weight of an equal bulk of Water 
 :: 10-5 : 1, and so for other substances, — tables have been formed in 
 which the numbers 1, 7'8, 105, &c, are placed opposite the words 
 "Water" "Iron" "Silver" &c. By means of these tables, (as will 
 be shewn), the weight of any bulk of any of the substances so registered 
 can be determined, if only the weight be known of some particular 
 magnitude (such as a cubic inch) of some one of them. 
 
 [The numbers given in these Tables are called, by some writers, 
 the " Specific Gravities" of the several substances registered ; but the 
 enunciation of Prop. vn. Art. 14, prevents their being so called here.] 
 
HYDROSTATICS. 53 
 
 14. Pitop. vii. If M be the Magnitude of a 
 body, S its Specific Gravity, and W its Weight, 
 W = MS. 
 
 Let M= number of cubic inches in the body. 
 
 S = Specific Gravity of the body; i. e. let S be the 
 number of grains that one cubic inch of it 
 weighs. 
 
 W = number of grains that the whole body weighs. 
 
 grains grains c. inches c. inch 
 
 Then since W : S :: M : I; 
 
 .'. W=Mx S. Q. E.D. 
 
 tt The Tables of Specific Gravities" give 
 
 Platina 21*5 Zinc 6'9 
 
 Gold 19-4 Sea Water 1-027 
 
 Mercury 13-6 Water 1- 
 
 Lead 11-4 Proof Spirit 0*930 
 
 Silver 10-5 Pure Alcohol 0825 
 
 Copper 8-9 Air,atthesurfaceofthe ) n nnio . 
 
 Iron 7*8 Earth,_the average < U UU1J5 
 
 Tin 7-3 
 
 Hence it may be shewn that " the weights of any two substances 
 of equal bulk are in the ratio of the numbers given by the Tables as 
 corresponding to the substances." 
 
 For w, w', w" being the respective weights of equal bulks of Water, 
 Iron, and Silver, since, as explained above, 
 
 w' : to :: 7'8 : 1, and to : w" :: 1 : 10-5; 
 
 therefore, compounding these proportions, 
 
 w' : w" :: 7'8 : 10"5. 
 
 So that if it be required to find the weight of a cubic foot of Iron, 
 having given that the weight of 10 cubic inches of Silver is 01 ounces 
 nearly, 
 
 V Weight of a cubic foot (or 12 x 12 x 12 cubic inches) of Silver, 
 
 = 12 x 12 x 12 x jTj ounces; 
 
 ttt • r, „ c . . - , T 12x12x12x01 7-8 
 .-. Weight of a cubic foot of Iron = ^ x ~- oz. 
 
 12xl2x 12x61x7-8 „__. 
 
 jQg = /830 ounces, nearly. 
 
 5—3 
 
54 HYDROSTATICS. 
 
 15. Pnoi\ viii. When a body of uniform density 
 floats on a fluid, the part immersed : the whole body 
 :: the specific gravity of the body : the specific gravity 
 of the fluid. 
 
 Let a solid of uniform density float on a fluid with M 
 cubic inches of it above the horizontal yj\ 
 
 plane of the surface, and N cubic inches y — ----- — ^ ggp 
 
 below that surface. " z==r ^^ Jf r - ,^JiSS^ 
 
 Let S=S.G. (Specific Gravity) of the solid; S'=S.G. 
 of the fluid. 
 
 Then, (M+ N) x S = weight of the solid. Prop. vu. 
 N x S'= fluid displaced. 
 
 But, because the body floats, these two weights are 
 equal, Prop. vi. ; 
 
 .:N.Sr={M+N)-Si and.-. ^^J, 
 
 or N 1 : M + N :: S : S'; 
 
 i. e. ; The part of it immersed : the whole of the body 
 :: S. G. of the body : S. G. of the fluid, q. e. d. 
 
 16. Prop. ix. When a body is immersed in a 
 fluid, the weight lost : whole weight of the body :: 
 the specific gravity of the fluid : the specific gravity 
 of the body. 
 
 Let M be the number of cubic inches contained in a 
 body of uniform density which is wholly 
 immersed in a fluid ; S the S. G. of the body, 
 S' the S. G of the fluid. 
 
 Then the pressure downwards of the solid 
 is its weight; and if the solid be removed, and 
 the space it filled be occupied by an equal bulk 
 of the fluid, equilibrium will still remain. And if the fluid 
 so added become solid, the equilibrium will continue, and 
 the pressure upwards of the surrounding fluid will remain 
 the same as before. 
 
HYDROSTATICS. 55 
 
 Now the pressure downwards produced by the fluid 
 that becomes solid is its weight. And since the pressure 
 upwards of the surrounding fluid supports this weight, that 
 pressure must be exactly equal and opposite to it. 
 
 The pressure downwards, therefore, before it was im- 
 mersed, of the original solid, (i. e. its weight MS, Prop, vu.), 
 must, by the solid having been immersed, have been di- 
 minished by a pressure upwards, arising from the surround- 
 ing fluid, exactly equal to the weight of the fluid displaced; 
 which weight, by Prop. vu. is equal to MS'. 
 
 .-. Weight lost by the body : the whole weight of the 
 body :: MS' : MS 
 
 :: S' : S 
 
 :: S. G. of fluid : S. G. of body. Q. e. d. 
 
 17. [[It appears from the proof of the last Proposition, 
 that the pressure of a fluid on a body wholly immersed in it 
 acts vertically upwards, and is equal to the weight of the 
 fluid which the body displaces. 
 
 If this pressure be less than the weight of the body, — 
 (that is, if the S.G. of the fluid be less than that of the 
 solid) — the pressure downwards arising from the weight of 
 the solid will be greater than the pressure of the surround- 
 ing fluid upwards, and the body will therefore sink to the 
 bottom of the vessel. 
 
 But if the pressure of the fluid upwards be greater than 
 the weight of the body immersed, — (that is, if the S.G. of 
 the fluid be greater than that of the solid) — the pressure 
 upwards will be greater than the pressure downwards, and 
 the body will therefore rise until the conditions of Propo- 
 sition viii are fulfilled, and will then float] 
 
 18. Prop. x. (1) To describe the Hydrostatic 
 Balance ; and (2) to shew how the Specific Gravity 
 of a body may be found by it,— first, when its Spe- 
 cific Gravity is greater than that of the fluid in which 
 it is weighed, — and second, when it is less. 
 
56 HYDROSTATICS. 
 
 (1) The Hydrostatic Balance is the common balance 
 with a hook attached to the under C 
 part of one of its scales, so that / v ' 
 bodies may be weighed either by 
 putting them into the scale, or by 
 suspending them from it and letting 
 them be immersed in a fluid, as here 
 represented. 
 
 (2) First, let the S. G. (Specific Gravity) of the body 
 be greater than that of the fluid. 
 
 Since the S.G. of the solid is greater than that of the 
 fluid, the body will sink in the fluid. Prop. ix. Cor. 
 
 Let S = S. G. of the solid, S' = S. G. of the fluid. 
 
 W— weight required to balance the body when 
 placed in the scale. 
 
 W'= weight required when the body is immersed. 
 
 .•. W- W f = diminution of the body's weight in consequence 
 of immersion. 
 
 Therefore, by Prop, ix., 
 
 W : W- W :: S : S'; 
 
 # TVS' 
 
 " * W-W 
 
 Whence, W and W being known weights, S may be de- 
 termined if S' be given. 
 
 Next, let the S. G. of the body be less than the S. G. 
 of the fluid. 
 
 When the body, in this case, is forced under the sur- 
 face of the fluid, the pressure downwards on it (which is 
 the weight 3 W of the body) being less than the pressu e 
 upwards (which is the weight of a quantity of fluid equal 
 in magnitude to the fluid displaced) the body must on the 
 whole be acted on by a pressure upwards, which is equal to 
 the weight of the fluid displaced minus the weight of the 
 solid. 
 
 Let there be taken a body Q, of greater S.G. than that 
 of the fluid, and large enough to sink both itself and the 
 body P whose S.G. is required when P is attached to 
 
 tk^SCA-le^t vrLldL is jLilMZA- !*-</ 
 
HYDROSTATICS. 
 
 57 
 
 it. Let Q be first immersed by itself in the fluid, and 
 balanced by weights in A. Next, leaving the weights in A, 
 let P be attached to Q and both of them be immersed. 
 The scale A will now preponderate, and to restore the 
 equilibrium a portion (TV') of the weights in it will have 
 to be removed. Wherefore, {W being the weight of the 
 body P when placed in the scale B), 
 
 TV = whole effective pressure on P when immersed; 
 which acts upwards; 
 = weight of an equal bulk of fluid minus the weight 
 of P (TV); 
 .-. weight of a bulk of fluid equal to that of P= TV + TV. 
 Now since by Prop. vn. TV=3IxS, therefore when 
 the magnitudes of the bodies are the same, TV oc S ;j£ w i^/ilxu 
 .'. weight of P : weight of an equal bulk of fluid w * • 
 :: S. G. of solid : S. G. of fluid 
 
 or, TV : W+ TV 
 
 £ ; 
 
 TVS' 
 
 " ~- TV+ TV' 
 
 Whence, (JFand TV being known weights), S can be found 
 if S' be given. 
 
 19. Prop. xi. (l) To describe the common 
 Hydrometer, and (2) to shew how to compare the 
 Specific Gravities of two fluids by it. 
 
 (1) The common Hydrometer consists of two spheres 
 attached to each other, and a long uniform slender 
 stem, which, if produced, would pass through the 
 centers of both of them. The upper sphere is 
 hollow; and the lower, being filled with small shot 
 or mercury, serves as ballast for the instrument, 
 and makes it float steadily in a vertical position 
 when put into a fluid. The stem is graduated by 
 divisions of equal lengths. 
 
 (2) The Hydrometer is made lighter than a 
 bulk equal to that of the instrument of any of 
 the fluids whose Specific Gravities it is used to 
 compare. 
 
 Suppose the bulk of the portion of the stem 
 included between every two graduations to be one 
 four thousandth part of the bulk of the whole 
 
58 HYDROSTATICS. 
 
 instrument. When the Hydrometer is put into a fluid 
 whose S. G. is S, let 20 divisions be above the surface ; 
 and when it is put into a fluid whose S. G. is £', let there 
 be 30 divisions out. 
 
 Now the weights displaced of the fluids are in both cases 
 the same, namely the weight of the instrument, (Prop. vi.). 
 If M and M , therefore, be the magnitudes of the fluids 
 displaced, by Prop, vn., 
 
 M 
 
 = weight of the 
 
 Hydrometer = 
 
 = M'xS'; 
 
 .-. S : S' 
 
 :: M : M 
 
 
 
 :: 4000-30 : 
 
 4000-20 
 
 
 :: 3970 : 
 
 3980; 
 
 and the ratio of the Specific Gravities of the two fluids is 
 thus determined. 
 
 £ A mark P is made at the point in the stem to which 
 the instrument sinks in a fluid called " Proof Spirit", 
 which is a mixture consisting of equal weights, — (?wt equal 
 magnitudes), — of pure Alcohol and of Water. Alcohol 
 being lighter than Water, if a mixture of these two fluids 
 contain a greater weight of the former than it does of the 
 latter, it will be lighter than Proof Spirit, and the Hydro- 
 meter therefore will displace a greater bulk of it than it 
 does of Proof Spirit, that is, it will sink deeper in the mix- 
 ture than in Proof Spirit. Wherefore the surface of such a 
 mixture will rise to a higher point in the stem than P, and 
 in such case the mixture is said to be " above proof." 
 
 On the other hand, if the weight of the Water contained 
 in the mixture be greater than that of the pure Alcohol, the 
 Hydrometer will not sink so low as to the point P, and the 
 fluid is then said to be " below proof." 
 
 When equal weights of pure Alcohol and of Water are 
 mixed, the ratio which the bulk of the Alcohol bears to that 
 of the Water does not differ much from 3:2. Out of 100 
 cubic inches of Proof Spirit, 56.36 are pure Alcohol, and 
 43.64 are Water.] 
 
CHAPTER IV. 
 
 ELASTIC FLUIDS. 
 
 20. Prop. xii. Air has Weight. 
 
 This is proved by these experiments. 
 
 If a close vessel be weighed after the air has been ex- 
 hausted from it, and again when the air has been admitted, 
 it is found that the weight is greater in the latter than in 
 the former case. 
 
 If a bladder be weighed in a vessel from which all the 
 air has been exhausted £by a method which will be de- 
 scribed hereafter], first when the bladder contains no air, 
 and again after air has been forced into it, the weight 
 required to balance the bladder is greater in the latter than 
 in the former case. 
 
 Whence it is concluded that 'Air has weight.' 
 
 21. Prop. xiii. The elastic force of Air at a 
 given temperature varies as the density. 
 
 This is proved by experiment. 
 
 Let ABCD be a tube, of which the bore is uniform (its 
 perpendicular section being an area of a square inches), 
 and the legs AB and CD are vertical. The end A of the 
 tube is open, and D is closed. 
 
 A quantity of air is retained in the shorter leg by some 
 mercury which rises to E and F in the two legs respectively. 
 Through E is drawn the horizontal line Ee. 
 
 Then, The weight of the mercurial column Fe 
 
 = pressure downwards at e on a surface a (by Prop, iv.), 
 the portion of the tube being vertical, 
 
 upwards at e ; Prop. i. 
 
 upwards at E ; Prop. u. 
 
 = pressure by the air in DE on the surface a of mercury 
 with which it is in contact; 
 
 ^because, the whole being at rest, the pressures upwards and 
 downwards on the same horizontal plane must be equal.] 
 
ID 
 
 E 
 
 C 
 
 60 HYDROSTATICS. 
 
 Similarly, when more mercury is poured A 
 in, if its surfaces stand at G and H in the 
 two legs, it follows, drawing Gg horizontally H 
 through G, 
 
 Weight of the mercurial column Hg = pres- 
 sure by the air in DG on a portion a of the j- 
 surface of mercury with which it is in contact. _ 
 
 Now if the lengths ED, Fe, GD, Hg be 
 measured, it is invariably found, however the 
 quantities of air and of mercury used in the ■** 
 experiment be altered, that 
 
 DE : DG :: Hg : Fe; 
 
 .'. content of DE : content of DG :: content of Hg 
 : content of Fe. 
 
 But since the same quantity of air is contained in DG 
 that there was in DE, 
 
 .-. Density of the air in DE : density of the air in DG 
 
 :: content of DG : content of DE 
 
 :: content of Fe : content of Hg 
 
 :: weight of mercury in Fe : weight of mercury in 
 Hg, 
 
 :: pressure of air in DE on a portion a of surface 
 : pressure of air in DG on an equal portion of surface; — 
 Qas shewn above.] 
 
 Whence it appears that 
 
 The pressure which air exercises on a given surface 
 varies as the density of the air. q. e. d. 
 
 [Note. In the experiment here described the longer leg of the tube 
 is placed in a vessel from which the air has been exhausted, so that there 
 is no pressure at all on the surface of the mercury in the longer leg. 
 
 If A communicated with the open air, the surrounding atmosphere 
 would press on the surface of the mercury in AB and would have to be 
 taken into consideration in estimating the whole amount of pressure sus- 
 tained by an area a placed at e or g.\ 
 
HYDROSTATICS. 61 
 
 22. Prop. xiv. The elastic force of the air is 
 increased by an increase of temperature. 
 
 This is proved by experiment. 
 
 If a bladder be partially filled with air and then brought 
 near the fire, the included air soon expands, and the 
 bladder becomes fully distended. On the included air 
 cooling down, the bladder returns to its original flaccid 
 state. 
 
 23. Defs. (1 ) A Valve is a kind of door which fits 
 an orifice, such that being pressed by a fluid on one side, 
 it opens and allows the fluid to pass through, but which 
 keeps the orifice tightly closed if the fluid press on the 
 other side. 
 
 [^Valves are of various forms, — a flap of leather (A) 
 fastened at one edge, — a frustrum ^. 
 of a cone (B) made of metal, — a — -j^f — ^_r~^_ 
 
 sphere (C), — or a plate of metal 
 (D) with an axis passing perpen- q •— -n 
 dicularly through it. By any of ' ~ \^y — 
 these contrivances the flow of a 
 fluid upwards would be prevented only by the weight of 
 the valve ; but a rush of fluid from above would carry the 
 valve along with it, and keep the orifice which the valve 
 fits completely closed. 
 
 ■af* 
 
 When the fluids employed are very rare, — like air or 
 gas,-— the valves are generally made of flaps of oiled or 
 varnished silk, which being attached at two or three points 
 to the surfaces in which the orifices are situated, are raised 
 by a very slight pressure, and so allow fluids of exceedingly 
 small densities to pass under them.] 
 
 (2) A Piston is a plug, mostly cylindrical, which, by 
 means of a rod attached to it, can be made to play within 
 a tube which it fits accurately. 
 
62 HYDROSTATICS. 
 
 24. Prop. xv. To describe the construction of 
 the common Air Pump and its operation. 
 
 Description. The Air Pump consists of a glass vessel 
 A, called the Receiver, made to 
 fit a table BC so as to be air 
 tight. A tube DE connects the 
 Receiver with a cylinder EF, 
 called the Barrel. At the bottom 
 of the Barrel there is a valve E ^- 
 opening upwards, and a piston F 
 (also furnished with a valve opening upwards) plays within 
 the Barrel. 
 
 Operation. Let the piston F be at its highest point, 
 and the instrument filled with air the same as that of the 
 surrounding atmosphere. 
 
 When the piston is driven down, the air at first in the 
 barrel is condensed and its pressure therefore increased. 
 The valve E is kept closed, and the valve in F being 
 pressed on the under surface more strongly than on the 
 upper, opens and allows the air in the barrel to escape 
 through it. 
 
 Next, on raising the piston, the external air keeps F 
 closed, and, there now being no air in EF, the pressure on 
 the under surface of the valve at E will open that valve, and 
 allow air from the receiver and pipe to flow into the barrel. 
 The figure represents the instrument during the ascent of 
 the piston. 
 
 When the piston descends again, the air in EF being 
 condensed closes the valve at E, and at length opens the 
 valve at F, and so escapes; and on the piston reascending, 
 more air rushes from the receiver into the barrel. 
 
 The process may be repeated until the air in the receiver 
 becomes so rare that its pressure is insufficient to overcome 
 the weight of the valve at E. 
 
 25. Prop. xvi. To describe the construction 
 of the Condenser and its operation. 
 
 Construction. The Condenser is a Barrel AB fur- 
 nished with a piston A that has a valve in it opening 
 downwards ; at the bottom of the Barrel there is a 
 
HYDROSTATICS. 63 
 
 like valve C. The neck of the Barrel commu- 
 nicates with an air-tight vessel -D, called the 
 Receiver. 
 
 Operation. Let the instrument be filled 
 with common air, and the piston be at its 
 greatest height. On the piston being forced 
 down, the air in the barrel is condensed, and 
 its pressure l^ing therefore increased, (Prop, 
 xni.) it keep- the valve A closed, opens the 
 valve C, and escapes into the Receiver. The 
 figure represents the instrument during the de- 
 scent of the piston. 
 
 On the piston's ascending, the air in the receiver keeps 
 C closed, and there now being no pressure on the under 
 surface of the valve A, the external air opens the valve, and 
 the barrel becomes filled with air, which may be driven into 
 the receiver by forcing down the piston again. And in 
 this manner the condensation of the air in the receiver 
 may be carried on to any extent required. 
 
 £The communication between the barrel and the re- 
 ceiver can be cut off at pleasure by means of a stopcock 
 at E ; and the barrel is made to screw off and on at its 
 lower end.] 
 
 26. Prop. xvii. To explain the construction of 
 the common Barometer, and to shew that the mercury 
 is sustained in it by the pressure of the air on the 
 surface of the mercury in the basin. 
 
 The Barometer is an instrument for measuring the pres- 
 sure of the air. It consists of a glass tube (see fig. Prop. 
 xviii.) of uniform bore, closed at one end, and not less than 
 33 or 34 inches long. This tube is filled with mercury, and 
 the open end, being first stopped with the finger, is placed 
 below the surface of some mercury in a basin, when the 
 tube being held in a vertical position and the finger being 
 withdrawn, the mercury in the tube subsides, and stands at 
 a height above the mercury in the basin which varies on 
 different days from about 28 to 32 inches. A scale of a few 
 inches is attached to the upper part of the tube, to shew the 
 height at which the mercury may be standing at any time. 
 
 
64 HYDROSTATICS. 
 
 That the column of mercury is supported in the tube by 
 the pressure of the atmosphere appears from this experi- 
 ment. When the whole is put into the receiver of an air 
 pump, the mercury in the tube sinks more and more for 
 every barrel of air that is pumped out, until at length it is 
 all but on a level with the surface of the mercury in the 
 basin. But on readmitting the air into the receiver the 
 mercury in the tube rises to its former level. 
 
 27. Prop, xviii. The pressure of the atmo- 
 sphere is accurately measured by the weight of the 
 column of mercury in the [vertical tube of the] 
 Barometer. 
 
 Let P be the surface of the mercury in the vertical 
 tube; AB the section of the tube made by the 
 horizontal surface of the mercury in the basin ; 
 CD a portion of area in that surface equal to 
 AB. 
 
 Then, Weight of mercury contained in PB 
 
 - pressure downwards on the area AB, Prop, iv ; 
 = pressure upwards on AB } 
 
 - pressure upwards on CD; since AB and CD 
 are equal areas situated in the same horizontal 
 plane of a fluid at rest, Prop. 11. 
 
 But the fluid being at rest, the pressure down- 
 wards on CD, (which arises from the pressure of 
 the air in contact with it), is equal to the pres- K-r-^ 
 sure upwards on it. L ' 
 
 Wherefore the pressure of the atmosphere on any area 
 CD is equal to the weight of the column of mercury sup- 
 ported in the [vertical] tube of the barometer, of a base AB 
 which is equal to that area. 
 
 Wherefore, " The Pressure, &c." q. e. d. 
 
 [Cor. 1. By Art. 8. the pressure of a fluid on a hori- 
 zontal plane immersed in it was shewn to be the weight of a 
 column of the fluid whose base is equal to the area of the 
 plane and whose height is the depth of the plane below the 
 surface of the fluid. Wherefore, the pressure exerted by 
 the atmosphere on such an area, — being measured by the 
 
HYDROSTATICS. 
 
 65 
 
 weight of the vertical column of mercury (of equal section) 
 that is supported in the barometer, — is equal to the weight 
 of a column of mercury whose base is equal to the plane 
 acted upon, and whose height is the same as that of the 
 column of mercury supported in the vertical tube of the 
 barometer.] 
 
 £Cor. 2. The pressure of the air being measured by 
 the weight of the column of fluid which it supports, the 
 barometer might be filled with any fluid whatever. But 
 mercury being by far the heaviest fluid known, the column 
 of mercury required to produce a given pressure is very 
 much shorter than if any other fluid were employed. 
 
 For example, if it took 30 inches of mercury to balance 
 the pressure of the air, then since mercury is 13*6 times 
 heavier than water (see Note. Art. 13.), it would take a 
 column of water 13*6 x 30 inches, or 34 feet high, to pro- 
 duce the same effect.] 
 
 £Cor. 3. The mercury in the barometer standing at 30 
 inches, the pressure of the air on a surface of the area of a 
 square inch is equal to the weight of a column of mercury 
 30 inches long and whose base is a square inch ; i. e. to the 
 weight of 30 x 1, or 30, cubic inches of Mercury. — To 
 Jiiid how much this pressure amounts to. 
 
 The weight of a cubic foot of Water is found to be 
 1000 ounces avoirdupois; 
 
 .*. the weight of a cubic foot of Mercury = 1000 x 13*6 oz. 
 (Note to Art. 13.) 
 
 cubic foot cubic inches cubic inches oz. 
 
 .-. 1, or 12 x 12 x 12 : 30 :: 1000 x 13*6 : weight 
 
 of 30 cubic inches of mercury, which .*. = 
 
 J 12x12x12 
 
 ounces = nearly 236 oz. or 14§lbs.] 
 
 [Cor. 4. In Chapters i and u the pressure of the air 
 on the surface of the fluid contained in an open vessel has 
 been left out of consideration ; in other words, the experi- 
 ments there described were supposed to be made in the ex- 
 hausted receiver of an air-pump. 
 
 From Cor. 2 it appears, that when Water is the fluid 
 employed, the pressure of the air on its surface is equiva- 
 lent, (when the mercurial barometer stands at 30 inches), 
 to that which would be produced by a head of water 34 
 
 6—3 
 
66 
 
 HYDROSTATICS. 
 
 feet deep. In estimating, therefore, the pressure on any 
 surface placed at a given depth below the surface of water, 
 this large additional pressure, (amounting by Cor. 3 to more 
 than 14 lbs. on every square inch of surface subjected to it,) 
 must not be left out of the account.] 
 
 28. Pkop. xix. To describe the construction of 
 the Common Pump and its Operation. 
 
 Construction. In the Common Pump two hollow 
 cylinders, AB and BH, having the same 
 axis, are connected together, and at their 
 junction a valve B opening upwards is 
 placed. The upper {AB) of these cylin- 
 ders is called "the Body of the Pump"; 
 and in it a piston C, containing a valve 
 opening upwards, plays by means of a 
 rod attached to the end E of a lever 
 EFG whose fulcrum is F. A spout D 
 is placed just above the highest point to 
 which this piston ascends. The lower 
 cylinder (CH), which is called "the Sue- 
 tion Pipe," reaches below the surface H 
 of a well of water. 
 
 Operation. Let the piston be at B and the suction 
 tube full of air. As C is raised, the pressure of the external 
 air keeps the valve in C closed, and a vacuum between B 
 and C being consequently made, the air in HB, pressing 
 against the under surface of the valve at B, opens it, and 
 escapes into BC. 
 
 The air, therefore, which before the ascent of the piston 
 occupied the space BH, now occupies the greater space 
 CBH, and so becoming less dense than before, exercises a 
 less pressure on the surface of the water at H\ Prop. xiii. 
 Wherefore, since the external air continues to exercise the 
 same pressure as before on the surface of the water in the 
 well, it will force up water into the suction pipe to a height 
 K such that the pressure of the air in CK, together with the 
 pressure arising from the column of water KH, produces 
 the same effect on the section of the water in the suction 
 pipe at H, as the external air does on an equal area situated 
 in the surface of the water in the well. 
 
 When C has come to the highest point of its ascent, and 
 equilibrium exists between the pressure of the external air on 
 
HYDROSTATICS. 67 
 
 the surface of the water in the well on the one hand, and the 
 pressure of the air in CK together with that arising from the 
 fluid' column KH on the other, the valve B, being equally 
 pressed on its upper and its under surfaces, will fall down 
 by its own weight. The piston C is then forced down ; the 
 air in CB is condensed until its pressure becomes greater 
 than that of the external air, when it opens the valve in 
 C and escapes. 
 
 When C is raised again, the same circumstances recur. 
 The water rises a little higher in the suction pipe at every 
 stroke of the piston, and at last flows through B ; and on 
 C descending again, it gets above the valve C, and is 
 brought up to the spout at D. 
 
 £As the average pressure of the atmosphere will not 
 support a column of water more than 34 feet high, if the 
 valve B be more than 34 feet above the surface of the 
 water in the well, the pressure of the external air in that 
 case not being sufficient to force the water up so high as B, 
 the pump will not work. 
 
 N. B. The figure represents the pump during an ascent 
 of the piston, when (air, or water, flowing through it) the 
 valve at B is open, and that at C is shut. ~\ 
 
 29. Prop. xx. To describe the construction of 
 the Forcing Pump, and its operation. 
 
 Construction. The Forcing Pump consists of a 
 Barrel AB in which a solid piston C works G 
 
 by means of a rod GC ; BD is a Suction 
 pipe reaching below the surface D of a well 
 of water ; BE a pipe connecting BC with a 
 vessel EF; at B and E valves are placed, 
 opening upwards. 
 
 Operation. Suppose the piston C to be 
 at its greatest height, and the pump full of 
 air. When C descends, the air in AB is con- ' — 
 densed, and its pressure being increased (Prop. H 
 
 xiii.), the valve E is opened, and the air that ^ —3— i 
 was at first in AB is forced into EF. 
 
 On the piston reascending, the air in EBC being now 
 rendered less dense than that in EF and BD, the valve 
 E is kept closed by the external air, the valve B is opened, 
 and a portion of the air which was at first in DB rushes 
 into the Barrel. The external air then forces some water 
 
 D_- 
 
68 HYDROSTATICS; 
 
 a little way up the pipe, (to H suppose), until the pressure 
 on the horizontal section of the suction pipe at D, (a 
 pressure which arises from the pressure of the rarified air 
 in HBC together with the pressure of the column of water 
 HD), is equal to the pressure of the external air on an 
 equal area in the surface of the water in the well. When 
 this has taken place, there is equilibrium; and the valve B, 
 being pressed equally on both its surfaces, closes by its 
 own weight. 
 
 The piston again descends, and the air in AB is driven 
 through E into EF ; it ascends again, and more water is 
 forced up the suction pipe : and these operations are re- 
 peated until the water rises above B, when on the next 
 descent of the piston the water above B is driven into EF. 
 
 As in the case of the common pump, unless BD be less 
 than 34 feet, the pressure of the atmosphere upon the 
 surface of the water in the well will not be enough to raise 
 the water above the piston at B; in which case the 
 machine will not work. 
 
 £N. B. The Figure represents the Forcing Pump 
 during the ascent of the piston, when the valve E is shut, 
 and the air (or water) is rushing from the suction pipe into 
 the Barrel through the open valve Br\ 
 
 30. Prop. xxi. To explain the action of the 
 Siphon. 
 
 Thk Siphon is an instrument for drawing fluids out of 
 vessels. It is a bent pipe or tube, with legs of unequal 
 lengths. 
 
 Suppose the Siphon ABCD to be of uniform bore, and 
 the two legs AB and CD parallel and con- 
 nected together by a pipe BC at right angles 
 to them. Let the instrument, after it is filled 
 with the same fluid as that in the vessel and 
 its ends are stopped, be placed with the legs 
 vertical, and the end A of the shorter leg 
 plunged below the surface E of the fluid in 
 the vessel. Let the area of the perpendicular 
 section of the pipe be a square inches, and 
 let F be in the same horizontal line as E. 
 Suppose the ends A and D to be opened, and thin films 
 of fluid to fit the pipe at E and D. 
 
 JJr 
 E 
 
 A 
 
 J... 
 
 C 
 
 
 
 D 
 
HYDROSTATICS. 69 
 
 Then, the pressure upwards on the film at D = pressure 
 of the atmosphere on the area a; and the pressure down- 
 wards on it = weight of the fluid-column CD. 
 
 Similarly, the pressure upwards on the film at E = pres- 
 sure of the atmosphere on an equal area in the surface of 
 the fluid in the vessel; and the pressure downwards on it 
 = weight of the column BE of fluid. 
 
 Wherefore, the whole pressure on the column EBCD 
 of fluid in the direction EBCD is the pressure of the at- 
 mosphere on an area a + weight of column CD of fluid ; 
 and the whole pressure on the same column in the direction 
 DCBE, is the pressure of the atmosphere on an area 
 a + weight of a column BE of fluid*. And since the dif- 
 ference of these is a pressure in the direction EBCD 
 on the column of fluid in the siphon, motion must ensue 
 in that direction, and the fluid must flow out at D. And 
 as fast as the fluid flows out, the pressure of the atmosphere 
 will force up more from the vessel into the siphon, and so 
 the motion will be continued. 
 
 fXoTE. If Water be the fluid employed, and EB be 
 greater than 34< feet, the pressure of the column BE on 
 the surface a placed at E will be greater than the pressure 
 upwards on that surface which arises from the pressure of 
 the atmosphere on the surface of the water in the vessel. 
 When, therefore, the end A of the siphon is opened, the 
 column BE will begin to sink down until its extremity 
 stands at a height above E in the pipe, such that the 
 weight of the column of water in the tube is just balanced 
 by the pressure of the atmosphere. If D be now opened, 
 the water in CD runs out at D, and, (no water from 
 the vessel flowing over B), the air will rush up the pipe 
 DC and press on the upper end of the column of fluid 
 supported in BE. This column, therefore, being now 
 equally pressed by the atmosphere at each of its extremites, 
 will descend by its .own weight into the vessel. The 
 Siphon therefore will not act, when water is the fluid used, 
 if EB be not less than 3 4- feel. .] 
 
 * The weight of the fluid in BC produces no pressure on the vertical 
 C »l? mi l S n , ■ i • £?' ? ut is su PPorted by the under surface of the pipe 
 BC ; the fluid m BC, however, serves to transmit the vertical pressures 
 on AB and DC from the one pipe to the other. 
 
70 
 
 HYDROSTATICS. 
 
 31. Prop. xxii. To graduate a common Ther- 
 mometer. 
 
 The Thermometer is an instrument for comparing 
 the temperatures (i. e. the intensities of the heats) of solids 
 or fluids. [See Figure to Prop, xxiii.] 
 
 It consists of a slender tube BA of uniform bore, closed 
 at the upper end, and terminating at the lower in a bulb. 
 The bulb and part of the tube are filled with Mercury, 
 Spirits of Wine, or any other fluid (not of a gaseous form) 
 which expands on being heated, and contracts with cold; 
 the remaining part of the tube is a vacuum. If the fluid 
 in the thermometer occupy the same space when the instru- 
 ment is plunged into two different fluids, the temperature of 
 those fluids is the same. 
 
 • To graduate a thermometer, plunge the bulb into melt- 
 ing snow and make a mark A at the point to which the 
 mercury falls in the tube. Next plunge the bulb into 
 boiling water, and make another mark B at the point to 
 which the mercury rises*. The distance AB may then be 
 graduated by equal divisions, and if it be required to com- 
 pare temperatures by the instrument which are greater than 
 that of boiling water and less than that of melting snow, 
 the graduation may be continued above B and below A. 
 
 32. Prop, xxiii. Having given the number 
 of degrees on Fahrenheit's thermometer, to find the 
 corresponding number on the Centigrade thermometer. 
 
 In Fahrenheit's graduation of the scale of 
 the thermometer, (which is that generally 
 made use of in England), 32 is the number 
 placed opposite to the freezing point A, and 
 AB being divided into 180 parts (called de- 
 grees), the number placed opposite to the 
 boiling point B is 32 + 180, or 212. The 
 graduation of Fahrenheit therefore begins 
 from a point in the scale which is be- 
 low A. 
 
 * The temperatures of melting snoiv and of boiling water are taken 
 to determine the fixed points A and B in the scale of the thermometer, 
 because it is found, by experiment, that these temperatures are fixed and 
 invariable; the mercury always falling to A when plunged into melting 
 snow, whether the snow be melting quickly or slowly, — and always rising 
 to B when plunged into boiling water, whether the fire applied to the 
 vessel containing the water be great or small. 
 
HYDROSTATICS. 
 
 71 
 
 In the Centigrade thermometer, (which is that generally 
 sed on the Continent), the graduation begins from A, and 
 \B is divided into 100 parts, or degrees. 
 
 Now let there be a thermometer furnished on one side 
 v-ith a scale graduated according to Fahrenheit, and on 
 he other with a Centigrade scale. Let the mercury rise 
 o any point D, and let F and C be the number of degrees 
 ?spectively marked opposite to D on the scales of Fahren- 
 eit and the Centigrade. 
 
 Then, since the tube is uniform, 
 The number of degrees of Fahrenheit in AB } (180), 
 
 AD, OF- 32), 
 
 of the Centigrade in AB, (100), 
 
 AD,(C); 
 
 .-. 1S0xC=100x(F-32); 
 
 .%C-{g.(*-«)-f.(*-M>. 
 
 Ex. If the mercury in Fahrenheit's thermometer stand 
 t 77, then the corresponding number of degrees on the 
 Centigrade will be 
 
 5. (77-32) = jj. 45-25. 
 
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